Eureka_The_Solver_Owners_Handbook_1987 Eureka The Solver Owners Handbook 1987
Eureka_The_Solver_Owners_Handbook_1987 Eureka_The_Solver_Owners_Handbook_1987
User Manual: Eureka_The_Solver_Owners_Handbook_1987
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Eureka: The Solver~ Owner's Handbook BORLAND INTERNATIONAL, INC. 4585 SCOTTS VALLEY DRIVE SCOTTS VALLEY, CALIFORNIA 95066 Copyright © 1987 All Rights Reserved, First Printing, 1987 Printed in U.s.A. 10 9 8 7 6 5 4 3 Eureka: The Solver Owner's Handbook Table of Contents Introduction ............................................................................................................. 1 What Can You Do With Eureka? ............................................................................ 1 Structure of This Manual .......................................................................................... 3 Typography .................................................... ;............................................................ 4 The Distribution Disk ............................................................................................... 4 Hardware and Software Requirements ................................................................... 5 Acknowledgements .................................................................................................... 5 CHAPTER 1: Overview ........................................................................................ 7 Command Menus and Windows .............................................................................. 8 The Editor .................................................................................................................. 9 Equation File Syntax ................................................................................................. 9 Variables ................................................................................................................. 9 Initializations ....................................................................................................... 10 Special Symbols .................................................................................................. 10 Reserved Words .................................................................................................. 10 Solving Equations ..................................................................................................... 11 How Eureka Calculates Solutions ..................................................................... 11 Evaluating Equations .............................................................................................. 12 Reporting .................................................................................................................. l2 CHAPTER 2: Getting Started ............................................................................ 13 Setting Up the Program .......................................................................................... 14 Copying to a Floppy Disk .................................................................................. 14 One Floppy Disk Drive .................................................................................. 14 Two Floppy Disk Drives ................................................................................ 15 Copying to a Hard Disk ...................................................................................... 15 Copying to a RAM Disk ..................................................................................... 15 Tutorial Part 1: Using the Menus and the Editor .................................................16 Starting Eureka .................................................................................................... 16 What to Do If You Want Out of Eureka ........................................................... 17 Making a Selection From the Main Menu ...................................................... 20 Creating a File ...........'.......................................................................................... 21 Using the Eureka Editor ............................................................................... 22 Saving A File ....................................................................................................... 23 Tutorial Part 2: Creating and Solving an Equation File ..................................... 24 Creating the Equation File ................................................................................ 24 Writing an Equation ...................................................................................... 25 Writing a Comment ...................................................................................... 25 Solving the Equation File .................................................................................. 26 Modifying the Equation File ............................................................................. 27 Plotting a Graph ................................................................................................. 28 Printing a Report ................................................................................................ 30 Tutorial Part 3: Using Special Features ............................................................... 30 Writing a Directive ." .......................................................................................... 31 Changing Default Settings ................................................................................. 31 Selecting the Variables Option and Finding Other Roots ............................. 32 Evaluating the Solution ...................................................................................... 33 Using the Window Menu .................................................................................. 33 Saving the Equation File ................................................................................... 34 Leaving the Program .......................................................................................... 34 Wrap Up ................................................................................................................... 35 Eureka Editor Summary ......................................................................................... 35 General Editing Techniques .............................................................................. 35 Basic Features of the Editor .............................................................................. 36 More About the Editor Commands .................................................................. 37 CHAPTER 3: Modeling Tips and Techniques ................................................. 39 How Eureka Works ................................................................................................. 39 Eureka's Problem-Solving Methods ................................................................. 40 Solving Techniques .................................................................................................. 40 Maximizing and Minimizing ............................................................................. 40 Continuing a Search ............................................................................................ 41 Modifying Constants and Variables ................................................................... 41 Finding Alternate Solutions .............................................................................. 42 Running Problems from Batch Files ................................................................ 42 Interface with Data Files: Sample BASIC Input File ................................... 43 Eureka: The Solver Owner's Handbook Modeling Tips for Successful Problem-Solving .................................................... 44 Helping Eureka Find Correct Solutions .......................................................... 44 About Error Messages ........................................................................................ 46 CHAPTER 4: Menu Commands and Settings ................................................. 47 Main Menu ............................................................................................................... 48 The File Commands ................................................................................................ 51 File ....................................................................................................................... 51 Load ..................................................................................................................... 51 New ...................................................................................................................... 52 Save ...................................................................................................................... 52 Write to ................................................................................................................ 53 Directory .............................................................................................................. 53 Change dir ........................................................................................................... 54 Rename ................................................................................................................ 54 OS shell ............................................................................................................... 55 Quit ...................................................................................................................... 55 The Edit Command ................................................................................................. 56 Edit ....................................................................................................................... 56 The Solve Command ............................................................................................... 57 Solve ..................................................................................................................... 57 The Commands Commands .................................................................................... 59 Commands ........................................................................................................... 59 Verif}r .................................................................................................................... 59 Calculator ............................................................................................................. 60 Find other ............................................................................................................ 61 Iterate .................................................................................................................. 61 The Report Commands ........................................................................................... 63 Report ................................................................................................................... 63 Go ......................................................................................................................... 64 Output .................................................................................................................. 64 Formatted ............................................................................................................ 65 Capture ................................................................................................................ 65 Log file name ...................................................................................................... 65 The Graph Commands ............................................................................................ 66 Graph .................... :.............................................................................................. 66 Plot ....................................................................................................................... 66 Output .................................................................................................................. 67 List ....................................................................................................................... 68 Function ............................................................................................................... 69 The Options Commands ......................................................................................... 70 Options ................................................................................................................. 70 Variables ............................................................................................................... 70 Settings ................................................................................................................ 72 Table of Contents iii Colors .................................................................................................................... 74 Directories ............................................................................................................ 74 Load setup ........................................................................................................... 75 Write setup ......................................................................................................... 75 The Window Commands ........................................................................................ 76 Window ............................................................................................................... 76 Open .................................................................................................................... 76 Close .................................................................................................................... 77 Next ...................................................................................................................... 77 Zoom .................................................................................................................... 77 Tile ....................................................................................................................... 77 Stack ..................................................................................................................... 78 Goto ...................................................................................................................... 78 CHAPTER 5: The Equation File: Syntax, Directives, and Functions ......... 79 Contents of the Equation File ................................................................................ 79 Equations ............................................................................................................. 80 Initialization Values ............................................................................................. 81 User-defined Functions ...................................................................................... 81 Directives ............................................................................................................ 82 Comments ........................................................................................................... 82 Notes On Equation File Syntax ............................................................................. 82 Variable Names ................................................................................................... 82 Values and Precision .......................................................................................... 83 Mathematical Operators and Order of Evaluation .......................................... 83 Syntax Rules ........................................................................................................ 84 Special Symbols .................................................................................................. 85 Directives ................................................................................................................. 86 accuracy ............................................................................................................... 88 casefold ................................................................................................................ 88 complex (numbers) ............................................................................................. 88 digits .................................................................................................................... 89 end ........................................................................................................................ 90 finanmode ............................................................................................................ 90 finansmooth ......................................................................................................... 90 include .................................................................................................................. 91 initval .................................................................................................................... 91 listdefault ............................................................................................................. 92 max(imize) ........................................................................................................... 92 maxtime ............................................................................................................... 93 min(imize) ........................................................................................................... 93 penalty ................................................................................................................. 93 plotdefault ............................................................................................................ 94 radius ................................................................................................................... 94 iv Eureka: The Solver Owner's Handbook rootsign ................................................................................................................ 94 settings ................................................................................................................. 95 solve ...................................................................................................................... 95 substlevel ............................................................................................................. 96 syntax ................................................................................................................... 96 units (conversion) ................................................................................................ 97 Eureka Functions ..................................................................................................... 99 Financial Functions ............................................................................................ 99 Variables ........................................................................................................ 100 Functions ...................................................................................................... 100 Example ........................................................................................................ 100 User-defined Functions ..................................................................................... 101 Relationship of Variables and Functions in Eureka .................................. 101 Writing User-defined Functions ................................................................ 102 Error Messages ............................................................................................ 103 CHAPTER 6: Worked Examples ..................................................................... 105 Problem 1 A Quick Demonstration of Eureka's Mathematical Capabilities 107 Statement of the Problem ................................................................................ 108 Classic Identities .......................................................................................... 108 Derivatives and Integrals ........................................................................... 108 Maximizations and Minimizations ............................................................. 108 Complex Variables ....................................................................................... 109 Equations ........................................................................................................... 109 Equation Files and Solutions ............................................................................ 111 Graphing the Functions .................................................................................... III Generating Reports ............................................................................................ III Problem 2 Loan Payoff ....................................................................................... 123 Statement of the Problem ................................................................................ 123 Equations ........................................................................................................... 124 Equation File and Solution .............................................................................. 124 Problem 3 Ladders in an Alley: Solving Simultaneous Nonlinear Equations 127 Statement of the Problem ................................................................................ 127 Equations ........................................................................................................... 127 Equation File and Solution .............................................................................. 127 Problem 4 Distribution of Line Charges .......................................................... 130 Statement of the Problem ................................................................................ 130 Equations ............................................................................................................ 131 Equation File and Solution .............................................................................. 132 Problem 5 Radioactive Decay ............................................................................ 134 Statement of the Problem ................................................................................ 134 Equations ............................................................................................................ 134 Equation File and Solution .............................................................................. 135 Problem 6 Least-Squares Fit ............................................................................. 137 Table of Contents v Statement of the Problem ................................................................................. 137 Equations ............................................................................................................ 137 Additional Techniques: Inputting Data From Another Program ................. 139 Equation File and Solution ............................................................................... 139 Problem 7 Motion of a Projectile in a Gravitational Field with Air Resistance .............................................................................................. 141 Statement of the Problem ................................................................................. 141 Equations ............................................................................................................. 141 Equation File and Solution ............................................................................... 142 Problem 8 Charged Masses in a Gravitational Field ...................................... ,144 Statement of the Problem ................................................................................. 144 Equations ............................................................................................................ 145 Equation File and Solution ............................................................................... 146 Problem 9 Using the Built-in Polynomial Function to Find Real and Complex Roots ............................................................................................... 148 Statement of the Problem ................................................................................. 148 Equations ............................................................................................................ 148 Graphing the Functions .................................................................................... 149 Equation File and Solutions ............................................................................. 149 Problem 10 Maximization: Designing a Maximum-Square-Footage House Plan While Satisfying Several Constraints ............................................ 152 Statement of the Problem ................................................................................. 152 Equations ............................................................................................................. 152 Equation File and Solution ............................................................................... 154 Problem 11 Maximization: Linear Programming in Manufacturing .............. 158 Statement of the Problem ................................................................................. 158 Equations ............................................................................................................ 158 Equation File and Solution ............................................................................... 159 Problem l2 Equity in a House: the paymt, pval and fyal functions .............. 161 Statement of the Problem ................................................................................. 161 Equations ............................................................................................................ 161 Equation File and Solution ............................................................................... 162 Problem 13 Probability in a Card Game ........................................................... 165 Statement of the Problem ................................................................................. 165 Equations ............................................................................................................ 165 Equation File and Solution ............................................................................... 166 Problem 14 Income Distribution: The integ Function .................................... 169 Statement of the Problem ................................................................................. 169 Equations ............................................................................................................ 169 Equation File and Solution ............................................................................... 170 Problem 15 Chemical Calculations .................................................................... 172 Statement of the Problem ................................................................................. 172 Equations ............................................................................................................ 173 Equation File and Solution .; ...................... ;...................................................... 175 vi Eureka: The Solver Owner's Handbook Using the $ include Directive ................................................................. ,.. 175 Statement of the Problem ................................................................................ 178 Equations .................................................................. ;........................................ 179 Equation File and Solution .............................................................................. 179 Equations ............................................................................................................ 181 Equation File and Solution .............................................................................. 182 Equations ........................................................................................................... 184 Equation File and Solution .............................................................................. 185 APPENDIX A: Eureka Quick Reference Table ............................................. 187 APPENDIX B: Eureka Editor Commands ...................................................... 191 Cursor Movement Commands ............................................................................. 192 Insert and Delete Commands .............................................................................. 193 Block Commands ................................................................................................... 194 Miscellaneous Editing Commands ...................................................................... 195 APPENDIX C: Built-in Functions ................................................................... 197 abs ........................................................................................................................... 198 atan2 ........................................................................................................................ 198 cos ............................................................................................................................ 198 cosh .......................................................................................................................... 199 deriv ......................................................................................................................... 199 exp ........................................................................................................................... 200 fact ........................................................................................................................... 200 Hoor ......................................................................................................................... 200 frac ........................................................................................................................... 201 fyal ........................................................................................................................... 201 im .............................................................................................................................. 201 integ ........................................................................................................................ 201 In .............................................................................................................................. 202 log10 ........................................................................................................................ 202 msqrt ....................................................................................................................... 203 ncum ........................................................................................................................ 203 paymt ....................................................................................................................... 204 pi .............................................................................................................................. 204 polar ........................................................................................................................ 204 poly .......................................................................................................................... 205 pos ........................................................................................................................... 205 pval .......................................................................................................................... 206 re .............................................................................................................................. 206 sgn ........................................................................................................................... 206 sin .................................................................................. ;......................................... 206 sinh .......................................................................................................................... 207 Table of Contents vii sqrt .......................................................................................................................... 207 sum .......................................................................................................................... 207 tan ........................................................................................................................... 208 tanh ......................................................................................................................... 208 APPENDIX D: A DOS Primer ........................................................................ 209 What Is DOS? ........................................................................................................ 209 How to Load a Program ........................................................................................ 210 Directories ............................................................................................................... 211 Subdirectories ......................................................................................................... 212 Where Am I? The $p $g Prompt .......................................................................... 212 The AUTOEXEC.BAT File ................................................................................... 213 Changing Directories ............................................................................................. 214 APPENDIX E: Error Messages ........................................................................ 217 Assignment Must Be to a Variable or Function .................................................. 217 Bad Function Call .................................................................................................. 217 Cannot ComplexifY Floor or Frac ......................................................................... 218 Cannot Have Complex Derivatives or Integrals ................................................. 218 Circular Function Definition ................................................................................ 218 Comma Expected ................................. ""''',, ......................................................... 218 Command Not Understood .............................·...................................................... 219 Constant Expected ................................................................................................. 219 Dynamic Memory Depleted ................................................................................. 219 Equations are Inconsistent .................................................................................... 219 File Already Exists ................................................................................................. 219 File Not Found ...................................................................................................... 220 File Too Large ........................................................................................................ 220 Function is Multiply Defined .............................................................................. 220 Function is Undefined .......................................................................................... 220 Identifier Expected ................................................................................................ 221 Identifier Too Long ................................................................................................ 221 Incorrect Number of Function Arguments .......................................................... 221 Invalid Function Argument ................................................................................... 221 Invalid Function Definition ................................................................................. 222 Invalid Number ..................................................................................................... 222 Memory Heap Overflow ....................................................................................... 222 Must Differentiate With Respect to a Variable ................................................. 222 Numeral or Operator Expected ........................................................................... 222 Polynomial is Invalid ............................................................................................. 223 Printer Not Ready ................................................................................................. 223 Quote Mark Expected .......................................................................................... 223 Relational Operator Expected .............................................................................. 223 Second Argument Must Be a Variable ......................... ;...................................... 223 viii Eureka: The Solver Owner's Handbook String Space Overflow .......................................................................................... 224 Sum is Invalid ........................................................................................................ 224 Sum is Too Large ................................................................................................... 224 Syntax Error for Unit Conversion ........................................................................ 224 Too Few Arguments to Poly ................................................................................. 224 Too Many Active Variables ................................................................................... 225 Too Many Constants .............................................................................................. 225 Too Many Formulas ............................................................................................... 225 Too Many Functions .............................................................................................. 225 Too Many Unit Conversions ................................................................................. 225 Too Many Variables ............................................................................................... 226 Unable to Create File ........................................................................................... 226 Unexpected End of File ........................................................................................ 226 Unexpected $ End Directive ............................................................................... 226 Unmatched Parentheses ....................................................................................... 226 Unrecognized Character ....................................................................................... 227 Unrecognized Setting Value ................................................................................. 227 Use of Uninitialized Variable ............................................................................... 227 Variable Expected .................................................................................................. 227 Variable Has Not Been Initialized ....................................................................... 227 APPENDIX F: Customizing Eureka ............................................................... 229 Running EINST ..................................................................................................... 229 The Eureka Directory Option ......................................................................... 230 The Text Editor Command Option ................................................................. 230 The Default Edit Mode Option ...................................................................... 231 The Screen Mode Option ................................................................................ 232 Default Display Mode ................................................................................. 232 Color Display Mode .................................................................................... 232 Black and White Display Mode ................................................................. 232 Monochrome Display Mode ....................................................................... 233 Quitting the Program ............................................................................................ 233 GLOSSARY INDEX .......................................................................................................... 235 .................................................................................................................. 240 Table of Contents ix x Eureka: The Solver Owner's Handbook Introduction Welcome to the world of streamlined problem solving. Eureka: The Solver is an equation solver with a difference; this program solves systems of equations (linear, nonlinear, and transcendental), and it also: • Solves high-order polynomials • Evaluates derivatives and definite integrals • Minimizes and maximizes functions with or without constraints Eureka makes it easy for you to solve your real-world mathematical problems very quickly, without having to first learn numerical approximation techniques. You don't have to be a computer specialist to use Eureka. It was designed for people like you - scientists, engineers, financial analysts, and all other professionals and students - people who need to solve equations and who would rather use a computer instead of a calculator to do the solving. What Can You Do With Eureka? Eureka is a versatile tool designed for solving any problem that can be expressed as a system of linear or nonlinear equations. Its capabilities range from solving the simple to the very complex. Without any special training, you can use Eureka to solve: • Simple linear equations in one variable • Systems of linear equations • Nonlinear equations or systems of equations • Maximizations or minimizations of a variable • High-order polynomials (real and complex) • Derivatives and definite integrals Eureka also: • Works with inequalities • Plots and prints the graphs of functions • Converts units automatically • Generates reports Eureka contains several built-in functions, including: • The familiar trigonometric functions • The logarithmic and exponential functions • Several useful statistical and financial functions The program also includes a Calculator Mode, which has all the features of a powerful hand-held calculator, that can also access Eureka's built-in functions. Eureka is very easy to work with. You don't need programming knowledge or experience to use it. You simply type in an equation file containing the equation(s) to be solved, then use the Solve command to solve for the variables in your equations. The equations you enter use standard mathematical representations, very similar to those you enter in a calculator when solving the same problems the hard way. If you get stuck while in Eureka, you can ask the program for on-screen help by simply pressing a key. The Eureka text editor, which you use to enter the equation file, is virtually identical to Borland's SideKick and Turbo Pascal editors. If you are familiar with one of these editors, Eureka's editor will be no problem for you. Or, if you have ever used WordStar or a WordStar-like editor, you are already familiar with the editing syntax and commands. For those who do not have experience with a similar ASCII text editor, we have included several useful tables, a tutorial, and an appendix covering the Eureka editor. Eureka displays solutions on the screen in their own windows, along with information about how confident it is that the solution is correct. 2 Eureka: The Solver Owner's Handbook After Eureka solves the equation file, you can plot graphs of the functions·in the file by simply specifying the function and the range you want plotted. Eureka will display a graph of the function, with the coordinate axes scaled appropriately. You can also print out the equation file, the solutions, and the graphs in the form of a written report. Structure of This Manual This chapter, "Introduction," tells you the basics about Eureka: what you can use the program for, how to use the manual, and what hardware and software you need to run the program. Chapter 1, "Overview," gives an overall picture of the different parts of Eureka and how they fit together. Experienced software users can probably read this chapter and get started right away with Eureka. Chapter 2, "Getting Started," shows you how to load Eureka, then gives a progressive tutorial designed to familiarize you with using the editor, creating and solving problems, and incorporating some special techniques into your files. Chapter 3, "Modeling Tips and Techniques," summarizes the tasks that Eureka performs, describes the program's advanced features, and gives some tips on modeling. ' Chapter 4, "Menu Commands and Settings," describes each of the commands in the main menu and the submenus, and explains the settings that you can change with the Settings command. Chapter 5, "The Equation File: Syntax, Directives, and Functions" describes what you put into an equation file, shows you how to formulate the entries, tells what directives are and shows how to use them in the file, introduces Eureka's built-in functions, provides detailed explanations of the built-in financial functions, and explains what user-defined functions are and how to write them. Chapter 6, "Worked Examples," is a collection of solved equation files taken from typical scientific, engineering, and financial problems. Appendix A, "Quick Reference Table," briefly explains many of Eureka's functions. Appendix B, "Eureka Editor Commands," summarizes the Eureka editor commands. Appendix C, "Built-in Functions," describes Eureka's built-in functions. Appendix D, "A DOS Primer; contains basic information about DOS, with a focus on creating and using directories. Introduction 3 Appendix E, "Error Messages," lists all Eureka error messages and their meanings. Appendix F, "Customizing Eureka," tells you how to use the EINST program to custom-install Eureka. The Glossary gives definitions of key terms used in this manual. The Index lists page references for program features, commands, and windows. Typography The body of this manual is set in this typeface - Roman. Special typefaces are used for the following purposes: Alternate Type This typeface is used to show something as it appears on the screen or to show anything you must type. < > Angle brackets indicate data that depends on your system and should not be typed verbatim. Italics Italics are used to introduce a new term; all new terms are defined in the Giossary. Mathematical expressions are set in Roman type according to standard textbook notation except when they represent what you actually type into a Eureka equation file; these are set in alternate type, and some mathematical operators appear differently. For instance, the multiplier symbol, x, is replaced with * in alternate type, exponents (e.g.l) are represented by a caret (xA2), and s and ~. are represented by < = and> =. The Distribution Disk The Eureka distribution disk contains: • EUREKA.EXE, the program itsel£ • PROB} through PROB12, sample equation files. • READ ME, a message you should read before loading Eureka. • README.COM, a program to assist you in reading README. • EINST.COM, a program to customize Eureka keystrokes, set a path to your files, set the default edit mode, and set the screen mode. • EUREKA.MAC, SuperKey macros for mathematical symbols (SuperKey is a keyboard enhancer program from Borland International). 