A Practical Guide to Splines (Applied Mathematical Sciences Vol.27) (Rev.)


A Practical Guide to Splines (Applied Mathematical Sciences Vol.27) (Rev.)

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  • 製本 Hardcover:ハードカバー版/ページ数 360 p.
  • 商品コード 9780387953663


New in hardcover. Softcover was published in 1978.

Full Description

This book is based on the author's experience with calculations involving polynomial splines, presenting those parts of the theory especially useful in calculations and stressing the representation of splines as weighted sums of B-splines. The B-spline theory is developed directly from the recurrence relations without recourse to divided differences. This reprint includes redrawn figures, and most formal statements are accompanied by proofs.

Table of Contents

Preface                                            v
Notation xv
Polynomial Interpolation
Polynomial Interpolation: Lagrange form 2 (1)
Polynomial Interpolation: Divided 3 (5)
differences and Newton form
Divided difference table 8 (1)
Example: Osculatory interpolation to the 9 (1)
Evaluation of the Newton form 9 (2)
Example: Computing the derivatives of a 11 (1)
polynomial in Newton form
Other polynomial forms and conditions 12 (3)
Problems 15 (2)
Limitations of Polynomial Approximation
Uniform spacing of data can have bad 17 (3)
Chebyshev sites are good 20 (2)
Runge example with Chebyshev sites 22 (1)
Squareroot example 22 (2)
Interpolation at Chebyshev sites is nearly 24 (1)
The distance from polynomials 24 (3)
Problems 27 (4)
Piecewise Linear Approximation
Broken line interpolation 31 (1)
Broken line interpolation is nearly optimal 32 (1)
Least-squares approximation by broken lines 32 (3)
Good meshes 35 (2)
Problems 37 (3)
Piecewise Cubic Interpolation
Piecewise cubic Hermite interpolation 40 (1)
Runge example continued 41 (1)
Piecewise cubic Bessel interpolation 42 (1)
Akima's interpolation 42 (1)
Cubic spline interpolation 43 (1)
Boundary conditions 43 (5)
Problems 48 (3)
Best Approximation Properties of Complete 51 (8)
Cubic Spline Interpolation and Its Error
Problems 56 (3)
Parabolic Spline Interpolation 59 (254)
Problems 64 (5)
A Representation for Piecewise Polynomial
Piecewise polynomial functions 69 (3)
The subroutine PPVALU 72 (2)
The subroutine INTERV 74 (3)
Problems 77 (2)
The Spaces II<k,ζ,v and the Truncated
Power Basic
Example: The smoothing of a histogram by 79 (3)
parabolic splines
The space II<k,ζ,v 82 (1)
The truncated power basis for 82 (3)
II<k,ζ and II<k,ζ,v
Example: The truncated power basis can be 85 (1)
Problems 86 (1)
The Representation of PP Functions by
Definition of a B-spline 87 (2)
Two special knot sequences 89 (1)
A recurrence relation for B-splines 89 (2)
Example: A sequence of parabolic B-splines 91 (2)
The spline space $k,t 93 (1)
The polynomials in $k,t 94 (2)
The pp functions in $k,t 96 (3)
B stands for basis 99 (2)
Conversion from one form to the other 101(2)
Example: Conversion to B-form 103(3)
Problems 106(3)
The Stable Evaluation of B-Splines and Splines
Stable evaluation of B-splines 109(1)
The subroutine BSPLVB 109(4)
Example: To plot B-splines 113(1)
Example: To plot the polynomials that make 114(1)
up a B-spline
Differentiation 115(2)
The subroutine BSPLPP 117(3)
Example: Computing a B-spline once again 120(1)
The subroutine BVALUE 121(5)
Example: Computing a B-Spline one more time 126(1)
Integration 127(1)
Problems 128(3)
The B-Spline Series, Control Points, and Knot
Bounding spline values in terms of 131(2)
``nearby'' coefficients
Control points and control polygon 133(2)
Knot insertion 135(3)
Variation diminution 138(3)
Schoenberg's variation diminishing spline 141(1)
Problems 142(3)
Local Spline Approximation and the Distance
from Splines
