A practical guide to splines:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2001
|
| Edition: | Rev. ed., 1. hardcover print. |
| Series: | Applied mathematical sciences
27 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 346 S. graph. Darst. |
| ISBN: | 0387953663 |
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MARC
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| 100 | 1 | |a De Boor, Carl |d 1937- |e Verfasser |0 (DE-588)109311507 |4 aut | |
| 245 | 1 | 0 | |a A practical guide to splines |c Carl de Boor |
| 250 | |a Rev. ed., 1. hardcover print. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 2001 | |
| 300 | |a XVIII, 346 S. |b graph. Darst. | ||
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Record in the Search Index
| _version_ | 1805083652748476416 |
|---|---|
| adam_text |
Contents
Preface v
Notation xv
I ¦ Polynomial Interpolation
Polynomial interpolation: Lagrange form 2
Polynomial Interpolation: Divided differences and Newton form 3
Divided difference table 8
Example: Oscillatory interpolation to the logarithm 9
Evaluation of the Newton form 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 17
Chebyshev sites are good 20
Runge example with Chebyshev sites 22
Squareroot example 22
Interpolation at Chebyshev sites is nearly optimal 24
The distance from polynomials 24
Problems 27
ix
x Contents
III • Piecewise Linear Approximation
Broken line interpolation 31
Broken line interpolation is nearly optimal 32
Least-squares approximation by broken lines 32
Good meshes 35
Problems 37
IV • Piecewise Cubic Interpolation
Piecewise cubic Hermite interpolation 40
Runge example continued 41
Piecewise cubic Bessel interpolation 42
Akima's interpolation 42
Cubic spline interpolation 43
Boundary conditions 43
Problems 48
V • Best Approximation Properties of Complete Cubic Spline
Interpolation and Its Error 51
Problems 56
VI • Parabolic Spline Interpolation 59
Problems 64
VII ¦ A Representation for Piecewise Polynomial Functions
Piecewise polynomial functions 69
The subroutine PPVALU 72
The subroutine INTEBV 74
Problems 77
VIII • The Spaces II fci(iV and the Truncated Power Basis
Example: The smoothing of a histogram by parabolic splines 79
The space U kx,u 82
The truncated power basis for II fe ^ and H k.£.v 82
Example: The truncated power basis can be bad 85
Problems 86
Contents xi
IX • The Representation of PP Functions by B-Splines
Definition of a B-spline 87
Two special knot sequences 89
A recurrence relation for B-splines 89
Example: A sequence of parabolic B-splines 91
The spline space $kt 93
The polynomials in $^ t 94
The pp functions in $k,t 96
B stands for basis 99
Conversion from one form to the other 101
Example: Conversion to B-form 103
Problems 106
X • The Stable Evaluation of B-Splines and Splines
Stable evaluation of B-splines 109
The subroutine BSPLVB 109
Example: To plot B-splines 113
Example: To plot the polynomials that make up a B-spline 114
Differentiation 115
The subroutine BSPLPP 117
Example: Computing a B-spline once again 120
The subroutine BVALUE 121
Example: Computing a B-Spline one more time 126
Integration 127
Problems 128
XI • The B-Spline Series, Control Points, and Knot Insertion
Bounding spline values in terms of "nearby" coefficients 131
Control points and control polygon 133
Knot insertion 135
Variation diminution 138
Schoenberg's variation diminishing spline approximation 141
Problems 142
XII • Local Spline Approximation and the Distance from Splines
The distance of a continuous function from $fct 145
The distance of a smooth function from $fc,t 148
Example: Schoenberg's variation-diminishing spline approximation 149
Local schemes that provide best possible approximation order 152
Good knot placement 156
xii Contents
The subroutine NEWNOT 159
Example: A failure for NEWNOT 161
The distance from $^ n 163
Example: A failure for CUBSPL 165
Example: Knot placement works when used with a local scheme 167
Problems 169
XIII • Spline Interpolation
The Schoenberg-Whitney Theorem 171
Bandedness of the spline collocation matrix 173
Total positivity of the spline collocation matrix 169
The subroutine SPLINT 175