4 Eureka: The Solver Owner's Handbook • HELP.EKA, Eureka's on-line help text. • NDP.COM, a program that determines if an 8087 math coprocessor chip is present in your computer. • Sample equation files corresponding to the worked problems in Chapter 6 (various file names ending in .EKA) The file README contains important information about using Eureka. Before loading and using EUREKA.EXE, you should look over the README file. Hardware and Software Requirements Eureka runs on the IBM PC family of computers, including the XT and AT, along with most IBM compatibles. Eureka requires: • DOS 2.0 or higher • At least 384K of RAM Eureka will run on any 80-column monitor. A math co-processor chip (an 8087) significantly enhances performance, but is not required. Eureka will display plots in graphics mode if you have a eGA, EGA, or Hercules graphics card but a graphics card is not required. The Eureka program is in a file called EUREKA.EXE. It is not copy-protected, so you can easily transfer it to a hard disk or RAM disk. However, you should read Borland's No-Nonsense Licep.se Agreement at the front of this manual for an explanation of your responsibilities with respect to copying Eureka, and then sign it and mail to us . Ackoowledgements In this manual, references are made to several products: • SideKick, Turbo Basic, and Turbo Pascal are trademarks of Borland International, Inc. • WordS tar is a trademark of MicroPro, Inc. • IBM PC, XT, and AT are trademarks of International Business Machines, Inc. Introduction 5 6 Eureka: The Solver Owner's Handbook c H A p T E R 1 Overview This chapter takes a brief look at Eureka: The Solver. If you are an experienced software user, this chapter may be all you need to get going with Eureka. Check Appendix A, "Quick Reference Table," for more information. If you are new to computers and software, you'll probably want to read this chapter and then turn to Chapter 2 for the step-by-step tutorials. This chapter summarizes the four typical steps that you and Eureka go through to solve a problem, and explains features of each step: • Edit (create an equation file for a problem) • Solve (find a solution) • Verify (verify the solution's accuracy) • Report (print the results) All Eureka problems begin with the preparation of an equation file using the Eureka editor. Chapter 5 details the contents, syntax, and format of an equation file. Chapter 2 gives tutorials designed to familiarize you with preparing an equation file. 7 Cammand Menus and Windows Eureka uses pull-down command menus and windows. If you have not used pulldown menus before, you will find the following few comments helpful. The main menu runs across the top of the screen. When the menu is active, there is always one highlighted item. You can select an item by moving to it with the arrow keys, then pressing Enter, or simply press the key corresponding to the first letter in the item. For instance, to select Solve, you can simply press S. Each time you select a menu item, Eureka either opens up a new menu or executes a command. If the main menu is not active, it is because some window or submenu is active. Generally, when a window is active, all keyboard actions affect only that window. In most cases, you can move around in the windows with the cursor keys. The most important windows are: • Edit • Progress • Solution • Verify " Plot • List • Report • Help • Error message These windows are fully described in Chapters 2 and 4. The following keys affect windows: 8 Esc Either closes the active window or moves it to the background, making another window or menu active. Arrow Keys With Scroll-Lock on, move windows. With Scroll-Lock on and Shift or Num-Lock on, resize windows. FI Calls up a Help window that gives context-sensitive help. Wherever you are in Eureka, you can press FI and a help message appears on the screen. Eureka: The Solver Owner's Handbook The Editor Eureka features an ASCII editor for editing problem files. You can create and modify problem files with this editor while in Eureka, or with the ASCII editor of your choice. The command syntax of the Eureka editor is described in Chapter 4. To call up the editor and create a new file, select Edit from the main menu or select New from the File menu. To call up the editor and modify an existing file, select the File menu from the main menu, select Load, and specify an existing equation file name. When you press Enter, Eureka brings the named file into the Edit window. To save a file for later modification or examination, select Save from the File menu. Equation File Syntax An equation file consists of a collection of formulas, typically one per line, written in standard mathematical notation. Eureka recognizes the follOwing symbols: Relational operators Arithmetic operators Functions = < > + _ */A() exp, In, sin, cos, abs as well as variable names, numerical constants, and some special operators, functions, and directives. The caret symbol is the exponentiation operator, as in n xA2 = x * x or 5A3 = 5 * 5 * 5 = 125 Variables A variable name may be any sequence of letters, digits, and periods, starting with a letter. Uppercase and lowercase are distinguished in variable names, but not in built-in functions or file names. Overview 9 I nitializations If you know an approximate solution in advance, we recommend that you enter the approximate values as initializations. The notation x := 12.75 means that Eureka should initially assign the value 12.75 to the variable x. Eureka will use its own default initialization (x : = 1) if you give no value. More information on the equation :file syntax can be found in Chapter 5. Special Symbols The symbols that follow have special meanings within the Eureka program. The symbols and their meanings are listed, along with reference to where you can find a detailed explanation of each. Symbol $ # {} Where to Find an Explanation Initialization or user-defined function, Chapter 5 Directives, Chapters 4 and 5 Commands/Calculator, Chapter 4 Comments, Chapter 5 Comments, Chapter 5 Reserved Words Eureka's reserved words call up built-in functions. Look in Chapter 5 for directives, and in Appendix C for most built-in functions. Refer to Chapter 5 for definitions of the financial functions, marked here with F. Directives are marked here with a dollar sign ($). 10 Eureka: The Solver Owner's Handbook $ $ $ $ $ $ $ F Eureka Reserved Words im abs $ include accuracy atan2 $ initval integ casefold $ listdefault complex In cos log 10 cosh digits $ max deriv $ maxtime $ min end ncum exp fact F paymt finanmode $ penalty pi finansmooth floor $ plotdefault polar frac fval poly pos F pval $ radius re $ rootsign $ settings sgn sin sinh $ solve sqrt $ substlevel sum $ syntax tan tanh $ units Solving Equations After preparing the equation file, you select Solve from the main menu. Eureka solves the problem and displays the solutions to the equation file's functions and equations in the Solution window. How Eureka Calculates Solutions When Eureka solves a problem, it searches for a solution and then displays resultant values for all of the relevant variables in the Solution window. If there are too many variables to fit into the window, you can scroll through the Solution window with PgUp and PgDn. To determine the accuracy of a minimization or maximization solution, Eureka computes a confidence level between 0 and 100. This is a rough indicator of how well Eureka has performed. 100 is perfect. A confidence level between 93 and 100 means that Eureka successfully found a solution. In some cases a solution may not exist or may not be unique. Read Chapters 3 and 5 for techniques that will help you avoid giving such problems to Eureka. Overview II Evaluating Equations You use Verify from the Commands menu to evaluate expressions in the equation file, using data from the Solution window. Use Verify to verify the accuracy of a previously computed result. When you select Verify, Eureka evaluates all expressions in the equation file. The program ignores directives and initializations, solves no equations, and changes no variable values. Eureka displays the results of the evaluation in the Verify window. When an equation or inequality appears in the equation file, Eureka evaluates the left- and right-hand sides using data from the Solution window, then calculates the difference between the two sides of the equation (or inequality). If the difference indicates that the equation or inequality is invalid, the program gives an error message. Reporting When you need a hard copy of the solution and the equation file, select Report from the main menu. From the Report menu, select the output device and appropriate format, then select Co. The report contains a copy of the equation file, the Solution window information, the last plot or table graphed, and the Verify window information. 12 Eureka: The Solver Owner's Handbook c H A p T E R 2 Getting Started This chapter helps you get going with Eureka: The Solver through a progressive tutorial. It also contains tables that summarize Eureka's main menu, special function keys, and the editor commands. If you are a novice personal computer user or are unfamiliar with menu-driven software, we recommend that you take a few minutes to sit down at your computer with this manual and follow through the tutorial. When you complete it, you will know: • How to load (and leave) Eureka • How to create, solve, modify, and print out a problem • How to work with some of the program's special features In the tutorial, you will first set up (load) the program so you can use it when and how you want. Then you will read the text and follow the instructions given in the tutorial, which gives you hands-on experience with using Eureka to solve a real problem. The tutorial first introduces you to Eureka's menus and text editor, takes you step-by-step through creating and solving a problem, then shows you how to implement some of Eureka's special features. Note: Any time you need help while in Eureka, just press FI to bring up contextsensitive help screens. 13 Setting Up the Program Before you begin working with Eureka, you should make a copy of the distribution disk. Use the copy as your working disk, and store the original in a safe place in case anything should happen to the copy. You don't need to know much about your computer to use Eureka: All you need to know is: • How to turn the system on • How to handle and format a floppy disk • How to start DOS Appendix D, "A DOS Primer," contains useful information about DOS. Copying to a Floppy Disk One Floppy Disk Drive To copy the files from the distribution disk to another floppy disk using a system with one disk drive: 1. Insert your DOS disk in the disk drive. Type DISKCOPY Enter Your system will respond with Insert Source diskette in Drive A and press Enter when ready 2. Take the DOS disk out of the disk drive and insert the Eureka disk, then press Enter. Mter reading the contents of the Eureka disk into memory, your system will prompt you to Insert Target diskette into Drive A and press any key when ready 3. Insert a blank, formatted floppy disk into the drive and press any key. When your system finishes copying the contents of the Eureka disk from memory to the new disk, it will ask if you want to Copy another? Type N Enter 14 Eureka: The Solver Owner's Handbook Two Floppy Disk Drives To copy the files from the Eureka disk to another floppy disk using a system with two disk drives: 1. Insert the Eureka disk in Drive A and a blank, formatted disk in Drive B. 2. Type COPY A:*.* B: Enter Copying to a Hard Disk To copy the files from the distribution disk to a hard disk: 1. Insert the Eureka disk in Drive A. If you use directories, note the full path name of the directory in which you want the program to reside. 2. Type COPY A:*.* C:Enter (assuming your hard disk is Drive C). Copying to a RAM Disk If your system has a RAM disk, you can use it to speed up the program's access time. This means you must copy the Eureka program to the RAM disk each and every time you boot the computer, unless you use an AUTOEXEC.BAT file. (See the next paragraph for how to modify the AUTOEXEC.BAT file.) Type COPY A:EUREKA.EXE 0: Enter (assuming that your RAM disk is on Drive D) each time you want to load Eureka. If you already have an AUTOEXEC.BAT file, you can automate this process by putting a reference to EUREKA.EXE in the AUTOEXEC.BAT file. For instance, assuming your hard disk is Drive C, 1. Use your word processor to bring the AUTOEXEC.BAT file to the screen. (It should be in the root directory.) 2. Find the line that copies programs from their home directories to Drive D. Getting Started 15 3. Add C: \EUREKA.EXE to this line. For example, if EUREKA.EXE is in a directory called EUREKA, the line will look like this: COpy C:\EUREKA\EUREKA.EXE 0: 4. Save the updated AUTOEXEC.BAT file. You can also create an AUTOEXEC.BAT file if you don't already have one. For more information, see Appendix D. Tutorial Part 1: Using the Menus and the Editor This first part of the tutorial explains how to use Eureka's menus and submenus, and how to create and save a file. It tells you how to: • Start Eureka • Understand Eureka's program screens • Write a new file • Save the file Starting Eureka Eureka requires no installation. Once the program is in your computer (either on a floppy disk in the disk drive, or on your hard disk), all you need to do is call it up. For a floppy-disk system, make sure your working disk (the floppy disk with the copied version of Eureka on it) is in the desired drive (typically Drive A, which we will refer to in this manual). Then to start the program, log on to the drive or directory that contains the working disk and type EUREKA Enter 16 Eureka: The Solver Owner's Handbook After a few moments, the Eureka opening screen appears (Figure 2-1). I , - - - - - - - - - Bare": TIle Soluer - - - - - - - - - , IDI!l JUt So)w "port c.~,. O,tion IIi...... CoIIIIA... . - - - - - - liit Solutio. - - - - . leport !&-HeI, III-z- ~xt IJeriry II lli-Goto . . . . .Size/llO\fe Figure 2-1 _ _ Exit Opening Screen What to Do If You Want Out of Eureka Before you begin pressing keys on the keyboard and getting into Eureka, it's a good idea to know how to get out. The Escape (Esc) key always returns you to Eureka's previous active state. Since you are starting at the opening screen, pressing Esc from within one of Eureka's program screens will eventually bring you back to the opening screen. The main menu bar is always visible; you can't escape past it (unless you exit the program by pressing F Q.) As Figure 2-1 shows, the opening screen is made up of two features: the main menu and the status line.Once the opening screen appears, you can use any Eureka feature: select an item from the main menu and then make a selection from a submenu. Look over the screen on your display and compare it to Figure 2-1. Note that the main menu remains on the screen virtually all the time you use Eureka; so does the status line. What appears on the status line varies from time to time; this is explained later. Getting Started 17 A menu presents a list of options from which you can choose. When you select an item from the main menu, that item's pull-down menu extends from the main menu. Each pull-down menu contains a group of related functions. The File menu, for instance, lists operations concerned with files - making them, saving them, and so on. The status line lists keys that perform special functions in Eureka. For instance, • FI displays a window of context-sensitive help text. • F5 "zooms" the active window to fill the screen. Different keys are active under different circumstances, so the keys displayed on the status line change from time to time. You'll see this in the tutorials and in the worked examples in Chapter 6. Table 2-1 summarizes Eureka's main menu options, and Table 2-2 lists the special function keys. 18 Eureka: The Solver Owner's Handbook Table 2-1 Main Menu Summary Entry Option Description FILE Load New Save Write to .. Bring an existing file to the screen Open a new file for editing Write the current file to disk Save the current file under the file name you specify List files in the specified directory Change the active drive/directory Rename a file Temporarily exit to DOS Leave Eureka and return to DOS Create a new file or modify an existing file Find the value of the specified variable(s) Evaluate both sides of the equation and compare the result Open a window in which arithmetic functions can be solved Search another area for the value of the specified variable Continue iterating the value of the specified variable Print/save a report about an equation file Specify output device Put report into formatted form Log your work to a file on disk Specify name for capture file Plot a graph of a user-defined function Specify output device Generate a table of values for a user-defined function Enter function to allow plotting or graphing Change value of variable(s) or constants Open a submenu of user-modifiable settings Change window colors Specify where Eureka looks for its files Let a predefined group of settings be used Save a group of settings as a file Open the specified window Close the active window Open and make active the next window Expand the active window to fill the screen Make all open windows visible and of equal size Stack open windows at largest possible size Go to the active window Directory Change dir Rename OS shell Quit EDIT SOLVE COMMANDS Verify Calculator Find other Iterate REPORT GRAPH OPTIONS WINDOWS Getting Started Go Output Formatted Capture Log file name Plot Output List Function Variables Settings Colors Directories Load setup Write setup Open Close Next Zoom Tile Stack Goto I 19 Table 2-2 Summary of Special Function Keys Function Keys Description FI F2 F3 F5 Alt-F5 F6 F7 FB Alt-X Display help text Save current file In editor, open new file Zoom active window Text zoom for plot (graph) window Change active window Go to active window (in editor, Block begin) In editor, Block end Exit Eureka Go to Edit window Solve the equation file Activate the calculator Insert pi character Return to previous activity Interrupt a solve Move/scroll through screen Alt-E Alt-S Alt-C Alt-P Esc Ctrl-Break Arrow keys and PgUp, PgDn Scroll-Lock, Num-Lock, and arrow keys Scroll-Lock and arrow keys Resize window Move window to new position Making a Selection From the Main Menu There are two ways to select a menu item: the cursor method and the initial letter method. • The cursor metlwd: Move to the right or left with the Right arrow and Left arrow keys. When the item you are interested in is highlighted, press Enter to open a pull-down menu containing details about that item. Use the Up arrow and Down arrow keys to move the highlight bar up and down in the menu; press Enter to select an item from the menu. • The initial letter metlwd: Press the key for the first letter of the item. For instance, to select File, press F (lowercase or uppercase); the File pull-down menu will appear (Figure 2-2). To select Save from the File menu, press S. 20 iEureka: The Solver Owner's Handbook lIP! 1m Uit Solve IUPen: TJoe Soluer CO....HS leport Gn,. Options Uit Solution lepopt Uel"ify lIiHolO !lew SAve lIrite to .. lirectol"Y Cllaqe .ul" lellile OS s'ell lluit ..... 1. II!-Zoo• .....xt II IZ-Goto BlIB-Size/.oue Figure 2-2 r.ilJE-lxit File Menu Creating a File Although Eureka allows you to use any ASCII text editor you like, we will use the one supplied with the program to write a sample file. To create a file, you need to start at the main menu. Select New by pressing F N. Figure 2-3 shows the Edit window as it should appear after you have entered some text into the file. Getting Started 21 , - - - - - - - - - Eureka: Tile Solver - - - - - - - - - , File EUt Solve Co••allis Bepopt Cpa,. Options lIi1ldoll r===== Uit = = = = = , r - - - Solution --------. Line 2 Coil !!ere is so.e text in tile r iJe. [l}-Hel, !B-Save li-New ID-Zoo. iD-lIext iZ-Ieg Ilk lii}--E.... Ilk I!IID-Size/_ve '~~----------------------------/ Figure 2-3 Edit Window You can think of a window as a place where things happen - where you interact directly with the program. • Some windows display a prompt, requesting more information; • Others display the results of a particular operation; • Still others, such as the Edit window, let you enter data or text. Using the Eureka Editor Notice the information at the top of the window on your screen: • "C:NONAME" displays the active drive and the file's name. • "Line I» and "ColI» refer to the position of the cursor; it is on the first line, in the first space or column. • "Insert" tells you the editor is in Insert mode, not Overwrite typing mode. • "Indent" shows that automatic line indention is available. To write something into your new file, you type text just as you would if you were using a typewriter, remembering to press Enter at the end of every line. (Eureka's editor has no word-wrap feature.) Then, to move the cursor around in the text, you can use the cursor keys on the numeric keypad or the PgUp and PgDn keys. Table 2-3 summarizes selected editor commands you may find useful while working through the tutorials. The editor is discussed in greater detail at the end of this chapter and in Appendix B. 22 Eureka: The Solver Owner's Handbook Table 2-3 Summary of Editor Commands To Move the Cursor One space right To first letter of word on right To right end of line One space left To first letter of word on left To left end of line One line up, same column One screen up, same column To top of present screen Beginning of file One line down, same column One screen down, same column To bottom of present screen To end of file Ctrl-D or Right arrow Ctrl-F or Ctrl-Right arrow Ctrl-Q D or End Ctrl-S or Left arrow Ctrl- A or Ctrl-Left arrow Ctrl-Q 5 or Home Ctrl-E or Up arrow Ctrl-R or PgUp Ctrl-Home Ctrl-Q R Ctrl-X or Down arrow Ctrl-C or PgDn Ctrl-End Ctrl-Q C To Delete Text One character, wh~re cursor rests One character, immediately left of cursor Whole word, immediately right of cursor Whole line, where cursor rest~ Ctrl-G or Del Backspace Ctrl-T Ctrl-Y To Insert Text In between existing text Write over (replace) existing text Ctrl-Vor Ins (toggle Insert ON) Ctrl-Vor Ins (toggle Insert OFF, Overwrite ON) Now that you understand how to use the Eureka Editor, write something in your file. For example, you could type I'm learning how to use Eureka, and soon my problem-solving problems will be over. Enter Enter Enter Saving A File With some text in your file (although it doesn't contain any equations yet), this is a good time to save it and then continue working: Select Save from the File menu. When prompted, enter TRIAL as the name for your file. A Eureka file name follows standard DOS conventions: one to eight characters with an optional one-to-three-Ietter extension. In a file name, you can use any letters or numbers, and the symbol _. Getting Started 23 You can use either uppercase or lowercase letters in the file name; when it comes to file names, Eureka is not case sensitive. (That means that Eureka doesn't know the difference between uppercase and lowercase letters; for example, the file names TEST, test, and Test all look the same to Eureka.) Although the file doesn't disappear from the screen, a copy of it has been saved to disk, under the name TRIAL. To bring this saved file to the screen, you would use the Load command on the File menu. Tutorial Part 2: Creating and Solving an Equation File This part of the tutorial demonstrates how to use Eureka to solve a real problem and print a report. It tells you how to: • Create a problem file • Solve the problem • Modify the equation file to contain a user-defined function • Plot the graph of the function • Print a report about the problem The first step is to write an equation file. When you create an equation file, you assemble the various parts of a problem in a way that Eureka understands. An equation file contains one or more equations and any initializations, directives, functions, and comments that are required. You will write an equation file that contains an equation and a comment. You will then solve it, modify it to contain a user-defined function, and plot a graph of that function. Creating the Equation File To write the equation file, ask Eureka to load in the file named TRIAL; type F L Eureka will ask you the Load file name Enter the name of the file you created earlier: TRIAL 24 Enter Eureka: The Solver Owner's Handbook A window appears containing the file you previously saved. However, you can only look at the file from this window; you can't make any changes to what you see because you are not yet in the Edit window. To edit the file, select the Edit option from the main menu: press E. In the next step, you will create a typical equation file containing an equation and a comment. Writing an Equation You write equations, one per line, in standard mathematical syntax and notation, with a few special symbols. For now, all you need to know about equation syntax is that: • The exponentiation operator is the caret symbol, power is written as x 3. A. For instance, x to the third A • Multiplication is not performed implicitly. • Multiplication is denoted with an asterisk, * (3 times z is written as 3 * z). (For more information about equation file syntax, refer to Chapter 5.) The equation you will solve resembles a polynomial; it mayor may not have three roots. After solving for the first root, you will use some special techniques to determine if there are other roots. Enter the following equation to be solved: x-3 + 3 * x-2 - 2 * x - cos (x) = 0 Double check that you have typed the equation correctly. If you make a mistake, use the editor commands given in Table 2-3 to correct it. Writing a Comment Comments must either be set off by a semicolon (;) or appear between braces ({ }). To enter a comment, press Enter (to skip a line, improving legibility), then type { Test of equation solving} You can set off the comment with a semicolon instead: ; Test of equation solving The text you entered into the file during the first part of the tutorial can also be converted to a comment. Eureka can't solve the equation file if you just leave the text as is, so if you don't want to set it off as a comment, delete it. To erase or delete lines of text, see the editor commands in Table 2-3. Getting Started 25 Solving the Equation File When the file contains only comments (appropriately set off) and the equation, return to the main menu and select Solve. Eureka will first look directly, then iteratively, for an answer to the problem. While Eureka searches for an answer, a Progress window appears on the screen. It displays the amount of time the program is using to find the solution, the current amount of error, and the current solution. As Eureka iterates, it modifies the current variable values, then checks to see if the modified values yield the desired function value. The current amount of error is a measure of the difference between the desired function value and the function value calculated by using the current variable values. When the program finds a solution, the Progress window disappears and the Solution window (Figure 2-4) takes its place. The Solution window displays the values Eureka has found for the indicated variables. For the equation file TRIAL, Eureka determines x to be equal to 0.77442927. The Progress window is useful if you want to interrupt a lengthy solving process. To interrupt a Solve and examine the current variable values: • Press Ctrl-Break • Examine the current solution • Resume solving by selecting Iterate from the Commands menu Notice that the keys listed on the status line change as soon as the Solution window appears. The new keys affect only the currently active window. To move the cursor in the Solution window ("scroll the window"), use the arrow and PgUp and PgOn keys (just as in the Edit window). After the Solution window appears, you can still tell Eureka to continue looking for solutions by selecting Iterate from the Commands menu. Often the solution process is so quick that you do not see the Progress window. For minimization and maximization problems, after you select Solve, you will see interim iterative solutions in the Progress window. 26 Eureka: The Solver Owner's Handbook ----------------------._" , - - - - - - - - Eureka: The Solver - - - - - - - - - , rile Edit Solve Co.nruls Report Graph Options lIindm• . - - - - - - - EoJit - - - - - - - - , r = ' = = Solution = = = = j C:TRlAL. Li ne 3 Col 27 C:SOLUTION. Line 1 [olation: x"3 t 3 • x"Z - 2 • x - cos( x) = II {fest or e'lllatioB solving} Ual'iables Uailles .17442927 Itaxina. error is 3.!i2341138e-12 L -_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ ~==============~ I ._---------------------.//" Figure 2-4 Solution Window Modifying the Equation File You have determined one root of the equation to be 0.77442927. But what about the other roots this equation may have? You need to examine a graph of the equation to see if the other roots exist. The Eureka plotting feature allows you to graph the equation and determine if there are other values of x for which the functionf(x) = O. If you try to use either the plotting or table (list) feature now, you will find they are inactive; verify this by pressing Esc then G to select the Graph menu. Look closely at the Graph menu. Plot, Output, and List are dim, meaning that you cannot use them yet. You can only access the Function option; Plot, Output, and List are unavailable because they have no user-defined function to work with. In order for you to use Plot (or List), an equation file must • include at least one user-defined function and • have been solved or else you must enter a user-defined function directly into the program using the Function option. Let's modify the file to incorporate a user-defined function and then solve it. Getting Started 27 First get back into the Edit window by pressing F6 until the Edit window is active (indicated by double bars around it). Then press F7 to go to the window. Use the editing commands listed in Table 2-3 to change your equation file so it looks like this: f(x) := xA3 + 3 * xA2 - 2 * x - cos (x) f(x) :; 0 { Test of equation solving} (The symbol: = indicates thatf(x) is a user-defined function.) Return to the main menu screen, then solve the problem again by pressing Esc, then S. Once you've verified that the solution is the same (x = 0.77442927), continue on to the next step and plot the function. Plotting a Graph Re-select the Graph option from the main menu by pressing G. All the options are now available because the equation file contains a user-defined function. (None of the options are dim, and you can move the highlight bar up and down throughout the whole menu.) Select the Plot option by pressing P. Now set the left and right endpoints of the plot's horizontal (x) axis; when the Left endpoint prompt appears, type - 4 Enter; when the Right endpoint prompt appears, type 4 Enter. (If you hadn't responded to these prompts, Eureka would have plugged in the default values for Plot ( - 2 and 10), and would have plotted and displayed a different section of the function.) Since Eureka automatically scales the vertical (y) axis, a graph of the function appears immediately (Figure 2-5). Notice that several different sized point-like characters have been used to draw the plot. This has the effect of smoothing out the curve: If one type of character were used exclusively, the plot would appear to be much more jagged. If you have a graphics card in your computer, you can press the Zoom key (FS) to see the plot in graphics mode. To see a larger-scale version of the plot, with scaled axes, press Alt-F5 (To then return to the smaller version, press Alt-F5 again.) 28 Eureka: The Solver Owner's Handbook , - - - - - - - - - lureb: TIle Solver - - - - - - - - - - , rila JUt Solve CO••~MS laport GP~p. Options lIi"oli lilt Line 2 C:TRIAL. f(x) := x"3 = • t 3 • x 168. '(x) or (Test Solution Line 1 C:SOLlITIO". II Col 411 'Iot , U~lua. .77442927 e'lution • is 3.5Z34838e-12 e ~ . .....:.:..:..:.:..:•••.........• .. -4." -12.9 Figure 2-5 4 •• I :=J First Plot Screen This plot shows that there are at least two roots to the equation. One root is close to - 4. A region from about - 1.5 to 1.5 may contain two roots, the 0.77442927 value already solved for, as well as a third root. To examine this region more closely, select the Plot option again, but this time use - 4 and 1 as the left and right endpoints. This generates the plot shown in Figure 2-6. This second graph shows what the two roots must be. In the next part of this tutorial, after printing a report, you will select Variables from the Options pulldown menu to solve for the other two roots more exactly. r---------rila IUt I i it Line 2 C:TRlAL. rex) := x"3 f(x) t IoIl'8b: TIle Solver - - - - - - - - - - , Co_M. laport ~p. Options lIi..o.. Sollole 3 • x Col 411 'Iot f = • {Test or e'lu~tion • . .. .. - -.. '. . e +-.' -4." 0 00 .. . . DOL Uallles .77442927 i. 3.5Z34838e-lZ .... .... I -8.16 1..··l :=J _ _ Exit Figure 2-6 Getting Started Solution Line 1 C:SOLlITlO". II Second Plot Screen 29 Printing a Report Now you will print a report that includes the original equation file with the latest solution and plot. Select Report from the main menu. The Report menu offers several ways to store or print an equation file: Output lets you send the report to one of three places: to a file for saving, or to the printer or the screen for printing. Formatted lets you specify a left margin and page breaks. Use the arrow keys to select Output and then press Enter to display a menu of the output choices. To select the printer, press P. To select a formatted report, move the cursor to Formatted and toggle to yes with Enter. To print the report, use the Up arrow key to highlight Go and then press Enter. This sends the formatted report to your printer. Tutorial Part 3: Using Special Features This part of the tutorial gives instructions for using some of Eureka's special features on the equation file. This tutorial tells you how to: • Use the editor to add to the equation file, write a directive, and tailor settings to fit the current equation file • Change default settings • Select the Variables option and find other roots • Evaluate the equation • Save the equation file • Use the Window menu to manipulate windows • Leave the program 30 Eureka: The Solver Owner's Handbook Writing a Directive Now you will add a directive to the equation file TRIAL. A directive is a command that overrides the defaults for various conditions. For instance, although the default for solved variables is eight digits, you can use the digits directive to cause Eureka to display up to 13 digits (the internal limit). All directives follow the general format of the example directive shown in this tutorial. To find out more about directives, refer to Chapter 5. To change the number of displayed digits from eight (the default value) to ten, for example, add the following line to your equation file: $ digits = 10 Changing Default Settings This is a good time to change additional settings. Most of the settings that you can change with a directive can also be changed directly on the screen, using the main menu Options command. In general, you change the settings by entering a number or by pressing Enter to toggle between yes and 00. Specific conditions that apply to the individual settings are described on page 72 and in Chapter 5 under Directives. The accuracy setting is an example of one you can change with a directive or with the Settings command. Accuracy affects Eureka's iterative solution searches by defining how much a solution is subject to its attendant constraints. A constraint is a condition that a solution must meet in order to be valid. For instance, in our equation file TRIAL, f(x} = 0 is a constraint. Eureka will solve for x iteratively, and the smaller the accuracy setting is, the closer Eureka will try to make the left-hand side to the right-hand side of the equation. (The penalty setting also affects the relative importance of the constraints in an equation file; see the description of penalty on page 93.) Getting Started 31 To change the accuracy setting in TRIAL, first select Options from the main menu, then select Settings from the Options pull-down menu. Next, select Accuracy setting: Highlight Accuracy and press Enter. Then type 1.0e-10 Enter This new setting replaces the default value of l.Oe - 7. This smaller number increases the accuracy of the solution when Eureka resolves any constraints that may exist in the equation file. Now select Solve from the main menu. Eureka will solve the modified equation file TRIAL, taking into account the directive to display ten decimal places and the new, tighter accuracy requirement. The Solution window should read x = .7744292653 Selecting the Variables Option and Finding Other Roots With Variables from the Options menu, you can restart Eureka's search for a solution to TRIAL, starting at a value you specify. This is helpful if you have found a local minimum or maximum and want Eureka to search other regions. Select the Variables option: press Esc 0 V. The variables window lists the current solution for x as 0.7744292653. To start Eureka searching for a root in a different region of the x axis, you must enter a new value for the variable x. The new value must be close to the value you believe the alternate root to be. When you plotted the equation file, you saw that there is another root to the equation very close to - 4. To find the root that lies just above - 4, type Enter -4 Enter, then press Esc Esc C to activate the Commands menu and select Find other. This second solution should be x = -3.620186934 You can solve for the third root by repeating the substitution procedure just described, using a value for x close to - 1. The third solution should be x = -0.3288122319 32 Eureka: The Solver Owner's Handbook Evaluating the Solution You can use Verify to check the validity of anyone of the solutions. This command evaluates each side of the solved equation individually and then compares the two sides; the difference between the two results should ideally be O. Eureka may find an unwanted or inappropriate solution (such as a negative root, when the value you need must be greater than zero). With the Verify command, you can identify such solutions. Knowing which solutions do not apply, you can modify the file so that Eureka finds the appropriate solutions. To use Verify, press Esc, then type C V to select Verify from the Commands menu. The Verify window appears (Figure 2-7). The Verify command does not find a new solution; rather, it evaluates each side of the already-solved equation, then compares and displays the results. Notice that the left-hand side of the equation has been evaluated to be a very small number, which results in an error of the same amount. (You may need to move the cursor down to display the amount of difference.) In this case, the amount of error is minuscule, so you can disregard it. . - - - - - - - - - - lureb: TIle SolveI' - - - - - - - - - - , File JUt Solve Co••anis Repol't Gra,. Options DII!!I!l r------C:TRIAL. Uit - - - - - - - , Line 5 Col 1 Solution - - - - , Line 1 III Sol.tion: C: SOLUTIO". I(X) := x"3 t 3 • x"Z - Z • x - cos(x) I(X) = II $ .i,its = 111 Vel'ily Val'iahles C:VERIF'I'. Line 8 {Test 01 equation solv valuation 01 lOl'llUlas: .1111l1li11IIIIIIII1 f(x) 411 f el'ence = I]-Hel, ID-Zoo. -.3288122319 Values FOI'.ulas '--------111 Values = .IIIIIIIIIIIIIIIIIIII 1 - - - - - - - - ' .11111111111111l1li1 iD-llext lif-Goto Bmm!-Size/.ove riJlE-Exit ',,----------------------_._-,. Figure 2-7 Verify Window Using the Window Menu The Window menu allows you to manipulate Eureka's windows. You can open, close, rearrange, and move back and forth between windows by selecting items on this menu. Getting Started 33 Open and Close do exactly what they say: Open opens and makes active the window you specify, and Close closes the active window and removes it from the screen. Next is also easy to use; it moves from window to window, making each . window active in turn. (The function key F6 does exactly the same thing.) Goto simply moves to the active window. The other items on the Window menu - Zoom, Tile, and Stack - allow you to enlarge, shrink, and rearrange your windows. Let's play around a little with the four windows that are currently on screen: Edit, Solution, Plot, and Verify. Suppose you'd like the Plot window to be larger. Press F6 until the Plot window is active. Press the Scroll-Lock and Num-Lock keys. Now press the Up arrow key until the window is as big as you want it, then press Esc. If you just want a window to be as big as the entire screen, select Zoom and the window will expand to fill the screen. (The function key F5, the Zoom key, does the same thing.) To "dezoom" the window, press F5 again. Now let's move the Verify window to a different spot on the screen. Press F6 until the Verify window is active, then press the Scroll-Lock key. Use the four arrow keys to push the window around the screen. Finally, you can arrange all the windows at once. Select Tile, and all the windows will line up neatly side by side in a tile-like pattern, each the same size. Now select Stack. All the windows will pile atop each other in a neat stack in the center of the screen. When windows are either stacked or tiled, you can move from one window to the other by pressing F6 as usual. When windows are stacked, press F6 to "shuffle" them until the window you are interested in is on top. Saving the Equation File This marks the end of the tutorial. If you want to experiment with the equation file TRIAL (perhaps change other settings, or add directives, or print a report containing all three solutions), you should save it now. To save the file, press Esc to return to the main menu, then type F S to select Save from the File menu. Leaving the Program Finally, to leave the Eureka program, select Quit from the main menu to return to DOS, or press Alt-X 34 Eureka: The Solver Owner's Handbook Wrap Up This tutorial has introduced basic features of Eureka: • You have learned how to create, edit, solve, and save an equation file; how to move between and manipulate windows; and how to return to DOS. • You have learned that the usual sequence of operations is to write a problem, set options, solve the problem, and then evaluate the solution. • You have learned how to produce both a graph and a written report. But these are just basics; you will find that Chapter 3 describes useful special features, and Chapter 6 offers several worked examples. The example files can also be found on the Eureka disk. In addition, there are twelve other example files on disk under the file names PROBl through PROB12. (Note: PROBlO is an example that demonstrates the use of special IBM characters. If you have the APL character ROM installed, you will not be able to use this example.) You might want to experiment with these files since they illustrate some of Eureka's important mathematical capabilities, such as maximizations and minimizations, not shown in the tutorial. In addition, a close reading of Chapters 4, 5, and 6 will show you how to fine-tune your equation file. When you feel fairly proficient with the program, you may want to try running some problems in batch mode. See page 42 for information. Eureka Editor Summary This section explains some of the editor's basic features, and gives some tips about using the editor. For descriptions of all the editor commands, refer to Appendix B. To change any of Eureka's default editor keystrokes, use the EINST program (described in Appendix F). Eureka's editor is an ASCII text editor. It does not introduce non-ASCII characters into the file being worked on. For this reason, files created with Eureka are compatible with other programs. Getting Started 35 General Editing Techniques If you have used a word processing program before, using the editor to create and modify a Eureka file will be very simple. If you have never used a word processing program, Eureka's text editor will be easy for you to learn. One purpose of the tutorials in this chapter is to give you some practice with Eureka's editor before you dig in and start solving problems. Basically, you use Eureka's editor like you do a typewriter, except that what you type appears in the Edit window rather than on a piece of paper. When you want to move the cursor to a new location in your file, you use Control commands. • Ctrl-C means to hold down Ctrl on your keyboard while you press C. • Ctrl-Home means to hold down Ctrlon your keyboard while you press Home. The Eureka editor works very much like the Turbo Pascal editor or like WordStarin non-document mode, using many of the same Control commands. Basic Features of the Editor Two typing modes The editor has two typing modes: Insert and Overwrite. In Insert mode, anything you type is inserted at the cursor position. In Overwrite mode, pressing a key replaces the current character at the cursor position with the new typed character. By default, the editor operates in Insert mode (you can change the default with the EINST program; see Appendix F). To switch modes, press Ins or Ctrl-V. No word-wrap Unlike a more standard word processing program, the editor does not have wordwrap. You must press Enter at the end of every line. Autoindent The editor has an autoindent feature that allows subsequent lines to automatically indent. If you leave several spaces at the start of a line, use Ctrl-O I to indent subsequent lines the same amount. This command sequence is a toggle, so you only need to press Ctrl-O I again to stop indenting. The F5 ZOOM key Sometimes you may want to see more than the area covered by the current Eureka window. You can "zoom" the screen, temporarily blanking out all windows and menus by pressing F5. Zooming the Edit window converts the entire screen into an editor environment. To return to the previous screen, with all windows and menus intact, press F5 again. 