The distance of a continuous function from 145(3)
The distance of a smooth function from 148(1)
Example: Schoenberg's variation-diminishing 149(3)
spline approximation
Local schemes that provide best possible 152(4)
approximation order
Good knot placement 156(3)
The subroutine NEWNOT 159(2)
Example: A failure for NEWNOT 161(2)
The distance from $k,n 163(2)
Example: A failure for CUBSPL 165(2)
Example: Knot placement works when used 167(2)
with a local scheme
Problems 169(2)
Spline Interpolation
The Schoenberg-Whitney Theorem 171(2)
Bandedness of the spline collocation matrix 173
Total positivity of the spline collocation 169(6)
The subroutine SPLINT 175(5)
The interplay between knots and data sites 180(2)
Even order interpolation at knots 182(1)
Example: A large ∥I∥ amplifies noise 183(2)
Interpolation at knot averages 185(1)
Example: Cubic spline interpolation at knot 186(3)
averages with good knots
Interpolation at the Chebyshev-Demko sites 189(4)
Optimal interpolation 193(4)
Example: ``Optimal'' interpolation need not 197(3)
be ``good''
Osculatory spline interpolation 200(4)
Problems 204(3)
Smoothing and Least-Squares Approximation
The smoothing spline of Schoenberg and 207(4)
The subroutine SMOOTH and its subroutines 211(3)
Example: The cubic smoothing spline 214(6)
Least-squares approximation 220(3)
Least-squares approximation from $k,t 223(1)
The subroutine L2APPR (with BCHFAC/BCHSLV) 224(4)
L2MAIN and its subroutines 228(4)
The use of L2APPR 232(1)
Example: Fewer sign changes in the error 232(3)
than perhaps expected
Example: The noise plateau in the error 235(2)
Example: Once more the Titanium Heat data 237(2)
Least-squares approximation by splines with 239(1)
variable knots
Example: Approximation to the Titanium Heat 239(1)
data from $4.9
Problems 240(3)
The Numerical Solution of an Ordinary
Differential Equation by Collocation
Mathematical background 243(3)
The almost block diagonal character of the 246(5)
system of collocation equations; EQBLOK,
The subroutine BSPLVD 251(2)
COLLOC and its subroutines 253(5)
Example: A second order nonlinear two-point 258(3)
boundary-value problem with a boundary layer
Problems 261(2)
Taut Splines, Periodic Splines, Cardinal
Splines and the Approximation of Curves
Lack of data 263(1)
``Extraneous'' inflection points 264(1)
Spline in tension 264(1)
Example: Coping with a large endslope 265(1)
A taut cubic spline 266(9)
Example: Taut cubic spline interpolation to 275(1)
Titanium Heat data
Proper choice of parametrization 276(1)
Example: Choice of parametrization is 277(2)
The approximation of a curve 279(1)
Nonlinear splines 280(2)
Periodic splines 282(1)
Cardinal splines 283(1)
Example: Conversion to ppform is cheaper 284(1)
when knots are uniform
Example: Cubic spline interpolation at 284(1)
uniformly spaced sites
Periodic splines on uniform meshes 285(2)
Example: Periodic spline interpolation to 287(2)
uniformly spaced data and harmonic analysis
Problems 289(2)
Surface Approximation by Tensor Products
An abstract linear interpolation scheme 291(2)
Tensor product of two linear spaces of 293(4)
Example: Evaluation of a tensor product 297(1)
The tensor product of two linear 297(2)
interpolation schemes
The calculation of a tensor product 299(2)
Example: Tensor product spline interpolation 301(4)
The ppform of a tensor product spline 305(1)
The evaluation of a tensor product spline 305(2)
from its ppform
Conversion from B-form to ppform 307(2)
Example: Tensor product spline 309(1)
interpolation (continued)
Limitations of tensor product approximation 310(1)
and alternatives
Problems 311(2)
Postscript on Things Not Covered 313(18)
Appendix: Fortran Programs
Fortran programs 315(1)
List of Fortran programs 315(3)
Listing of SOLVEBLOK Package 318(13)
Bibliography 331(10)
Index 341