A Practical Guide To Spline

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A Practical Guide to Spline
Book in Mathematics of Computation · January 1978
DOI: 10.2307/2006241

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Carl de Boor

A Practical Guide to Splines
Revised Edition

With 32 figures

Springer

Contents

Preface
Notation

xv

I • Polynomial Interpolation
Polynomial interpolation: Lagrange form
Polynomial Interpolation: Divided differences and Newton form
Divided difference table
Example: Osculatory interpolation to the logarithm
Evaluation of the Newton form

2
3
8
9
9

Example: Computing the derivatives of a polynomial in Newton form 11
Other polynomial forms and conditions
12
Problems
15
II • Limitations of Polynomial Approximation
Uniform spacing of data can have bad consequences
Chebyshev sites are good
Runge example with Chebyshev sites
Squareroot example
Interpolation at Chebyshev sites is nearly optimal

17
20
22
22
24

The distance from polynomials
Problems

24
27

IX

x

Contents

III • Piecewise Linear Approximation
Broken line interpolation
Broken line interpolation is nearly optimal
Least-squares approximation by broken lines
Good meshes
Problems

31
32
32
35
37

IV • Piecewise Cubic Interpolation
Piecewise cubic Hermite interpolation
Runge example continued
Piecewise cubic Bessel interpolation
Akima's interpolation
Cubic spline interpolation

40
41
42
42
43

Boundary conditions
Problems

43
48

V • Best Approximation Properties of Complete Cubic Spline
Interpolation and Its Error
Problems

51
56

VI • Parabolic Spline Interpolation
Problems

59
64

VII • A Representation for Piecewise Polynomial Functions
Piecewise polynomial functions
The subroutine PPVALU
The subroutine INTERV
Problems

69
72
74
77

VIII • The Spaces H
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