36 Eureka: The Solver Owner's Handbook Editing ASCII files In addition to the equation file, you can edit files created by another ASCII editor, the report files Eureka generates, and any setup files you create. Scroll-Lock and the F6 key To move the Edit window, you must first activate the window by pressing F6. To move the window, press Scroll-Lock then use the cursor arrow keys. When the Edit window is in its new position, reactivate the window by again pressing Scroll-Lock. More About the Editor Cammands Cursor movement commands control the position of the cursor in the file. You can edit your file by using only the cursor keys, PgUp and PgDn; the rest of the commands listed here are handy but not critical for small files. Word left Ctrl-Left arrow or Ctrl-A Moves the cursor to the beginning of the word to the left. Ctrl-Right arrow or Ctrl-F Word right Moves the cursor to the beginning of the word to the right. Insert and delete commands let you insert and delete text and control the typing mode. Backspace Delete character to left This is the "backspace" key immediately above the Enter key. It moves one character to the left and deletes the character there. Delete character under cursor Del or Ctrl-G Deletes the character under the cursor and moves any characters that are to the right of the cursor over one position to the left. This command does not work across line breaks. Delete word right Ctrl- T Deletes the word to the right of the cursor. This command works across line breaks. Delete line Ctrl- Y Deletes the line containing the cursor and moves any lines below one line up. Insert mode on/off Ins or Ctrl-V This command lets you toggle between Insert mode (the default) and Overwrite mode while entering text. In Insert mode, new text is tucked in at the cursor position and existing text is moved over to the right. In Overwrite mode, new characters replace existing characters as you type over them. Getting Started 37 38 Eureka: The Solver Owner's Handbook c H p A T E R 3 Modeling Tips and Techniques This chapter covers some of the techniques you can use with Eureka to solve problems. It also discusses Eureka's problem-solving method. The last section gives some tips for successful modeling. The techniques covered include: • maximizing and minimizing • continuing a search • modifying constants and variables • finding alternate solutions • running problems from batch files • interface with data files (sample BASIC input file) How Eureka Works This section discusses the inner workings of Eureka, covering both the program's strengths in solving many types of difficult problems and its limitations with regard to the phrasing of equations. For examples of the solutions Eureka finds for real problems, refer to Chapter 6. For more information about error messages, turn to Appendix E. 39 Eureka's Problem-Solving Metlwds Mter reading in an equation file, Eureka sorts out the relevant variables and formulas. It then performs symbolic manipulations and substitutions to eliminate any extraneous variables. If the problem cannot be solved directly, Eureka recasts it as a minimization problem. Eureka minimizes functions by the steepest-descent method. This method involves starting at some initial point, and proceeding along a path which is always in the direction of the most rapid decrease for the function. If there are constraints, the solution path goes from the initial point to the constraint manifold, and then lies on the constraint manifold until Eureka finds a solution. Mathematically speaking, the solution process is actually more complicated than this description suggests. The manifold of points has a Riemannian metric which is derived from the Hessian of the function being minimized. The constraints are enforced by adding a penalty function. Many problems have multiple solutions. Eureka's search method finds one at a time. The program can find other solutions if you insert different starting values or if you add constraints that exclude undesirable solutions. You can also use the Find other command, which directs Eureka to sample distant points in hopes of finding another promising search region. There is no known efficient algorithm that always yields the solutions to all problems. Eureka does its best, but its solution path will occasionally get trapped in a region far from the actual solution. Because some variables are eliminated before the search begins, it is not necessary to initialize them. Only the active variables influence the search process. Solving Techniques This section covers the various solving techniques you can use with Eureka. Maximizing and Minimizing In addition to solving equations, Eureka can solve maximization and minimization problems. This means that, instead of just finding any values for the variables which satisfy the given constraints, Eureka can find values for the variables which 40 Eureka: The Solver Owner's Handbook satisfy the constraints, and maximize or minimize some variable among all possible values of the variables which satisfy the constraints. Maximization or minimization is thus an additional requirement imposed upon a set of equations or inequalities. You impose this requirement by inserting either the max or min directive in the equation file. Max and min work on variables, not on functions; if you need to maximize or minimize a function, you first need to define a variable equal to the function, then direct Eureka to max (or min)imize the variable. For instance, suppose you want to find the maximum'value of a profit function, where the relationships between the variables are expressed by Profit(x,y) := 4 * x + 2 * x * y - 3 * y + 10 Before Eureka can maximize this relationship, you need to add the proviso that P = Profit(x,y) Then the program can follow the directive $ max(P) Continuing a Search If Eureka is unable to solve a set of equations directly, it uses an iterative method. With the iterative method, the program obtains better and better approximations until a solution satisfies certain preprogrammed convergence criteria. If you think Eureka gave up too soon on a given problem, you can select Iterate from the Commands menu, and the program will continue to look for solutions. You can interrupt a search by pressing Ctrl-Break. Eureka will then give the current values of the variables as the solution. Modifying Constants and Variables Sometimes Eureka finds a solution other than the intended one, or gets off the track looking for a solution. Usually this is due to an inadequate initialization. It is possible to restart the iterative procedure with new initial values by using the Variables command on the Options menu. When you select the Variables command, a window opens showing the equation file variables and constants that can be modified. You select one with the cursor keys, then type in a new value to replace the current value. Modeling Tips and Techniques 41 Finding Alternate Solutions Many systems of equations have several distinct solutions. Normally, Eureka is satisfied when it finds one solution. To find other solutions, select Find other from the Commands menu. This causes. Eureka to try to find a solution different from the one already obtained. You can use the Find other command repeatedly to find multiple solutions. You can also use the Find other command to find other local minima to a minimization problem or to find other local maxima to a maximization problem. Running Problems from Batch Files In solving a large number of problems, it may be more convenient to run them directly from DOS or from a batch file rather than from the Eureka pull-down menus. The syntax for doing this is >Eureka ProblemFile OutputFile where is the DOS prompt, and ProblemFile is the problem file name. OutputFile is the output file to be created. The output is the same as an unformatted Eureka report. Some interactive capabilities, such as plotting, are not available when you run Eureka from DOS. 42 Eureka: The Solver Owner's Handbook Interface with Data Fil£s: Sampl£ BASIC Input Fil£ Because Eureka uses standard ASCII files, it can take input from a wide variety of ASCII programs. For example, suppose you are working in Borland's Turbo Basic, and have twenty points in an x-y plane. These are stored in arrays X(I) and Y(I), where I runs from 1 to 20, the X(I) are the points' x-coordinates, and the Y(I) are the respective y-coordinates. The following Turbo Basic program will create a file called PROBLEM and write the data on that file in a format suitable for Eureka to solve: cls defdbl a-z dim X(20), Y(20) -A = 5 B = 1.2 for i = 1 to 20 X(i) = i/10 - 1 Y(i) = A * exp(-B * X(i)~2) + 0.001 * rnd next i open "0", 1, "PROBLEM" print #1, "$ substlevel = 0" print #1, "F(X) := A * exp(-B * X~2)" print #1, "A := 4" print #1, "B := 1" for i = 1 to 20 print #1, "F("; X(i); ") = "; Y(i) next i close #1 This Turbo Basic program will create the following Eureka equation file: $ substlevel = 0 F(X) := A * exp(-B * X~2) A := 4 B := 1 F(-.9 ) = 1.892596768501327 F(-.8 ) = 2.319901816869821 F(-.7 ) = 2.777704759459417 Refer to Chapter 6, Problem 6, for an example problem in which this technique can be quite useful. Modeling Tips and Techniques 43 Modeling Tips for Successful Problem-Solving This section provides some tips on how to set up problems in ways that Eureka can handle well. Helping Eureka Find Correct Solutions Sometimes Eureka does not find the correct solution or gives an inaccurate solution. When this happens, you can 1. Try a different initialization. 2. Use the Find other command. This is especially useful for finding other local minima to a minimization problem, or other local maxima to a maximization problem. 3. Adjust the accuracy setting (select Options/Settings). Often more accurate solutions can be obtained by merely changing the accuracy to be closer to zero. 4. Reformulate the problem. More general considerations to keep in mind when formulating your problem include: 1. When setting up an equation file, it is important to remember to include all the constraints. Variables are not always positive. For instance, if you want to minimize (x + l/x), you probably want to require that x > O. 2. Put in starting values when possible. Otherwise, Eureka may start its search at some random point that is very far from the solution and thus may not find the solution. 3. Use soft constraints where appropriate. A hard constraint is a condition that absolutely must be met for the solution to be useful; a soft constraint is more flexible. The follOwing minimization has only a hard constraint: $ min(z) 5 ; x +w+y + z x <; 10 w >; 0 y >; 0 The constraint on x, to be equal to or less than 10, corresponds to a curve with infinite slope. If this were a cost curve, it would be untenable. 44 Eureka: The Solver Owner's Handbook To alleviate this difficulty, you might recast the problem as: $ min(z) 5 = x +w+y + z x < = 10 + z w>=0 y >= 0 The slope now has a definable limit. 4. Assign names to constants. Instead of this user-defined function containing numeric constants, f(x) := 4.7 * exp(-2.1 * x + 1) use this modification, inserting letters for the constants: f(x) := A A = 4.7 B = 2.1 * exp(-B * x + 1) This does not create extra work for Eureka, and it makes it easier for you to rerun the equation file with different parameters. 5. Try setting the substlevel setting to different values. Eureka tries to be clever about making symbolic substitutions before it does a numerical search. Usually, this speeds up the numerical search, but sometimes the substitutions convert the problem to a form in which the search is more difficult. In these cases, sometimes it helps to lower substlevel- the number of substitutions Eureka is allowed to make during a search. Substlevel can have any value from 0 to 6, with 6 as the default. If substlevel = 0, Eureka performs no substitutions. This can be useful (see Chapter 6, Problem 6 for an example where substlevel = 0 produces the solution to a complicated problem). 6. Avoid using badly scaled variables and functions. For example, x = 7 and 0.0000001 * x = 0.0000007 have the same solution mathematically, but computers sometimes cannot handle equations such as the second one very well because the difference in scale between the constant (0.0000001) and the variable (x = 7) is several orders of magnitude. Eureka might conclude that x = 1 is a good approximation, since the difference between the right- and left-hand sides is only 0.0000006. Refer to Chapter 6, Problem 15, for an example in which poorly scaled variables and functions cause Eureka great difficulty in solving a problem. Modeling Tips and Techniques 45 About Error Messages If Eureka encounters a syntax: error in the equation file, it displays an appropriate error message. If this happens when you tell Eureka to Solve an equation file, refer to Appendix E for an explanation of the error message, then modify the equation file as necessary to correct the error. Sometimes floating-point errors occur in the process of finding a solution. Floating-point errors do not necessarily invalidate a solution; they may mean only that Eureka used inappropriate numbers in the search. For example, in solving l/x = 10 Eureka may try x = 0 before the correct solution of x = 0.1. Because division by zero results in floating-point overflow, the solution appears with a message: Warning: floating point overflow Similarly, Eureka may try to take a square root of a negative number while searching for a solution and issue a warning to that effect. If a floating-point error does occur, it's a good idea to use the Verify or Iterate commands to validate the solution. 46 Eureka: The Solver Owner's Handbook c H A p T E R 4 Menu Commands and Settings This chapter describes Eureka's commands and settings that you select from the main menu or a submenu. The descriptions are arranged first by main menu item (Commands, Graph, Report, and so on). Then, under each menu item heading, the descriptions are arranged in the order they appear on the menu. Each description includes the name of the command or setting, followed by a discussion of the command's or setting's purpose, effect, and notable features. The menu commands provide a way for you to direct the program and to make things happen. A command specifies a particular kind of operation that the computer is to perform. For instance, the File/Save command instructs Eureka to incorporate all newly entered text into the current file in memory and then save it (write it to disk). Settings are those program parameters that you can change in the equation file, such as the number of digits displayed for values. Settings remain in effect throughout the entire work session. When you exit Eureka, the settings revert to their default values. You can save default settings with the Write setup command. 47 Main Menu The main menu displays the fundamental program commands and setting groups: File a group of commands that perform operations on files; also allows you to access DOS. Edit allows you to create a new file, or change an existing one. Solve finds the value of the variable(s) in the equation file. Commands a group of commands that perform operations that supplement Solve. Report a group of commands that produce a report based on the current equation file. Graph a group of commands that generate plots or tables of values for the current equation file. Options allows you to change default settings or selected variables. Windows a group of commands that allow you to move and manipulate windows. To select a command, type either the first letter of its name or cursor to the command you want and press Enter. In most cases you select a setting in the same way. However, in some instances you select a setting by toggling a word (such as YES or NO) with the Enter key. The screen menu will prompt you for the correct response. Table 4-1 summarizes the functions of the menu commands and settings, also in order of screen appearance. Following Table 4-1, complete descriptions of each command and setting round out this section. 48 Eureka: The Solver Owner's Handbook Tabk 4-1 Main Menu And Submenu Summary Entry Option Description FILE Load New Save Write to .. Directory Change dir Rename OS shell Quit Bring an existing file to the screen Open a new file for editing Write the current file to disk Save the current file under the file name you specify List files in the specified directory Change the active drive/directory Rename a file Temporarily exit to DOS Leave Eureka and return to DOS Create a new file or modify an existing file Find the value of the specified variable(s) Evaluate both sides of the equation and compare the result Open a window in which arithmetic functions can be solved Search another area for the value of the specified variable Continue iterating the value of the specified variable Print/save a report about an equation file Specify output device Put report into formatted form Log your work to a file on disk Specify name for capture file Plot a graph of a user-defined function Specify output device Generate a table of values for a user-defined function Enter function to allow plotting or graphing Change value of variables(s) or constants Open a submenu of user-modifiable settings Set stopping point for iterative searches Let upper and lowercase letters define different variables Let variables be complex numbers Set number of displayed digits Set odd period as beginning or end Make financial functions smooth Initialize variable Set first table value Set increment for table Number of values in table Override the default list settings EDIT SOLVE COMMANDS Verify Calculator Find other REPORT GRAPH OPTIONS Iterate Go Output Formatted Capture Log file name Plot Output List Function Variables Settings accuracy casefold complex digits finanmode finansmooth initval lisLfirst lisUnc lisLnum listdefault Menu Commands and Settings 49 Table 4-1 Main Menu And Submenu Summary, continued WINDOWS 50 maxtime penalty ploLleft ploLright plotdefault radius rootsign substlevel syntax Colors Directories Load setup Write setup Open Close Next Zoom Tile Stack Goto Set time limit for solving Determine relative weight of constraints Set left endpoint of plot Set right endpoint of plot Override the default plot endpoints Set range of solve search Set roots all negative or positive Set level of internal substitution in solving Set syntax for arithmetic expressions Change window colors Specify where Eureka looks for its files Let a predefined group of settings be used Save a group of settings as a file Open the specified window Close the active window Open and make active the next window Expand the active window to fill the screen Make all open windows visible and of equal size Stack open windows at largest possible size Go to the active window Eureka: The Solver Owner's Handbook The File Commands , - - - - - - - - - - - lureh: file Solver - - - - - - - - - , Edit Solve Co••~"'s leport Gr~.11 Options Uhdoll m 1m Uit - - - - - - , . . - - - - S o l u t i o n - - - - - - - , !lew Save Urite to •• lirectory Cllnge llir Ie.~.e os sllell Quit Uerify leport II!-Help '\ jE-zoo. !D-JIext IZ-Goto II I!Iiim!!-s i xe/.ove riBE-lxit '- Figure 4-1 The File Menu File Press F Function Enables you to select one of the File commands from the File menu. Description The File commands, in order of appearance in the File menu, are: Load, New, Save, Write to ... , Directory, Change dir, Rename, as shell, and Quit. The File commands are described in the order they appear on the menu. Looo Press FL Function Brings a copy of an existing file from disk to the screen, or names the edit buffer with a new file name. Description When you select Load, a prompt appears Load Fil e name Menu Commands and Settings 51 Type the name of the desired file, followed by Enter. You can specify a file on another drive or in another directory by specifying the full path name (for example, B:\PROBLEMS\FILENAME). You can also use wildcards (* or ?) to see a list of files; you can then choose the file you want to load. For example, you could type *.EKA to see all files with the .EKA extension. Once you have entered the file name, Eureka reads the file from disk (or opens a new file if there is no existing file by that name) and displays it in the Edit window. Select Edit to activate the Eureka editor. New Press FN Function Opens a new unnamed file for editing and activates the Eureka editor. Description When you select New, Eureka opens the Edit window. You can then create a new equation file. When you want to save the file, select Save or Write to ... Save Press FS Function Incorporates your latest changes to the named file in the Edit window and writes a copy of the changed file to disk. Description When you select Save, Eureka writes a copy of the file in the Edit window to the current directory or to another directory/drive if you specify a path name. The file remains on the screen so you can continue editing or solve the equations. Save does not save plots or tables that you generate with the Graph command. To save a copy of a plot or table, use the Report command. Save creates a backup file with the extension .BAK each time you save a file. If for some reason your original file is corrupted, you can always load the .BAK file. 52 Eureka: The Solver Owner's Handbook Write to... Press FW Function Allows you to give a name to the file you are editing and save it to disk. Description Write to... offers an easy way to make files that are variations of each other. When you select Write to ... a prompt appears New name To save a file incorporating your current changes, type a name and then press Enter. You can make changes to your file as many times as you like, each time saving the variations to disk under a different file name. Directory Press F0 Function Displays all or some of the files and subdirectories on the current directory or disk. Description When you select Directory, a prompt appears Enter mask The term mask, as used in Eureka, refers to a group of characters that you type in order to search selectively for a file (or group of files) or some other object. To view the contents of the current directory, press Enter. A window appears, listing all the files in the directory. To view only certain files in the directory, type an appropriate mask followed by Enter. Eureka displays a selected listing to be displayed. Pressing Esc returns you to the main menu. With a mask, you can screen the contents of a directory by using the two DOS wildcards: the asterisk (*) and the question mark (?). The * wildcard tells the computer to find all matching character strings, starting from the position of the *. For example, you could use * to display all files starting with the letters AN by typing AN*.* as the mask. Or, to search for a file that starts with Math and ends with the extension CAL, you would type Math*.CAL as the mask. Menu Commands and Settings 53 The ? wildcard works similarly to the * wildcard, except ? finds only one character at a time. For example, to search for the following files AMT120PA AMT430PB AMT890PA you would enter the mask AMT??OP?; Eureka will list these files (and any others with the same characters in the first, second, third, sixth, and seventh positions). You can also view a list of files in any other directory by specifying the path name of the directory. To view all the files in a subdirectory, move the cursor to the subdirectory name in the file list and press Enter. Change dir Press Fe Function Changes the current drive/directory. Description When you select Change dir, a prompt appears Enter dir name To change the current drive/directory, type the full path name of the desired directory (and the drive name if you are changing drives), followed by Enter. The usual drives are A, B, or C, unless you have a RAM disk or are using a network (in that case, the directory can be D: or higher). Eureka will now look for files on the new directory. If the requested directory does not exist, Eureka will display an error message. Rename Press FR Function Renames the file in the Edit window. Description When you select Rename, a prompt appears: Enter name 54 Eureka: The Solver Owner's Handbook Enter a new name for the file in the Edit window, then press Enter. Eureka renames the file in memory only. Use Save to save the file with the new name to disk. as shell Press F0 Function Allows you to access DOS without removing Eureka from memory. Description When you select OS shell, Eureka disappears from the screen (but not from memory), and the DOS prompt (» comes up. To execute any DOS command, type the command followed by Enter. To return to Eureka, type EXIT. Quit Press FQ Function Leaves Eureka and returns to DOS. Description When you select Quit, you will be returned to DOS. If you have not saved the file in the Edit window, Eureka will ask you if you want to save it. If you don't save, any changes you've made to it will be lost. Menu Commands and Settings 55 The Edit Command /-----------------------..,. \, r - - - - - - - - - Eureka: TIle Soiller - - - - - - - - - , File EoIit Sollie Co ••aMB .eport Gnp. Options mmm r = = = = = = Uit = = = = = = = j I C:TRIAL. I(x) := x"3 t 3 I(x) = 8 $ ii,its = III Line 5 It x"z - Z It X Coil - cos(x) {fest of equation solllhlg} IIJ-llel, !D-Zoo. rD-Next [ii-Goto !!mlm!!-Size/.o\le r.JIE-Exit \'-.._----------------------) Figure 4-2 Typical Edit Screen Edit Press E Function Activates the Eureka editor. Using the editor, you can either modify an existing file or create a new one. Description When you select Edit, the Edit window becomes active. You can create a new equation file or edit an existing one. To create a new file, just type in your file. When you want to save it, select Write To ... from the File menu. To edit an existing file, select Load from the File menu, then type the file's name (using its full path name if necessary) followed by Enter. The editor uses a combination of standard control character sequences and predefined function keys for editing. It is virtually identical to Borland's Turbo Pascal and SideKick editors. To switch between Insert and Overwrite modes, press Ins. More detailed information about the editor can be found in Chapter 2 and Appendix B. 56 Eureka: The Solver Owner's Handbook The Solve Command ,r-.-----------------------... - - - - - . - - - - . - - - - - Eurek.: TIle Soluer - -..- -..- ..---.-.. --.. - . - - - rile .. it Solue Co ••• n.ls leport Gnp. Options 311m!! r = = = = = Uit = = = = = 9 1 5 = = = = Solution =====;0 C:TRIAL. Line 5 CoIl I(x) := x"3 t 3 • x"2 - 2 • x - cos(x) I(x) = 8 $ 'i,its = 111 C:SOLUTIOH. Sol.Uo.: Vari.hles Line 1 Values - _3288122319 {Test 01 equation soluhg} _ _ Exit '~-.-------------------------/ Figure 4-3 " Typical Solve Screen Solve Press s Function Finds a value for each variable in an equation file that satisfies the constraints of the equation file. Description For each variable, Solve finds the first value that satisfies the constraints of an equation. If a problem cannot be solved directly (by plugging in a formula), it is recast as a minimization. Eureka then checks for a user-supplied initialization value (using the value 1 if none is supplied) and begins to iterate toward a minimum. Once Eureka finds a solution, the Solution window displays all variables and their solved values. / You can interrupt a search for a solution by pressing Ctrl-Break. Eureka then gives the current values of the variables. To continue the search, press Esc and select Iterate from the Commands menu. Many problems have multiple solutions. Eureka's search method finds only one at a time. You can find other solutions by using different starting values, by adding constraints that exclude undesirable solutions, or by using the Find other command on the Commands menu. Menu Commands and Settings 57 Eureka can find the roots of polynomials. Consider an equation file containing a function defined as a polynomial in one variable: p(x) : = poly(x,l,O,O,O,l) Applying Solve to the file causes Eureka to find all of the roots of p(x) as well as to solve for whatever other variables appear in the file. 58 Eureka: The Solver Owner's Handbook The Commands Commands r---------------------------------------------~~ rile Ult Sollie -,- lareb: TIle SollieI' "port I!lIIIII Ult Line 5 CaICtlI.tor ri.. oUel' F(x) := x"3 • 3 • x"Z - Z • Iterate f(x) = I $ ii,its = .1 C:TRIAL. Gr.,. O,tio.. 111.._ Sol.tion Line 1 c: sOLlm Solation: 0". Unldl.. x U.laes = -.3288lZZ319 {Test 01 e.-at ion sollli.,} rDE-hit \.,---------~--------------------./ Figure 4-4 The Commands Menu Commands Press C Function The Commands menu contains commands that perform mathematical operations on elements in the equation file. The commands, in order of appearance in the Commands menu, are: Verify, Calculator, Find other, and Iterate. Description Verify Press CV Function Evaluates all expressions in an equation file and compares the results on an equation-by-equation basis. Description You use the Verify command primarily to verify the accuracy of a previously computed result. A typical command cycle for solving problems is: • Edit (to create a problem) • Solve (to find a solution) Menu Commands and Settings 59 • Verify (to verify the accuracy) • Report (to print the results) When you select Verify, Eureka goes through all equations and inequalities in the equation file and evaluates every expression, substituting in the previously solved values for variables. During an evaluation, Eureka ignores all directives and initializations, solves none of the equations, and changes none of the variable values. When the evaluation is complete, Eureka displays the results with any discrepancies in the Verify window. If the difference indicates that an equation or inequality is invalid (the difference is greater than that allowed by the accuracy setting), the difference is flagged as an error. For example, if a file contains the following obviously incorrect equation 1/3 = 10 * exp(l) the Verify window shows this result: 1/3 = 0.33333 10 * exp(l) = 27.18282 difference (error) = -26.84948 Calculator Press CC Function Lets you perform arithmetic calculations in display-calculator fashion. Description The Calculator command acts as a shortcut: use it to evaluate an arithmetic expression when you don't want to make a new equation file. Eureka's Calculator can handle any built-in function as long as it does not require an undefined variable. When you select Calculator from the Commands menu, an empty Calculator window appears on the screen. Type in an expression such as 10 * exp(1) - 7 followed by Enter, and Eureka displays the value of the expression. 60 Eureka: The Solver Owner's Handbook An arithmetic expression is limited to one line of 29 characters, unless you expand the window with the arrow keys after pressing Scroll-Lock and Num-Lock. If there is a syntax error in the arithmetic expression, an error message appears. If you have solved an equation file, you can insert variable names from the file into expressions in the Calculator, such as the variable Depth shown in the example below: sin(Depth~2) The pound symbol (#) has a special meaning in the Calculator: it contains the value of the previously evaluated expression. Find other Press CF Function Tells Eureka to find another solution to the equation file, different from the one that Solve just generated. Description While Eureka is normally satisfied when it finds one solution to an equation file, many systems of equations have several distinct solutions. The solution you need may not be the first one that Eureka finds. When you select Find other, Eureka tries to find a solution that is different from the solution(s) already obtained. You can use Find other repeatedly to find multiple solutions. For example, Find other can find additional local minima to a minimization problem. (Note: In such a minimization problem, you can also use the Variable setting to help Eureka start a new search in a different range of numbers.) Iterate Press CI Function Continues solving a problem that has been Solved at least once. Description Sometimes, Eureka is unable to solve a set of equations directly. If this occurs, the program uses an iterative method to obtain better and better approximations until the solutions satisfy certain convergence criteria. However, you may think that the program has given up too soon, settling for a solution that meets the accuracy require- Menu Commands and Settings 61 ments, but stops short of the best solution. If you think this has happened, select the Iterate command and Eureka will continue solving where it left off. You can also use the Iterate command to resume a search that has been interrupted with Ctrl-Break. 62 Eureka: The Solver Owner's Handbook The Report Commands ~=====Uit C:TRIAL. Line 5 Coil fIx) := x"3 • 3 • x"2 - 2 • x - cos(x) fIx) = II $ ii,its = 111 Screen VIS orr EOIIIA.LOG - .3288122319 {Test of equation sO)IIill!J} IlE-lxit Figure 4-5 The Report Menu Report Press R Function Enables you to select one of the Report commands from the Report menu. The Report commands prepare a report of the solved equation file and then send it to a specified output device. Reports are formatted for a standard 66-line, 80-column printer. A report consists of: • a header with date, time, and file name information • the equation file • the solution (as displayed in the Solution window) • the evaluation (as displayed in the Verify window) • the most recently generated graph or table Description The Report commands, in order of appearance in the Reports menu, are: Go, Output, Formatted, Capture, and Log file name. Menu Commands and Settings 63 Go Press Function Description RG Sends a report in the specified format to the previously specified output device. You use Go after you have set the format and output device for the report (use the Formatted and Output commands to set these report features). See Chapter 6 for examples of finished reports. Output Press Function R 0 < s, f, or p > Determines where Eureka sends a report: to the screen (s), to a file (f), or to a printer (p). Description Select one of three possible values for output by highlighting the option you want, then pressing Enter. The report destinations are: • Screen - (the default) sends the report to the screen • File - sends the report to a disk file • Printer - sends the report to a printer When you select File, then select Go, a prompt appears: Output file name Type a file name just as you would for an equation file. The report is saved to disk. Once saved as a file, you can edit a report like an equation file with the Eureka editor or any ASCII text editor. However, no mathematical operations (such as Solve) can be performed on a report file. 64 Eureka: The Solver Owner's Handbook Formatted Press R F Function Determines whether or not a report is formatted. Default is yes. Description A formatted report is designed to go to a printer from your computer and has the following features: • l/2-inch left margin (rather than printing Hush with the edge of the paper) • incorporates page breaks If you intend to edit the report with a different text editor before printing it, you should select Formatted/no. To toggle between Formatted/yes (the default) and Formatted/ no, press Enter. When Eureka sends a formatted report to a printer, printing starts at the top of the page. If you want blank lines at the top of the page, position the printer appropriately. Capture Press RC Function Automatically writes work session into a file periodically. Description Capture is useful for automatically saving your work to guard against power failures. It is also useful when you want to solve a series of problems and save all your work to a file. When you select Capture, you toggle Capture on and off. With Capture on, your work will be continuously saved to the file name specified by Log file name. wgfile name Press RL Function Specifies a file name to be used by the Capture command; the default name is EUREKA.LOG. Menu Commands and Settings 65 The Graph Commands I . - - - - - - - - - - - EUl'en: TIle SollieI' - - - - - - - - - - , File Edit Sollie Co....... 'epol't iII!Ill Options lIiNOIl r====== C:TRIAL. Llit Line 5 I,m Coli Sol o.t,.t: List FallCtioa Screen r u.'-------------'I rex) := x"3 t 3 • x"Z - Z • x - cos(x) rex) = 8 $ ii!Jits = 111 -.32118122319 {Test of e'luation solllhg} !Il-Hel, ID-Zoo. iD-lIext li-Goto BDl-Size/1tOVe Figure 4-6 IJE-Exit The Graph Menu Graph Press Function Description G Enables you to select one of the commands from the Graph menu. The Graph commands generate and display on-screen plots of functions and tables of values. The Graph commands, in order of appearance in the Graph menu, are: Plot, Output, List, and Function. Plot Press GP Function Plots a function on the screen. Description Eureka can plot functions on the screen, even on a computer without a graphics card. Eureka can only plot user-defined functions of exactly one variable. For example, the function sin(x) + 0.1 X x can be plotted if the following definition of the function appears in the equation file: f(x) := sin(x) 66 + 0.1 * x Eureka: The Solver Owner's Handbook When you select the Graph menu, the Plot command is available only if at least one user-defined function of a single variable appears in the just-solved equation file. If the Plot command is not available, it is displayed in half intensity on the Graph menu. If no appropriate function exists in the equation file or if there is no just-solved equation file, you can use the Function command to write a function directly to Eureka. If more than one user-defined, single-variable function appears in the equation file, a list of choices appears, prompting you to select the name of the function you intend to plot. When you have selected the function to be plotted, a prompt appears immediately, requesting the left and right endpoints (lower and upper limits of the variable). Enter the left value, then press Enter and repeat for the right value. If you hit Enter without selecting lower and upper limits to the variable, Eureka either resorts to its default values, - 2 and 10, or whatever values you have specified in the Settings menu for ploLleft and ploLright. Eureka scales the vertical axis automatically. Using your selection ofleft and right endpoints for the horizontal axis, Eureka displays the plot in a window on the screen; you can save the plot in a report with the Report commands. You can change the default values for several settings that govern the plot (see ploLleft, ploLright, and plotdefault under the Options/Settings menu). If you have a graphics card in your computer, you can see a graphic display of your plot by pressing the Zoom key (FS). Output Press GO Function Determines where Eureka sends a plot: to the screen (s) or the printer (p). Description By default, Eureka will send your plot to the screen. If you press P, Eureka sends the plot to a printer (must be Epson-compatible). Menu Commands and Settings 67 List Press GL Function Computes a table of values for a function, then displays the table on screen. Description The List command creates a two-column table (a set of ordered pairs of numbers) for a function. The table represents the same information that would appear in a graph of the function but with greater precision. The List command, like Plot, acts on user-defined functions of one variable. Eureka prompts you for: 1. the first value of the function argument 2. the increment 3. the number of points (200 is the maximum allowed) 2 For example, given the function fix) : = x , with: First point: 3 Increment: 1 Number of values: 5 Eureka returns the following table: fix) x 3 9.000 4 16.000 5 25.000 6 36.000 7 49.000 You can use the Report menu to save your tables in report files or print them as hard copy. You can change the default values for several settings that govern the list (see lisLfirst, lisLinc, and lisLnum on page 73 ). 68 Eureka: The Solver Owner's Handbook Function Press Function GF Allows you to enter a function of one variable that can then be plotted or listed. Description If an equation file does not contain a user-defined function, this command provides an alternate way to list or plot an equation. You can use this feature by itself or in conjunction with an equation file. When you select Function, a prompt appears: Enter function Type the appropriate function name; for example, j(x), followed by Enter. Another prompt appears: Enter function definition Type the definition (for example, x-3), followed by Enter. The function name and the definition are each limited to 18 characters. When you have entered the function name and definition, use the cursor keys to move to the Plot or List options. If your equation file contains a user-defined function, you can bypass the Function command and go directly to Plot or List. Menu Commands and Settings 69 The Options Commands r - - - - - - - - - Eurek.. : TIM! Solller - - - - - - - - - , File Edit Sollie Co....nis leport Gr.,. mID lIiMoN ltJMIiI'· C:TRIAL. f(x) := x""3 t 3 • x"Z - Z • x - cos(x) rex) = 8 $ digits = 18 Settings Colors Directories Load set.p IIrite set.p {Test of equation SOlllhlg} Version 0.98h ~Ip ID-Zoo. iD-lIext II-Goto IiIilIiD!-Size/.o\le DJE-Exit ,,~------------------------j Figure 4-7 The Options Menu Options Press 0 Function Enables you to select one of the commands from the Options menu. The Options commands allow you to temporarily change some of Eureka's internal information and screen appearance, as well as write or read a setup file. Settings remain in effect throughout the work session. When you exit Eureka, the settings revert to their default values (unless you select Write setup). Description The Options commands, in order of appearance in the Options menu, are: Variables, Settings, Colors, Directories, Load setup, and Write setup. Variables Press ov Function Lets you restart Eureka's solving procedure using a new initial value; it also lets you change the value of a constant. 70 Eureka: The Solver Owner's Handbook Description After Eureka solves an equation file, select Variables to display a window of any active variables and constants. To modify any of the displayed items: 1. Use the cursor keys to select the desired item. Press Enter. 2. Type in a new value. 3. Press Enter. To reactivate the main menu and re-solve the problem, press Esc Esc, then press S. If the problem has already been solved, press Esc Esc C I to reactivate the main menu. The Variables command is useful when the first solution Eureka finds is not the one you want. (Problems often have more than one solution, though only one may be appropriate.) For example, in the case of a polynomial, Eureka starts from some initial solution and iterates toward a final solution. While most equations are readily solved this way, others are not. This is because an initialization value might be selected that leads Eureka off into a local minima that is not the true solution. Whenever you suspect this is the case, use Variables to re-examine the problem. This is an example of a situation where you could use the Variables command: Suppose that Eureka is to minimize (exp(x) - a X 2) for various values of a. You could type this into the equation file: y = (exp(x) - a * 2) a =3 $ min(y) After the solution appears, choosing Variables reveals that x and yare active variables and that a is a constant. Note: If a had been initialized to 3 (a : = 3) rather than set equal to 3, it would appear as an additional active variable and not as a constant. If you change a to some other value and select Iterate, Eureka can find a minimum value for y using the new value of a. Using the Variables command, a whole sequence of problems can be solved without changing the equation file. Menu Commands and Settings 71 Settings Press Function OS Lets you alter selected settings for the current equation file. When you select Settings, a submenu appears, listing each modifiable setting, the setting's current default value, and the value's type (real number, integer, and so on). One of the settings is highlighted. To make a selection from this menu: 1. Use the arrow keys to highlight the setting you plan to change. 2. Type the new value for the setting (or press Enter for yes/no toggles). 3. Press Esc. Enter the new value in the same format as the default value, which is the middle item in the highlighted line. For example, to set complex, press Enter to toggle it to yes. Description A number of settings can be changed from the Options/Settings menu item. You can also change most of these ,settings by inserting a directive in the equation file. There are six settings that cannot be changed via this menu: include, maximize, minimize, settings, solve, and unit conversions. See Chapter 5, "Directives," for information about these directives. The settings that can be controlled from the Settings menu are: accuracy casefold complex digits finanmode finansmooth initval lisLflrst lisLinc lisLnum 72 listdefault maxtime penalty ploLleft ploLright plotdefault radius rootsign substlevel syntax Eureka: The Solver Owner's Handbook All of these settings except lisLfirst, lisLinc, lisLnum, ploLleft, and ploLright can be used as directives (see "Directives" in Chapter 5 for more information about these settings). The seven non-directive settings are described as follows: listdefault Options: < yes or no > Default: no Determines whether values specified for lisLfirst, lisLinc, and lisLnum are used. When set to yes, these values are used; when 00, you are prompted to enter values. lisLfirst Options: < any negative or positive number> Default: 0.000000 Sets a new default value for the first variable to be solved for in a table of values for the listed function. lisLinc Options: < any negative or positive number> Default: 0.500000 Sets a new default value for the increment between values to be solved for in a table of values for the function. lisLnum Options: < the integers 1 through 200 > Default: 10 Sets a new default value for the number of values to be solved for in a table of values for the function. plotdefault Options: < yes or no> Default: no Determines whether values specified for ploLleft and ploLright are used. When set to yes, these values are used; when 00, you are prompted to enter values. ploLleft Options: < any negative or positive number> Default: -2 Sets a new default value for the left-most point at which the plot of the function begins. ploLright Options: < any negative or positive number> Default: 10 Sets a new default value for the right-hand side of the plot of the function. The value must be greater than the value set for ploLleft. Menu Commands and Settings 73 Colors Press 0 C Function Lets you change the window colors. Description When you select Colors, Eureka calls up a windows menu to the center of the screen. Move the highlighted bar up and down in the window with the arrow keys. When the block highlights the name of the window to be changed, press Enter. A blank sample version of the selected window appears on the screen. To change the window's colors, use the up and down arrow keys to select background, text, or margin (items in the status line), then use the left and right arrow keys to toggle the color of the selected item in the sample window. When the item displays in the color you want, press Enter. To save these changes in a setup file, select Write setup. Directories Press 0 D Function Determines where Eureka stores its files. Description When you select Directories, Eureka presents you with a list of two kinds of files: Eureka dir and Problem files. Use the arrow keys to highlight a selection and press Enter. Eureka then prompts you for the path name of the directory where you want to store your files. For instance, suppose you want to store your equation files in a subdirectory called C:\EUREKA\PROBLEMS and the Eureka program itself in a directory C:\EUREKA. Enter the full path name for each type of file. From now on, your equation flIes will be automatically saved in the C:\EUREKA \PROBLEMS directory when you select Write to .. or Save on the Files menu, and Eureka will look for its program, help, and setup files in the Eureka directory. 74 Eureka: The Solver Owner's Handbook Inad setup Press Function 0 L Brings a file (created with the Write setup command) containing various setup parameters to the screen. These setup parameters will be used until you exit Eureka. Description When you select Load setup, a prompt appears Enter file name Type the full path name of the desired file, followed by Enter. If you specify no extension, the extension .EKA is automatically appended. Write setup Press OVV Function Saves a file containing various screen and printer parameters. Description When you select Write setup, Eureka saves the current setup file to disk. By default, Eureka looks for a file named INITIAL.EKA when it is first loaded. If you want to use a standard set of setup parameters when you load Eureka, name your file INITIAL.EKA. If you specify no extension, the file name extension .EKA is automatically appended. The modifiable Setup parameters include those for window size, placement, and screen colors. Menu Commands and Settings 75 The Window Commands .. , - - - - - - - - - Eoreb: Tile Solver - - - - - - - - - - - . rile I.lit Solve CO ••AM. leport GrAph Options JIIIII!! llext ZTile Stack Goto riJlE-lxit ~~-----------------------------------./ Figure 4-8 The Window Menu Window Press VV Function Selects one of the commands from the Window menu. These commands allow you to move and manipulate on-screen windows. Description The Window commands, in order of appearance on the Window menu, are: Open, Close, Next, Zoom, Tile, Stack, and Goto. Open Press VV 0 Function Opens the specified window. Description Opens one of Eureka's windows. When you select Open, Eureka presents you with a menu of all of Eureka's windows. Move the cursor down to the window you want to open, then press Enter. If several windows are currently displayed on the screen, you can move between them by pressing F6. 76 Eureka: The Solver Owner's Handbook Close Press Function Description VVC Closes the currently active window. When you select Close, Eureka closes the currently active window and removes it from the screen. If you have several windows open, you can select Close repeatedly until all on-screen windows are closed. Next Press Function Description VVN Opens and makes active the next on -screen window. Use Next to move back and forth between on-screen windows. If you select Next repeatedly, Eureka will activate each window in turn, moving in a clockwise direction. The function key F6 does exactly the same thing as the Next command. Zoom Press Function Description VV Z Expands the active window to fill the screen. Use Zoom when you want the window you are currently working with to fill the screen. This might come in handy when you are working with an equation file that is too long to see with the default size Edit window. To "dezoom" the window, select Zoom again. The F5 key does exactly the same thing as the Zoom command. Tile Press Function Description VV T Makes all open windows visible and of equal size. The Tile command rearranges all on-screen windows so that that are aligned side by side in a tile-like pattern, each the same size. Menu Commands and Settings 7,7 Stack Press Function WS Stacks all on-screen windows on top of each other. Description The Stack command stacks all on-screen windows atop each other in the center of the screen. Windows are expanded to fill the entire screen. You can move back and forth between windows by pressing F6 or by selecting the Next command. This "reshufHes" the windows so that the selected window is on top. Goto Press Function WG Puts the cursor in the last active window. Description The F7 key does the same thing as Coto. 78 Eureka: The Solver Owner's Handbook c H A p T E R 5 The Equation File: Syntax, Directives, and Functions This chapter describes the equation file: • what goes into an equation file • what the syntax of an equation file must be • what the equation file directives are and what they do • what built-in and user-defined functions are; how they are alike, how they differ • what causes Eureka to return an error message Chapter 2 explains the Eureka editor and how to create the equation file; it includes a discussion of predefined keys and illustrations of the Edit window, the Solution window, and the Verify window. Chapter 3 provides tips for writing good equation files. Appendix C describes all the built-in functions. Contents of tlw Equation Fik An equation file must contain one or more equations, as well as initialization values and user-defined functions when needed. It may also contain directives and comments. 79 Equations Equations are mathematical statements that consist of two expressions (sometimes more) connected by an equal sign (=) or a relational operator « or ». The following are examples of equations that can be used in a Eureka equation file: +2 * x +x - 4 sin«x + 2 * pi)/3) > cos«x - pi)/4) y-2 - x < x-2 + 2 * x * y y = x-3 An expression is composed of one or more terms connected by an algebraic operator. A term is either a numeric value, a variable name, or a function. The argument of a function may in turn be a variable, a constant, or a term. Algebraic operators are: * .multiply / divide + add subtract raise to the power Typically, you write one equation per line. However, you can use a colon (:) to separate equations appearing on the same line. The colon is useful when you want to define the constraints that limit a variable. For example, you could enter the constraint o,< >,< =, > = 2 3 4 5 (last) The Equation File: Syntax, Directives, and Functions 83 For example, Eureka evaluates the following expression: - U * V- x * y/ z + AABAC as See the description of the syntax directive in this chapter for more information about parentheses in order of evaluation. Syntax Rules Eureka is fairly lenient about what it considers to be a syntactically correct equation file. You need not enter the different parts of the equation file in any specific order as long as each equation, comment, directive, initialization, and user-defined function is properly set up. Eureka follows these syntax rules: 1. All spaces and tabs are ignored. Blank lines are permitted. 2. Multiple formula relations are allowed, as with t = rA2 = xA2 + yA2 or o< x < 3. A colon (:) can be used to separate formulas on the same line. For example, the above multiple relations are equivalent to T = r A2 : r A2 = xA2 + yA2 and o in $end 84 : x / 2.54 Eureka: The Solver Owner's Handbook 5. Comments may either be inserted to the right of a semicolon (;) on any line or enclosed in braces ({ }), as in r A2 = xA2 + yA2 ; Thi sis a conment or { This file solves the ideal gas law, } { using mks units} Mixing the two notations for comments can have unpredictable results and is not recommended. 6. An underscore character (_) means that the formula continues to the next line, as in z = (x - a)A2 _ + (y - br2 The underscore character is unnecessary if the upper line is an incomplete formula, as in z = (x - a)A2 (y _ b)A2 + 7. A tilde (-) gives the complex conjugate of a function, as in z = re(z) + im(z) * sqrt(-l) -z = re(z) - im(z) * sqrt(-l) abs (z) = sqrt (z * - z) 8. Variables may be initialized with any valid arithmetic expression, such as x := exp(l) * 3.14159/2 9. A directive is a line that has a dollar sign ($) as its first nons pace character. Special Symbols Most equation files consist of ordinary ASCII characters, but there are a few special IBM graphics characters that Eureka recognizes. These special symbols and their decimal values are: • square root sign (251) • greek letters (224 through 238, except 236) • divide sign (246) • greater-than-or-equal, less-than-or-equal (242, 243) • two (253) • one half, one quarter (171, 172) • pi (227) (You can also get the pi symbol by pressing Alt-P.) The Equation File: Syntax, Directives, and Functions 85 To use a special symbol in an equation file, hold down Aft and Shift while typing the symbol's value on the numeric keypad to the right of the keyboard (the one set up like a lO-key adding machine). For SuperKey users, the file EUREKA.MAC on the distribution disk contains macros for these special symbols. Directives A directive is a command to the Eureka program; it is embedded within an equation file and affects only that file. Most directives in Eureka affect the settings that are user controlled. There are a few others that perform different kinds of functions. You can also change some settings from the Options menu. The settings stay in effect until you exit Eureka. You can also use the Load setup and Write setup commands to save and load settings. You can only make permanent setting changes within a given equation file by using directives. The only directives (settings) that canrwt be changed from the Options menu are: include, maximize, minimize, settings, solve, and unit conversions. A directive consists of the symbol $ followed by the directive name, an equal sign (usually), and some value. For example, you can set the setting syntax to 2 in an equation file with the directive $ syntax = 2 You can use the settings directive to enter a subgroup of the directives (those which affect Eureka's internal default settings) in an alternate format, illustrated here: $ settings syntax = 2 digits = 10 accuracy = le-13 penalty = 50 $ end This format is convenient for changing multiple settings, since only two $ symbols are required. You can include as many directives as you need between settings and end. If you have a group of non-default settings common to a number of problems, you can store them in a separate file. You can then insert this separate settings file into an equation file with the include directive. 86 Eureka: The Solver Owner's Handbook For instance, the following directive would include a settings file called DEFAULTS: $ include DEFAULTS II II The directives are individually described in alphabetical order on the following pages. Each description includes the directive name, syntax (exactly how to type it), default value, function, and any pertinent remarks. The Equation File: Syntax, Directives, and Functions 87 accuracy SY11tax $ accuracy = Default .0000001 (tOe - 7) Function Specifies a value that must be met before an iterative search stops. Description When Eureka reports a solution, it is not necessarily exact. In most cases, Eureka's iterative procedure searches for a solution until the accuracy matches the accuracy setting value. The default value (.0000001) results in an answer that is accurate to six or seven decimal places. Eureka can achieve greater accuracy if you set the accuracy directive to a smaller value, such as $ accuracy = 1.De - 12 However, there are tradeoffs to using very small values for the accuracy. Smaller values increase computer processing time and could undermine the overall reliability of the solution procedure. casefold Syntax $ casefold Default No Function Changes Eureka's sensitivity to the case of variable names. Description The default allows uppercase and lowercase letters to be used as different, unique characters; by default, Eureka considers pressure :md Pressure to be two different variables. If you change casefold to yes, Eureka will consider the uppercase and lowercase versions of any letter as only one variable. Key words (directives, and so on) and built-in functions are never affected by casefold and are not case sensitive. For example, the exponential function can be written exp(x), Exp(x), or EXP(x). complex (numbers) Syntax $ complex = Default No Function Makes all variables complex. 88 Eureka: The Solver Owner's Handbook Description When complex = no, Eureka cannot solve equations whose solutions have a non-zero imaginary component, except for polynomials solved with the poly function. For example, it is impossible to solve the following equation without using complex numbers, since its solutions are x = (3 + i) and x = (3 - i): However, Eureka can find a complex solution if you change the mode to complex, either from the Settings menu or by putting the complex directive in the equation file. While Eureka is operating in complex mode, the functions exp, ln, sin, cos, tan, and sqrt refer to their analytic continuations. (A branch cut is used for in, sqrt, and exponentiation.) Re and im are the real and imaginary parts, respectively. The tilde ( - ) gives the complex conjugate. By this definition, z = re(z) + im(z) * sqrt(-1) - z = re ( z ) - i m(z ) * s q rt ( -1) abs (z) = sqrt (z * - z) When maximizing or minimizing a complex function, Eureka uses only the real part. Similarly, <, >, and polar refer only to the real parts of the expressions they operate on. Usually, however, these expressions are arranged so as to be real, as in abs(z) < 100 The best way to obtain complex constants is to assign an imaginary value to some variable (typically i or I). The following example illustrates this technique: = sqrt( -1) z=2+3*i i digits Syntax $ digits Default 8 Function Sets the number of digits displayed for solved variables. Description The digits directive tells Eureka to display only the specified number of digits for values in the Solution window. The Equation File: Syntax, Directives, and Functions 89 This directive only affects the way in which Eureka displays real numbers and is unrelated to the underlying accuracy of the solutions. end Syntax $ end Default None Function Completes a list of directives begun with either a settings or units directive. Description See the descriptions for settings and units. finanrrwde Syntax $ finanmode = Default End Function Sets the odd financial period (if there is one) to be at either the end or the beginning of the series of periods used for calculations made with the financial functions. Description Changingfinanrrwde to begin puts the odd period at the beginning and results in slightly different values for the functions pval, paymt, andfval. finansrrwoth Syntax $ finansmooth = Default No Function Determines whether the pval, paymt, and fval functions are smooth. Description The financial functions, pval, paymt, and fval, are discontinuous as functions of time. This is because the value of a loan contract changes at the end (or beginning) of each period, every time a payment is made. These discontinuities make solving for the time variable difficult or impossible. 90 Eureka: The Solver Owner's Handbook You can use the finansrrwoth directive to circumvent this problem. When you set finansrrwoth to yes, pval, paymt, and fval become smooth functions. These smoothed functions are exactly correct if the time is an integer; if time is not an integer, they return interpolated values which may differ somewhat from the true values. include Syntax $ include " " Default None Function Inserts a second file into the body of a first. Description The way files are included in Eureka is similar to the way they are commonly included in programming languages. This directive is provided as an editing convenience. When Eureka processes an equation file with an include directive, it replaces the directive with the named file. This keeps your equation files from becoming unnecessarily large and unwieldy. Include files are primarily useful for storing unit conversions and set-up options separately, so they can easily be used in a number of problems. They are also useful for storing frequently used data, such as chemical molecular weights for a series of stoichiometry problems. Include files can be nested. initval Syntax $ initval Default 1 Function Provides an alternative way to initialize the starting value for a variable. Description To start the solution procedure, Eureka requires a starting value for every variable. If a variable is not given an explicit initialization in the problem file, it is initialized with the default value, 1. If some variables in the equation file are not explicitly initialized in the file, and if you include initval in the file, Eureka will use the initval value to initialize those variables. The Equation File: Syntax, Directives, and Functions 91 listdefault SY1ltax $ listdefault = Default No Function Lets you override the default values for the List function. Description If you set listdefault to no, a series of prompts appears in response to the List command. These prompts allow you to override the default values set for List. If you set listdefault to yes, Eureka automatically plugs in the default values for lisLfirst, lisLinc, and lisLnum, and generates the corresponding table. Note: lisLfirst, lisLinc, and lisLnum are not directives. See Chapter 4 under the Settings menu. max (imize) SY1ltax $ max (variable) Default None Function Maximizes the specified variable. Description When the max directive is in an equation file and you select Solve, Eureka finds a maximum for the specified variable. For example, with the following in an equation file y = -x~2 - 4 * x +3 $ max (y) Eureka will find the value of x that yields the maximum value of y. Without a max directive, Eureka solves for the roots of y: the value(s) of x that yield a zero value for y [ - 2 ± sqrt(7)]. You can use the max directive only once in an equation file, and you cannot use both max and min in the same file. 92 Eureka: The Solver Owner's Handbook maxtime Syntax $ maxtime = (positive integer) Default 200 Function Sets a limit, in seconds, to the amount of time Eureka spends solving an equation. Description When you place the maxtime directive in an equation file, you set a limit to the amount of time Eureka spends seeking a solution. min (imize) Syntax $ min (variable) Default None Function Minimizes the specified variable. Description When you place the min directive in an equation file, it supersedes the Solve command (which searches for roots-zeroes-of equations). Eureka will find a minimum value for the specified variable or function rather than a solution for all variables that make equations in the file equal zero. For example, with the following in an equation file y = x~2 +4 * x +3 $ min (y) Eureka will find the value for x that yields the minimum value of y. Without the min directive, Eureka would solve for those values of x which make y = 0 (-3 and -1). You can use the min directive only once in an equation file, and you cannot use both max and min in the same file. penalty Syntax $ penalty = Default 30 Function Sets a weight (relative importance) for the constraints affecting an equation. Description Eureka uses the value of penalty to calculate the effect of the constraints in an equation file. Changing the default affects the relative strength of the constraint criteria. The Equation File: Syntax, Directives, and Functions 93 plotdefault SYJltax $ plotdefault = Default ~o Function Lets you override the default values for the Plot command. Description If you set plotdefault to no, a series of prompts appears in response to the Plot command, allowing you to override the default values set for Plot. If you set plotdefault to yes, Eureka automatically plugs in the default values for ploLleft and ploLright, and generates the corresponding plot. The default values for ploLleft and ploLright are - 2 and 10, respectively. Note: ploLleft and ploLright are not directives. See Chapter 4, page 72 for more information. radius Syntax $ radius = Default ~one Function Sets a limit to the range over which Eureka seeks a solution. Description If you set radius to no, Eureka will search an indefinite range of numbers, limited only by its internal constraints, accuracy, and the maxtime directive. If you set radius to a real number, Eureka will only search a range of numbers that are equal to or less than the number speci .. fied. rootsign SYJltax $ rootsign = Default Pos Function Toggles between displaying the roots of an even-powered number as negative or positive numbers. Description If you set rootsign to pos, Eureka displays roots of even powers as all positive values; if you set it to neg, Eureka displays the roots as all negative values. 94 Eureka: The Solver Owner's Handbook settings Syntax $ settings $ end Default None Function Allows you to enter several directives in a series without having to set each one off with a $ symbol. Description You can think of the settings directive as the first, or opening, step in a three-step process for changing multiple settings. The second step is to type in the directives for the settings that are changing (without typing in the initial $ symbol each time). The final step is to close the change session by typing in the end directive. If you have a group of frequently used settings, you can store them in a separate file. You can then introduce this separate Settings file into an equation file with the include directive. Refer to the description of the include directive in this chapter for more information. solve Syntax $ solve (variable) Default All Function Specifies which variable in an equation file Eureka will solve for. Description By default, Eureka solves for all variables in an equation file. However, you can use the solve directive to tell Eureka to solve for only the designated variable and ignore the rest. The Equation File: Syntax, Directives, and Functions 95 substlevel Syntax $ substlevel Default 6 Function Sets the maximum amount of internal variable substitutions Eureka can perform when solving an equation file. Description The default, 6, allows Eureka to make the maximum number of substitutions during a Solve process. Substitution level 5 allows Eureka to make fewer substitutions; level 4 allows less than level 5, and so on down to substitution level 0, which allows no substitutions. <0, 1, 2, 3, 4, 5, or 6> syntax Syntax $ syntax Default 0 Function Specifies one of three possible options for the syntax of arithmetic expressions used in the equation file. Description There are three options for the syntax of arithmetic expressions: = <0, 1, or 2> 0: The default syntax, which requires the * symbol to represent multiplication. The default syntax allows scientific notation. 1: Does not allow scientific notation, because a number placed before a variable is treated as a multiplier. 2: Does not allow scientific notation, for the same reason as syntax level 1. Also restricts variables to one letter each, rather than the default choice of up to 60 alphanumeric characters. Examples of each syntax level Syntax = 0: The default syntax is the standard used throughout this manual. Syntax = 1: Scientific notation (such as 2.3e - 4 to represent 0.00023) is not allowed because Eureka interprets it as 2.3 X (a variable called e) - 4. A number before a variable name is interpreted as a multiplier. 96 Eureka: The Solver Owner's Handbook For example, when syntax = 1, Eureka treats the expression 2.7 water as an abbreviation for 2.7 * water As when syntax = 0, Eureka ignores spaces. However, the program still treats a character string (such as dog34) as a variable name. Syntax = 2: Variables are restricted to a single letter. All juxtapositions of numbers and variables are treated as products. For example, a problem to find the intersection of two hyperbolas in the x - y plane could be formulated as: $ syntax = 2 x-2 - 4xy + 3y-2 = 7 2xy - 5y + 3 = 0 { exact solution is x = 2, y = 3 } When syntax = 2, variable names must be distinct from function names; otherwise, an expression such as f(x + 1) is ambiguous (Le., is f a variable multiplying x + 1, or is it a function of x + I?). If the equation file contains a user-defined function, the function definition must precede the first use of the function. Refer to the section "Notes On Equation File Syntax" (page 82) for more information about writing equations using the default syntax level. units (conversion) Syntax $ units -> $ end Default No automatic conversion Function Defines unit conversions. Description Eureka can automatically associate units to variables, and then convert the values of the variables to other units in the Solution window. The Equation File: Syntax, Directives, and Functions 97 For example, the variable dist can appear in the equation file expressed in feet, but can be converted to meters and yards when Eureka solves for its value. The units conversion directive lets you define as many unit conversions as needed. For example, $ units em -> in : x/2.54 F. -> C. : (x - 32) * 5/9 feet -> yds : x/3 km/hr -> mi/hr : 0.6214 * x $ end Each line after $ units gives a formula for converting from one unit of measurement to another. The nrst formula converts from centimeters (em) to inches (in); x is simply a dummy variable. The formula states that if x is in centimeters, dividing x by 2.54 will convert it from centimeters to inches. The names for the units can be any sequence of characters, including punctuation and math symbols, as long as the symbol does not include a hyphen (-) or a colon (:). Eureka ignores blanks. The syntax for expressions is the standard Eureka syntax. You associate units with the variables in the initialization statement. For example, the initialization dist := 122.7 [em] initializes the variable dist to 122.7 centimeters. When solving for dist, Eureka nrst gives the solution in centimeters, then converts the value in centimeters to any other units for which a units conversion directive exists. Here is a typical solution for this example: dist = 110.0235 em = 43.3163 in Eureka can invert any simple unit conversion. If the equation :6le contains the formula for converting from centimeters to inches, it is not necessary to also give the formula for converting from inches to centimeters. 98 Eureka: The Solver Owner's Handbook Eureka Functions Besides providing a wide variety of standard mathematical and special built-in functions, Eureka also allows you to define and use your own functions. Most of the built-in functions need little explanation; they are described in Appendix C. However, the financial functions require some in-depth explanation; they are discussed in detail in this chapter. After the section about financial functions in this chapter, we give an explanation of user-defined functions. Financial Functions Eureka has three functions for performing interest-rate calculations: ivaI, paymt, and pval. These functions are designed for situations in which a person enters into a contract by which a fixed sum is paid at regular intervals. We assume that money earns interest at some fixed rate, with the interest compounded in each interval. In general, there are five variables involved in these financial calculations: n pval paymt ivaI number of payment intervals interest rate per payment interval present value of the contract payment at the end of each interval future value of the contract Given any four of these five variables, Eureka can solve for the fifth. The variables are related by the financial functions in the following ways: pval = pval(i,n,paymt!val) paymt = paymt(i,n,pval!val) ivaI = ival(i,n,pval,paymt) nand i can be derived directly from the values of the other variables using Solve. The Equation File: Syntax, Directives, and Functions 99 Variables The number of payment intervals, n, is normally an integer, but it can be any positive number. If n is not an integer, there is a fractional period that is assumed to occur before all other periods. Interest is compounded at the end of, but not during, the fractional period. The finanrrwde directive can place the fractional period at the beginning or end of the series. i, the interest rate, should not be converted to a percentage rate. For example, i = 0.12 if the interest rate is 12%. The interest is compounded at the end of each payment interval. Functions pval, paymt, andfval all represent money that changes hands. By convention, the number is positive for money received and negative for money paid out. pval, the present value, is defined as the principal: the money received (loaned) at the beginning of the contract. In the case of a home mortgage, pval is the price of the home minus the down payment. paymt, the payment, is the money received at the end of each payment interval. If the person pays the money out, as with a home mortgage, paymt is a negative amount. fval, the future value, is the money received at the end of the contract. In the case of a home mortgage, fval is either zero or a negative number that represents the balloon payment. Example A typical use of the financial functions would be to compute the payments on a mortgage. For example, suppose a house costs $100,000; the interest rate is 12%; and the mortgage is for a period of 30 years. Assume that the down payment is 10%, there is no balloon payment, and payments and interest compounding are monthly. Using paymt, the monthly payment can be calculated as follows: paymt(0.12/12,30 X 12,90000,0) where 0.12/12 is the monthly interest rate, 30 X 12 is the number of payments, 90000 is the amount of principal, and 0 is no balloon. Note: The last argument would be non-zero only if there were a balloon payment at the end of the 30 years. 100 Eureka: The Solver Owner's Handbook By default, Eureka assumes that payments are made at the end of each interval rather than at the beginning. This is because the settings parameter finanmode has the following default value: finanmode = end You can change this setting to begin either from the Options/Settings pull-down menu or by directive in the equation file. Because banks often have their own unique methods for computing interest rates, Eureka's built-in financial functions may vary slightly from any particular bank's methods. User-defined Functions Eureka deals primarily with variables and the relationships among them. It is often convenient to think of variables as functions of one another, and Eureka does so with its set of built-in functions. You can add to Eureka's set of functions by defining your own for a given equation file; these are known as user-defined functions. The following discussion describes the relationship between variables and functions and then explains various aspects of user-defined functions. Relationship of Variables and Functions in Eureka Consider an equation with one or more variables in which it is possible to solve for one variable in terms of the others. This solved variable could, in effect, be expressed as a function of the other variables. However, to Eureka, it is still just a variable. For example, the function sin(x) where + cos(x) can be maximized by maximizing f f = sin(x) + cos(x) Here, f can be conceived as a variable that happens to be related to x. In another example, the equation x2 + y2 = 25 might be transformed by Eureka internally to either x = sqrt(25 _ y2) or The Equation File: Syntax, Directives, and Functions 101 In other words, x can be considered as a function of y, or y as a function of x. To a user, however, x and yare simply related variables. Eureka recognizes two types of functions: • built-in functions (such as cos and sin) • user-defined functions Built-in functions have no values unless arguments are supplied, and it makes no sense for Eureka to solve for them. User-defined functions describe configurations of variables that you wish to capture in shorthand fashion. Note: User-defined functions affect only the equation files in which they are defined. Writing User-defined Functions The symbol : = Hags a user-defined function as in trig(x) := sin(x) + cos(x) This example defines a new function called trig that acts in the same way as a built-in function. The variable x is a dummy variable that merely serves to define the function and is unrelated to any other x that may occur elsewhere in the equation file. When an equation file contains this user-defined function, trig(x), you can insert tri 9(7) or trig(a * b + 3) into an equation rather than inserting the lengthier expressions (sin(7) + cos(7» or (sin(a * b + 3) + cos(a * b + 3» The lexical rules regarding function names are the same as those for variable names (begin with a letter or certain special characters, limited to 60 alphanumeric characters). Although you can give the same name to a variable and a function, this practice is not recommended. You can define a function of several variables (or even no variables). 102 Eureka: The Solver Owner.'s Handbook For instance, defining a function by gas(x,y) := x +y * + 10 Z causes the expression 2 2 * (3 + (a + 1) * Z * gas(3,a + 1) to be treated as + 10) For most equation solving purposes, user-defined functions are essentially editing conveniences. However, several Eureka features require user-defined functions: • making plots or tables • finding roots of polynomials • solving differential calculus problems Error Messages When writing user-defined functions, keep in mind that Eureka returns an error if: 1. You use a function without defining it somewhere in the equation file. 2. You redefine a function in the equation file. 3. A function calls for a certain number of arguments, but you use it with a different number of arguments than those defined, as in f(x,y,z) := x~2 - y~2 + z~2 f(a,b) = 100 4. You define a function in a circular way, as in f(x) := x~2 g(x) := f(x + 3 * g(x) + 1) - 7 * x The Equation File: Syntax, Directives, and Functions 103 104 Eureka: The Solver Owner's Handbook c H A p T E R 6 Worked Examples This chapter provides several worked problems. These examples are designed to demonstrate Eureka's power in solving a variety of mathematical problems. The problems are contained on your distribution disk under the following file names: Problem 1 Quick Demo Problem 2 Loan Payoff Problem 3 Ladders in an Alley Problem 4 Distribution of Line Charges Problem 5 Radioactive Decay Problem 6 Least-Squares Fit Problem 7 Motion of a Projectile Problem 8 Charged Masses Problem 9 Using the Poly Function Problem 10 Maximization (House Design) Problem 11 Maximization (Linear Program) Problem 12 Equity in a House Problem 13 Probability in a Card Game CLASSICS.EKA, DERINT.EKA, MAXMIN .EKA, COMPLEX.EKA CARLOAN .EKA LADDERS.EKA LINECHG.EKA DECAY.EKA LSQRFIT.EKA PROJCTL.EKA MASSCHG.EKA POLYNOM.EKA HOUSE.EKA LINPROG.EKA EQUITY.EKA CARDS.EKA 105 Problem 14 Income Distribution (Integ Function) Problem 15 Chemical Calculations INTEC.EKA CHEMBAL.EKA, CHEMEQl.EKA, CHEMEQ2.EKA, CHEMEQ3.EKA The first series of problems demonstrate Eureka's basic features: • solving systems of nonlinear (and linear) equations • solving and plotting user-defined functions • calculating a least-squares fit • solving transcendental functions Later problems incorporate some of Eureka's special capabilities and built-in functions: • maximizations and minimizations • definite integrals • factorials • polynomial solutions • financial calculations If you want to get an idea of how to set up and solve problems with Eureka, or if you want to see how we have incorporated some of Eureka's capabilities and features into an assortment of problems, we suggest you look over and study some or all of the examples in this chapter. They include a sampling from a variety of fields and topics: chemistry, finance, probability, physics, electrostatics, and economics, to name a few. Each worked problem consists of: • a statement of the problem • an explanation of the model equations • a copy of the report generated after Eureka solves the problem All of the reports contain the equation file and the contents of Eureka's Solution window for the problem. Some reports also include the contents of Eureka's Verify window, comparing the solution values with the original equations and constraints. Other reports include a two-column table (generated with Eureka's Graph/List command) that compares values of a specified function from the equation file with the function variable. Remember - Eureka: The Solver is designed to help you solve a whole world of mathematical problems: This chapter presents but a small sample of what Eureka can do! 106 Eureka: The Solver Owner's Handbook Problem 1 A Quick Demonstration of Eureka's Mathematical Capabilities This problem (CLASSICS.EKA, DERINT.EKA, MAXMIN.EKA, and COMPLEX.EKA on the distribution disk) highlights some of Eureka's capabilities. It demonstrates how Eureka: • quickly solves basic mathematical identities • calculates and plots definite integrals and derivatives • solves integral problems • maximizes or minimizes functions over a given range • solves for complex variables This example is broken into four sets of problems, covering four topics: • classic identities • integrals and derivatives • minimizations and maximizations • complex variables In the first set of problems, Classic Identities, Eureka solves a few trigonometric and transcendental equations. You already know the solutions to the equations; our purpose in demonstrating how Eureka handles them is to familiarize you with how fast (and how well) the equation solver works. The second set of problems, Derivatives and Integrals, illustrates how you use Eureka's built-in integ and deriv functions, and shows how to set up user-defined functions (delimited with the: = symbol) so Eureka can plot them. Maximizations and Minimizations, the third set of problems, demonstrates how to incorporate Eureka's max and min directives in your equation files. Maximizations (and minimizations) are not an easy task for a computer program, especially when the variables are subject to certain constraints. However, as you will see in this set of problems, Eureka solves them with ease. The final set of problems, Complex Variables, shows how Eureka solves for both the real (re) and imaginary (im) part of problems such as y2 + 1 = 0 and e iXpi = theta. Worked Examples 107 Statement of the Problem Suppose you were given the following (admittedly simple) problems to solve with a hand-held calculator. How long would it take you to solve for the given variables, verifY that your answers were appropriate, and plot the functions requested? Classic Identities 1. Solve for x where In(x) = 1; plot the function for 0.1 < x < 5 =1 =1 2. Solve for y where sin(y/2) 3. Solve for z where tan(z/4) 4. Plot F(a) = sin(a)/a for -10 < a < 10 Solve for j where F(j) = 1/2 Derivatives and Integrals 5. Given the following functions A, B, C, and D: A(x) = (COS(X))2 B(x) = sin(x) C(x) = cos(x/2) D(x) = A(x) + B(x) + C(x) a. Plot D(x) for -10 < x < 10 b. Calculate the integral of A(x) for 0 < x < 5 c. Calculate the integral of D(x) for 0 < x < 5 d. Find a zero of dA(x)/dx e. Find a zero of dB (x)/dx £ Find a zero of dC(x )/dx g. Find two zeroes of dD(x)/dx: one negative, one non-negative Maximizations and Minimizations 6. For the function F(a) in problem 4: a. Find the minimum value of F(a) over the range 0.1 < a < 10 b. Find the maximum value of F(a) over the range -10 < a < - 4 108 Eureka: The Solver Owner's Handbook 7. At what point x > 0 does the integral of [sin(a) da] reach the value I? 8. Maximization: at what point x > 0 does the integral of [sin(a) da] reach a maximum value? 9. Maximization: at what point c on the range - 4 to 4 does sin(c) reach a maximum? 10. Minimization: at what point a on the range - 4 to 4 does sin(a) reach a minimum? 11. Plot sin(a) over the range - 4 < a < 4 Complex Variables 12. Solve for the variables x, y, and theta in the following equations: x2 = -1 y2 + 1 = 0 exp(x X pi) = theta (pi = 3.1415926) Would you look forward to solving, evaluating and plotting these functions with only the aid of a calculator and a pencil? Probably not. Not only can Eureka solve for the given variables; the equation solver can also plot the requested functions and evaluate the solutions (flagging any that don't satisfy the given constraints). Not only that, Eureka will generate complete, formatted written reports giving the original problems, the solutions, the evaluations and the plots. And you never have to lift a pencil. Equations For the given problems, Eureka will solve these corresponding equations: 1. L(x) : = In(x) L(A) = 1.00000 2. S(x) : = sin(x/2.0) S(B) = 1.00000 T(x) : = tan(x/4.0) T(C) = 1.00000 F(x) : = sin(x)/x F(j1) = 0.50000 : j1 < 0 F(j2) = 0.50000 : j2 > 0 3. 4. Worked Examples 109 5. A(x) : = (COS(X))2 B(x) : = sin(x) C(x) : = cos(x/2) D(x) : = A(x) + B(x) + C(x) D(ddl) = < ddl < 10 D(dd2) = dd2 : = - 1 °°::° Ap(x) : = Bp(x) : = Cp(x) : = Dp(x) : = deriv((cos(x))2, x) deriv(sin(x), x) deriv(cos(x/2), x) deriv(D(x), x) °° °° Ap(ap) = Bp(bp) = Cp(cp) = Dp(dp) = Dintl = integ(D(x),x,0,5) Aintl = integ(A(x),x,0,5) Eureka will solve for ap, bp, cp, dp, ddl, dd2, Aintl, and Dintl. Those equations containing the: = symbol are "user-defined" functions; refer to Chapter 5 for more information about these and other special Eureka functions. 6. F(z) : = sin(z)/z $ min (K) = F(z) OJ < .z < 10 K $ max (L) L = F(z) -10 < z < -4 7. 1.000 = integ(sin(a), a, 0, w/2) w/2 =x 8. $ max (1) 1 = Maxlnt(x) Maxlnt(a) : = integ(sin(a), a, 0, x) 9. $ max (z) z = S(c) S(c) : = sin(c) -4 < c < 4 d = 2c 10. $ min (y) y = S(a) S(a) : = sin(a) 110 Eureka: The Solver Owner's Handbook -4 < a < 4 b = 2a 11. 12. S(a) : = sin(a) x2 = -1 y2 + 1 = 0 exp(x X pi) = theta: pi = 3.1415926 To solve these problems, select one of the equation files (use File/Load from the main menu), then select Solve. Equation Files and Solutions The print-outs that follow show the equation files and Eureka's solutions for the examples in this problem. Graphing the Functions Eureka also offers the ability to plot functions of one variable. To graph anyone of the functions in these problems: • Solve the equation file. • Select the Graph option from the main menu. • From the Graph menu, choose Plot. • Select one function from the mini-menu listing of all user-defined functions. • Enter values for the left and right x axis endpoints. Eureka will automatically plot the function. Generating Reports You can also direct Eureka to generate your own written reports; after solving an equation file (also evaluating the solutions, if you choose) and plotting a function, select Report from the main menu. Eureka will place the report on screen, write it to a disk file, or send it directly to your printer, as you choose. Note: We used Borland's SideKick to capture the "zoomed" plots and insert them in the reports that follow. Worked Examples III ******************************************************************* Eureka: The Solver, Version 1.0 Thursday February 12, 1987, 10:13 am. Name of input file: CLASSICS.EKA ******************************************************************* These examples show how Eureka quickly solves some classic problems L(x) := In(x) L(A) = 1.00000 plot L(x) for 0.1 < x < 5 Sex) := sin(x/2.0) S(B) = 1.00000 T(x) T(C) := tan(x/4.0) = 1. 00000 After solving this equation file, plot the function F(x) for -10 < x < 10 F(x) := sin(x)/x F(j1) 0.50000 j1 < 0 F(j2) = 0.50000 : j2 > 0 ******************************************************************* Solution: Variables Values A 2.7182818 B 3.1524423 C 3.1416042 j1 -1.8954943 j2 1.8954943 Maximum error is .000014714291 ************************************************************************* 112 Eureka: The Solver Owner's Handbook L 1.87 -c~·------------~I .111.5.11 -2.51 ******************************************************************* T 18.2 .. . . . . t--.....••--. - ...... ---. - ......•• __._-.....•• __ .--.. ...•• --+ -38.8 • • •• • 38.8 -10.2 ******************************************************************* Worked Examples 113 r e - - . . .e . l·--·-----+------·--·l -11.1 - e 11.1- -.289 ******************************************************************* r 1.16~ .- f-.---- -18.8 -----'.-+ 111.8· ******************************************************************* 114 Eureka: The Solver Owner's Handbook ******************************************************************* Eureka: The Solver, Version 1.0 Thursday February 12, 1987, 11:23 am. Name of input file: DERINT.EKA ******************************************************************* These problems demonstrate how to use Eureka's INTEG and OERIV functions, and include examples of plotting the functions A(x) : = (cos (x»~2 Sex) := sin(x) C(x) . - cos (x/2) O(x) := A(x) + Sex) + C(x) 0(dd1) 0 < dd1 < 10 D(dd2) = dd2 := -1 °: ° Ap(x) := deriv«cos(x»~2, x) Sp(x) .- deriv(sin(x), x) Cp(x) .- deriv(cos(x/2), x) Op(x) .- deri v(0 (x), x) Ap(ap) Sp(bp) Cp(cp) Op(dp) Ointl Aintl 0 °°0 integ(D(x),x,0,5) integ(A(x),x,O,5) ******************************************************************* Solution: Variables Values Aintl 2.3639947 ap 1.5707964 bp 1.5707964 cp -1.8097622e-10 dd1 3.6612084 dd2 -2.2077911 Ointl dp Maximum error is Worked Examples 4.2772768 -349.75174 .067092381 115 ******************************************************************* Evaluation of formulas: Formulas Values D(dd1) -6.1014156e-08 .00000000 difference (error) -6.1014156e-08 o o dd1 difference .00000000 3.6612084 -3.6612084 10 dd1 difference = 10.000000 3.6612084 -6.3387916 D(dd2) = 4.0720587e-08 .00000000 difference (error) = 4.0720587e-08 o dd2 -2.2077911 Ap(ap) 8.7422780e-08 .00000000 difference (error) = 8.7422780e-08 o Bp(bp) = -4.3711390e-08 .00000000 difference (error) = -4.3711390e-08 o Cp(cp) o = 4.5244055e-11 .00000000 difference (error) = 4.5244055e-11 Dp(dp) -.067092381 .00000000 difference (error) -.067092381 o Dintl integ(D(x) ,x ... difference = 4.2772768 4.2772768 .00000000 Aintl integ(A(x) ,x ... difference = 2.3639947 2.3639947 .00000000 Maximum error is .067092381 116 Eureka: The Solver Owner's Handbook ******************************************************************* • 1-. ------. -- 4-------.--+ -11.1 - - -1. 92 - 18.8 - ******************************************************************~ D ~.-----.----- -18.8. .--~r------------------- ------------+1 18.8 -2.81 ******************************************************************* Worked Examples 117 ******************************************************************* Eureka: The Solver, Version 1.0 Thursday February 12, 1987, 11:48 am. Name of input file: MAXMIN.EKA ******************************************************************* These examples illustrate how to use Eureka's $max and $min directives to maximize and minimize any variable that is defined to be the result of a given function. One of these examples also demonstrates how to solve an integral equation with Eureka. NOTE: To solve anyone of these examples, remove the ";" symbol at the beginning of each equation line for that example. Eureka will only solve for ONE $min or $max in an equation file. To solve a different example, first reinsert the ";" symbols from the previously-solved example. Part 1: A well-known example: F(z) ::: sin(z)/z Problem I-a: $min (K) K :: F(z) 0.1 < z < 10 Problem I-b: $ max (L) L :: F(z) -10 < z -4 Part 2: Integral of Sin(x) dx Problem 2-a: Solving an integral equation: at what point x > 0 does the integral of [sin(a) da] reach the value I? (Let w/2 equal x.) 1.000 :: integ(sin{a), a, 0, w/2) w/2 :: x Problem 2-b: Maximization: at what point x > 0 does the integral of [sin(a) da] reach a maximum value? $max (I) I :: Maxlnt(a) Maxlnt(a) ::: integ(sin(a), a, 0, x) Part 3: Max and Min of Sin{x) 118 Eureka: The Solver Owner's Handbook Problem 3-a: Maximization: at what point c on the range -4 to 4 does sin(c) reach a maximum? (Note: The function S(c) is included for plotting purposes - Eureka does not need it to solve for c or z.) $max (z) z = S(c) S(c) := sin(c) -4 < c < 4 d = 2*c Problem 3-b: Minimization: at what point a on the range -4 to 4 does sin(a) reach a minimum? (Note: The function Sea) is included for plotting purposes - Eureka does not need it to solve for a or y.) $min (y) y = Sea) Sea) := sin(a) -4 < a < 4 b = 2*a ******************************************************************* Solution: (I-a) Variables Values K -.21723363 z 4.4934066 Confidence level 98.9% All constraints satisfied. ******************************************************************* r 1·16t - .-. .-. t·-·---+-----·-·t -11.1 - - 11.1- -.289 ******************************************************************* Worked Examples 119 r 1.1rtf- .- __----C-.-+ 1------' 11.1· -II.' -.311 ******************************************************************* Solution: (2-a) Variables Values w 3.1415882 x 1.5707941 Maximum error is 2.2415074e-06 ******************************************************************* Evaluation of formulas: Formul as Values 1.000 integ(sin(a) ••• difference (error) w/2 x difference 1.0000000 .99999776 2.2415074e-06 1.5707941 1.5707941 .00000000 Maximum error is 2.2415074e-06 ******************************************************************* 120 Eureka: The Solver Owner's Handbook Solution: (2-b) Variables Values 2.0000000 3.1415927 x Confidence level 100.0% All constraints satisfied. ******************************************************************* Solution: (3-a) Variables Values c 1.5707963 d 3.1415927 z 1.0000000 Confidence level 98.7% All constraints satisfied. ******************************************************************* s 1. 111 -- - 4---.---- ---_.-+ -4.11- 4. • • -1.11 ******************************************************************* Solution: (3-b) Variables Values a -1.5707963 b -3.1415926 y -1.00000000 Confidence level = 98.7% All constraints satisfied. ******************************************************************* Worked Examples 121 ******************************************************************* Eureka: The Solver, Version 1.0 Thursday February 12, 1987, 1:35 pm. Name of input file: COMPLEX.EKA ******************************************************************* This example demonstrates how Eureka solves for complex variables $ complex = yes theta exp(x*pi) pi = 3.1415926 ******************************************************************* Solution: Variables re pi im pi Values 3.1415926 .00000000 re theta im theta -1.00000000 5.3589793e-08 re x im x 6.1257423e-17 1.0000000 re y 1.078372ge-27 im y 1.0000000 ******************************************************************* 122 Eureka: The Solver Owner's Handbook Problem·2 Loan Payoff This problem (CARLOAN.EKA on the distribution disk) demonstrates how to set up equations for a financing situation and use Eureka's Graph/List capability to generate a list of function values vs. the variable, time. Statement of the Problem The car you purchased cost $10,900. When you bought it, you made a down payment of $2,900 and financed the rest ($8,000) for 48 months at 13.5% annually. Now it is a year and a half later, and you are thinking about buying a new car. • How much would it cost to pay off the loan? • How long should you continue to make payments? These questions can be answered by a few mathematical calculations. To start with, define some terms: Pay = periodic payment Int = interest per period Prin = principal amount borrowed ntot = the total number of periods for the original loan n = the number of periods actually paid before payoff FofP = the future value of the principal at month n, given a present value FofA = the future value of a payment series at month n, given a periodic payment Using these terms, you first need to calculate the original payment. You could use the built-in function paymt, but this time you are going to use the actual mathematical formula to see how it is done. Once you know the monthly payment, you need to find the remainder of the principal at a given month, then subtract the value of the sum of remaining payments. To do this, use the time at which you borrowed the money as a reference point. From this point, find the future value of the principal (FofP) at the time of the payoff and the future value of the payment series (FofA) at the time of the payoff. After subtracting FofA from FofP, you have the payoff amount. By making the formula for PayOff a function of n, you can generate a list of 20 possible payoff points with the Eureka Graph/List function. This can help you make your decision. Worked Examples 123 Equations The first step when setting up the equations for this problem is to define the variables and assign values to them. The variables are: Monthly interest (IntM) is 13.5 annual percentage rate divided by 12 months per year: IntM = (13.5/100)/12 = 0.01125 Principal, the amount borrowed: Prin = 8,000.00 Total number of payments for the loan: ntot = 48 Monthly payment (to be calculated from given variables): PayM The equations required for solving the two questions posed by this problem are: Calculate the monthly payment: PayM = Prin X IntM X ((1 + IntMttot - 1)/((1 + IntM) -1) Calculate the future value of the original loan amount at the specified time n, where n indicates one of the 48 months: FofP(n) : = (Prin X (1 + IntM)n) Caiculate the future value of the payments at the same specified month, n: FofA(n} = (PayM X ((1 + IntMt -1)/IntM) (Note that these two calculations are made from user-defined functions.) Finally, for a given month, n, calculate the Payoff: What is the principal that remains outstanding after the payment n? Pay()jJ{n} : = FofP(n} - FofA(n) Equation File and Solution The equation file and solution for this problem appear in the print-out that follows, along with the table generated using Eureka>s Graph/List command. The table lists the function, Payoff, vs. months 15 through 34. 124 Eureka: The Solver Owner's Handbook ******************************************************************* Eureka: The Solver, Version 1.0 Thursday November 6, 1986, 5:47 am. Name of input file: CARLOAN.EKA ******************************************************************* $ finansmooth = yes IntM .01125 Prin 8000.00 ntot 48 n = (l+IntM)-ntot PayM = Prin * IntM * «l+IntM)-ntot) / ({l+IntM)-ntot - 1) FofP{n) := (Prin * (ltIntMrn) FofA(n) : = {PayM * ({l+IntMrn - 1) / IntM) PayOff{n) := FofP{n) - FofA{n) ******************************************************************* Solution: Variables Values IntM .011250000 ntot 48.000000 PayM 216.61058 Prin 8000.0000 n 1.7108410 ******************************************************************* Worked Examples 125 List of function values. n PayOff(n) 15.000000 16.000000 17.000000 18.000000 19.000000 20.000000 21.000000 22.000000 23.000000 24.000000 25.000000 26.000000 27.000000 28.000000 29.000000 30.000000 31.000000 32.000000 33.000000 34.000000 5943.7392 5793.9957 5642.5676 5489.4359 5334.5814 5177.9849 5019.6267 4859.4869 4697.5455 4533.7823 4368.1768 4200.7082 4031.3556 3860.0977 3686.9133 3511.7804 3334.6774 3155.5819 2974.4716 2791.3239 ******************************************************************* 126 Eureka: The Solver Owner's Handbook Problem 3 Ladders in an Alley: Solving Simultaneous Nonlinear Equations This problem (LADDERS.EKA on the distribution disk) demonstrates Eureka's ability to quickly and accurately solve simultaneous nonlinear equations. Statement of the Problem You have two ladders, one 35 feet long, the other 45 feet long. You need to arrange them in an alley between two skyscrapers so that they cross 10 feet above the ground. How wide should the alley be? Equations First, assign variables to all the known and desired quantities: x = width of the alley a = height of the 45-foot ladder at the wall b = height of the 35-foot ladder at the wall h = the point on the ground immediately below wher~ the ladders meet. Equation File and Solution The print-out that follows shows the equation file and solution for this problem. Notice the initialization values for the four variables. If you have a general idea about the region where the solution(s) lie, it's a good idea to start Eureka off within or close to that region. Eureka solves problems such as this one by iteration, getting closer and closer to an error-free solution. However, the default value for all initializations is 1. When you are confident that the solution(s) are significantly larger (or smaller) than 1, you can initialize the variables with some plausible values, then tell Eureka to Solve. This way, Eureka will not spend a lot of time covering unproductive ground. Worked Examples 127 When you are confident that the solution(s) are significantly larger (or smaller) than 1, you can initialize the variables with some plausible values, then tell Eureka to Solve. This way, Eureka will not spend a lot of time covering unproductive ground. If Eureka's first round of iterations produces an unsatisfactory solution, you can use other techniques to get the program re-solving in a different region. These other techniques are covered in Chapters 3 and 5 of this manual. 128 Eureka: The Solver Owner's Handbook ******************************************************************* Eureka: The Solver, Version 1.0 Thursday November 6, 1986, 6:47 am. Name of input file: LADDERS.EKA ******************************************************************* This example demonstrates how Eureka can solve a system of nonlinear equations iteratively, after you initialize the iteration. System of nonlinear equations: x~2 x~2 + a~2 + b~2 = 45~2 = 35~2 y/10 = x/b (x-y}/10 = x/a Initializations: x := 10 y .- 10 a := 20 b := 20 constraints: a>O b>O x>O x>y y>O Exact solutions are: x = 31.817459 , Y and a = 31.822151 , b = 14.582500 21.818934 ******************************************************************* Solution: Variables Values a 31.822151 b 14.582500 x 31.817459 y 21.818934 Maximum error is 6.8212103e-13 ******************************************************************* Worked Examples 129 Problem 4 Distribution of Line Charges This problem in electrostatics (LINECHG.EKA on the distribution disk) illustrates Eureka's ability to solve several simultaneous nonlinear equations. Statement of the Problem A system of several parallel positive line charges is aligned along an axis within an electric field so that the total force exerted on each line charge is zero~ As shown in Figure 6-1, a line charge is fixed at the origin, lying on the z axis. Three free-line charges lie on the x - z plane (positive x) parallel to the first. The electric field is constant in magnitude and aligned in the negative x direction. From physics, you know that the force on a charge is proportional to the electric field and like charges repel. The problem is to distribute the line charges along the x axis so that the sum of forces on each of the three free charges equals zero. ¢. I Y axis I Figure 6-1 Line Charges in Equilibrium This problem requires you to solve a set of nonlinear equations. (An interesting note: The same equations which govern this problem also apply to the distribution of dislocations under uniform shear stress in the study of the fracture of solids.) 130 Eureka: The Solver Owner's Handbook Equations From basic electrostatic theory, you know that the electric field produced by a line charge varies • inversely with the first power of the distance from the line, and • linearly with the charge density of the line The field produced by a line charge at some point p (or infinitesimal length p) has the form EOinecharge) = 2l!r where 1 = charge density along the line (assumed to be invariant) and _ r = distance from point p to the line The force F from a line charge at point p is the field at p multiplied by the probe's line charge density: F = 2l!r X 1 In this example, you have an array of line charges. Each line charge produces a field that affects all its neighbors; the line charge, in tum, is affected by all its neighbors and the constant field, E(constant). The force acting on any of the movable lines, Fi, is the sum of each other line's individual force acting on that line (where Xi - Xj = r(i - j)): Fa = 1 X Ea(l) = 1 X (2l!r(a - 0) + 2l!r(a - b) + 2l!r(a - e)) Fb = 1 X Eb(l) = 1 X (2l!r(b - 0) + 2l!r(b - a) + 2l!r(b - e)) Fe = 1 X Ee(l) = 1 X (2l!r(e - 0) + 2l!r(e - a) + 2l/r(e - b)) Fi = (sum Une i, 0 S 2 j S 3] 2l /(X j - X)) - E(constant) X 1 for each infinitesimal length of the line (assuming each line has the same charge density, l). To find the solution to this problem, derive a set of coupled equations that must be solved if the force on each line charge is zero. Using the notation in Figure 6-1, derive the following equations: lI(a - b) + lI(a - e) + l/a = E(constant)/l2 lI(b - a) + lI(b - e) + lib = E(constant)/l2 lI(e - a) + 1/(e - b) + lie = E(constant)/l2 For simplicity, select values for E(constant) and 1 such that E(constant)/l2 = 1, then solve the resulting set of equations. Worked Examples 131 Equation File and Solution The print-out that follows shows the equation file and solution for this problem. Notice that although Eureka generates solutions for a, b, and c, the order of the variables is not the same in the solution as it is in the equation file. This is because the equations are symmetrical with respect to interchange of the variables, and Eureka's solution process happens to produce the given order. 132 Eureka: The Solver Owner's Handbook ******************************************************************* Eureka: The Solver, Version 1.0 Saturday November 8, 1986, 1:28 pm. Name of input file: LINECHG.EKA ******************************************************************* This example demonstrates Eureka's ability to solve a system of nonlinear equations. These equations represent the net force on each line charge in a system of 4 parallel, coplanar line charges aligned in an electric field. l/(a-b) l/(a-c) l/a l/(b-a) l/(b-c) l/b l/c l/(c-b) l/(c-a) ******************************************************************* Solution: Variables Values a b c 3.8793852 .46791111 1.6527036 Maximum error is -1.1102230e-15 ******************************************************************* Worked Examples 133 Problem 5 Radioactive Decay In this problem (DECAY.EKA on the distribution disk), you use Eureka to solve a transcendental function of time. Statement of the Problem Radioactive elements, such as uranium, decay according to an exponential law. This means that the rate of decay is constant and proportional to the amount of the material. In the case of uranium-238, a given amount of uranium will decay into lighter elements at a rate such that half of it will have decayed every 10 million years. If 100 kilograms of uranium-238 sits for 10 million years, at the end of that period only 50 kilograms of radioactive material will remain. At the end of another 10 million years, 25 kilograms of radioactive uranium will be left. This time period for the uranium to be reduced to half its original amount is called the half-life. Given 100 kilograms of uranium-238, how much radioactive material will remain after 1,000,000 years? Equations The rate of mass decrease as a function of time is proportional to the mass of radioactive substance at that time. Symbolically, in terms of differential calculus, dm/dt = -km where m = mass t = time k = proportionality constant According to the laws of calculus, the only functions to satisfy this formula are of the form m = a X exp( - kt) where a is some constant. HalfLife and k are related by the formula k = In(2)IHalfLife 134 Eureka: The Solver Owner's Handbook Mathematically, the following formula gives the amount of a radioactive substance at any particular time: Mass(time) = InitMass X 2(-timeIHa lfLife) where InitMass = initial mass (or weight) of the substance Mass (years ) = mass of the substance after a given time; in this case, years HalfLife = half-life of the substance Equation File and Solution The print-out that follows shows the equation file and solution for this problem. Because a user-defined function is entered for mass as a function of years, you can use Eureka's Graph/List capability to generate a list of the remaining radioactive mass vs. time. This list, covering the period from 500,000 to 5,250,000 years elapsed in increments of 250,000 years, completes the print-out. Worked Examples 135 ******************************************************************* Eureka: The Solver, Version 1.0 Wednesday January 28, 1987, 11:30 am. Name of input file: DECAY.EKA ******************************************************************* This problem demonstrates how to solve a user-defined transcendental function of time. The problem is to calculate MassUnknwn, the mass of 100 kg Uranium 238 after one million years. Mass(Years) := InitMass * 2 InitMass = 100 Half Life = 10 * 1e6 MassUnknwn = Mass(le6) A ( - Years / Half Life) ******************************************************************* Solution: Variables Values Half Life 10000000. InitialMass 100.00000 MassUnknwn 93.303299 List of function values. Mass x 500000.00 750000,00 1000000.0 1250000.0 1500000.0 1750000.0 2000000.0 2250000.0 2500000.0 2750000.0 3000000.0 3250000.0 3500000.0 3750000.0 4000000.0 4250000.0 4500000.0 4750000.0 5000000.0 5250000.0 Mass(x) 96.593633 94.934212 93.303299 91. 700404 90.125046 88.576752 87.055056 85.559503 84.089642 82.645032 81.225240 79.829839 78.458410 77 .110541 75.785828 74.483873 73.204285 71.946679 70.710678 69.495911 ******************************************************************* 136 Eureka: The Solver Owner's Handbook Problem 6 Least-Squares Fit This problem (LSQRFIT.EKA on the distribution disk) demonstrates how you can direct Eureka to find values for the constants in a function so that the function closely fits predetermined data points. You do this by limiting the program's substitution level (using the substlevel directive) during a Solve. Statement of the Problem You have a set of points (x,y) from empirical data, and you know that the data ideally should fit a curve of the form y = f(x) = e(Ax N + B) where A, Band N are constants. You need to determine those values for A, B, and N that produce the best-fit curve for your data. The data points are: Xn 1 2 3 4 5 Yn 1.49 2.35 4.26 8.59 19.80 Equations Normally, in the process of solving a problem, Eureka makes symbolic substitutions in the equation file. However, you can limit the number of times the program will substitute for terms in the file by adding a substlevel directive to the file. Setting substlevel to 0 directs Eureka to make no substitutions; this is the technique you will use here. In this problem, you need to tell Eureka three things: • the general form of the equation • the data points (x,y) • the level of substitution (none) Worked Examples 137 This user-defined function represents the general form of the equation: f(x) = exp(AxN + B) These equations give Eureka the empirical data points with the y values given as functions of x: f(l) f(2) f(3) f(4) f(5) = 1.49 = 2.35 = 4.26 = 8.59 = 19.80 This directive sets the level of substitution to zero: $ substlevel = 0 For Eureka to successfully find a unique solution for the problem, the number of equations (data points) must equal or exceed the number of unknown constants in the user-defined equation. Notice that there are five equations (f(1),f(2),f(3),f(4), andf(5)) in this problem and three unknowns (A, B, and N). If you do not set the substitution level to zero, Eureka will solve for those values of the three unknowns (A, Band N) which satisfy the first three equations (f(1) = 1.49,f(2) = 2.35, andf(3) = 4.26) without looking at the equations for f(4) and f(5). With the substitution level at zero, Eureka looks at all the given equations (f(x) = y) and automatically minimizes the least-squares function F where F(x,y) = ((f(x) - y/ + (f(x 2 ) - Y2)2 + (f(x 3 ) + (f(x) - y/ + (f(x s) - y/) - Y3)2 This function, F, represents the sum of the vertical (ordinate) distances of the points (xn'y) from the points (x(n),f(x(n)) on the plot of the sought function. Minimizing this sum is equivalent to finding the best-fitting curve to match the plotted data. You could enter up to 20 data points into a problem of this type, although the computations involved would make the Solve process quite lengthy. Ten data points (or less, where possible) is a more practical limit. 138 Eureka: The Solver Owner's Handbook Additional Techniques: Inputting Data From Another Program This is the type of problem where you can make good use of Eureka's ability to solve problems created in another software environment. For example, if you have a Turbo Basic program that generates the points xn'Y n used in this problem and those point values are stored in two arrays, X(I) and Y(I), you can write a Turbo Basic program that creates the Eureka equation file for this problem. The following Turbo Basic program will create a file called PROBLEM and write the data from the arrays into the file in a format suitable for Eureka to solve. OPEN "0", 1, "PROBLEM" PRINT #1, "$ substlevel = 0" PRINT #1, "f(x) : = exp(A * (x~N) + B)" FOR 1= 1 TO 5 PRINT #1, using "\\#\ \#11.##"; "f("; X(I);"):="; Y(I) NEXT I CLOSE #1 END Equation File and Solution The equation file, the solution, and a list of calculated values for this problem appear in the print-out that follows. Worked Examples 139 ******************************************************************* Eureka: The Solver, Version 1.0 Sunday November 9, 1986, 3:33 pm. Name of input file: LSQRFIT.EKA ******************************************************************* This example demonstrates Eureka's ability to find values for constants in a function that make the function best fit empirical data. Because the equation file includes a $ substlevel = 0 directive, Eureka will perform a Least Squares Fit to find the function (of the required form) that best matches the points (x, f(x» given. In this example, the function is f(x) : = EXP(A * x~N + B) where the ideal solution is A = 0.25, B N = 1.5 f(x) .- EXP(A * f(1) f(2) f(3) f(4) f(5) x~N 0.15 and + B) 1.49 2.35 4.26 8.59 19.01 $ substlevel = 0 ******************************************************************* Solution: Variables Values A .25247836 B .14432763 N 1.4951057 Maximum error is -.00011769096 ******************************************************************* List of function values. x f(x) .00000000 1.0000000 2.0000000 3.0000000 4.0000000 5.0000000 6.0000000 7.0000000 8.0000000 9.0000000 10.000000 1.1552625 1.4870674 2.3537876 4.2597419 8.5891228 19.010163 45.722423 118.62542 330.10583 980.76072 3099.2752 ******************************************************************* 140 Eureka: The Solver Owner's Handbook Problem 7 Motion of a Projectile in a Gravitational Field with Air Resistance This is a problem in elementary dynamics (PROJCTL.EKA on the distribution disk). You are to find the motion of a projectile in the atmosphere; air resistance and gravity eventually bring the projectile to earth. This problem demonstrates Eureka's ability to solve transcendental functions as well as automatically convert units of measurement. Statement of the Problem Consider a projectile that is fired into the air at an angle A with an initial velocity Vinit. If you neglect air resistance, it is easy to show that the path of the projectile is a parabola. With air resistance, the path is a more complicated function (which you derive below). Eventually, the projectile will hit the ground after it has travelled a distance R (the range). In this problem, you find the range (R) and the total flight time (T). Equations You will consider the x and y coordinates separately. The initial conditions are: x(t) = 0 when t = 0 y(t) = 0 when t = 0 x'(O) = Vinit(O)cos(A) = U y'(O) = Vinit(O)sin(A) = V where x' and y' are the first time derivatives (that is, the velocity). The equations of motion are: mx" = -kmx' my" = -kmy' - mg (the y-deceleration) The first of these equations states that the x-deceleration is proportional to the velocity; this is due to air resistance. The second equation states that the y-deceleration (remember, the projectile is fired upwards) has two terms - an air resistance term and a gravitational term. The constant k, the drag coefficient, measures the strength of the resistance; g is the acceleration of gravity; and m is the mass of the projectile. Worked Examples 141 These differential equations can each be solved by integrating twice; the results are: x(t) = (U/k) X (1 - e -kt) y(t) = gt/k + ((kV + g)/k2) X (1 - e- kt ) where U and V are the horizontal and vertical components, respectively, of the projectile's initial velocity, Vinit. The total flight time, T, can be calculated from the equation for y(t) by noting that y = 0 at the end of the flight; therefore, y(t = T) = O. Solving for T, you get y(T) = 0 = (gT/k) + (((kV + g)/e) T = - ((kV + g)/gk) X (1 - e -kT) X (1 - e- kT )) The range, R, is the distance travelled in the x direction at t R = (U/k) X (1 - e- kT = T, so ) For typical ballistics problems involving artillery: • drag coefficient = 0.1 X (l/sec) • 1,000 ft/s < Vinit < 3,000 fUs (30,000 cm/sec < Vinit < 90,000 cm/sec) Let Vinit 2 = 980 cm/(sec = 50,000 cm/sec, and let A = • acceleration of gravity ) 45 degrees, then U = V = 35,355 cm/sec 1 T = ((0.1 X 35,355) + 980)/(980 X 0.1) X (1 - e-· x T) sec R = (35,355/0.1) X (1 - e -0.1 x T) cm You will use Eureka to solve for T and R. Equation File and Solution The print-out that follows shows the equation file and solution for this problem. To simplify the equation for T, we show the calculation of an intermediate constant c = (k * V + g)/(g * k) as a separate line. We also use Eureka's unit conversion directive, so the solution will be displayed in several convenient units. You can experiment with this problem and see how varying the initial parameters alters the solution. In particular, notice that a significant change to the drag coefficient changes the range of the projectile significantly, while Hight time changes slowly. For example, changing k from 0.1 to 0.001 increases the range by more than 7 times (to 15 miles), but only increases the Hight time to 71 seconds. 142 Eureka: The Solver Owner's Handbook ******************************************************************* Eureka: The Solver, Version 1.0 Saturday November 8, 1986, 2:22 pm. Name of input file: PROJCTL.EKA ******************************************************************* This example demonstrates solving a typical ballistics problem, and Eureka's unit-conversion capability. c ((k*V + g)/(g*k)) k 0.1 V 35355 V := 35355 [cm/sec] g 980 intermediate constant to simplify equation for T drag coefficient define units of initial velocity force of gravity solve for total flight time T c*(l - exp(-k*T)) T : = 40 [sec] R = (V/k)*(l - exp(-k*T)) ; solve for projectile's range R : = 1 [cm] $ units cm -> feet: x / (2.54*12) cm/sec -> feet/sec: x / (2.54*12) feet -> miles: x / 5280 feet/sec -> miles/hour: x / (5280/3600) $ end ******************************************************************* Solution: Variables Values c 46.076531 g 980.00000 k .10000000 R 349848.99 cm 11477 . 985 feet 2.1738608 miles T 45.594195 sec V 35355.000 cm/sec 1159.9409 feet/sec 790.86883 miles/hour Maximum error is 7.1054274e-15 ******************************************************************* Worked Examples 143 Problem 8 Charged Masses in a Gravitational Field This is a typical problem in electrostatics (MASSCHG.EKA on the distribution disk), illustrating Eureka's ability to find the minimum of a function. Statement of the Problem Four negatively charged mass points (for example, pith balls) are constrained to a vertical line (for example, by a non-conducting wire) in a gravitational field. The lowest ball rests on the ground. The problem is to find the equilibrium positions of the other three balls. As shown in Figure 6-2, the balls lie along the y axis and gravity acts in the negative y direction. The charges repel each other, but the balls are also pulled together by gravity. The problem is to find the positions where the gravitational pull and electrostatic repulsion balance. , r ...... 9 ball #2 t""t"\ x(3) :: Z \toii~ x(2) :: Y ball #1 6:[') 'liliiP' ball #0 fl) I x(1):: X Figure 6-2 144 Charged Balls Eureka: The Solver Owner's Handbook Equations You will solve this problem by minimizing the total potential energy of the system. The total potential energy of the system is the sum of the gravitational and electrostatic potential energies of each ball. The gravitational potential energy of a ball is equal to its mass multiplied by its distance from the ground multiplied by the acceleration of gravity. This value is zero at the ground and increases with increasing height. The electrostatic potential energy of a ball due to the charge of another is just the charge of the first ball multiplied by the electrostatic potential produced by the other one. The total potential energy of a ball is the sum of the potential energies due to each of the other balls in the system. This value is defined to be zero for the ball on the ground and positive for all other balls. Thus, if the four balls are numbered 0 to 3, with number 0 being the ball on the ground, and if • the masses are mo to m3 • the charges are eo to e3 • the distances from the origin are Xo to X3 (with Xo being zero), then the following equations express the electrostatic potential energies E of each ball: j Eo = 0 EI = eJejx I + e/(x2 E2 = e2 [ejx 2 + eJ(x2 E3 = e3[ejx3 + e/(x3 - - + e/(x3 X) + e/(x3 Xl) + e/(x 3 Xl) x)] x2)] x2)] - - The individual gravitational potential energies, G(i), have the following form: G = mjgxj j 2 where g is the acceleration of gravity (980 cm/sec ). The total potential energy of the system, T, is just the sum of all the E and G j j : 3 T= LE + j i = G j 0 In order to find the solution, you will use Eureka to find the values for x(i) that minimize T. Worked Examples 145 Assume that the balls all have a mass of 1 gm and a charge of 80 esu. To avoid the use of subscripts, we will use x, y, and z for the distances Xi. Mter combining common terms, the expression for T is: T = 980 X (x + y + z) + 6400 + 2/(z - x) + 2/(z - y)] X [lIx + lIy + liz + 2/(y - x) Since we are using the CGS system, T is measured in ergs, and x, y, and z are in centimeters. Equation File and Solution The print-out that follows shows the equation file used to minimize T. Notice that you must supply the additional constraints z > y > x > 0 and T > 0, and initial values for x, y, and z. Without these constraints and initial values, Eureka can get lost. 146 Eureka: The Solver Owner's Handbook *************************************************************** Eureka: The Solver, Version 1.0 Sunday December 7, 1986, 6:23 pm. Name of input file: MASSCHG.EKA **************************************************************** $ min (T) T = 980*(x + y + z) + 6400*(1/x + l/y + l/z + 2/(y~x) 2/(z-x) + 2/(z-y)) T>0 T := 29000 x >0 x : = 2 [em] y >x y : = 5 [em] z >y z := 10 [em] y-x > 0 z-x > 0 z-y >0 **************************************************************** Solution: Variables Values T 29900.376 x 1.5793519 em y 4.6531267 em z 9.0228156 em 96.6% Confidence level All constraints satisfied. **************************************************************** Worked Examples 147 Problem 9 Using tire Built-in Polynomial Function to Find Real and Complex Roots This problem (POLYNOM.EKA on the distribution disk) demonstrates how to use Eureka's built-in poly function to find the roots of nth degree polynomials in one variable and illustrates methods for plotting polynomial functions. The poly function can find real and complex roots, and, with the Graph/Plot option, can plot the function over a specified, definite range. Statem£nt of the Problem Consider the following functions of x: R{x) = X4 - 2x3 S{x) = x2 13x2 + 14x + 24 - 2x + 5 - T{x) = 2l + 3x + 9 U{x) = 2X4 - x3 + 13x2 V{x) = 2x 3 5x - W{x) = 3x + 5x 5 5 P{x) = x + 2 - 3x + 45 - 3x - 36 - 7x + 24 3 1 The problem is to find all roots of these polynomials. One of these functions has all real roots, two have all complex roots, and four have a combination of real and complex roots. When you use poly, it is not necessary to change the Complex setting to yes in order to find the complex roots; Eureka can solve all these problems with a single Solve command. Equations To use the poly function in Eureka, you only need to enter one equation for each polynomial function to be solved; these are user-defined functions designated by the symbol : = . Each user-defined function states, in terms that Eureka understands, that" some function of x is defined as the polynomial in x that has the following listed coefficients." 148 Eureka: The Solver Owner's Handbook When setting up a poly function, remember to: • first list the variable (typically x, but can be any valid variable) • then list all coefficients of x; start with the highest degree of x, then the next highest, and so on to the constant. ° If a polynomial has no coefficient for a given degree of x, then you must enter into the poly function for that degree's position, so that Eureka can keep track of each degree of the variable. For example, using the functions of x given above, the user-defined functions would be written as: R(x) : = poly(x,l, - 2, -13, 14, 24) S(x) : = poly(x,l, - 2, 5) T(x) : = poly(x, 2, 3, 9) U(x) : = poly(x,2, -1, 13, - 3,45) V(x) : = poly(x, 2, - 5, - 3, - 36) W(x) : = poly(x, 3, 0, 5, 0, - 7, 24) P(x) : = poly(x, 1, 0, 0, 0, 1) Graphing the Functions Eureka also offers the ability to plot functions of one variable. To graph one of the functions in this problem: • Solve the equation file. • Select the Graph option from the main menu. • From the Graph pull-down menu, choose Plot. • Select one function from the mini-menu listing of all user-defined functions. • Enter values for the left and right x axis endpoints. Eureka will automatically plot the function. Equation File and Solutions The print-out that follows shows the equation file and Eureka's solutions for the seven polynomials in this problem. Worked Examples 149 ******************************************************************** Eureka: The Solver, Version 1.0 Saturday November 8, 1986, 8:44 pm. Name of input file: POLYNOM.EKA ******************************************************************** This problem demonstrates Eureka's built-in Poly function, which finds the roots of n-th order polynomials of one variable. For polynomials in one variable with complex roots, you do not need to change the Complex setting to yes - Eureka can automatically find complex roots when using the Poly function. all real roots ; R(x) = (1*x A4 - 2*xA3 - 13*xA2 + 14*x + 24) R(x) := poly(x,l,-2,-13,14,24) ; all complex roots - Eureka solves for the real part ; of the root being < or = 1 Sex) := poly(x,l,-2,5) T(x) .- poly(x,2,3,9) ; complex and real roots U(x) := poly(x,2,-l,13,-3,45) Vex) := poly(x,2,-5,-3,-36) ; some coefficients are zero - they still must be entered ; into the list for Poly W(x) := poly(x,3,O,5,O,-7,24) P(x) := poly(x,l,O,O,O,O,l) ******************************************************************* 150 Eureka: The Solver Owner's Handbook ******************************************************************* Roots to the polynomial R Real part Imaginary part # 1 2 3 4 -1.0000000 -3.0000000 2.0000000 4.0000000 .00000000 .00000000 .00000000 .00000000 Roots to the polynomial S Real part Imaginary part # 1 2 1.0000000 1.0000000 2.0000000 -2.0000000 Roots to the polynomial T Real part Imaginary part # 1 2 -.75000000 -.75000000 1.9843135 -1.9843135 Roots to the polynomial U Real part Imaginary part # 1 2 3 4 1.0000000 1.0000000 -.74999999 -.74999999 2.0000000 -2.0000000 1.9843135 -1.9843135 Warning: Some roots inaccurate. Roots to the polynomial V Real part Imaginary part # 1 2 3 4.0000000 -.75000000 -.75000000 .00000000 1.9843135 -1.9843135 Roots to the polynomial W Real part Imaginary part # 3 2 1 4 5 -1.4464361 -.28746570 -.28746570 1.0106838 1.0106838 .00000000 1.7610191 -1.7610191 .84598193 -.84598193 Roots to the polynomial P Real part Imaginary part # 1 2 3 4 5 -1.0000000 -.30901699 -.30901699 .80901699 .80901699 .00000000 .95105652 -.95105652 .58778525 -.58778525 ******************************************************************** Worked Examples 151 Problem 10 Maximization: Designinc ~ Maximum-Square-Footage House Plan While Satisjying Several Constraints This problem (HOUSE.EKA on the distribution disk) shows how to use Eureka to maximize a variable that is a nonlinear function of several other variables, while satisfying constraints placed on the function and on the variables. Statement of the Problem In this problem, you are to design an L-shaped house with as much square footage (Hoor area) as possible, within the limitations imposed by city codes and certain aesthetic considerations. A rectangular second story will be built above the largest part of the first Hoor, extending from the back wall but only overlapping half of the front wing. Limitations and Constraints 1. The foundation must encompass no more than 3000 square feet. 2. The width of the front wing (dimensions xy, where x is width) must be within one-third to one-half of the total house width. The total house width is b, and the dimensions of the back wing are abo 3. The front and back wings will not be disproportionately sized. 4. The house will sit on a 90 x I50-foot lot with 10 foot minimum setbacks on either side and 25 foot minimum setbacks front and back. 5. Leave at least 1,500 square feet in the inside corner of the L for a pool and patio. Equations These equations state (1) the purpose of the problem (to maximize the total.Hoor area of the house you design) and (2) the limitations and constraints. Areas Area of first Hoor (covered by foundation): B = (xy 152 + ab) sq ft Eureka: The Solver Owner's Handbook Maximum foundation area allowed by city code: B < 3,000 sq ft Area of second floor (located above front wing of first floor, leaving room for a balcony at front of house): A2 = xz sq ft Total area of house (sum of areas of first and second floors): A = (xy + ab + xz) = B + xz sq ft Aesthetic constraints Front wing width is to be between one-third and one-half of total house width b: x > b/3 x < b/2 The patio/pool area will occupy the area bordered by the inner walls of the two wings. The dimensions of this area are (b - x) feet by (y) feet, and the total patio/ pool area must be at least 1,500 sq ft: y X (b - x) > 1,500 The house width (b) must be less than the lot width (90 ft) minus the two 10-foot setbacks, but it must be at least 40 feet wide: b < 68 b > 40 The total house length (y + a) must be less than the lot length (150 ft) minus the two 25-foot setbacks, but it must be at least 56 ft long: + a < 100 y + a > 56 y The second story, length (z), must leave half of the extending part of the front wing uncovered: z =a+ y/2 The smaller, back-wing extension must extend out at least half as far as the frontwing extension (so the wings will not be disproportionate): a > y/2 Worked Examples 153 Equation File and Solution The print-out that follows shows the equation file and solution for this maximization problem, along with an evaluation of all the solutions found. Note the directive $ max (A); that is, the command for Eureka to find the maximum combined first- and second-Hoor area possible within the listed constraints. Also note that the equation file includes initial values for band y. Since this problem contains many conditions that the few variables must satisfy, it is a good idea to get Eureka started in the right direction with some good first guesses; If you run this problem on your computer, you may notice that it seems to take a relatively long time to reach the solutions. This is normal for a problem like this, which contains many constraints for just a few variables. Watch the Progress window for information about how Eureka is handling the problem. 154 Eureka: The Solver Owner's Handbook ******************************************************************* Eureka: The Solver, Version 1.0 Saturday November 8, 1986, 9:02 pm. Name of input file: HOUSE.EKA ******************************************************************* This problem illustrates maXlmlzlng a nonlinear function while satisfying several constraints on the function and on the variables. Notice that we maximize a variable, A, which is related to the other variables x, y, z, a and b according to the first equation. $ max (A) ; maximize area within limits of city code A (x*y + a*b + x*z) = B + x*z B (x*y + a*b) B < 3000 aesthetic constraints x > b/3 x < b/2 ; leave room for patio/pool y*{b-x) > 1500 ; set-backs and more aesthetics b < 68 b > 40 y+a < 100 y+a > 56 z a + y/2 a > y/2 initial values b:=68 y:=56 ******************************************************************* Worked Examples 155 ******************************************************************* Solution: Variables Values A 4500.0000 a 23.677311 B 3000.0000 b 63.351788 x 31.675894 y 47.354622 z 47.354622 Confidence level = 92.3% All constraints satisfied. Evaluation of formulas: Formulas A (x*y+a*b+x*z) difference B+x*z Values 4500.0000 4500.0000 .00000000 difference 4500.0000 .00000000 (x*y+a*b) difference 3000.0000 3000.0000 .00000000 B 3000.0000 3000.0000 difference -4. 9408300e-08 ******************************************************************* B 3000 156 Eureka: The Solver Owner's Handbook ******************************************************************* x b/3 difference = x b/2 31.675894 21.117263 10.558631 31.675894 31.675894 difference = -2.9878677e-10 y*{b-x) 1500.0000 1500 1500.0000 difference (error) = -4.0927262e-12 difference = 63.351788 68.000000 -4.6482119 63.351788 40.000000 23.351788 difference 70.031933 100.00000 -28.968067 b 68 difference b 40 y+a 100 y+a 56 difference = 70.031933 56.000000 15.031933 difference = 47.354622 47.354622 .00000000 z a+y/2 a y/2 23.677311 23.677311 difference (error) = -1.0976464e-10 b y 63.351788 47.354622 Maximum error is -1.0976464e-10 ******************************************************************* Worked Examples 157 Problem 11 Maximization: Linear Programming in Manufacturing This example, a problem in linear programming (LINPROG.EKA on the distribution disk) demonstrates Eureka's ability to solve simultaneous linear equations in multiple variables and to maximize the objective function in order to reach an optimum solution. Statement of the Problem Suppose a company makes two types of widgets, one wooden and one plastic. To make a wooden widget, the producer uses machine A for 2 hours, machine B for 1 hour and machine C for 1 hour. To make a plastic widget requires 1 hour of machining on A, 2 hours on B, and 1 hour on C. The maximum amount of time (per month) available on machine A is 180 hours; on B, 160 hours; and on C, 100 hours. The profit on each wooden widget is $4.00 and on each plastic one $6.00. Assuming that all widgets made will sell, how many of each type should the company make in order to maximize the monthly profit? Equations Machine Hours To make wooden widgets To make plastic widgets Available per month A Machine B Machine C Profit (per unit) 2 1 180 1 2 160 1 1 100 $4.00 $6;00 Profit Function (the Objective Function) First, assign variables to each model of widget, with x being the monthly manufactured amount of the wooden variety and y being the same for the plastic widgets. Monthly profit is then the sum of the individual profits: Profit 158 = 4x + 6y Eureka: The Solver Owner's Handbook This is the objective function that you are going to maximize. To maximize the objective function, use Eureka's built-in maximization function. Add the max directive to the file: $ max (Profit) Remember that the max directive can be used only once per equation file. To avoid a solution which says that only the higher-profit plastic model will be manufactured, set x > 0 ; wooden widgets y > 0 ; plastic widgets Since the total time on machine A can't exceed 180 hours per month and the manufacturing processes require 2 hours of machine A's time for a wooden widget or 1 hour for a plastic widget, write the following constraint to represent the time limitation: 2 * x + Y < 180 Similarly, write constraints for the manufacturing time on machines Band C: x + 2 * Y < 160 ; total time on machine B can't exceed 160 hours per month x + y < 100 ; total time on machine C can't exceed 100 hours per month Equation File and Solution The print-out that follows shows the equation file and solution for this problem. Notice that we have started Eureka off with some initial values for x and y. We chose initial values that represented the maximum possible value for each variable that would satisfy the manufacturing-time constraints. Although this is not necessary, it may help to speed up Eureka's search for a solution; the default value for both x and y at the beginning of the search is 1.0. Notice also that we did rwt set the Profit function up as a user-defined function; we used the equal sign ( =) for an assignment operator, not the colon-equals operator (: =). This is an important distinction; if you were to write the objective function as Profit := (4 * x +6 * y), Eureka would interpret that as meaning that Profit is a variable to be maximized (due to the max directive) and it merely has a starting point value of 4 * x +6 * y. The result would be that Eureka maximizes Profit to the rather high value of 1.1529215 * 10 20 (dollars). A Worked Examples 159 ******************************************************************* Eureka: The Solver~ Version 1.0 Saturday November 8, 1986, 9:06 pm. Name of input file: LINPROG.EKA ******************************************************************* This example, a problem in linear programming, demonstrates Eureka's ability to solve simultaneous linear equations in multiple variables, and to maXlmlze the objective function in order to reach an optimum solution. Equations Profit function (the objective function) Profit = (4*x) + (6*y) $ max (Profit) x >0 y > 0 wooden widgets plastic widgets x := 90 y := 80 2*x + Y < 180 ; total time on machine A can't exceed 180 x + 2*y < 160 ; total time on machine B can't exceed 160 x + y < 100 total time on machine C can't exceed 100 ******************************************************************* Solution: Variables Values Profit 520.00000 x 40.000000 y 60.000000 Confidence level = 93.1% All constraints satisfied. ******************************************************************* 160 Eureka: The Solver Owner's Handbook Problem 12 Equity in a House: the paymt, pval and foal functions This example (EQUITY.EKA on the distribution disk) is a simple problem in finance. Using some of Eurekas built-in functions, you will calculate the growth in the equity in a house purchased with a mortgage. Statement of the Problem Suppose you purchase a house on January 1 for $200,000, using a $20,000 down payment and a 13% 30-year fixed-rate mortgage for $180,000. Assume that the mortgage interest compounds monthly and that your payments are made monthly. Use Eureka to calculate: 1. Your monthly payments. 2. How long it will take for your equity in the house to increase to $100,000. Equations Eureka has several built-in functions that are useful in financial calculations. These functions compute the key values in interest-rate calculations. In this example, you will use the paymt and fval functions. paymt calculates the payment in each interval on the loan, using as inputs the interest rate, the number of intervals, and the present and final value of the loan. fval calculates the future value of a loan, using as inputs the present value, the payment per interval, the number of intervals until the future date, and the interest rate. These are the input parameters: number of payment intervals interest rate per interval price of house down payment value of mortgage (principal) amount of balloon payment due Worked Examples N = 30 X 12 I = .13/12 P = 200,000 D = 20,000 pval = 180,000 B =0 = 360 months 161 You need to calculate the size of the monthly payments (mp) and the number of months (time) until the equity is $100,000. The monthly payment can be found by simply plugging the input parameters· into paymt in the following order: mp mp = paymt(J, N, P-D, B) which leads to = paymt(0.13/12, 30 * 12, 180000, 0) ° where the final is the final value of the mortgage after 30 years (in this example, there is no balloon payment). Equity in the house is simply the difference between what the house is worth and what you still owe the bank. The equity will be $100,000 at some future time that can be calculated using the fval function: Future Value of Mortgage = fval(J, time, L, mp) which gives 100,000 = 200,000 + fval(.13/12, time, 180000, mp) You can now solve these two equations for mp and time to determine the regular monthly payment and the elapsed time (in months) until the equity reaches $100,000.00. Equation File and Solution The Eureka equation file corresponding to these equations is shown in the printout that follows. Notice that the first entry in the equation file is a directive (set off with the $ symbol) to set finansmooth to on. This is necessary because financial functions are typically discontinuous in time. With finansmooth on, Eureka automatically smooths out the discontinuities and converts the financial function to a smooth curve instead of a series of step functions. 162 Eureka: The Solver Owner's Handbook ******************************************************************* Eureka: The Solver, Version 1.0 Tuesday February 10, 1987, 5:33 pm. Name of input file: EQUITY.EKA ******************************************************************* This problem shows how to use Eureka's built-in financial function PAYMT to calculate monthly payments for a 30-year, $180,000.00 mortgage, with a $20,000.00 down payment and a yearly interest rate of 13%. This problem also demonstrates using the built-in function FVAL to calculate when the equity reaches $100,000.00, and calculates the balance of the loan at any given time. Note: this problem incorporates zero market appreciation. $ finansmooth = yes variables assigned to the given data: DownPay = 20000 Loan = 180000 Duration = 30 * 12 determining monthly interest rate from the annual rate Yrate = 0.13 Mrate = Yrate / 12 finding the monthly payment Payment PAYMT(Mrate, Duration, Loan,O) formula for determining Equity at a given time: FutureValue(GivenMonth):=FVAL(Mrate,_ GivenMonth,Loan,Payment) Equity(GivenMonth) := DownPay + Loan + FVAL(Mrate,_ GivenMonth, Loan, Payment) Equity(HowSoon) = 100000 formula for determining total amount paid at the end of the life of the loan TotalPaid = Payment * Duration Amount still owed on loan at any given month AmtOwed(GivenMonth) := Loan + DownPay - Equity(GivenMonth) Worked Examples 163 Solution: Variables Values Down Pay 20000.000 Duration 360.00000 HowSoon 287.10773 Loan 180000.00 Mrate .010833333 Payment -1991.1591 Total Paid -716817.29 Yrate .13000000 Maximum error is .011900878 List of function values. Equity x 12.000000 24.000000 36.000000 48.000000 60.000000 72.000000 84.000000 96.000000 108.00000 120.00000 132.00000 144.00000 156.00000 168.00000 180.00000 192.00000 204.00000 216.00000 228.00000 240.00000 252.00000 264.00000 276.00000 288.00000 300.00000 312.00000 324.00000 336.00000 348.00000 360.00000 Equity(x) 20524.427 21121.243 21800.438 22573.385 23453.023 24454.080 25593.315 26889.801 28365.245 30044.348 31955.222 34129.859 36604.665 39421.076 42626.243 46273.827 50424.896 55148.947 60525.071 66643.275 73605.989 81529.784 90547.320 100809.57 112488.34 125779.17 140904.55 158117.74 177706.90 200000.00 ******************************************************************* 164 Eureka: The Solver Owner's Handbook Problem 13 Probability in a Card Game This problem (CARDS.EKA on the distribution disk) illustrates Eureka's built-in factorial function, fact, and demonstrates how to set up a user-defined function. Statement of the Problem In a game of Bridge, find the probability that a trump suit with six trumps outstanding will split three-three. For those who do not play Bridge, here is a synopsis of how to play the game: There are 13 cards dealt to each of 4 players. Of these, 13 of the cards are trumps. When a player says s/he is hoping for a three-three trump split, it is because s/he controls 2 of the hands and 7 of the trumps, with the remaining 6 trumps divided between the remaining 2 hands in an unknown manner. The trumps are said to split three-three if there are exactly 3 trumps in each of the unknown hands. Equations Stating the problem in different terms, we can ask how many combinations of N clwose M exist for this problem, where N choose M represents the number of unique ways to choose a subset of size M from a set of N items. In standard mathematical notation, this is expressed as a "combinatorial coefficient'" (CofNandM) where _ CofNandM - (MI) X (NI) ((N - M)I) If you (perhaps naively) assume that the probability of distributing six things randomly yields a three-three split, the equation for the (approximate) probability is: ProbApprox = C(6,3) /2 Worked Examples 6 165 If you assume that the probability of random distribution does not yield a threethree split, the exact probability for an even split is: ProbExact = C(26,13) C(20,lO) X C(63) , Equation File and Solution The print-out that follows shows the equation file and solution for this problem. 166 Eureka: The Solver Owner's Handbook ******************************************************************* Eureka: The Solver, Version 1.0 Saturday November 8, 1986, 10:46 pm. Name of input file: CARDS.EKA ******************************************************************* This example demonstrates how to use Eureka's built-in FACT function (FACT(x) = x factorial). The problem is: In a Bridge game, one player controls 7 trump cards. We must determine the probability that the remaining 6 trump cards are evenly distributed (3 and 3) among the other two players. We calculate both the approximate and the exact probability in this example. A(x) : = FACT (x) C6 = A(6) C3 = A(3) C10 A(10) C20 A(20) C26 A(26) C13 A(13) The combinatorial coefficients for total available cards and cards per remaining hand: Cof6and3 = A(6) / (A(3) * A(6-3» Cof20and10 = A(20) / (A(10) * A(20-10» Cof26and13 = A(26) / (A(13) * A(26-13» ; approximate probability of 3-3 split: ProbApprox = Cof6and3 / 2~6 exact probability of 3-3 split: ProbExact = Cof6and3 * Cof20and10 / Cof26and13 The correct answers are ProbApprox = .3125 ProbExact = .355279503 Worked Examples 167 Solution: Variables Values Cl0 3628800.0 C13 6.2270208e+09 C20 2.4329020e+18 C26 4.0329146e+26 C3 6.0000000 C6 720.00000 Cof20andl0 = 184756.00 Cof26and13 10400600. Cof6and3 20.000000 ProbApprox .31250000 ProbExact .35527950 ******************************************************************* 168 Eureka: The Solver Owner's Handbook Problem 14 Income Distribution: The integ Function This problem (INTEG.EKA on the distribution disk) demonstrates how to use Eureka's built-in integ function to calculate the definite integral of functions. Statement of the Problem In the study of income distributions, one useful quantitative measure is the Lorentz Curve, a function relating the cumulative percentage of income recipients (x) to the cumulative percentage of income (y). The equality of income distribution given by the line x = y represents the ideal state, where income is evenly distributed among the whole population: 10% of the people receive 10% of the income, 30% receive 30%, and so on. A more realistic, non-ideal income distribution is the function y = (20/21)x 2 + (l/21)x The coefficient of inequality measures the degree of deviation from equality; this coefficient is defined as the area between the income-distribution curve and the diagonal (x = y), divided by the area under the diagonal. When all incomes are evenly and equally distributed, the coefficient of inequality becomes zero. The problem: For the income-distribution function given above, what is the coefficient of inequality? Equations To solve this problem, you must integrate two functions from 0 to 1 (0 to 100%), then calculate a quotient using the two integrals. With Eureka's built-in integ function, this is an easy exercise. From the definition of the coefficient of inequality given above, define the numerator (Num) and denominator (Venom) as the integrals of the two functions given: Num is the difference between the definite integral of the diagonal (x the definite integral of the function (y(x)): = y) and Num = [integral(x) - integral((20/21)x2 + (l/21)x)] from x = 0 to x = 1 Venom is the definite integral of the function: y = x from x = 0 to x = 1 Worked Examples 169 The coefficient of inequality is the quotient CoefflnEq = Num/Denom Using the integ function, this is how you set up the two integrals: Num = integ(x, x, 0, 1) - integ«20/21)*x 2 + (1/21)*x, x, 0, 1) Denom = integ(x, x, 0, 1) ft Equation File and Solution The print-out that follows shows the equation file and the solution for the problem. The table at the end of the print-out shows the result of selecting Graph/List. 170 Eureka: The Solver Owner's Handbook ******************************************************************* Eureka: The Solver, Version 1.0 Saturday November 8, 1986, 11:40 am. Name of input file: INTEG.EKA ******************************************************************* This problem demonstrates how to use Eureka's built-in INTEG function to perform definite integrals. The example comes from the study of income distributions. income distribution function (Lorentz curve - non-ideal) y(x) := (20/21)*x-2 + (l/21)*x Num = INTEG(x, x, 0, 1) - INTEG«20/21)*x-2 + (1/21)*x, x, 0, 1) Denom = INTEG(x, x, 0, 1) coefficient of inequality: CoeffInEq = Num/Denom ******************************************************************* Solution: Variables Values CoeffInEq 0.31746032 Denom 0.5000000 Num 0.15873016 List of function values. x .00000000 .10000000 .20000000 .30000000 .40000000 .50000000 .60000000 .70000000 .80000000 .90000000 1.0000000 y(x) .00000000 .014285714 .047619048 .10000000 .17142857 .26190476 .37142857 .50000000 .64761905 .81428571 1.0000000 ******************************************************************* Worked Examples 171 Problem 15 Chemical Calculations This problem (CHEMBAL.EKA, CHEMEQ1.EKA, CHEMEQ2.EKA, and CHEMEQ3.EKA on the distribution disk) illustrates how you can use Eureka to solve problems in chemistry, from the simple-but-tedious (balancing chemical equations) to the more sophisticated (calculating equilibrium concentrations). The first part of this problem: • demonstrates Eureka's ability to solve some simple systems of linear equations • identifies a situation where the include directive would be useful • gives a quick method for balancing chemical equations and determining unknown stoichiometric quantities The second part of the problem gives three examples of calculating equilibrium concentrations, including a classic difficult problem where poorly-scaled variables hamper Eureka's solution process. Part I: The Reaction of Iron Sulfide with Oxygen to Produce Iron Oxide and Sulfur Dioxide This problem in stoichiometry (CHEMBAL.EKA on the distribution disk) asks you to find out how much of a reactant is needed for a chemical reaction to produce a certain amount of product. Statement of the Problem Consider the chemical reaction between FeS and 02' written here as an unbalanced chemical equation: + 02 ... Fe20 3 + S02 How much iron sulfide (in grams) is required to produce 100 grams of iron oxide in this reaction? FeS To solve this problem, you will first need to balance the chemical equation. In addition, you will need to convert the given mass information (in this case, 100 grams of Fe2 0) into its molar equivalent. A mole is defined as the formula weight (in grams) of an element or compound. It is also, by definition, the mass of a specified number of molecules of a substance. 23 The specified number is Avogadro's number (6.0221 X 10 molecules.) 172 Eureka: The Solver Owner's Handbook Once you know the ratios between the different substances in the reaction and the given number of moles (of product, in this case), you can determine the number of moles of other substances that satisfy the given requirement. Equations You can assign variables for all the unknown amounts in the unbalanced chemical equation and set up the following algebraic expression to represent the balanced state: a * (Fe + S) + b * (O * 2) - c * (Fe * 2 + 0 * 3) - d * (S + 0 * 2) = 0 Since elements cannot just appear or disappear, the net amounts of each element must be the same on each side of the balanced equation. This means that the variables for each of the elements have the following relationships: for Fe, a = 2c for S, a = d for 0, 2b = 3c + 2d The solutions to these variables will yield a balanced chemical equation, which specifies the number of moles of reactants used in the chemical reaction and the number of moles of products that result. The next step is to write an equation for converting the given information (100 grams of Fe20 3 ) into moles. Remember that chemical symbols (Fe, K, Ca, Na, 0, S, and so on) are just thatsymbols - and can be manipulated just like variables in equations. You can combine these symbols with each other and with constants representing mass, valence, atomic number, or any other property that you need to factor in when solving chemical equations. For example, consider Fe 20 3 and the following definitions: mFe zFe vFe mO zO = mass of one mole of Fe (55.847 grams) = atomic number of Fe (26) = valence of Fe (2 or 3; depends on particular compound) = mass of one mole of (15.9994 grams) ° ° = atomic number of (8) = valence of (typically, - 2) ° vO mFe20 3 = mass of one mole of Fe20 3 = (mFe X 2) + (rnO X 3) similarly, mFeS = mFe + mS mS0 2 = mS + (mO Worked Examples X 2) 173 To convert measured amounts of substances into their molar equivalents, you divide the measured amount by the mole weight: 100.00 gm of Fe20 3 100.00 159.69 gm/mole = 159.69 = 0.6262 moles of Fe2 0 3 The amount of each substance used or produced in the chemical reaction can be represented by a "how much P" variable, written by adding h to the beginning of each chemical formula. In this example, "how much Fe2 0}" is represented by hFe2 0 3 and is the given quantity-100 grams. Similarly, hFeS represents "how much FeSP", the amount (in grams) ofFeS used, and h02 represents the amount of 2 consumed. ° In terms of the variables in this problem, the number of moles of Fe20 3 is calculated by howmuch Fe2 0 3 mass of one mole of Fe2 0 3 where mFe2 0 3 hFe2 0 3 mFe20 3 = 159.69 and hFe20 3 is the given, 100 grams. You can see that the number of moles of Fe20 3 produced is equal to the number of moles of FeS supplied. This can be rephrased as: a moles of FeS produce c moles of Fe20 3 so the ratio of (moles of FeS):(moles of Fe2 0) is the ratio a:c. The equation for this is moles of FeS a ----------= -which leads to the equation c X (moles of FeS) = a X (moles of Fe20 3) By substituting the number of moles equation from above, you get c hFeS x--- = mFeS a hFe2 0 3 x-~--=- mFe2 0 3 Eureka will solve for the variables a and c in the first part of this problem; mFeS and mFe20 3 are constants and hFe20 3 is the given, so you can easily solve for the unknown quantity, hFeS. 174 Eureka: The Solver Owner's Handbook Equation File and Solution The print-out that follows shows the Eureka equation file and solution for this problem. Eureka will solve this system of linear equations and give values for a, b, c, and d that balance the chemical equation. To get Eureka started, you assign a value to one of the variables and see what solutions the program finds. This is not necessary, but it helps to keep the solutions for a, b, c, and d as whole numbers. You can see from the equation for the variable b that c should be an even number so b will be an integer; that is why we set c equal to 2. Besides giving a value to one of the variables, you must include the values for the given amount of product (100 grams of Fe20 3) and the constant values. In this case, the constant values are the mole weights of the elements Fe, 0, and s. Using the $ include Directive You can experiment with this problem, or similar ones, to explore Eureka's usefulness in stoichiometry. If you plan to be solving many problems such·as this one, you will probably find it helpful to create a separate file containing the mole weights for all the elements that you will be dealing with. Keeping such a file as a permanent part of your Eureka problem disk, you could use the include directive in your equation file to call up the mole weight information when it is needed instead of entering it over and over. Worked Examples 175 ******************************************************************* Eureka: The Solver, Version 1.0 Thursday November 6, 1986, 6:11 am. Name of input file: CHEMBAL.EKA ******************************************************************* This is an example of balancing a chemical equation and then finding the amount of reactant, FeS, needed to yield 100.00 grams of Fe203 product the unbalanced chemical equation: FeS t 02 --> Fe203 t S02 algebraic expression for the balanced equation: a*FeS t b*02 --> c*Fe203 t d*S02 a = 2*c d =a b*2 = 3*c t 2*d c = 2.000 ; mole weights (grams/mole) mFe mS mO mFeS mFe203 m02 mS02 55.847 28.086 15.9994 mFe t mS mFe*2 t mO*3 mO*2 mS t mO*2 ; amount of product required is 100 grams hFe203 = 100.00 ; ratio of moles of FeS to moles of Fe203 is a:c c*(hFeS/mFeS) = a*(hFe203/mFe203} ; correct answer is a = d = 4, b = 7, c 176 2 Eureka: The Solver Owner's Handbook ******************************************************************* Solution: Variables Values a 4.0000000 b 7.0000000 c 2.0000000 d 4.0000000 hFe203 100.00000 hFeS 105.11847 rnFe 55.847000 rnFe203 159.69220 rnFeS 83.933000 rnO 15.999400 rn02 31.998800 rnS 28.086000 rnS02 60.084800 ******************************************************************* Worked Examples 177 Part II: Calculating Chemical Equilibrium Concentrations The second part of this problem (CHEMEQ1.EKA, CHEMEQ2.EKA, and CHEMEQ3.EKA on the distribution disk) demonstrates Eureka's ability to solve high-order polynomials by finding equilibrium concentrations for the reactants and products in some sample reversible chemical reactions. Statement of the Problem Many chemical reactions do not proceed to completion (that is, all the available reactants are not converted into products) because the reaction is reversible. Some of the products actually react chemically and convert back into their original constituents. Some examples of such reversible reactions are: + 12(gas) N2(gas) + 3H2(gas) H 2(gas) <=> 2HI(gas) <=> 2NH3(gas) When the rate of forward reaction (in these examples, the combining action of the gases) and the rate of decomposition reach a balance, the chemical reaction is said to be in equilibrium. At equilibrium at a given temperature, such a reversible chemical reaction can be characterized by an equilibrium constant, Keq' The equilibrium constant is the ratio of two numbers: The first is a product of the concentrations of all the substances on the right-hand side of the chemical equation, and the second is a product of the concentrations of all the substances on the left-hand' side. (If a substance in the chemical equation has a coefficient a, then the concentration for that substance is raised to the ath power when calculating Keq for the reversible reaction.) Laboratory scientists have determined Keqs for many reactions at specific temperatures. If you know the Keq for a reaction and have enough additional information about the reactants or the products, you can determine the equilibrium concentrations of all the substances involved in the reaction. First Example Consider the reaction of carbon monoxide (CO) and steam: H 20 <=> CO2 +' H2 At 800°C, the Keq for this reaction is 1.2. If you put 2.0 moles of CO and 2.0 moles of H 20 into a 10-liter container at 800°C, what will be the equilibrium concentrations of all the substances? CO 178 + Eureka: The Solver Owner's Handbook Equations The first step in setting up this problem is to write out a table that lists the starting and equilibrium concentrations for the substances involved. Starting Concentration Equilibrium Concentration 0.2 moles/L 0.2 moles/L 0.2 - X moles/L 0.2 - X moles/L X moles/L X moles/L o o By definition of the equilibrium constant, cco X CH2 cco X cH20 2 = --=---.;;. K eCJ Substituting in the values for KeCI and the four concentrations from the table yields 12 = . (X)(X) . (0.2 - X)(0.2 - X) 1.2 = (0.2 _ X)2 You will rewrite these equations as a secOnd-order polynomial and let Eureka solve for X; then, knowing X, Eureka will calculate the equilibrium concentrations of all the reactants and products. Equation File arul Solution The print-out that follows shows the equation file and the solution for this problem. Note that we have given an initial value to Xin the equation file (X: = 0.19). We made an assumption that at least 5% of the original substances were converted into products. Although it was not theoretically necessary to provide Eureka with that starting point, it turned out to be a practical action. Without a reasonable initial value for X, Eureka is capable of finding some solutions that are unique butwrong (such as concentrations less than 0). The easiest way to avoid such solutions is to provide Eureka with a reasonable starting point of where you expect the answer to be, if possible. Worked Examples 179 ******************************************************************* Eureka: The Solver, Version 1.0 Thursday November 6, 1986, 6:23 am. Name of input file: CHEMEQl.EKA ******************************************************************* Chemical Equilibrium Calculations - First Example This problem demonstrates solving a second-order polynomial in order to calculate equilibrium concentrations in a reversible chemical reaction The reaction is: CO + H20 <===> C02 + H2 At 800 degrees C, the Keq for this reaction is 1.2 cC02 = X cH2 = X cCO = 0.2 - X cH20 = 0.2 - X equil. equil. equil. equil. conc. conc. conc. conc. of of of of C02 H2 CO H20 ; Keq = (cC02 * cH2) / (cCO * cH20) 1.2 = X~2 / (0.2 - X)~2 : initial value given for X X:= 0.19 correct answers are: cC02 = cH2 cCO = cH20 = 0.2 - X = .0954451 X .104555 ******************************************************************* Solution: Variables Values cCO .095445115 cC02 .10455488 cH2 .10455488 cH20 .095445115 X .10455488 Maximum error is -1.092463ge-l0 ******************************************************************* 180 Eureka: The Solver Owner's Handbook Second Example Consider the reversible reaction of nitrogen (N2) and hydrogen (H2) to form ammonia: N2(gas) + 3H2(gas) ~ 2NH3(gas) Empirical evidence shows that log(base 10) of Kf' the equilibrium constant for the forward reaction of N2 and H2 to form NH3 , is 2.914 at 25°C. K = 102 .914 "f (cNH/ = 102 .914 cN2 X (CH2)3 In this problem, you start with .1 mole of N2 and .3 moles of H2 in a 10-liter flask at 25°C: Starting Concentration Final Concentration 0.1 moles/L 0.3 moles/L 0.0 moles/L (0.1 - X) moles/L (0.3 - 3X) moles/L 2X moles/L What will be the concentrations of all the constituents at equilibrium? Equations The following polynomial results from substituting the concentration values from the table into the equation for IS-: 2 (2X) (0.1 - X)((0.3 - 3X)3) = 102.914 =K which simplifies to o= (K X 27(0.1 - X)4) - (2X)2 Worked Examples 181 Equation File and Solution The equation file and solution for this problem appear in the print-out that follows. Notice the constraints placed on the value X (which represents the equilibrium concentration of NH3): • X > 0 (There must be some NH3 produced) • X < 0.1 (You can't have more products than the reactants) 182 Eureka: The Solver Owner's Handbook ******************************************************************* Eureka: The Solver, Version 1.0 Thursday November 6, 1986, 6:27 am. Name of input file: CHEMEQ2.EKA ******************************************************************* Chemical Equilibrium Calculations - Second example This is an example of solving a fourth-degree polynomial, in order to find the equilibrium concentrations of N2, H2 and NH3 in the reversible chemical reaction N2 + 3H2 <===> 2NH3 at 25 degrees C, where log(base 10) of K(f) (the equilibrium constant for the forward reaction) is 2.914. equil. cone. of N2 equil. conc of H2 equil. cone. of NH3 cN2 = 0.1 - X cH2 = 0.3 - 3*X cNH3 = 2*X equation for equilibrium concentrations: K = (cNH3)A2 / (c~!2* (cH2r3) 4*(XA2) = K * (0.1 - X) * (0.3 - 3*X)A3 constraints so Eureka finds appropriate roots o < X < 0.1 ******************************************************************* Solution: Variables Values cH2 .091649822 cN2 .030549941 cNH3 .13890012 K 820.35154 X .069450059 Maximum error is -4.234442ge-10 ******************************************************************* Worked Examples 183 Third Example This example is a classic case of poorly scaled variables and equations and clearly demonstrates where and how things could go wrong in your equation files. Consider the dissociation of hydrogen peroxide at 800°C to yield water (H2 0) and oxygen (02): 2 X H 20 2 <=> 2 X H 20 + ° 2 At 800°C, the Keq for this reaction is 3.6 X 1013 • If you start with 0.1 mole of H 20 2 in a 10-liter container at 800°C, what will be the equilibrium concentrations of all the substances? Equations Starting Concentration Equilibrium Concentration 0.01 moles/L 0.01 - 2X moles/L X moles/L 2X moles/L o moles/L o moles/L 3.6 X 1013 = (X) (2X)2 (0.01 _2X)2 (0.01 - 2X)2 13 Since 3.6 X 10 is a very large number with respect to the other quantities in this problem, the equation is poorly scaled. Eureka runs into difficulty when trying to solve an equation that spans so many orders of magnitude (14 in this case). For example, Eureka might find a solution that yields an evaluation difference of 106 • 6 13 Because 10 is relatively tiny compared to 10 , Eureka might be satisfied with the solution, even though that solution would be patently wrong. Given these problems inherent with spanning large orders of magnitude, it would seem that you could convert both sides of the equation to the natural logs and ask Eureka to solve the following: 1n (3.6 184 * 10A13) = 1n (4) + 1n A (X 3) - 1n «0.001 - (2 * X))A2} Eureka: The Solver Owner's Handbook However, rewriting the original equation to this form does not work well, because Eureka must calculate each of the natural logs used; the problem in this form becomes unworkable. Your best approach to this quandary is to formulate an intermediate equation that spans fewer orders of magnitude. To accomplish this, create the following set of intermediate variables and equations: 1. substitute Keq = W3 so W= 36 Y = (0.01 - (2 * W3 = 4X 3/Y 2 so W= X * «2/Y) A A (1/3) * 10 4 A X» A A A A (2/3» 2. substitute B = (Y/2V(2/3) so B * W= X 3. set constraints and initializations o < X < 0.005, W> 0, B > 0, X := 0.0001 By redefining the problem as B* W= X, you create a situation where the difference between the largest and smallest values is on the order of 107 (from 104 to 10- 3), which Eureka is able to solve in just a few seconds. Note: It is important to set up the equations in the order shown on the next page. If you present the equations in a different order, Eureka gives an "Equations are inconsistent" error message and cannot solve the problem. Equation File and Solution The print-out that follows shows the equation file and Eureka's solution for this problem. Worked Examples 185 ******************************************************************* Eureka: The Solver, Version 1.0 Thursday November 6, 1986, 6:32 am. Name of input file: CHEMEQ3.EKA ******************************************************************* Chemical Equilibrium Calculation - Third Example This example demonstrates how to handle a poorly-scaled problem. The original equation was Keq = 3.6 * 10A13 Y = (0.01 - (2*X» 4*XA3 / yA2 = Keq To reduce the total span of 14 orders of magnitude, substitute W (= KeqA(1/3» and B (= (Y/2)A(2/3» W 36A(1/3) * 10A4 Keq W3 A B*W B Y X (Y /2)A(2/3) (0.01 - (2*X» X 0 X 0.005 W> 0 X := 0.0001 B >0 Solution is X = 0.0043698957, Y = 0.0012602085 A B = 1.323437ge-07, W= 33019.272, Keq = 3.6 * 10 13 ******************************************************************* Solution: Variables Values B 1.323437ge-07 Keq 3. 6000000e+ 13 W 33019.272 X 0.0043698957 Y 0.0012602085 Maximum error is 1.5178830e-18 Warning: floating point overflow ******************************************************************* 186 Eureka: The Solver Owner's Handbook A p p E N o x A Eureka Quick Reference Table This appendix provides brief explanations, in alphabetical order, for many of Eureka's features. msqrt(x) ncum(x) paymt(i,n,pval,fval) piO polar(x 1,x2) poly(x,series) pos(x) pval(i,n,paymt,fval) re(x) sgn(x) sin(x) sinh(x) sqrt(x) sum(exp,var,xl,x2 < n > ) tan (x) Built-in functions abs(x) atan2(xl,x2) cos (x) cosh(x) deriv(x 1,x2) exp(x) fact(x) Hoor(x) frac(x) fval(i,n,pval,paymt) im(x) integ(exp, var,x 1,x2 < n > ) In(x) loglO(x) Calculator operations Press C C to select Calculator from the Commands menu. Press Num-Lock, then use the number pad keys located at the right of your keyboard to make entries, then press Enter. Eureka works just like a hand-held scientific calculator to solve arithmetic and built-in functions. 187 Default settings Change default settings in one of two ways: 1. Embed directives in the equation file. 2. Change settings from the Options menu: Press 0 S to select Settings from the Options menu. Scroll through the menu and change the desired settings by pressing Enter to toggle between yes and no, or by entering an appropriate number. Then press Esc. Load and save setup files with Load setup and Write setup. Directives accuracy, casefold, complex, digits, finanmode, finansmooth, include, initval, listdefault, max, maxtime, min, penalty, plotdefault, radius, rootsign, setting, solve, substlevel, syntax, unit Directory Press F D from the main menu, enter optional directory mask, and press Enter (* and? are wildcards). By default, Eureka will list all files in the directory. Select Edit to input the equation you want to solve. The editor works like the Turbo Pascal and SideKick editors or WordStar in non-document mode. A brief listing of editor commands follows: Edit an equation file Word left Word right Delete character to left Delete character under cursor Delete word to right of cursor Delete line Insert mode on/off Equation file format Equations. Equations are written one per line in standard mathematical syntax and notation. Multiplication is not implicit. * := 188 Ctrl-Left arrow or Ctrl-A Ctrl-Right arrow or Ctrl-F Backspace Ctrl-G Ctrl-T Ctrl-Y Ins or Ctrl-V multiplication operator exponentiation operator user-defined functions and initializations used to separate formulas on the same line indicates formula continued on following line Eureka: The Solver Owner's Handbook Directives. Directives begin with a dollar sign and follow the general format of the directive shown here: $ digits = 6 Comments. Comments must be set off by a semicolon (;) or appear between braces ({ }). File names Press F L to select Load from the File menu, then enter a file name. File names follow DOS conventions (up to eight characters, and may also include an optional 1 to 3 character extension). Use the Write to ... option on the File menu to make an alternate version of the current file. Find other roots, modify Press 0 V from the main menu to select Variables from variables and constants the Options menu. Enter a value, and press Enter. Press Esc twice to return to the main menu, then choose Iterate from the Command menu to solve for the new variable. In seeking another root, the value should be close to but not greater than that of the second root. Press Esc, then press S to return to the main menu and Solve. Function keys FI F2 F3 F5 Alt-F5 F6 F7 FB Esc Ctrl-Break Up arrow, Down arrow, Left arrow, Right arrow, Display help text Save the current file New file Zoom to full screen editor Text zoom in plot window Change active window Go to active window or block Begin (edit function) Block end (edit function) Select menu Interrupt solve Move/scroll through the screen PgUp, PgDn, Alt-X List a table of values Eureka Quick Reference Table Exit Eureka Press G L from the main menu to select List from the Graph menu. Respond to the prompts by entering the function (if necessary) and the initial value, increment, and number of values; then press Enter. See comment under "Plot a graph." 189 Load a file Plot a graph Press F L from the main menu to select Load from the File menu. At the prompt, type in the name of the file to be loaded; press Enter. Press G P from the main menu to select Plot from the Graph menu. Respond to the prompts by entering the left and right endpoints and then press Enter. (Note: A file must have at least one user-defined function or you must enter one with the Function command before you can select Plot or List.) Print a report Press R 0 to select the Output option from the Report menu. Indicate the desired output device and whether you want a formatted header (Enter toggles between the various choices). Select Go and press Enter. Quit the program Press Q from the File menu, or press Alt-X. Return to DOS Press F 0 to select OS Shell from the File menu. Return to main menu From a window or submenu, press Esc. Save a file Press F S from the main menu to select Save from the File menu. This saves a copy of your file on your active drive. Select menu items Cursor Method Move cursor to desired menu item and press Enter. Initial Letter Method Type the capital letter of the desired option. For example, to select the Calculator option on the Commands menu, you would press Cc. Settings In addition to those listed under "Directives:" lisLfirst, lisLnum, lisLinc, ploLleft, ploLright. Start the program Log on to the drive or directory containing the program and type EUREKA Enter. Solve a problem Press S from the main menu to select Solve. To solve the problem another time, press C I from the main menu to select the Iterate command from the Commands menu. To look in a different region for an answer, press C F from the main menu to select the Find other command from the Commands menu. Verify a solution 190 Press C V from the main menu to select Verify from the Commands menu. Eureka: The Solver Owner's Handbook A p p E N D x B Eureka Editor Commands All Eureka editor commands are described in this appendix. Each description consists of: • a heading defining the command • the keystrokes that activate the command • a brief explanation of what the command does If there are two ways of giving a command, that is, one using the IBM cursor keys and the other using Ctri, both ways are listed, with the IBM cursor key method listed first. If you are familiar with WordStar or the Turbo Pascal and SideKick editors, you should have no trouble with the Eureka editor, since it is virtually identical to these editors. For hands-on practice with using the editor, see the tutorials in Chapter 2. To change default editor keystrokes, use the EINST program, described in Appendix F. 191 Cursor Movement Commands These commands control the position of the cursor in the file. You can manage fairly well with just the cursor keys and PgUp and PgDn; the other commands are handy but may not be critical for small files. Character left Left arrow or Ctrl-S Moves the cursor one character to the left without affecting the character there. This command does not work across line breaks; when the cursor reaches the left edge of the window, it stops. Character right Right arrow or Ctrl-D Moves the cursor one character to the right without affecting the character there. This command does not work across line breaks; when the cursor reaches the right edge of the window, the text starts scrolling horizontally until the cursor reaches the extreme right of the line in column 250, where it stops. Line down Down arrow or Ctrl-X Moves the cursor to the line below. If the cursor is on the second-to-the-Iast line, the window scrolls up one line. Line up Up arrow or Ctrl-E Moves the cursor up to the line above. If the cursor is on the top line, the window scrolls down one line. Page down PgDn or Ctrl-C Moves the cursor one page (window) down with an overlap of one line. The cursor moves one window less one line forward in the text. Page up PgUp or Ctrl-R Moves the cursor one page (window) up with an overlap of one line. The cursor moves one window less one line backward in the text. To end of file Moves the cursor to the last character of the text. Ctrl-PgDn or Ctrl-Q C To top of file Moves the cursor to the first character of the text. Ctrl-PgUp or Ctrl-Q R To last cursor position Ctrl-Q P Moves the cursor to the last position it occupied. This command is useful, for example, to move back to the last position after a Find/Replace operation. To left margin Home or Ctrl-Q S Moves the cursor all the way to the left edge of the window (column 1). To right end of line End or Ctrl-Q D Moves the cursor to the end of the line; in other words, to the position following the last printable character on the line. Trailing blanks are always removed from all lines to save space. 192 Eureka: The Solver Owner's Handbook To top of window Moves the cursor to the top of the editor window. To bottom of window Moves the cursor to the bottom of the editor window. Ctrl-Home or Ctrl-Q E Ctrl-End or Ctrl-Q X Ctrl-Left arrow or Ctrl-A Word left Moves the cursor to the beginning of the word to the left. A word is defined as a sequence of characters delimited by one of the following characters: space < > , ; • () [ ] A , * + _ / $. Word right Ctrl-Right arrow or Ctrl-F Moves the cursor to the beginning of the word to the right. A word is defined as a sequence of characters delimited by one of the following characters: space < > , ; * + _/ $. • () [ ] A , Insert and Delete Commands These commands let you insert and delete text by character, word, and line, and control the Insert/Overwrite typing mode. Delete character to left Backspace This is the backspace key just above Enter. It moves one character to the left and deletes the character there. Any characters to the right of the cursor are moved to the left. Delete character under cursor Del or Ctrl-G Deletes the character under the cursor and moves any characters to the right of the cursor one position to the left. This command does not work across line breaks. Delete word right Ctrl- T Deletes the word to the right of the cursor. A word is defined in "Word left" under cursor movement commands. Delete to end of line Deletes all text from the cursor position right to the end of the line. Ctrl-Q Y Insert mode on/off Ins or Ctrl-V This command lets you switch between insert, the default, and overwrite mode while entering text. The current mode is displayed on the status line of the editor window. In Insert mode, new text is tucked in at the cursor position, and existing text is moved to the right. Overwrite mode is convenient for replacing old text with new. In this mode, existing characters are replaced by the new characters typed over them. Eureka Editor Commands 193 Line deletion Ctrl- Y Deletes the line containing the cursor and moves any lines below one line up. The cursor moves to the left edge of the window. Be careful; there is no way to restore a deleted line. Line insertion Inserts a line break at the cursor position. The cursor remains stationary. Ctrl-N Block Commands Block commands allow you to maneuver chunks of text. A block can be any amount of text from a single character to an entire file. A block is marked by placing a "begin block" marker before the first character and an "end block" marker after the last character of the desired portion of text. So marked, a block can be copied, moved, deleted, or written as a file. Ctrl-K C Block copy Places a copy of a marked block starting at the cursor position. The original block is left unchanged. Ctrl-K V Block move Moves a marked block from its original position to the cursor position. The block disappears from its original position. Block delete Ctrl-K Y Deletes a marked block. Be careful? no provision exists for restoring a deleted block of text. Hide/display block Ctrl-K H This command causes the visual marking of a block to be toggled off and on. The copy, move, delete, and write to a file block manipulation commands work only when the block is visibly marked. Mark block begin Ctrl-K B Marks the beginning of a block. The marker itself is not visible on the screen, and the block only becomes visibly marked (displayed in lower-intensity text) when the end block marker is set. You can also use the begin block marker as a reference point in your text and jump directly to it with the Ctrl-Q B command. Mark block end Ctrl-K K Marks the end of a block. As above, the marker itself is not visible on the screen, and the block only becomes visible after the begin block marker is set. You can also use the begin block marker as a reference point in your text and jump directly to it with the Ctrl-Q K command. 194 Eureka: The Solver Owner's Handbook Mark single word Ctrl-K T Mark a single word as a block. If the cursor is placed within a word, the word is marked; if the cursor is not on a word, the word to its left is marked. A word is as defined in "Word left" under cursor commands. Read block from disk Ctrl-K R This command is used to read a file into the current text at the cursor position, exactly as if it were a block that was being copied or moved. Write block to disk Ctrl-KW This command is used to write a previously marked block to a file. The block is left unchanged, and the markers remain in place. When you use this command, you are prompted for the name of the file to write to. If the file name specified already exists, a warning is given before the existing file is overwritten. Miscellaneous Editing Commands Abort operation Ctrl-U Abort the current editing operation when it pauses for input (for example, when Search and replace asks "Replace YIN?" or during the entry of a file name for a block read). . Ctrl-P Control character prefix The editor allows you to enter control characters into the file by prefixing the desired control character with Ctrl-P. Control characters are displayed as halfintensity capital letters. For instance, to cause an Epson printer to print condensed type, enter Ctrl-P Ctrl-O. Consult your printer's manual for more information about printing control commands. You can also use Ctrl-B (for bold) or Ctr/~S (for underline) to emphasize text. These are toggle characters, so you turn off bold, for example, by entering Ctrl-P Ctrl-B after the last character that should be printed in bold. Find Ctrl-Q F This command lets you search for a string of characters. A prompt appears above the status line, requesting a search string. If you make an error while typing the string, use the Backspace key to back up and correct it. The string may contain any characters, including control characters. Control characters are entered into the search string with the Ctrl-P prefix. To find a line break, search for Ctrl-M Ctrl-J. After the search string has been specified, you are asked for search options. Enter the required options (if any), then press Enter to start the search. If the text contains a target matching the search string, the cursor is positioned at the end of Eureka Editor Commands 195 the target. To repeat the search operation, use the repeat last find command, Ctrl-L. The following options are available (and need not be entered in uppercase): B Search backwards from the current cursor position towards the beginning of the file n n = an integer. Type the number of occurrences of the string for which the search should take place, counted from the current cursor position. U Ignore uppercase or lowercase. Regard uppercase and lowercase alphabeticals as equivalent. W Search for whole words only. Skip matching patterns that are embedded in other words. Find and replace Ctrl-Q A This command lets you search for a string of characters, and then replace it with another. A prompt appears above the status line, requesting a search string. A further prompt requests the replacement string. In nearly every other respect, this commands functions like Find, described previously. There are two additional options: G Global search and replace. Search and replace throughout the entire text, irrespective of the current cursor position. N Replace without asking. Does not stop and query whether or not the replace should be carried out at every occurrence of the search string. Page break Inserts a page break (an ASCII form feed) at the cursor position. Ctrl-P Ctrl-L Ctrl-L Repeat last find Repeats the last Find or Find and replace operation exactly as if all the information had been re-entered. Restore error message Redisplay last error message. Ctrl-Q W Restore line Ctrl-Q L Restores a line to its original state regardless of what changes you have made; however, you cannot have left the line for the command to be effective. Tab or Ctrl-I Tab Moves the cursor eight spaces to the right (hard tabs) or moves cursor to next column based on spacing in the line above (soft tabs). Tab mode toggle Toggles tabs on and off. Ctrl-O T Set tab Sets tab at cursor position. Ctrl-O I 196 Eureka: The Solver Owner's Handbook p A p E N D x c Built-in Functions Eureka's built-in functions are described in alphabetical order in this appendix. The standard mathematical and other functions are listed below: Standard Mathematical Functions abs atan2 cos cosh exp fact In log 10 pi () sin sinh tan tanh Other Functions deriv floor frac fval im integ ncum paymt polar poly pos pval re sgn sqrt sum 197 The following notations are used in describing the Junctions: expr series var x abs must be an expression series of values or expressions variable identifier value or expression Absolute Value abs(x) This function returns the absolute value of x. If x is positive, x is returned. If x is negative, - x is. returned. Examples abs(5) returns 5 abs(-15.5) returns 15.5 atan2 Arc Tangent atan2(xl,x2) This trigonometric function returns the inverse tangent of xl/x2. This is the angle, in radians, formed by x2 and xl. This function offers an extension to the usual Arc Tangent in that it applies to all four quadrants. The rules that one used· in determining e = atan2 (xl, x2) are as follows: Quadrant I ifxl II if xl III if xl IV if xl and x2 > 0, then 0 < e < -rr/2 > 0 and x2 < 0, then -rr/2 < e < -rr and x2 < 0, then--rr < e < -'IT /2 < 0 and x2 > 0, then - -rr /2 < e < 0 The quadrants progress counterclockwise, with I at the upper right-hand corner. Example atan2(1,-1) returns 2.35619449 cos Cosine cos (x) This trigonometric function returns the cosine of x. x is assumed to be the angle in radians. Example cos(1) returns 0.54030231 198 Eureka: The Solver Owner's Handbook cosh Hyperbolic Cosine cosh(x) This function returns the hyperbolic cosine of x. Example cosh(2) returns 3.76219569 deriv Partial Derivative deriv (xl, x2) This function returns the partial derivative of the first argument with respect to the second. The second argument must be a variable. For example, solving z = deriv(x~3 + 5 * x,x) x =1 gives x = 1.00000000 z = 8.00000000 The derivative of x + 5x is 3x + 5, which has the value 8 when x = 1. If other variables besides x were to appear in the expression, they would be treated as constants in the derivative to be evaluated. For example, solving 3 2 z = deriv(x~3 + y~2 + 5 * x,x) x =1 y =2 gives x = 1.00000000 = 2.00000000 z = 8.00000000 y However, while Eureka makes substitutions internally in order to solve equations, such substitutions do not affect derivatives. For instance, solving y = x~3 + 5 * x z = deriv(y,x) x =1 gives x = 1.00000000 y = 6.00000000 z = 0.00000000 Built-in Functions 199 This result occurs because deriv treats x and y as independent variables. To differentiate an expression that occurs frequently, you should define an appropriate function. Here this function could be f(x) : = x-3 + 5 * x z = deriv(f(x),x) x =1 gives x = 1.00000000 z = 8.00000000 exp Exponentiation exp(x) This exponential function raises the number e to the power x. Examples exp(O} returns 1.00000000 exp(l) returns 2.71828183 fact Factorial fact(x) This function calculates the factorial of the positive integer specified by x. The integer x must be less than or equal to 170. Examples fact(6) returns 720.000000 fact(3) returns 6.00000000 floor Floor floor(x) This function rounds x down to the nearest integer. Examples floor{-2.5) returns -3.00000000 floor(2.999 + 2.999} returns 5.00000000 200 Eureka: The Solver Owner's Handbook frac frac(x) Fraction This function returns as a positive number the part of x that is not an integer. Examples frac(2.5) returns 0.50000000 frac(1.33 + 1.55) returns 0.88000000 Future Value fval fval(i, n,pval,paymt) This financial function returns the future value of a mortgage or similar contract (the money received at the end of a contract). n is normally an integer representing the number of payment intervals; i is the interest rate per payment expressed in hundredths; pval and paymt are described elsewhere in this section. fval is either zero or a negative number representing the balloon payment. See page 99 in Chapter 5 for more details. Example If you borrow $250,000 at 10% and make ten annual payments of$35,000, how much money do you owe the lender at the end of the contract? fval(0.10,10,250000,-35000) returns -90625.75 im Imaginary Number im(x) This function returns the imaginary part of a complex number. (Note: The complex number setting must be set to yes.) Example $ complex = yes im(3 + 2 * sqrt(-l» integ Integral returns 2 integ(exp,var,xl,x2 < n > ) This function returns the integral of the first argument with respect to the second. The second argument must be a variable. The third argument is the lower limit of integration, and the fourth argument is the upper limit. Improper integrals are not allowed. Built-in Functions 201 This function returns the definite integral for the variable (var) varied in the specified expres~ion between the limits xl and x2. Integ uses Simpson's Rule to compute the integral. < n > is the number of subdivisions used for the integral. If n is not specified, 10 subdivisions are used. (See deriv for more discussion of differential calculus.) Expressions can also be integrated with respect to a given variable. For example, solving z pi = = integ(sin(x),x,O,pi) 3.14159265359 gives the integral of sin(x) as x goes from 0 to pi: pi x z = = = 3.14159265359 1.23900000 2.00000000 In this case, the value for x is meaningless because it is not subject to any constraints. As with derivatives, variables other than the variables of integration are treated as constants, but functions can be used. The last two arguments to integ, lower and upper limits of integration, can be arbitrary expressions. Example integ(x-2 + 3 * x + 4,x,2,7) = y returns the area under the function x 2 + 3x + 4, between x = 2 and x = 7, ory = 199.16666667. In Natural Logarithm In{x) This function returns the natural logarithm (base e greater than o. = 2.71828183) of x. x must be Examples 1n(exp(l» returns 1 1n(15) returns 2.70805020 lnglO Logarithm base 10 logl 0 (x) This function returns the logarithm base 10 of x. x must be greater than o. Example 10g10(l) = 0 10g10(12) = 1.0791812 202 Eureka: The Solver Owner's Handbook msqrl Negative Square Root msqrl(x) This function returns the negative (minimum or minus) square root of x. Example msqrt(9) returns -3 ncum Normal Cumulative ncum(x) This function returns the probability that a normal random variable will be below its mean plus x standard deviations. It can also be expressed as ncum(x) = '\f(~'TrT J~X~( -t'/2) dt ncum is the basic tool used in analyzing the variance of normal distributions in statistics and probability. The importance of the normal distribution is primarily due to the Central Limit Theorem. This theorem states that any sum of independent random variables approximates a normal random variable, with the approximation becoming more accurate as more variables are summed. In the experimental sciences, the deviation from theory and experiment can be explained as an error which is the sum of error contributions from many sources. The resulting error is usually quite well approximated as a random variable. As an example of the use of ncum, consider a chemist who is measuring pollutants in the air. After taking a number of readings, s/he computes the average pollen concentration at 150 parts per million (ppm), with a standard deviation of 10 ppm. The probability of making a reading less than 170 ppm is ncum( + 2), since 170 is 2 standard deviations above 150. Also, the chemist can be very confident that a reading will be less than 150 + lOx where ncum(x) = 0.99 Example ncum{l) returns 0.84134456 Built-in Functions 203 paymt PaY111£nt paymt(i,n,pval,jval) This financial function returns the payment on a mortgage or similar contract. n is normally an integer representing the number of payment intervals; i is the interest rate per payment expressed in hundredths; pvai and fvai are described elsewhere in this section. See the section on financial functions in Chapter 5 for more details. pi pi pi () This function returns the value of pi (3.14159265359) polar Polar Coordinate polar(xl,x2) This function returns the angle in the rectangular to polar coordinate conversion in radians. The conversion x = r y = r X X cos (theta) sin(theta) is inverted by r = sqrt(x2 + y2) theta = polar(x,y) except for the restriction < theta < - 'TT' 'TT' = 3.14159265359 'TT' If z is complex, these transformations are equivalent to x = re(z) y = im(z) r = abs(z) theta = im(ln(z)) Polar(x,y) is the same as the Fortran function atan2(y,x), and is the same as the arc tangent of y/x with the quadrant chosen carefully (see arctan2 for more details). Example polar(-1,O) returns 3.14159265 204 Eureka: The Solver Owner's Handbook poly poly (x, series) Polynomial This function returns the value of the polynomial of x using the specified series of coefficients. Coefficients are listed in descending order of the variable exponent. For example, the following polynomial equation X4 + 3x 3 2 - x + 2x + 1 would be expressed as follows: p(x) := poly(x, 1, 3, -1, 2, 1) Applying Solve to a file containing poly causes Eureka to find all the roots, as well as solve for whatever other variables are in the file. A root to a polynomial is a complex number that yields zero when plugged into the polynomial. A polynomial of degree n always has exactly n roots, counting repetitions. Since the polynomial here has real coefficients, the roots are either real or occur in complex conjugate pairs. The above polynomial has the following roots: No. Real Part - 0.29105149 0.50677653 0.50677653 - 2.72250157 1 2 3 4 pos Positive Imaginary 0.00000000 1.00258931 - 1.00258931 0.00000000 pos(x) This function returns the positive value of x. Examples pos(I.2) returns 1.20000000 pos(-12) returns 0.00000000 Built-in Functions 205 pval pval(i, n,paymt,jval) Present Value This financial function returns the present value on a mortgage or similar contract. (For a mortgage, this is the amount of the loan minus the down payment.) n is normally an integer representing the number of payment intervals; i is the interest rate per payment expressed in hundredths; paymt and fval are described elsewhere in this section. See page 99 in Chapter 5 for more details. re Real Number re(x) This function returns the real part of a complex number. (Note: Complex numbers must be set to yes.) Example $ complex:: yes re(3 + 2 * sqrt(-1) returns 3 Sign sgn sgn(x) This function returns 1 if x > 0 or if x = 0; otherwise, it returns -1 if x < O. Examples sgn(10) returns 1 sgn(O) returns 1 sgn(-I) returns -1 sin Sine sin (x) This trigonometric function returns the sine of x. x is the angle in radians. Example sin(1) returns 0.84147098 206 Eureka: The Solver Owner's Handbook sinh Hyperbolic Sine sinh (x) This function returns the hyperbolic sine of x. Example s;nh(l} returns 1.17520119 sqrt Square Root sqrt(x) This function returns the positive square root of the argument x. Example sqrt(81} returns 9.00000000 Sum sum sum (exp, var, xl, x2) This function returns the sum of the series defined by replacing the variable in the second argument in the function of the first argument over the range expressed in the third and fourth arguments, or n Lf(x) x = 1 where f(x) = exp x I = var = xl n = x2 The second argument must be a variable. The third argument is the lower limit of the summation, and the fourth argument is the upper limit. Example sum(4 * x,x,I,3} returns 24 Built-in Functions 207 tan Tangent tan (x) This trigonometric function returns the tangent of x, where x is an angle measured in radians. Examples tan(O) returns 0.00000000 tan (1) returns 1.55740772 tanh Hyperbolic Tangent tanh (x) This trigonometric function returns the hyperbolic tangent of x. Example tanh(l) returns 0.76159416 208 Eureka: The Solver Owner's Handbook A p p E N D x D A DOS Primer If you are new to computers or to DOS, you may have trouble understanding certain terms used in this manual. This appendix provides you with a brief overview of the following DOS concepts and functions: • What DOS is and does • The proper way to load a program • Directories, subdirectories, and the path command • Using AUTOEXEC.BAT files This information is by no means a complete explanation of the DOS operating system. If you need more details, please refer to the MS-DOSTM or PC_DOSTM user's manual that came with your computer system. Eureka runs under the MS-DOS or PC-DOS operating system, version 2.0 or later. What Is DOS? DOS is shorthand for Disk Operating System. MS-DOS is Microsoft's version of DOS, while PC-DOS is IBM's rendition. DOS is the traffic coordinator, manager, and operator for the transactions that occur between the parts of the computer system and the computer system and you. DOS operates in the background, taking 209 care of many of the menial computer tasks you wouldn't want to have to think about - for instance, the How of characters between your keyboard and the computer, between the computer and your printer, and between your disk(s) and internal memory (RAM). Other DOS transactions are ones that you initiate by entering commands on the DOS command line; in other words, immediately after the DOS prompt. Your DOS prompt looks like one of the following: A> B> C> The capital letter refers to the active disk drive (the one DOS and you are using right now). For instance, if the prompt is A>, it means you are working with the files on drive A, and that commands you give DOS will refer to this drive. When you want to switch to another disk, making it the active disk, all you do is type the letter of the disk, followed by a colon and Enter. For instance, to switch to drive B, just type .B:fnter. There are a few commands that you will use often with DOS, if you haven't already, such as DEL or ERASE DIR COPY EUREKA To erase a file To see a list of files on the logged disk To copy files from one disk to another To load Eureka: The Solver DOS doesn't care· whether you type in uppercase or lowercase letters, or a combination of both, so you can enter your commands however you like. We'll assume you know how to use the first three commands listed; if you don't, refer to your DOS manual. Next, we will explain the proper way to load a program like Eureka: The Solver, and that involves the last command-EUREKA. How to Load a Program On your distribution disk, you'll find the main Eureka program, under the file name EUREKA.EXE. This program file is necessary for all functions, so you always need it when you first start the program. A file name with the "last name" or extension .COM or .EXE means a program file that you can load and run (use) by typing only its "first name" on the DOS command line. So, to invoke Eureka, you simply type EUREKA and press Enter, and Eureka will be loaded into your computer's memory. 210 Eureka: The Solver Owner's Handbook There's one thing you need to remember about loading Etireka and other similar programs: You must be logged onto the disk and directory where the program is located in order to load it; otherwise, unless you have set up a DOS path (described shortly), DOS won't know where to find the program. (In DOS 3.0, however,you can load a program by giving the full path name.) For instance, if your distribution disk with the EUREKA.EXE program is in drive A but the prompt you see on your screen is B >, DOS won't know what you're talking about if you type EUREKA and press Enter, and will give you the message "Bad command or file name." It's as if you were shufHing through the "School Records" :(ile in your file cabinet looking for information about your home finances. You're in the wrong place. So if you happen to get that DOS message, simply switch to drive A by typing A: and then press Enter. Then type EUREKA and press Enter to load Eureka. You can set up a "path" to the Eureka files so that DOS can find them, using the DOS path command. See the section on the AUTOEXEC.BAT file for more information. Directories A directory is a convenient way to organize your floppy or hard disk files. Directories allow you to subdivide your disk into sections, much the way you. might put groups of manila file folders into separate file boxes. For instance, you might want to put all your file folders having to do with finance - for instance, a bank statement file, an income tax file, or the like - into a box labeled "Finances." On your computer, it would be convenient to make a directory to hold all your Eureka files, another for your SideKick files, another for your letters, and so on. That way, when you type DIR on the DOS command line, you don't have to wade through hundreds of file names looking for the file you want. You'll get a listing of only the files on the directory you're currently logged onto. Although you can make directories on either floppy or hard disks, they are used most often on hard disks. This is because a hard disk can hold a greater volume of data, so there is a greater need for organization and compartmentalization. When you're at the DOS level, rather than in Eureka or ,another program, you can tell DOS to create directories, move files around between directories, and display which files are in a particular directory. In the examples that follow, we assume you are using a hard disk system, and that you are logged on to the hard disk so that the prompt you see on your screen is C >. If you want to create directories on your floppy disks, just substitute A or B A DOS Primer 211 To make a directory for your Eureka files, do the following: 1. At the C > prompt, type MKDIR EUREKA and press Enter. The MKDIR command tells DOS to make a directory called EUREKA. 2. Type CHDIR EUREKA and press Enter. The CHDIR command tells DOS to move you into the EUREKA directory. 3. Now, put the Eureka disk you want to copy from into one of your Hoppy drives -let's say A for this example - and type COPY A:*. * Enter. (The asterisks are wildcards that stand for all files.) The COpy command tells DOS to copy all files on the A drive to the EUREKA directory on the C drive. As each file on the disk is copied, you will see it listed on the screen. That's all there is to it. Treat a directory the same way you would a disk drive: To load Eureka, you must be in the EUREKA directory before typing Eureka and pressing Enter, or DOS won't be able to find the program. Subdirectories If you are someone who really likes organization, you can further subdivide your directories into subdirectories. You can create as many directories and subdirectories as you like - just don't forget where you put your filesl A subdirectory is created the same way as a directory. To create a subdirectory from the EUREKA directory (for instance, for storing your equation files), do the following: 1. Be sure you are in the EUREKA directory. 2. Type MKDIR FILES Enter. 3. Type CHDIR FILES. You are now in the FILES subdirectory. 4. Copy your equation files to the new subdirectory. If you do put your files in a subdirectory, you can let Eureka know where they are by using the EINST program (see Appendix F). Where Am I? The $p $g Prampt You probably have noticed that when you change directories, you still see the C > prompt; there is no evidence of what directory or subdirectory you are in. This can be confusing, especially if you leave your computer for a while. It's easy to forget where you were when you left. 212 Eureka: The Solver Owner's Handbook DOS gives you an easy way to find out. Just type prompt=$p $g and from now on (until you turn your computer off or reboot), the prompt will show you exactly where you are. Try it. If you are still in the FILES subdirectory, your DOS prompt should look like C:\EUREKA\FILES > The AUTOEXEC.BAT Fik To avoid typing the prompt command (discussed in the previous section) to see "where you are" every time you turn on your computer, you can set up an AUTOEXEC.BAT file to do it for you each time you turn on your computer. The AUTOEXEC.BAT file is a useful tool to set your computer to do things automatically. There are many more things it can do, but rather than go into great detail here, we suggest referring to your DOS manual for more information. We will show you how to create an AUTOEXEC.BAT file that will automatically change your prompt so you know where you are in your directory structure, set a path to the Eureka directory, and change to the EUREKA directory. The DOS path command tells your computer where to look for commands it doesn't recognize. DOS only recognizes programs in the current (logged) directory, unless there is a path to the directory containing pertinent programs or files. In the following example, we will set a path to the EUREKA directory. If you have an AUTOEXEC.BAT file in your root directory, your computer will do everything in that file when you first turn your computer on. (The root directory is where you see the C> or C:\ prompt, with no directory names following it.) Here's how to create an AUTOEXEC.BAT file. 1. Type CHDIR \ Enter to get to the root directory. 2. Type COPY CON AUTOEXEC.BAT Enter. This tells DOS to copy whatever you type next into a file called AUTOEXEC.BAT. 3. Type PROMPT=$P $G PATH=C:\EUREKA CHOIR EUREKA Ctrl-Z The Ctrl-Z sequence saves your commands in the AUTOEXEC.BAT file. To test your new AUTOEXEC.BAT file, reboot your computer by holding down the Ctrl and Alt keys and then pressing Del. You should see C:\EUREKA>. A DOS Primer 213 Clwnging Directories How do you get from one directory to another? It depends on where you want to go. The basic DOS command for changing directories is CHDIR. Use it like this: • To move' from one directory to another: For example, to change from the EUREKA directory to one called WP, type the following from the EUREKA directory: C:\EUREKA> CHOIR \WP Enter Notice the backslash (\) before the directory name. Whenever you are moving from one directory to another unrelated directory, type the name of the directory, preceded by a backslash. • To move from a directory to its subdirectory: For example, to move from the EUREKA directory to the FILES subdirectory, type the following from the EUREKA directory: C:\EUREKA> CHOIR FILES Enter In this case, you did not need the backslash, because the FILES directory is a direct offshoot of the EUREKA directory. In fact, DOS would have misunderstood what you meant if you had used the backslash in this case. If you had included the backslash, DOS would have thought that FILES was a directory off the main (root) directory. • To move from a subdirectory to its parent directory: For example, to move from the FILES subdirectory to the EUREKA directory, type the following from the FILES subdirectory: C:\EUREKA\FILES> CHOIR •. Enter DOS will move you back to the EUREKA directory. Any time you want to move back to the parent directory, type two periods after the CHDIR command. • To move to the root directory: The root directory is the original directory. It is the parent (or grandparent) of all directories (and subdirectories). When you are in the root directory, you see this prompt: C:\ >. To move to the root directory from any other directory, simply type CHOIR \ Enter The backs lash without a directory name signals DOS that you want to return to the root directory. 214 Eureka: The Solver Owner's Handbook This appendix has presented only a quick look at DOS and some of its functions. Once you're familiar with the information given here, you may want to study your DOS manual and discover the many things you can do with your computer's operating system. There are many other DOS functions not mentioned here that can simplify and enhance your computer use. A DOS Primer 215 216 Eureka: The Solver Owner's Handbook A p p E N o x E Error Messages The various error messages you may encounter while using Eureka are described in alphabetical order in this appendix. These messages appear in the Error window. When applicable, a line reference may also appear with the error message. The line reference specifies the line in the equation file that contains the error. However, when you use the Calculator, only the word ERROR appears, although the cause of the problem may be one described in this appendix. Assignment Must Be to a Variable or Function Cause: Syntax problem. Attempting to solve an equation file in which an assignment (using: =) has been made to something other than a recognizable variable or function. Solution: Edit the file to provide a legitimate variable or function, as required. Bad Function Call Cause: Internal error. Generated if Eureka attempts to solve an incorrectly defined function. Solution: Quit Eureka, then reactivate it. 217 Canrwt Complexify Floor or Frac Cause: Setting error. The active equation file includes the floor or frac function, but either the complex setting or complex directive has been set to yes. Solution: Change the setting or directive to no. Canrwt Have Complex Derivatives or Integrals Cause: Setting error. The active equation file includes the deriv or integ function, but either the complex setting or complex directive has been set to yes. Solution: Change the setting or directive to no. Circular Function Definition Cause: Attempting to solve an equation file that contains a tautological function definition; that is, one that does not produce an assignable value. An example of this is: f(x) := x~2 g(x) := f(x + 3 * g(x) + 1) - 7 * x Solution: Rewrite the incorrect definition. Comma Expected Cause: Syntax problem. In attempting to solve the equation file, Eureka has encountered an equation that lacks one or more commas between values or constants that are function arguments. Solution: Edit the equation so that it contains the correct number of commas. 218 Eureka: The Solver Owner's Handbook Command Not Understood Cause: Syntax error. In attempting to solve an equation file, Eureka has encountered an unintelligible directive (to the left of the equal sign); for example: $ ccomplex = yes Solution: Restore the directive to its proper form: $ complex = yes Constant Expected Cause: Attempting to solve an equation file in which something other than a recognizable constant appears where one is syntactically expected. Solution: Edit the file to provide a legitimate constant. Dynamic Memory Depleted Cause: Memory problem. The area allocated to Eureka has been filled. Solution: Quit Eureka, then reactivate it. Equations are Inconsistent Cause: Attempting to solve an equation file in which the equations do not follow the correct sequence. Solution: Edit the equations so they follow the correct syntax. File Already Exists Cause: In using the Rename command, you have entered a file name that already exists. Solution: Select Rename again, but enter a name not currently in use. Error Messages 219 File Not Found Cause: In attempting to use Load to bring a file to the screen, no file name has been found to match the one entered. Solution: Doublecheck the file name and try again, remembering extensions. File Too Large Cause: Attempting to use Save on a file that will not fit in the available disk space. Solution: Cut down the size of" the file, or if possible, save to another memory device (such as a different data disk). Function is Multiply Defined Cause: Syntax error. Attempting to solve an equation file in which the same function definition is used more than once. For example: f(a) := pos(x + y) feb) := pos(z) Solution: Edit the equation file to eliminate redundant version{s} of the function definition: f(a) := pos(x g(b) : = pos(z) + y) Function is Undefined Cause: Syntax problem. Attempting to solve an equation file that contains an equation with an undefined function. Solution: Edit the equation so that it contains an appropriate built-in function or correctly stated user-defined function. 220 Eureka: The Solver Owner's Handbook Identifier Expected Cause: Syntax problem. Attempting to solve an equation file that lacks a variable name where one should syntactically appear. Solution: Edit the file to include the necessary character. Identifier Too Long Cause: A variable name is too long. Solution: Edit the file to eliminate redundant characters or recast the formula as necessary. Incorrect Number of Function Arguments Cause: Syntax problem. Attempting to solve an equation file that contains a function with a different number of arguments than those indicated in the function definition. Solution: Edit the function to contain the correct number of arguments. Invalid Function Argument Cause: Attempting to solve an equation file containing a function with an improper argument. For example, replacing either of the plus signs in this equation with an equal sign: frac(x + 2.5 + 0.33). Solution: Edit the necessary function arguments. Error Messages 221 Invalid Function Definition Cause: Syntax problem. Attempting to solve an equation file that contains an improperly defined function, such as a numeral appearing between the parentheses. Solution: Edit the function definition so that it is properly formed. Invalid Number Cause: In attempting to solve the equation file, Eureka has encountered an illegal symbolic value. For instance, while e (indicating scientific notation) is allowed, it may be entered incorrectly. Solution: Edit the equation file so that it contains a legal symbolic value. Memory Heap Overflow Cause: Attempting to solve an equation file that contains an equation requiring more memory than is available in RAM on your computer. Solution: Recast the equation or get more RAM for your computer. Must Differentiate With Respect to a Variable Cause: Attempting to solve an equation file that uses the deriv function incorrectly. Solution: Recast the deriv function. Numeral or Operator Expected Cause: Attempting to solve an equation file that contains something other than a numeral or operator where, syntactically, one should appear. Solution: Edit the file to correct the character(s}. 222 Eureka: The Solver Owner's Handbook Polyrwmial is Invalid Cause: An internal error is generated if Eureka attempts to solve an invalid polynomial. Solution: Exit Eureka, then reactivate it. Printer Not Ready Cause: Hardware problem. In attempting to use the. Report Go command to print a report, some difficulty with the printer has been encountered. Solution: Check that the printer is on, select is on, ribbon and paper are properly feeding and in place, cables are tight, and so on. Quote Mark Expected Cause: Syntax error. Attempting to solve an equation file that lacks a quotation mark where one should appear around a file name specified by the include directive. Solution: Edit the file to insert the necessary quotation mark. Relational Operator Expected Cause: Syntax error. In attempting to solve the equation file, Eureka has encountered an equation lacking an expected relational operator « or ». Solution: Edit the incomplete equation so that it contains such a relational operator. Second Argument Must Be a Variable Cause: Attempting to solve an equation file containing a function in which the second argument must be a variable (such as deriv or integ) but is not. Solution: Recast the function argument as a variable. Error Messages 223 String Space Overflow Cause: The memory area allocated to storing strings is full. Solution: Quit Eureka, then reactivate it. Sum is Invalid Cause: Internal error. In attempting to solve the sum function, Eureka has encountered an internal problem. Solution: Quit Eureka, then reactivate it. Sum is Too Large Cause: Attempting to solve for the sum function when the spread between the initial and final endpoints overtaxes Eureka's internal capacity. Solution: Recast the function. Syntax Error for Unit Conversion Cause: Attempting to solve an equation file that contains an incorrectly formulated unit conversion. Solution: Reformulate the incorrect syntax. Too Few Arguments to Poly Cause: Syntax problem. Attempting to solve an equation file that includes the poly function but with only one argument (minimum requirement is two arguments). Solution: Edit the function to contain the correct number of arguments. 224 Eureka: The Solver Owner's Handbook Too Many Active Variables Cause: Attempting to solve an equation file having more than twenty active variables. Solution: Recast the file so that it contains twenty or fewer active variables. Too Many Constants Cause: Attempting to solve an equation file that causes Eureka to handle more than 200 constants. While an equation file may have only a relatively small number of constants, the solving process itself may sometimes cause Eureka to generate a large number of internal substitution constants and thus cause this message to appear. Solution: Set the substlevel setting to O. Too Many Formulas Cause: Attempting to solve an equation file with more than twenty equations. Solution: Recast the equation file so that it contains no more than twenty equations. Note: Solving in complex mode causes Eureka to actually handle three times the number of equations that appear in the file: the original equation, one for imaginary numbers, and one for real numbers. Too Many Functions Cause: Attempting to solve an equation file with more than ten user-defined functions. Solution: Recast the file so that it contains ten or fewer functions. Too Many Unit Conversions Cause: Attempting to solve an equation file having more than ten unit conversions. Solution: Recast the file so that it contains ten or fewer such conversions. Error Messages 225 Too Many Variables Cause: Attempting to solve an equation file having more than twenty variables. Solution: Recast the file so that it contains twenty or fewer variables. Unable to Create File Cause: Attempting to use Edit to start a new file when, for any of a number of reasons (for example, the data disk is read-only), this is impossible. Solution: Ascertain the reason for the failure, then use Edit again. Unexpected End of File Cause: Attempting to solve an equation file in which the expected terminating character (for example, a closing }) is missing. Solution: Edit the file to supply a recognizable termination character. Unexpected $ End Directive Cause: Attempting to solve an equation file that includes an end directive where, syntactically, one should not appear. Solution: Edit the file to reposition the end directive. Unmatched Parentheses Cause: Syntax problem. Attempting to solve an equation file containing an ex:pression or comment that lacks either an initial or terminal parenthesis or curly bracket. Solution: Edit the· file so that all parenthetical expressions are correctly enclosed by a pair of symbols. 226 Eureka: The Solver Owner's Handbook Unrecognized Character Cause: Attempting to solve an equation file that contains a character other than a letter, identifier, operator, or numeral in a position where it might be something other than part of a variable name or comment. Solution: Edit the file to supply a recognizable character. Unrecognized Setting Value Cause: Attempting to solve an equation file in which a directive appears that assigns an unknown setting value (for example, $ initval = four, where a numeral is expected). Solution: Edit the directive to include a legitimate setting value ($ initval = 4). Use of Uninitialized Variable Cause: Syntax problem. Attempting to solve an equation file in which no assignment of a value (using: =) has been made to a variable that must be initialized. Solution: Edit the file to provide an appropriate initialization. Variable Expected Cause: Attempting to solve an equation file that contains something other than an identifiable variable where, syntactically, one should appear. Solution: Edit the file to correct the erroneous variable. Variable Has Not Been Initialized Cause: Syntax problem. The active equation file contains a variable that has not been properly initialized. Solution: Edit the file so that the variable is initialized. Error Messages 227 228 Eureka: The Solver Owner's Handbook A p p E N D x F Customizing Eureka The program EINST.COM lets you do four things: • set up a path to your help and setup files • customize your keyboard to use with Eureka's editor • modify the default edit modes • set the screen mode (default, color, black and white, or monochrome) If you want to store your help (HELP.EKA) and/or setup (INITIAL.EKAcreated with Write setup on the Options menu) files on a directory other than the one where you have EUREKA.EXE, or if you are running DOS 2.0, you will need to use the Eureka directory option to set a path to those files. If you don't like Eureka's default editor keystrokes, or if you want the Eureka editor to behave more like your own text editor, you can use the Text editor command option to customize the editor keystrokes to your liking. Finally, you can use the Default editor mode option to set several defaults for the editor: insert or overwrite mode, tabs, and autoindenting. Running EINST To get started, type EINST at the DOS prompt. The first menu lets you select Eureka directory, Text editor commands, Default edit modes, or Quit. You can 229 either press the highlighted capital letter of the option you want, or use the Up and Down arrow keys to move to your selection and then press Enter. For instance, press D to modify the Default edit modes. Pressing Esc will eventually return you to the main screen. The Eureka Directory Option The Eureka directory option is really only useful for hard disk users. You'll use this option to specifY a path to your help and setup files (HELP.EKA and INITIAL. EKA), so that they will be accessible from wherever you call up Eureka. (INITIAL. EKA is the file created when you select Write setup from the Options menu.) When you select the Eureka directory option, you're prompted to enter the full path to the directory where you are storing your files. For example, if you want to keep the files in a subdirectory off a directory called EUREKA, you might type for your path name C:\EUREKA\FILES Mter entering a path, press Enter to accept it and the main menu will redisplay. When you exit the program, you're prompted whether or not to save the changes. Once you save the path, the location is written to disk. (Note that the status line at the bottom of the screen tells you which keystrokes to use when you're in this screen.) The Text Editor Command Option This option allows you to change the default keys that you use while in the Eureka editor. Press T to modifY the editor commands. The help line at the top of the screen shows you which keys to use to move around and make changes. Most of these commands are simply cursor movement commands; however, the R option is useful if you make a lot of changes, then decide you want to restore the keystrokes to their factory defaults. Notice that you can only modifY the secondary, or highlighted, keystrokes; the other keystrokes cannot be changed. To change a keystroke, move the selection bar to the key you want to change, then press Enter. You'll then see a selection bar next to the command you want to redefine. Press the key(s) you want to use. If you take another look at the top of the screen, you'll see the help line now lists the available commands: ~ 230 backspace C clear R restore ~ accept edit literal Eureka: The Solver Owner's Handbook Use the Backspace key to backspace and/or delete something in the keystroke box. The C option clears, or erases, the whole box. Use R to restore the original keystrokes before exiting from the screen. Press Enter to accept the keystroke modification you've made. And finally, the < Scroll Lock> is a toggle that lets you alternate between command and literal modes. To explain the < Scroll Lock> option, let's take a look at the Enter key, which is used to modify and accept the editing of a key command. But suppose you want to change the Find String command from < CtrlQ > < CtrlF > to < CtrlQ > < Enter> . To do so, you would have to toggle Scroll-Lock to literal mode, so that when you press the Enter key, it will be interpreted literally as part of the new keystroke you are entering. Follow these steps: 1. Make sure < Scroll Lock> is toggled to command mode (check the upper right-hand corner of your screen). 2. Then press Enter at the Find String command line. 3. Press Backspace to delete the < CtrlF > part of the string. 4. Now toggle Scroll-Lock> to literal and press Enter. 5. Again, toggle Scroll-Lock to command mode and then press Enter to accept. After you've defined a new keystroke(s) for a command, press Enter to accept it. If you're finished making changes, press Esc to exit. If you still have more changes to make, use the arrow keys to scroll up and down the list and select your next command. At this point, if you've accidentally assigned a keystroke sequence that's been used as a control character sequence in the primary command column, the message Command conflicts need to be corrected. Press Esc will flash across the screen. Any duplicated sequences will be highlighted, so you can easily search for any disallowed items and reselect a sequence. If you change your mind, you can use the R option to restore the factory default key definitions. The Default Edit Mode Option Press 0 to bring up the Default edit modes menu. There are three editor modes that can be installed: Insert mode, Autoindent mode, and Tabs. Use this menu to set the editor's modes to the settings you prefer. You'll still be able to toggle them ON/OFF from inside the editor; this menu is used to determine whether they start ON or OFF when you first load Eureka. Customizing Eureka 231 With Insert mode on, anything you enter at the keyboard is inserted at the cursor position, pushing any text to the right of the cursor further right. Toggling Insert mode off allows you to overwrite text at the cursor. With Autoindent mode on, the cursor returns to the starting column of the previous line. When toggled off, the cursor always returns to column one. Toggle on Tab mode when you want to insert tabs; toggle off and the tab is automatically set to the beginning of the first word in the previous line. When you load Eureka, the default values for all three modes are on. You can change the defaults to suit your preferences and save them back to Eureka. Of course, you'll still be able to toggle these modes from inside Eureka's editor. Either use the arrow keys to move the selection bar to the option and then press Enter or else press the key that corresponds to the highlighted capital letter of the option. The Screen Mode Option Press S to select Screen mode from the installation menu. A pull-down menu will appear from which you can select the screen mode Eureka will use during operation. Your options include • Default • Color • Black and white • Monochrome Default Display Mode By default, Eureka will always operate in the mode that is active when you load it. Color Display Mode Eureka will use color mode with 80 X 25 characters no matter what mode is active, switching back to the active mode when you exit. Black and White Display Mode Eureka will use black and white mode with 80 x 25 characters no matter what mode is active, switching back to the active mode when you exit. 232 Eureka: The Solver Owner's Handbook Morwchrome Display Mode Eureka will use monochrome mode no matter what mode is active, switching back to the active mode when you exit. Quitting the Program Once you have finished making all desired changes, select Quit (or press Esc) at the main menu. The message "Save changes to EUREKA.EXE?" will appear at the bottom of the screen. If you press Y(for Yes), all of the changes you have made will be permanently installed into Eureka. (Of course, you can always run this program again if you want to change them.) If you press N (for No), your changes will be ignored and you will be returned to the DOS prompt. If you decide you want to restore the original Eureka factory defaults, simply copy EUREKA.EXE from your master disk onto your work disk. 233 234 Eureka: The Solver Owner's Handbook Glossary absolute value: The value of a positive or negative number when the sign has been removed. For instance, the absolute value of both - 2 and + 2 is 2. argument: A variable or expression representing a value in the definition of a builtin or user-defined function. comment: In Eureka, statements used to help you identify or document various equation file components. Comments are set off by a semicolon (;) or are enclosed in curly brackets ({like this}); Eureka ignores comments when solving an equation file. confidence level: A rating that appears in the Solution window when Eureka solves a minimization or maximization problem. The confidence level reveals the degree of certainty Eureka feels about the found solution. constraint: A condition that must be met for an equation to be satisfactorily solved. The accuracy and penalty settings can alter the relative strength of a constraint. default: The value or instruction Eureka uses until you enter a different value. For example, Eureka automatically solves a variable to eight decimal places; that is the default value. You can change selected default values that are listed in the main menu's Options pull-down menu. You can change nearly all the defaults by including a directive in the equation file. directive: A command to Eureka, embedded within an equation file, that affects only that file. For example, $ complex = yes means variables are solved for as complex numbers. 235 equation file: The file that contains the problem to be solved. It must contain an equation or inequality and, where pertinent, initialization values and user-defined functions. It may also include directives and comments. error message: A message that appears in the error message window on the Eureka screen if you type something that is incorrect or inappropriate. Appendix E lists Eureka's error messages and describes how to correct or avoid errors. file name extension: An optional 1 to 3 character extension to a file name. As shown in these examples, a period separates the file name and the extension: FINANCE.TOM, FORMULA.l, and FORMULA.165. Extensions are convenient for identifying different file versions, or for identifying file ownership. floating-point error: A warning message that may appear when Eureka has solved for an extremely large number or when a number is divided by zero. It does not necessarily mean the solution is erroneous. header: A standardized format for the information appearing at the beginning of a Eureka report. The Eureka report headers include the date, time, and file name. help message: A message that appears in the Help window when you press Fl. Help messages explain the current area of the program, whether it is a highlighted menu item, window, or other Eureka feature. identifier: Name for a variable. initialization: The process of setting initial values for variables or the starting point of an iterative search. Initializations are flagged with the symbol : = . list: A table of values for a function, generated with Eureka's Graph/List command. mask: A way to search selectively for a file. In Eureka, wildcard characters (both the asterisk (*) and question mark (?)) are used in combination with fragments of file names as masks. The asterisk finds all character strings starting from the position of the wildcard character; the ? finds only single characters. math co-processor: A peripheral chip that is used in PCs to speed up mathematical operations. path name: A listing of the complete DOS path to a particular file, starting with the root directory and including all directories leading up to the file. For example, the path name \TOM\FINANCE\PROB.21 refers to a file called PROB.21 that is located in the finance directory. The finance directory is, in turn, a subdirectory of TOM, which is a subdirectory of the root directory (\). Progress window: A window that opens when you select Solve. It displays the amount of time the program is taking to find the solution, the current amount of error, and the current solution. When Eureka finds a solution, the Progress window disappears and the Solution window opens. 236 Eureka: The Solver Owner's Handbook RAM disk: A software program that causes a definable portion of RAM memory to be treated as an additional disk drive (usually D:). report file: A Eureka file that contains a report about an equation file. Such a file can be edited using the Eureka editor, but cannot be solved. Solution window: A window that displays the values Eureka has found for the indicated variables, as well as a rating expressing the degree of confidence Eureka has in the solution(s). status line: The line at the bottom of the program screen that provides information about the special function keys that are currently available. steepest descent method: A method of minimizing functions that starts at an initial point and proceeds along a path that is always in the direction of the most rapid decrease for the function. user-defined function: A one-variable function defined by the user in an equation file or with the Graph Function command. Such a function is essential for a plot to be generated. verify window: Displays the results of evaluating both sides of the equations in a file separately, and then comparing the results. Eureka must solve an equation file before it can evaluate it. Glossary 237 238 Eureka: The Solver Owner's Handbook ~ a Software For the dealer nearest you or to order by phone Call (800) 543-7543 4585 Scotts Valley Drive Scotts Valley, CA 95066 .II'.~~1I'fll IJ'~ r;n...~ rlE 'E'IT" : ••,AIIIE. ® Whether you're running WordStar,® Lotus,® dBASE,® or any other program, SideKick puts al/ these desktop accessories at your fingertips-Instantly! A lull-screen WordStar-like Editor to jot down notes and edit files up to 25 pages long. A Phone Directory for names, addresses, and telephone numbers. Finding a name or a number is a snap. An Autodialer for all your phone calls. It will look up and dial telephone numbers for you. (A modem is required to use this function.) All the SideKick windows stacked up over Lotus 1-2-3.e From bottom to top: SideKick's "Menu Window," ASCII Table, Notepad, Calculator, Appointment Calendar, Monthly Calendar, and Phone Dialer. A Monthly Calendar from 1901 through 2099. 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You'll soon become dependent on it." -Jerry Pournelle, BYTE Suggested Retail Price: $84.95 (not copy protected) Minimum system configuration: IBM PC, XT, AT, PCjr and true compatibles. PC-DOS (MS-DOS) 2.0 or greater. 128K RAM. One disk drive. A Hayes-compatible modem, IBM PCjr internal modem, or AT&T Modem 4000 is required for the autodialer function. SideKick is a registered trademark of Borland International, Inc. dBASE is a registered trademark of Ashton-Tate. IBM, XT, AT, and PCjr are registered trademarks of International Business Machines Corp. AT&T is a registered trademark of American Telephone & Telegraph Company Lotus and 1-2-3 are registered trademarks of Lotus Development Corp. WordStar is a registered trademark of MicroPro International Corp. Hayes is a trademark of Hayes Microcomputer Products, Inc. Copyright 1987 Borland International BOR0060C The Organizer For The Computer Age! Traveling SideKick is BinderWare,@ both a binder you take with you when you travel and a software program-which includes a Report Generator-that generates and prints out all the information you'll need to take with you, Information like your phone list, your client list, your address book, your calendar, and your appointments. The appointment or calendar files you're already using in your SideKickllJ can automatically be used by your Traveling SideKick. You don't waste time and effort reentering information that's already there. One keystroke prints out a form like your address book. No need to change printer paper; you simply punch three holes, fold and clip the form into your Traveling SideKick binder, and you're on your way. Because Traveling SideKick is CAD (Computer-Age Designed), you don't fool around with low-tech tools like scissors, tape, or staples. And because Traveling SideKick is electronic, it works this year, next year, and all the "next years" after that. 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SideKick, BinderWare and Traveling SideKick are registered trademarks of Borland International, Inc. IBM, AT, XT, and PCjr are registered trademarks of International Business Machines Corp. MS-DOS is a registered trademark of Microsoft Corp. Copyright 1987 Borland International BOR 0083A RAM-resident Increased productivity lor IBM~PCs or compatibles SuperKey's simple macros are electronic shortcuts to sueeess. By lelting you reduce a lengthy paragraph into a single keystroke of your choice, SuperKey eliminates repetition. SuperKey turns 1,000 keystrokes into 11 SuperKey can record lengthy keystroke sequences and play them back at the touch of a single key. Instantly. Like magic. In fact, with Super Key's simple macros, you can turn "Dear Customer: Thank you for your inquiry. We are pleased to let you know that shipment will be made within 24 hours. Sincerely," into the one keystroke of your choice! SuperKey keeps your confidential files-confidential! 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SuperKey also implements the U.S. government Data Encryption Standard (DES). ~ ~ ~ ~ ~ ~ ~ ~ RAM resident-accepts new macro files even while running other programs Pull-down menus Superfast file encryption Choice of two encryption schemes On-line context-sensitive help One-finger mode reduces key commands to single keystroke Screen OFF/ON blanks out and restores screen to protect against "burn in" Partial or complete reorganization of keyboard ~ Keyboard buffer increases 16 character keyboard "type-ahead" buffer to 128 characters ~ Real-time delay causes macro playback to pause for specified interval ~ Transparent display macros allow creation of menus on top of application programs ~ Data entry and format control using "fixed" or "variable" fields ~ Command stack recalls last 256 characters entered Suggested Retail Price: $99.95 (not copy protected) Minimum system conllguratlon: iBM PC, XT, AT, PCjr, and true compatibles. PC-DOS (MS-DOS) 2.0 or greater. 128K RAM. One disk drive. SuperKey is a registered trademark 01 Borland International, Inc. IBM, Xl, AT, and PCjr are registered trademarks 01 International Business Machines Corp. MS-DOS is a registered trademark 01 Microsoft Corp. BOA 0062C If you use an IBM® PC, you need T U R B 0 Lightning@ Turbo Lightning teams up with the Random House Concise Word List to check your spelling as you type! Turbo Lightning, using the BO,OOO-word Random House Dictionary, checks your spelling as you type. If you misspell a word, it alerts you with a "beep." At the touch of a key, Turbo Lightning opens a window on top of your application program and suggests the correct spelling. Just press one key and the misspelled word is instantly replaced with the correct word. Turbo Lightning works hand-in-hand with the Random House Thesaurus to give you instant access to synonyms Turbo Lightning lets you choose just the right word from a list of alternates, so you don't say the same thing the same way every time. Once Turbo Lightning opens the Thesaurus window, you see a list of alternate words; select the word you want, press ENTER and your new word will instantly replace the original word. Pure magic! If you ever write a word, think a. word, or say a word, you need Turbo Lightning You can teach Turbo Lightning new words You can teach your new Turbo Lightning your name, business associates' names, street names, addresses, correct capitalizations, and any specialized words you use frequently. Teach Turbo Lightning once, and it knows forever. Turbo Lightning is the engine that powers Borland's Turbo Lightning Library@ The Turbo Lightning Proofreader Turbo Lightning brings electronic power to the Random House Concise Word List and Random House Thesaurus. They're at your fingertips-even while you're running other programs. Turbo Lightning will also "drive" soon-to-be-released encyclopedias, extended thesauruses, specialized dictionaries, and many other popular reference works. You get a head start with this first volume in the Turbo Lightning Library. The Turbo Lightning Thesaurus Suggested Retail Price: $99.95 (not copy protected) Minimum system configuration: IBM PC, Xl, AT, PCjr, and true compatibles with 2 floppy disk drives. PC-DOS (MS-DOS) 2.0 or greater. 256K RAM. Hard disk recommended. BORLAND INTERNATIONAL Turbo Lightning and Turbo Lightning Library are registered trademarks of Borland International. Inc. IBM, XT, AT, and PCjr are registered trademarks of International Business Machines Corp Random House is a registered trademark of Randorn House, Inc Copyright 1987 Borland International BOA 0070B Your Development Toolbox and Technical Reference Manual for Thrbo Lightning 8 l I G H T N I N G Lightning Word Wizard includes complete, commented Turbo Pascal@ source code and all the technical information you'll need to understand and work with Turbo Lightning's engine." More than 20 fully documented Turbo Pascal procedures reveal powerful Turbo Lightning engine calls. Harness the full power of the complete and authoritative Random House@ Concise Word List and Random House Thesaurus. 6I Turbo Lightning's "Reference Manual" The ultimate collection of word games and crossword solvers! Developers can use the versatile on-line examples to harness Turbo Lightning's power to do rapid word searches. Lightning Word Wizard is the forerunner of the database access systems that will incorporate and engineer the Turbo Lightning Library® of electronic reference works. The excitement, challenge, competition, and education of four games and three solver utilities-puzzles, scrambles, spellsearches, synonym-seekings, hidden words, crossword solutions, and more. You and your friends (up to four people total) can set the difficulty level and contest the highspeed smarts of Lightning Word Wizard! Turbo Lightning-Critics' Choice "Lightning's good enough to make programmers and users cheer, executives of other software companies weep." Jim Seymour, PC Week "The real future of Lightning clearly lies not with the spelling checker and thesaurus currently included, but with other uses of its powerful look-up engine." Ted Silveira, Profiles "This newest product from Borland has it all." Don Roy, Computing Now! Minimum system configuration: IBM PC, Xl, AT, PCjr, Portable, and true compatibles. 256K RAM minimum. PC-DOS (MS-DOS) 2.0 or greater. Turbo Lightning software required. Optional-Turbo Pascal 3.0 or greater to edit and compile Turbo Pascal source code. Suggested Retail Price: $69.95 (not copy protected) Turbo Pascal. Turbo Lightning and Turbo Lightning Library are registered trademarks and Lightning Word Wizard is a trademark of Borland International, Inc. Random House is a registered trademark of Random House. Inc. IBM, XT, AT. and PCjr are registered trademarks of International BUSiness Machines Corp. MS-DOS is a registered trademark of Microsoft Corp. Copyright 1987 Borland International BOR0087B 1 '££1 £1®1IE IA1A'AIE ~r'~~ : IAIAIE. The high-performance database manager that's so advanced it's easy to use! Lets you organize, analyze and report information faster than ever before! If you manage mailing lists, customer files, or even your company's budgets-Reflex is the database manager for you! Reflex is the acclaimed, high-performance database manager you've been waiting for. Reflex extends database management with business graphics. Because a picture is often worth a 1000 words, Reflex lets you extract critical information buried in mountains of data. With Reflex, when you look, you see. The REPORT VIEW allows you to generate everything from mailing labels to sophisticated reports. You can use database files created with Reflex or transferred from Lotus 1-2-3,e dBASE,e PFS: File,e and other applications. Reflex: The Critics' Choice "... if you use a PC, you should know about Reflex ... may be the best bargain in software today." Jerry Pournelle, BYTE "Everyone agrees that Reflex is the best-looking database they've ever seen." Adam B. Green, InloWorld "The next generation of software has officially arrived." Peter Norton, PC Week Reflex: don't use your PC without it! Join hundreds of thousands of enthusiastic Reflex users and experience the power and ease of use of Borland's award~winning Reflex. Suggested Retail Price: $149.95 (not copy protected) Minimum system configuration: IBM PC, XT, AT, and true compatibles. 384K RAM minimum. IBM Color Graphics Adapter, Hercules Monochrome Graphics CArd, or equivalent. PC·DDS (MS·DOS) 2.0 or greater. Hard disk and mouse optional. Lotus 1·2·3, dBASE, or PFS: File optional. Reflex is a trademark of Borland/Analytica Inc. Lotus 1-2-3 is a registered trademark of Lotus Development Corporation. dBASE is a registered trademark of Ashton-Tate. PFS: File is a registered trademark of Software Publishing Corporation. IBM, XT, AT, and IBM Color Graphics Adapter are registered trademarks of International Business Machines Corporation. Hercules Graphics Card is a trademark of Hercules Computer Technology. MS-DOS is a registered trademark of Microsoft Corp Copyright 1987 Borland International BOR 0066e REFLEX Includes 22 "instant templates" covering a broad range of business applications (listed below). Also shows you how to customize databases, graphs, cross tabs, and reports. It's an invaluable analytical tool and an important addition to another one of our best sellers, Reflex: The Analyst 1.1. Fast-start tutorial examples: Learn Reflex8 as you work with practical business applications. The Reflex Workshop Disk supplies databases and reports large enough to illustrate the power and variety of Reflex features. Instructions in each Reflex Workshop chapter take you through a step-by-step analysis of sample data. You then follow simple steps to adapt the files to your own needs. 22 practical business applications: Workshop's 22 "instant templates" give you a wide range of analytical tools: Administration • Tracking Manufacturing Quality Assurance • • • • • Scheduling Appointments Planning Conference Facilities Managing a Project Creating a Mailing System Managing Employment Applications Sales and Marketing • • • • Researching Store Check Inventory Tracking Sales Leads Summarizing Sales Trends Analyzing Trends Production and Operations • Summarizing Repair Turnaround • Analyzing Product Costs Accounting and Financial Planning • • • • • • • • • • Tracking Petty Cash Entering Purchase Orders Organizing OutgOing Purchase Orders Analyzing Accounts Receivable Maintaining Letters of Credit Reporting Business Expenses Managing Debits and Credits Examining Leased Inventory Trend~ Tracking Fixed Assets Planning Commercial Real Estate Investment ~rether you're a newcomer learning Reflex basics or an experienced "power user" looking for tips, Reflex Workshop will help you quickly become an expert database analyst. Minimum system configuration: IBM PC, AT, and Xl, and true compatibles. PC-DOS (MS-DOS) 2.0 or greater. 384K RAM minimum. Requires Rellex: TIle Anllyst, Ind IBM Color Graphics Adapter, Hercules Monochrome Graphics Card or equivalent. Suggested Retail Price: $69.95 (not copy protected) Reflex is a registered trademark and Reflex Workshop is a trademark of Borland/Analytica, Inc. IBM, AT, and XT are registered trademarks of International BusineSS . Machines Corp. Hercules is a trademark of Hercules Computer Technology. MS-DOS is a registered trademark of Microsoft Corp. BOROO88A Version 3.0 with 8087 support and BCD reais Free MicroCa/c Spreadsheet With Commented Source Code! FEATURES: THE CRITICS' CHOICE: One-Step Compile: No hunting &fishing expeditions! Turbo finds the errors, takes you to them, lets you correct them, and instantly recompiles. You're off and running in record time. "Language deal of the century ... Turbo Pascal: it introduces a new programming environment and runs like magic." -Jeff Duntemann, PC Magazine "Most Pascal compilers barely fit on a disk, but Turbo Pascal packs an editor, compiler, linker, and run-time library into just 39K bytes of random access memory." -Dave Garland, Popular Computing Built-in Interactive Editor: WordStar®-like easy editing lets you debug quickly. Automatic Overlays: Fits big programs into small amounts of memory. MicloCalc: A sample spreadsheet on your disk with ready-to-compile source code. IBM® PC Version: Supports Turtle Graphics, color, sound, full tree directories, window routines, input/output redirection, and much more. "What I think the computer industry is headed for: well-documented, standard, plenty of good features, and a reasonable price." -Jerry Pournelle, BYTE LOOK AT TURBO NOW! ~ More than 500,000 users worldwide. ~ Turbo Pascal is the de facto industry standard. ~ Turbo Pascal wins PC MAGAZINE'S award for technical excellence. ~ Turbo Pascal named "Most Significant Product of the Year" by PC WEEK. 5l' Turbo Pascal 3.0-the fastest Pascal development environment on the planet, period. Suggested Retail Price: $99.95; CPIM®-80 version without 8087 and BCD: $69.95 Features for 16-bit Systems: 8087 math co-processor support for intensive calculations. Binary Coded Decimals (BCD): eliminates round-off error! A must for any serious business application. Minimum system configuration: 128K RAM minimum. Includes 8087 & BCD features for 16-bit MS-DOS 2.0 or later and CP/M-861.1 or later. CP/M-80 version 2.2 or later 48K RAM minimum (8087 and BCD features not available). 8087 version requires 8087 or 80287 co-processor. BORLAND INTERNATIONAL Turbo Pascal is a registered trademark of Borland International. Inc. CP/M is a registered trademark of Digital Research Inc. IBM is a registered trademark of International Business Machines Corp. MS-DOS is a registered trademark of Microroft Corp. WordStar is a registered trademark of MicroPro International. Copyright 1987 Borland International BOR 0061B VERSION 2.0 Learn Pascal From The Folks Who Created The Turbo Pascal® Family Borland International proudly presents Turbo Tutor, the perfect complement to your Turbo P~scal compiler. Turbo Tutor is really for everyoneeven if you've never programmed before. And if you're already proficient, Turbo Tutor can sharpen up the fine points. The manual and program disk focus on the whole spectrum of Turbo Pascal programming techniques. • For the Novice: It gives you a concise history of Pascal, tells you how to write a simple program, and defines the basic programming terms you need to know. • Programmer's Guide: The heart of Turbo Pascal. The manual covers the fine paints of every aspect of Turbo Pascal programming: program structure, data types, control structures, procedures and functions, scalar types, arrays, strings, pointers, sets, files, and records. • Advanced Concepts: If you're an expert, you'll love the sections detailing such topics as linked lists, trees, and graphs. You'll also find sample program examples for PC-DOS and MS-OOS.® 10,000 lines of commented source code, demonstrations of 20 Turbo Pascal features, multiplechoice quizzes, an interactive on-line tutor, and more! Turbo Tutor may be the only reference work about Pascal and programming you'll ever need! Suggested Retail Price: $39.95 (not copy protected) Minimum system configuration: Turbo Pascal 3.0. PC·DOS (MS·DOS) 2.0 or later. 192K RAM minimum (CP/M·aO version 2.2 or later: 64K RAM minimum). Turbo Pascal and Turbo Tutor are registered trademarks of Borland International Inc. CPIM is a registered trademark of Digital Research Inc. MS-DOS is a· registered trademark of Microsoft Corp. Copyright 1987 Borland International BOA 0064C Is The Perfect Complement To Turbo Pasca/~ It contains a complete library of Pascal procedures that allows you to sort and search your data and build powerful database applications. It's another set of tools from Borland that will give even the beginning programmer the expert's edge. THE TOOLS YOU NEED! TURBO ACCESS Using B+ trees: The best way to organize and search your data. Makes it possible to access records in a file using key words instead of numbers. Now available with complete source code on disk, ready to be included in your programs. TURBO SORT: The fastest way to sort data using the QUICKSORT algorithm-the method preferred by knowledgeable professionals. Includes source code. GINST (General Installation Program): Gets your programs up and running on other terminals. This feature alone will save hours of work and research. Adds tremendous value to all your programs. GET STARTED RIGHT AWAY.-FREE DATABASE! Included on every Toolbox diskette is the source code to a working database which demonstrates the power and simplicity of our Turbo Access search system. Modify it to suit your individual needs or just compile it and run. THE CRITICS' CHOICE! 'The tools include a B+ tree search and a sorting system. I've seen stuff like this, but not as well thought out, sell for hundreds of dollars." -Jerry Pournelle, BYTE MAGAZINE "The Turbo Database Toolbox is solid enough and useful enough to come recommended." -Jell Duntemann, PC TECH JOURNAL Suggested Retail Price: $69.95 (not copy protected) Minimum system configuration: 128K RAM and one disk drive (CP/M-80: 48K). 16-bit systems: Turbo Pascal 2.0 or greater for MS-DOS or PC-DOS 2.0 or greater. Turbo Pascal 2.1 or greater for CP/M-86 1.0 or greater. 8-bit systems: Turbo Pascal 2.0 or greater for CP/M-80 2.2 or greater. Turbo Pascal and Turbo Database Toolbox are registered trademarks of Borland International Inc. CP1M is a registered trademark of Dignal Research. Inc. MS-DOS is a registered trademark of Microsoft Corp. Copyright 1987 Borland International BOR 00630 A Library of Graphics Routines for Use with Turbo Pasca/® High-resolution graphics for your IBM" PC, AT," XT,'"' PC;,., true PC compatibles, and the Heath Zenith Z-100." Comes complete with graphics window management. Even if you're new to Turbo Pascal programming, the Turbo Pascal Graphix Toolbox will get you started right away. It's a collection of tools that will get you right into the fascinating world of high-resolution business graphics, including graphics window management. You get immediate, satisfying results. And we keep Royalty out of American business because you don't pay any-even if you distribute your own compiled programs that include all or part of the Turbo Pascal Graphix Toolbox procedures. What you get includes: • Complete commented source code on disk. • Tools for drawing simple graphics. • Tools for drawing complex graphics, including curves with optional smoothing. • Routines that let you store and restore graphic images to and from disk. • Tools allowing you to send screen images to Epson@l-compatible printers. • • • • Full graphics window management. Two different font styles for graphic labeling. Choice of line-drawing styles. Routines that will let you quickly plot functions and model experimental data. • And much, much more ... "While most people only talk about low-cost personal computer software, Borland has been dOing something about it. And Borland provides good technical support as part of the price." John Markov & Paul Freiberger, syndicated columnists. If you ever plan to create Turbo Pascal programs that make use 01 business graphics or scientific graphics, you need the Turbo Pascal Graphix Toolbox. Suggested Retail Price: $69.95 (not copy protected) Minimum system configuration: IBM PC, Xl, AT, PCjr, true compatibles and the Heath Zenith Z·100. Turbo Pascal 3.0 or later. 192K RAM minimum. Two disk drives and an IBM Color Graphics Adapter (CGA), IBM Enhanced Graphics Adapter (EGA), Hercules Graphics Card or compatible. BORLAND INTERNATIONAL Turbo Pascal and Turbo Graphix Toolbox are registered trademarks of Borland International. Inc. IBM, XT, AT, and PCjr are registered trademarks of International Business Machines Corporation. Hercules Graphics Card is a trademark of Hercules Computer Technology. Heath Zenith Z-100 is a trademark of Zenith Data Systems Epson is a registered trademark of Epson Corp. Copyright 1987 Borland International BOR 0068C 2VR8O PASCAl. .EDJ1OR 2txltSOX® It's All You Need To Build Your Own Text Editor Or Word Processor Build your own lightning-fast editor and incorporate it into your Turbo Pascal@ programs. Turbo Editor Toolbox gives you easy-to-install modules. Now you can integrate a fast and powerful editor into your own programs. You get the source code, the manual, and the know-how. Create your own word processor. We provide all the editing routines. You plug in the features you want. You could build aWordStar@-like editor with pull-down menus like Microsoft's@ Word, and make it work as fast as WordPerfect.@ To demonstrate the tremendous power of Turbo Editor Toolbox, we give you the source code for two sample editors: Simple Editor MicroStar A complete editor ready to include in your programs. With windows, block commands, and memory-mapped screen routines. A full-blown text editor with a complete pull-down menu user interface, plus a lot more. Modify MicroStar's pull-down menu system and include it in your Turbo Pascal programs. The Turbo Editor Toolbox gives you all the standard features you would expect to find in any word processor: • • • • • • • Wordwrap UN-delete last line Auto-indent Find and Find/Replace with options Set left and right margin Block mark, move, and copy Tab, insert and overstrike modes, centering, etc. MicroStar's pull-down menus. And Turbo Editor Toolbox has features that word processors selling for several hundred dollars can't begin to match. Just to name a few: [!] RAM-based editor. You can edit very large files and yet editing is lightning fast. ~ Memory-mapped screen routines. Instant paging, scrolling, and text display. ~ Keyboard installation. Change control keys from WordStar -like commands to any that you prefer. ~ Multiple windows. See and edit up to eight documents-or up to eight parts of the same document-all at the same time. ~ Multitasking. Automatically save your . text. Plug in a digital clock, an apPOintment alarm-see how it's done with MicroStar's "background" printing. Best of all, source code is included for everything in the Editor Toolbox. Suggested Retail Price: $69.95 (not copy protected) Minimum system configuration: IBM PC, XT, AT, 3270, PCjr, and true compatibles. PC-DOS (MS-DOS) 2.0 or greater. 192K RAM. You must be using Turbo Pascal 3.0 for IBM and compatibles. Turbo Pascal and Turbo Editor Toolbox are registered trademarks of Borland Inlernational, Inc. WordStar is a registered trademark of MicroPro International Corp. Word and MS-DOS are registered trademarks of Microsoft Corp. WordPerfect is a trademark of Satellite Software International. IBM, XT, AT, and PCjr are registered trademarks of International BUSiness Machines Cor~ OOROO~B ® Secrets And Strategies 01 The Masters Are Revealed For The First Time Explore the world of state-of-the-art computer games with Turbo GameWorks. Using easy-to-understand examples, Turbo GameWorks teaches you techniques to quickly create your own computer games using Turbo Pascal~ Or, for instant excitement, play the three great computer games we've included on disk-compiled and ready to run. TURBO CHESS Test your chess-playing skills against your computer challenger. With Turbo GameWorks, you're on your way to becoming a master chess player. Explore the complete Turbo Pascal source code and discover the secrets of Turbo Chess. "What impressed me the most was the fact that with this program you can become a computer chess analyst. You can add new variations to the program at any time and make the program play stronger and stronger chess. There's no limit to the fun and enjoyment of playing Turbo GameWorks Chess, and most important of all, with this chess program there's no limit to how it can help you improve your game." -George Koitanowski, Dean of American Chess, former President of the United Chess Federation, and syndicated chess columnist. TURBO BRIDGE Now play the world's most popular card game-bridge. Play one-on-one with your computer or against up to three other opponents. With Turbo Pascal source code, you can even program your own bidding or scoring conventions. ''There has never been a bridge program written which plays at the expert level, and the ambitious user will enjoy tackling that challenge, with the format already structured in the program: And for the inexperienced player, the bridge program provides an easy-to-follow format that allows the user to start right out playing. The user can 'play bridge' against real competition without having to gather three other people." -Kit Woolsey, writer of several articles and books on bridge, and twice champion of the Blue Ribbon Pairs. TURBO GO-MOKU Prepare for battle when you challenge your computer to a game of Go-Moku-the exciting strategy game also known as Pente.e In this battle of wits, you and the computer take turns placing X's and D's on a grid of 19X19 squares until five pieces are lined up in a row. Vary the game if you like, using the source code available on your disk. Suggested Retail Price: $69.95 (not copy protected) Minimum system configuration: IBM PC, XT, AT, Portable, 3270, PCjr, and true compatibles. PC·DOS (MS·DOS) 2.0 or later. 192K RAM minimum. To edit and compile the Turbo Pascal source code, you must be using Turbo Pascal 3.0 for IBM PCs and compatibles. TURBO PASCAL IUlfEIICAllfE1111B 1111BI,TN New .from Borland's Scientific &Engineering Division! A complete collection of Turbo Pascaf routines and programs New from Borland's Scientific & Engineering Division, Turbo Pascal Numerical Methods Toolbox implements the latest high-level mathematical methods to solve common scientific and engineering problems. Fast. So every time you need to calculate an integral, work with Fourier Transforms or incorporate any of the classical numerical analysis tools into your programs, you don't have to reinvent the wheel. Because the Numerical Methods Toolbox is a complete collection of Turbo Pascal routines and programs that gives you applied state-of-the-art math tools. It also includes two graphics demo programs, Least Squares Fit and Fast Fourier Transforms, to give you the picture along with the numbers. The Numerical Methods Toolbox is a must for you if you're involved with any type of scientific or engineering computing. Because it comes with complete source code, you have total control of your application. What Numerical Methods Toolbox will do for you now: • Find solutions to equations • Interpolations • Calculus: numerical derivatives and integrals • Fourier transforms • Matrix operations: inversions, determinants and eigenvalues • Differential equations • Least squares approximations 5 free ways to look at ilL east Squares Fit"! As well as a free demo "Fast Fourier Transforms," you also get "Least Squares Fit" in 5 different forms-which gives you 5 different methods of fitting curves to a collection of data pOints. You instantly get the picture! The 5 different forms are: 1. Power 4. 5-term Fourier 2. Exponential 5. 5-term 3. Logarithm Polynomial They're all ready to compile and run "as is." To modify or add graphics to your own programs, you simply add Turbo Graphix Toolbox@) to your software library. Our Numerical Methods Toolbox is designed to work hand-in-hand with our Turbo Graphix Toolbox to make professional graphics in your own programs an instant part of the picture! Suggested Retail Price: $99.95 (not copy protected) Minimum system configuration: IBM PC. XT, AT and true compatibles. PC-DOS (MS-DOS) 2.0 or later. 256K. Turbo Pascal 2.0 or later. The graphics modules require a graphics monitor with an IBM CGA, IBM EGA, or Hercules compatible adapter card, and require the Turbo Graphix Toolbox. MS-DOS generic version will not support Turbo Graphix Toolbox routines. An 8087 or 80287 numeric co-processor is not required, but recommended for optimal performance. Turbo Pascal Numerical Methods Toolbox is a trademark and Turbo Pascal and Turbo Graphix Toolbox are registered trademarks of Borland International, Inc. IBM, XT, and AT are registered trademarks of International Business Machines Corp MS-DOS is a registered trademark of Microsoft Corp. Hercules is a trademark of Hercules Computer Technology. Apple is a registered trademark of Apple Computer, Inc. Macintosh is a trademark of Mcintosh Laboratory, Inc. licensed to Apple ComputeLCopyright 1987 Borland International BOR 0219A TURBO the natural language of ArtirlCiallntelligence , Turbo· Prolog brings fifth-generation supercomputer power to your IBM®PC! Turbo Prolog takes Turbo Prolog provides programming into a new, a fully integrated pronatural, and logical gramming environment environment like Borland's Turbo With Turbo Prolog, Pascal,® the de facto because of its natural, worldwide standard. logical approach, both You get the people new to programming complete Turbo and professional programmers Prolog programming can build powerful applicasystem tions such as expert systems, You get the 200-page customized knowledge manual you're holding, bases, natural language software that includes interfaces, and smart the lightning-fast Turbo information management systems. Prolog six-pass Turbo Prolog is a declarative language which compiler and interactive editor, and the uses deductive reasoning to solve \ free GeoBase natural query language programming problems. database, which includes commented source code on disk, ready to compile. (GeoBase is a complete database de-stgned Turbo Prolog's development system and developed around U.S. geography. includes: You can modify it or use it "as is.") o o o o A complete Prolog compiler that is a variation of the Clocksin and Mellish Edinburgh standard Prolog. A full-screen interactive editor. Support for both graphic and text windows. All the tools that let you build your own expert systems and AI applications with unprecedented ease. BORLAND INTERNATIONAL Minimum system configuration: IBM PC, Xl, AT, Portable, 3270, PCjr and true compatibles. PC-DOS (MS-DOS) 2.0 or later. 384K RAM minimum. Suggested Retail Price: $99.95 (not copy protected) Turbo Prolog is a trademark and Turbo Pascal is a registered trademark of Borland Intemational, Inc. IBM, AT, XT, and PCjr are registered trademarks of International Business Machines Corp. MS-DOS is a registered trademark of Microsoft Corp. Copyright 1987 Borland International BOR oo16D Enhances Turbo Prolog with more than 80 tools and over 8,000 lines of source code Turbo Prolog, the natural language of Artificial Intelligence, is the most popular AI package in the world with more than 100,000 users. Our new Turbo Prolog Toolbox extends its possibilities. The Turbo Prolog Toolbox enhances Turbo Prolog-our 5th-generation computer programming language that brings supercomputer power to your IBM PC and compatibles-with its more than 80 tools and over 8,000 lines of source code that can be incorporated into your programs, quite easily. g g g g g g g g g Turbo Prolog Toolbox features include: Business graphics generation: boxes, circles, ellipses, bar charts, pie charts, scaled graphics Complete communications package: supports XModem protocol File transfers from Reflex,8 dBASE 111,8 Lotus 1-2-3,~ Symphony~ A unique parser generator: construct your own compiler or query language Sophisticated user -interface design tools 40 example programs Easy-to-use screen editor: design your screen layout and 1/0 Calculated fields definition Over 8,000 lines of source code you can incorporate into your own programs Suggested Retail Price: $99.95 (not copy protected) Minimum system configuration: IBM PC, XT, AT or true compatibles. PC-DOS (MS-DOS) 2.0 or later. Requires Turbo Prolog 1.10 or higher. Dual-floppy disk drive or hard disk. 512K. Turbo Prolog Toolbox and Turbo Prolog are trademarks of Borland International, Inc. Reflex is a registered trademark of Borland/Analytica, Inc. dBASE III is a registered trademark of Ashton-Tate. Lotus 1-2-3 and Symphony are registered trademarks of Lotus Development Corp. IBM, XT, and AT are registered trademarks 01 International Business Machines Corp. MS-DOS is a registered trademark of Microsoft Corp. BOA 0240 lUllI IABIC® The high-speed BASIC you've been waiting for! You probably know us for our Turbo Pascal® and Turbo Prolog.® Well, we've done it again! We've created Turbo Basic, because BASIC doesn't have to be slow. If BASIC taught you how to walk, Turbo Basic will teach you how to run! With Turbo Basic, your only speed is "Full Speed Ahead"! Turbo Basic is a complete development environment with an amazingly fast compiler, an interactive editor and a trace debugging system. And because Turbo Basic is also compatible with BASICA, chances are that you already know how to use Turbo Basic. Turbo Basic ends the basic confusion There's now one standard: Turbo Basic. And because Turbo Basic is a Borland product, the price is right, the quality is there, and the power is at your fingertips. Turbo Basic is part of the fast-growing Borland family of programming languages we call the "Turbo Family." And hundreds of thousands of users are already using Borland's languages. So, welcome to a whole new generation of smart PC users! Free spreadsheet included with source code! Yes, we've included MicroCalc,'" our sample spreadsheet, complete with source code. So you can get started right away with a "real program." You can compile and run it "as is," or modify it. B B B B B B A technical look at Turbo Basic Full recursion supported executable program, with separate windows for editing, messages, tracing, and execution Standard IEEE floating-point format Floating-point support, with full 8087 copro- B Compile and run-time errors place you in source code where error occurred cessor integration. Software emulation if no B Access to local, static and global variables 8087 present B New long integer (32-bit) data type Program size limited only by available memory (no 64K limitation) B Full 80-bit precision EGA, CGA, MCGA and VGA support B PUll-down menus Full integration of the compiler, editor, and B Full window management Suggested Retail Price: $99.95 (not copy protected) Minimum system configuration: IBM PC, AT, XT, PS/2 or true compatibles. 320K. One floppy drive. PC-DOS (MS-DOS) 2.0 or later. Turbo Basic, Turbo Prolog and Turbo Pascal are registered trademarks and MicroCalc is a trademark of Borland International, Inc. Other brand and product names are trademarks or registered trademarks of their respective holders. Copyright 1987 Borland International BOR 0265B .11,rll'''I® liE IEII1I1' In'.. : IJ'~ ~ III.flll Release 2.0 Macintosh T • The most complete and comprehensive collection 01 desk accessories available lor your Macintosh! Thousands of users already know that SideKick is the best collection of desk accessories available for the Macintosh. With our new Release 2.0, the best just got better. We've just added two powerful high-performance tools to SideKick-OutlookTM: The Outliner and MacPlan™: The Spreadsheet. They work in perfect harmony with each other and while you run other programs! Outlook: The Outliner • It's the desk accessory with more power than a stand-alone outliner • A great desktop publishing tool, Outlook lets you incorporate both text and graphics into your outlines • Works hand-in-hand with MacPlan • Allows you to work on several outlines at the same time • • • • • MacPlan: The Spreadsheet Integrates spreadsheets and graphs Does both formulas and straight numbers Graph types include bar charts, stacked bar charts, pie charts and line graphs Includes 12 example templates free! Pastes graphics and data right into Outlook creating professional memos and reports, complete with headers and footers. 1!1 1!1 1!1 1!1 1!1 1!1 1!1 1!1 1!1 1!1 SideKick: The Desktop Organizer, Release 2.0 now includes Outlook: The Outliner MacPlan: The Spreadsheet Mini word processor Calendar Phone Log Analog clock Alarm system Calculator Report generator Telecommunications (new version now supports XModem file transfer protocol) • 13.67'fi5 5.l1u,", ~ ::::: :~::~venQes I:J!Hl J~H1LI III • em em txP«IU'f o Q.ll9i1 Labor [0 4.66'iii1i Natwia1s ~ 6.219a Ov.rh"ad I] 11.IB~ Tol~IExpens(l$ El • "" leA!!; fletPn:>fil MacPlan does both spreadsheets and business graphs. Paste them into your Outlook files and generate professional reports. Suggested Retail Price: $99.95 (not copy protected) Minimum system configurations: Macintosh 512K or Macintosh Plus with one disk drive. One aOOK or two 400K drives are recommended. With one 400K drive, a limited number of desk accessories will be installable per disk. BORLAND INTERNATIONAL SideKick is a registered trademark and Outlook and MacPlan are trademarks of Borland International, Inc. Macintosh is a trademark of Mcintosh Laboratory, Inc. licensed to Apple Computer, Inc. Copyright 1987 Borland International BOR 0069D '~~III; 111 IA1AIA" r;,] • • AIAIEI 1 The easy-to-use relational database that thinks like a spreadsheet. Rellex lor the Mac lets you crunch numbers by entering lormulas and link databases by drawing on-screen lines. • A client billing application set up for a law office, but easily customized by any profeSSional who bills time. • A parts explosion application that breaks down an object into its component parts for cost analysis. • A sample 1040 tax application with Schedule A, Schedule B, and Schedule 0, each contained in a separate report document. • A portfolio analysis application with linked databases of stock purchases, sales, and dividend payments. • A checkbook application. Reflex for the Mac accomplishes all of these tasks without programming-using spreadsheet-like formulas. Some other Reflex for the Mac features are: • Visual database design. • "What you see is what you get" report and form layout with pictures. • Automatic restructuring ·of database files when data types are changed, or fields are added and deleted. • Display formats which include General, Decimal, Scientific, Dollars, Percent. • File Edit formet • Data types which include variable length text, number, integer, automatically incremented sequence number, date, time, and logical. • Up to 255 fields per record. • Up to 16 files simultaneously open. • Up to 16 Mac fonts and styles are selectable for individual fields and labels. De5tribellDlllll!llWlnclow ~~"~~T ,p.~j--J .__ ......... a~~~:~-~_~~J~g------.~"__ ----. POtt -. -----.- -- -'",-- ~ '" I F~~::~:~~,,::··mbi~-' After opening the "Overview" window. you draw link lines between databases directly onto your Macintosh screen. The link lines you draw establish both visual and electronic relationships between your databases. You can have multiple windows open simultaneously to view all members of a linked set-which are interactive and truly relational. Critic's Choice " ... a powerful relational database ... uses a visual approach to information management." InloWorld "... gives you a lot of freedom in report design; you can even import graphics." A+ Magazine "... bridges the gap between the pretty programs and the power programs." Stewart Alsop, PC Letter Suggested Retail Price: $99.95* (not copy protected) Minimum $Y$tem conll,uration: Macirtosh 512K or Macintosh Plus with one disk drive. Second external drive recommended. Reflex is a registered trademark of BorlandiAnalytica, Inc. Macintosh is a trademark of Mcintosh Laboratory, Inc. and is used with express permiSSion of its owner. Copyright 1987 Borland International BOR0149A The ultimate Pascal development environment Borland's new Turbo Pascal lor the Mac is so incredibly last that it can compile 1,420 lines 01 source code in the 7.1 seconds it took you to read this! And reading the rest of this takes about 5 minutes, which is plenty of time for Turbo Pascal for the Mac to compile at least 60,000 more lines of source code! Turbo Pascal for the Mac does both Windows and "Units" The separate compilation of routines offered by Turbo Pascal for the Mac creates modules called "Units," which can be linked to any Turbo Pascal program. This "modular pathway" gives you "pieces" which can then be integrated into larger programs. You get a more efficient use of memory and a reduction in the time it takes to develop large programs. Turbo Pascal for the Mac is so compatible with Lisa~ that they should be living together Routines from Macintosh Programmer's Workshop Pascal and Inside Macintosh can be compiled and run with only the subtlest changes. Turbo Pascal for the Mac is also compatible with the Hierarchical File System of the Macintosh. The 21-second Guide to Turbo Pascal for the Mac • Compilation speed of more than 12,000 lines per minute • "Unit" structure lets you create programs in modular form • Multiple editing windows-up to 8 at once • Compilation options include compiling to disk or memory, or compile and run • No need to switch between programs to compile or run a program • Streamlined development and debugging • Compatibility with Macintosh Programmer's Workshop Pascal (with minimal changes) • Compatibility with Hierarchical File System of your Mac • Ability to define default volume and folder names used in compiler directives • Search and change features in the editor speed up and simplify alteration of routines • Ability to use all available Macintosh memory without limit • "Units" included to call all the routines provided by Macintosh Toolbox Suggested Retail Price: $99.95* (not copy protected) Minimum system configuration: Macintosh 512K or Macintosh Plus with one disk drive. Turbo Pascal and SideKick are registered trademarks of Borland Internalional, Inc. and Reflex is a registered trademark of Borland/Analytica, Inc. Macintosh is a trademark of Mcintosh Laboratories, Inc. licensed to Apple Computer with its express permission. Lisa is a registered trademark of Apple Computer, Inc. Inside Macintosh is a copyright of Apple Computer, Inc. Copyright 1987 Borland International BOA 0167A ',IBB PABCAl® ','111 From the folks who created Turbo Pascal. Borland's new Turbo Pascal Tutor is everything you need to start programming in Turbo Pascal on the Macintosht It takes you from the bare basics to advanced programming in a simple, easy-to-understand fashion. M Using the Power of the Macintosh discusses the revolutionary hardware and software features of this machine. It introduces the 600-plus utility routines in the Apple Toolbox. Programming the Macintosh in Turbo Pascal shows you how to create true Macintosh programs that use graphics, pull-down menus, dialog boxes, and so on. Finally, MacTypist, a complete stand-alone application featuring animated graphics, builds on Turbo Typist and demonstrates what you can do with all the knowledge you've just acquired. The disk contains the source code for all the sample programs, including Turbo Typist, MacTypist, and Turbo Tutor. The Tutor's split screen lets you run a procedure and view its source code simultaneously. After running it, you can take a test on the procedure. If you're stuck for an answer, a Hint option steers you in the right direction. No gimmicks. It's all here. The manual, the Tutor application, and 30 sample programs provide a step-by-step tutorial in three phases: programming in Pascal, programming on the Macintosh, and programming in Turbo Pascal on the Macintosh. Here's how the manual is set up: Turbo Pascal for the Absolute Novice delivers the basics-a concise histor~ of Pascal, key terminology, your first program. . A Programmer's Guide to Turbo Pascal covers Pascal specifics-program structure, procedures and functions, arrays, strings, and so on. We've also included Turbo Typist, a textbook sample program. Advanced Programming takes you a step higher into stacks, queues, binary trees, linked structures, writing large programs, and more. Macintosh topics included are 5f 5f 5f g memory management resources and resource files QuickDraw events g windows 5f controls g menus g dialogs g debugging g desk accessory support g File Manager Suggested Retail Price: $69.95 Minimum sYltem requirements: Any Macintosh with at least 512K 01 RAM. Requires Turbo PascaL Turbo Pascal and Turbo TUlor are regislered Irademarks 01 Borland Inlernalional, Inc Other brand and prOduct names are trademarks or registered trademarks ot their respective hotders Copyright 1987 Borland tnlernational BOR 0381 I"cludes tree MicroCalc spreadsheet witll source code A complete interactive development environment With Turbo C, you can expect what only Borland delivers: Quality, Speed, Power and Price. And with its compilation speed 01 more than 1000 lines a minute, Turbo C makes everything else look like an exercise in slow motion. Turbo C: The C compiler for both amateurs and professionals If you're just beginning and you've "kinda wanted to learn C," now's your chance to do it the easy way. Turbo C's got everything to get you going. If you're already programming in C, switching to Turbo C will considerably increase your productivity and help make your programs both smaller and faster. Turbo C: a complete interactive development environment Like Turbo Pascale and Turbo Prolog," Turbo C comes with an interactive editor that will show you syntax errors right in your source code. Developing, debugging, and running a Turbo C program is a snap! Technical Specifications ~ Development Environment: A powerful "Make" is included so that managing Turbo C program development is easy. Borland's fast liTurbo Linker" is also included. Also includes pull-down menus and windows. Can run from the environment or generate an executable file. [iV' Links with relocatable object modules created using Borland's Turbo Prolog into a ,-,/ single program.. ~ ANSI C compatible. . ~ Start-up routine ~ource ~ode Included.. ~ Both. co~mand line and Integrated environment versions Included. [iV' Compiler: One-pass compiler generating native inline code, linkable object modules and assembler. The object module format is compatible with the PC-DOS linker. Supports small, medium, compact, large, and huge memory model libraries. Can mix models with near and far pOinters. Includes floating point emulator (utilizes 8087/80287 if installed). [iV' Interactive Editor: The system includes a powerful, interactive full-screen text editor. If the compiler detects an error, the editor automatically positions the cursor appropriately in the source code. "Sieve" benchmark (25 iterations) Turbo C Microsoft'" C Lattice C Compile time 3.89 16.37 13.90 Compile and link time 9.94 29.06 27.79 Execution time 9.51 13.79 Object code size 5.77 274 297 301 Price $99.95 $450.00 $500.00 Benchmark run on a 6 Mhz IBM AT using Turbo eversion 1.0 and the Turbo Linker version 1.0; Microsoft eversion 4.0 and the MS overlay linker version 3.51; Lattice eversion 3.1 and the MS object linker version 3.05. Suggested Retail Price: $99.95* (not copy protected) Minimum system configuration: IBM PC, XT, AT and true compatibles. PC-DOS (MS-DOS) 2.0 or later. One floppy drive. 320K. Turbo C and Turbo Pascal are registered trademarks and Turbo Prolog is a trademark of Borland International, Inc. Microsoft C and MS-DOS ere registered trademarks of Microsoft Corp. Lattice C is a registered trademark of Lattice. Inc. IBM. Xl, and AT are registered trademarks of International Business Machines Corp. BOA 0243 For the dealer nearest you or to order by phone Call (800) 543-7543 INTERNATIONAL 4585 Scotts Valley Drive Scotts Valley, CA 95066 Index A E Accuracy setting, 31-32, 44, 88 ASCII files, importing to Eureka, 43 AUTOEXEC.BAT, 15, 213 Edit command, 56 Editor, 2, 9, 21-23, 28, 35-37, 56, 191-196 customizing, 229-233 key summary, 23, 37, 192-196 EINST.COM, 229-233 .EKA files, 105-186, see also Problem files Equation file, capturing, 65 creating, 21, 24 directories for, 53-54 loading, 51, 52 renaming, 54 saving, 23, 34, 52-53 syntax, 9-10, 79-87, 96 comments, 25, 82, 85 directives, 31, 82, 86 functions, 102 initialization, 10, 40-41, 44, 71, 81 order of evaluation, 83 reserved words, 10 symbols, 9, 10, 25, 80, 83-85, 96 units conversion, 84, 97 values, 83 variables, 9, 82 Equations, accuracy of, 31-32, 43-46, 88 confidence level, Eureka's, 11 constraints on, 31, 41, 44, 93 directives in, 31, 72, 82, 86-98 evaluating, 12, 33, 59 functions in, 27-29, 41, 45, 58, 66-69, 81 graphing, 27 integral (example), 107-109 modeling tips for, 44-46 nonlinear, 127-129, 130-133 report of, 12 roots, finding other, 29, 32, 40, 42, 44, 61 roots, polynomial, 58, 148-151 simultaneous, 127-129, 130-133, 158-160 solving, 11, 26, 31, 40-42, 57, 93-95 batch problems, 42 iterative search, 26, 31, 41, 61 Progress window and, 26 range searched, 94 selected variables, 95 time spent, 93 B BASIC problem file, 43, 139 Batch file problems, 42 Branch cut, 89 C Calculator command, 2, 60 Capture command, 65 CARDS.EKA, 165-168 CARLOAN.EKA,123 Change dir command, 54 CHEMBAL.EKA, 172-177 CHEMEQ.EKA,177-186 Chemistry problems, 172-186 CLASSICS.EKA, 108 Close command, 77 Colors, changing, 74, 75 Colors command, 74 Commands menu, 59-62 Comments, 25, 82, 85 COMPLEX.EKA,109 Complex numbers, 89 examples, 107, 109, 148-151 Confidence level, Eureka's, 11 Constants, 45, 71 complex, 89 example, 137-140 Constraints on solving, 31-32, 41, 44, 93 D DECAY.EKA, 134-136 Decimal places displayed, 31, 89 DERINT.EKA, 108 Deriv function (example), 107, 108 Derivatives, 107, 108 Digits directive, 31, 89 Directives, 31, 72, 82, 86-98, 230 Directories, changing, 53, 54, 74, 211-213 Directories command, 74 Directory command, 53 DOS, 209-215 DOS shell, 55 Eureka: The Solver Owner's Handbook transcendental (examples), 108, 134-136, 141-l43 trigonometric (example), 108 Verify command and, 12, 33, 59 EQUITY.EKA,161-164 Error messages, 46, 217-227 EUREKA.LOG, 65 EUREKA.MAC, 86 Evaluating equations, see Verify command F Fact function (example), 165-168 Factorial, see Fact function File menu, 51-55 Files on Eureka disk, 4 Financial functions, 90, 99-101 examples, 123-126, 161-164 Finansmooth directive (example), 162 Find other command, 32, 40, 42, 61 Floating point error, 46 Formatted command, 65 Function command, 69 Functions built-in, 99-10l, 197-208 complex, 89 definite integrals, 107-109, 169-171 financial, 90, 99-10l examples, 123-l26, 161-164 lists, 68, 92 maximizing/minimizing, 41, 92-93 examples, 107, 144-147, 152-157, 158-160 nonlinear, 152-157 plotting, 27-29, 66-69 polynomial, 58, 148-151 poorly scaled, 45 transcendental, 134-136, 141-143 trigonometric, 89 user-defined,2~ 81, 101-103,107-108 variables and, 101, 152-157 Fval function (example), 161-164 G Go command, 64 Goto command, 78 Graphs, see Plots Graph menu, 27-29, 66-69 Graphic card, and plots, 28 Index H Hard copy, see Report Help, 13 HOUSE.EKA, 152-157 I Imaginary numbers, 89,107,109,148-151 Include directive (example), 175 Include files, 91 INITIAL.EKA, 75 Initialization values, 81, 91 Installation, 14-16 Floppy disk, 14 Hard disk, 15 RAM disk, 15 INTEG.EKA,169-171 Integ function (examples), 107-109, 169-171 Integrals (examples), 107-109, 169-171 Iterate command, 26, 41 Iterative searches, 31, 41 K Keys, in Eureka, 8, 18, 20 summary, 20, 23 L LADDERS.EKA, 127-l29 Least squares fit (example), 137-140 LINECHG.EKA, 130-133 Linear programming (example), 158-160 LINPROG.EKA, 158-160 List command, 68 Load command, 51 Load setup command, 75 Loading J;:ureka, 14-16 Loan problem, 123 LSQRFIT.EKA, 137-140 M Main menu, 48 MASSCH;G.EKA, 144-147 Math coprocessor chip, 5 Mathematical symbols in Eureka, 4, 9, 25, 83-85,96 Max directive, 41, 92, 107, 108 examples, 152-157, 158-160 Maximization, see Minimization/maximization problems MAXMIN.EKA,108 Menus, 8, 17, 19, 47-78 selecting from, 20 summary, 19, 49 Min directive, 41, 93, 107, 108 example, 144-147 Minimization/maximization problems, 11, 26, 40-42, 92-93, examples, 107, 108, 144-147, 152-157, 158-160 Modeling tips, 44-46 INTEG.EKA,169-171 LADDERS.EKA, 127-129 LINECHG.EKA,130-133 LINPROG.EKA, 158-160 LSQRFIT.EKA, 137-140 MASSCHG.EKA, 144-147 MAXMIN.EKA,108-121 POLYNOM.EKA, 148-151 PROJCTL.EKA, 141-143 Progress window, 26 PROJCTL.EKA, 141-143 Pval function (example), 161-164 N NOP.COM,5 New command, 52 Next command, 77 Nonlinear equations, simultaneous (examples), 127-129, 130-133 o Q Quick reference, 187-190 Quit command, 55 Quitting Eureka, 17, 34, 55 R Open command, 76 Operators, see Mathematical symbols Options menu, 31, 70-75, 86 Order of evaluation, 83 OS Shell command, 55 Output command, 64, 67 RAM disk, Eureka with, 15 Rename command, 54 Report, 12,30, 63-65 Report menu, 63-65 Reserved words, 10 Roots, alternate, 29-32, 42, 44 positive/negative, 94 p S Paymt function (example), 161-164 Penalty setting, 31, 93 Plot command, 66, 94 Plots, 27-29, 66-69 graphics mode, 28 zooming, 28 Poly function (example), 148-151 POLYNOM.EKA, 148-151 Polynomials, roots to, 58, 148-151 Printing, 12, 30, 63-65, 67-68 plots, 67 reports, 63 tables, 68 Problem files (.EKA) 105-186 CARDS.EKA, 165-168 CARLOAN.EKA, 123-126 CHEMBAL.EKA, 172-177 CHEMEQ.EKA,172-186 CLASSICS.EKA, 108-114 COMPLEX.EKA, 108-122 OECAY.EKA,134-136 OERINT.EKA,108-117 EQUITY.EKA, 161-164 HOUSKEKA, 152-157 Save command, 52 Scientific notation, 96-97 Screen mode, installing, 232 Settings command, 31, 47, 95 Setup, saving, 75 Solve command, 11, 26, 57 Solving equations, see Equations, solving Solving, Eureka's method, 40 Stack command, 78 Substitutions, symbolic, 45, 96 Substlevel setting, 45 example, 137-140 SuperKey macros, 86 Symbols, see Mathematical symbols T Tile command, 77 Transcendental, equations (examples), 108, 134-136, 141-143 Trigonometric functions, 89 example, 108 Turbo Basic problem file, 43, 139 Tutorials, 13-34 Eureka: The Solver Owner's Handbook u Units conversion, 84, 97 example, 141-143 User-defined functions, 27, 81, 101-103, 107-108 V Variables, complex, 89, 107, 109 functions and, 101, 152-157 initializing, 10, 40, 41, 44, 71, 81, 91 ~aximizing/minimizing, 92-93, 152-157 modifying, 41 names, 82, 88 polynomial, 58 poorly scaled, 45, 184 solving for selected, 95 Variables command, 32, 41, 70 Verify command, l2, 33, 59 W Window menu, 33-34, 76-78 Windows closing, 34, 77 expanding, 37, 77 moving, 34 opening, 34, 76 saving setup of, 75 tiling, 34, 77 zooming, 34, 77 Write setup command, 75 Write to ... command, 53 Z Zoom command, 77 Index I f you're a scientist, engineer, financial analyst, student, teacher, or any other professional working with equations, Eureka: The Solver can do your Algebra, Trigonometry and Calculus problems in a snap. Eureka also handles maximization and minimization problems, plots functions, generates reports, and saves an incredible amount of time. Even if you're not a computer specialist. Eureka, can help you solve your real-world mathematical problems fast, without having to learn numerical approximation techniques. Using Borland's famous pull-down menu design and context-sensitive help screens, Eureka is easy to learn and easy to use-as simple as a hand-held calculator. X + exp(X) = ID solved Instantly Instead of e'lentuallyl Imagine you have to "solve for X," where X + exp(X) = 10, and you don't have Eureka: The Solver. What you do have is a problem, because it's going to take a lot of time guessing at "X." With Eureka, there's no guessing, no dancing in the dark-you get the right answer, right now. (PS: X = 2.0705799, and Eureka solved that one in .4 of a second!) How to Ule Eureka: The SONer It's easy. 1. Enter your equation into the full-screen editor 2. Select the "Solve" command 3. Look at the answer 4. You're done You can then tell Eureka to • Evaluate your solution • Plot a graph • Generate a report, then send the output to your printer, disk file or screen • Or all of the above . Some of Eureka's key features You can key in: ~ A formula or formulas ~ A series of equations- and sdlve for all variables s' Constraints (like X has to be < or = 2) A function to plot Unit conversions Maximization and minimization problems s' Interesl Rale/Present Value calculations s' Variables we call "What happens?," like "What happens if I change this variable to 21 and that variable to 27?" i Eureka: The Sol'ler Includes: A full-screen editor Pull-down menus Context-sensitive Help On-screen calculator Automatic 8087 math co-processor chip support s' Powerful financial functions ~ Built-in and user-defined math and financial functions s' Ability to generate reports complete with plots and lists ~ Polynomial finder ~ Inequality solutions I Minimum system requirements: IBM PC. AT. XT. Portable. 3270 and true compatibles. PC-DDS (MS-DOS) ?O and later. 384K. Eureka: The Solver is a trademark of Borlald Inlernalional. Inc. IBM. AT. and XT are registered trademarks of International Business Machines Corp. MS· DOS is a registered trademark of Microsoft Corp. BOR 0236 Copyright 1986 Borland International
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