The interplay between knots and data sites 180
Even order interpolation at knots 182
Example: A large [|/|| amplifies noise 183
Interpolation at knot averages 185
Example: Cubic spline interpolation at knot averages with good knots 186
Interpolation at the Chebyshev-Demko sites 189
Optimal interpolation 193
Example: "Optimal" interpolation need not be "good" 197
Osculatory spline interpolation 200
Problems 204
XIV • Smoothing and Least-Squares Approximation
The smoothing spline of Schoenberg and Reinsch 207
The subroutine SMOOTH and its subroutines 211
Example: The cubic smoothing spline 214
Least-squares approximation 220
Least-squares approximation from $k,t 223
The subroutine L2APPR (with BCHFAC/BCHSLV) 224
L2MAIN and its subroutines 228
The use of L2APPR 232
Example: Fewer sign changes in the error than perhaps expected 232
Example: The noise plateau in the error 235
Example: Once more the Titanium Heat data 237
Least-squares approximation by splines with variable knots 239
Example: Approximation to the Titanium Heat data from $4,9 239
Problems 240
Contents xiii
XV ¦ The Numerical Solution of an Ordinary Differential Equation
by Collocation
Mathematical background 243
The almost block diagonal character of the system of collocation
equations; EQBLOK, PUTIT 246
The subroutine BSPLVD 251
COLLOC and its subroutines 253
Example: A second order nonlinear two-point boundary-value
problem with a boundary layer 258
Problems 261
XVI • Taut Splines, Periodic Splines, Cardinal Splines and
the Approximation of Curves
Lack of data 263
"Extraneous" inflection points 264
Spline in tension 264
Example: Coping with a large endslope 265
A taut cubic spline 266
Example: Taut cubic spline interpolation to Titanium Heat data 275
Proper choice of parametrization 276
Example: Choice of parametrization is important 277
The approximation of a curve 279
Nonlinear splines 280
Periodic splines 282
Cardinal splines 283
Example: Conversion to ppform is cheaper when knots are uniform 284
Example: Cubic spline interpolation at uniformly spaced sites 284
Periodic splines on uniform meshes 285
Example: Periodic spline interpolation to uniformly spaced data
and harmonic analysis 287
Problems 289
XVII • Surface Approximation by Tensor Products
An abstract linear interpolation scheme 291
Tensor product of two linear spaces of functions 293
Example: Evaluation of a tensor product spline. 297
The tensor product of two linear interpolation schemes 297
The calculation of a tensor product interpolant 299
xiv Contents
Example: Tensor product spline interpolation 301
The ppform of a tensor product spline 305
The evaluation of a tensor product spline from its ppform 305
Conversion from B-form to ppform 307
Example: Tensor product spline interpolation (continued) 309
Limitations of tensor product approximation and alternatives 310
Problems 311
Postscript on Things Not Covered 313
Appendix: Fortran Programs
Fortran programs 315
List of Fortran programs 315
Listing of S0LVEBL0K Package 318
Bibliography 331
Index 341 |
| any_adam_object | 1 |
| author | De Boor, Carl 1937- |
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| discipline | Mathematik |
| edition | Rev. ed., 1. hardcover print. |
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| indexdate | 2024-07-20T07:51:58Z |
| institution | BVB |
| isbn | 0387953663 |
| language | English |
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| spelling | De Boor, Carl 1937- Verfasser (DE-588)109311507 aut A practical guide to splines Carl de Boor Rev. ed., 1. hardcover print. New York [u.a.] Springer 2001 XVIII, 346 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 27 Spline theory Spline-Approximation (DE-588)4182394-1 gnd rswk-swf Spline-Interpolation (DE-588)4182396-5 gnd rswk-swf Spline-Funktion (DE-588)4056332-7 gnd rswk-swf Spline-Funktion (DE-588)4056332-7 s DE-604 Spline-Approximation (DE-588)4182394-1 s Spline-Interpolation (DE-588)4182396-5 s 1\p DE-604 Applied mathematical sciences 27 (DE-604)BV000005274 27 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009646183&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | De Boor, Carl 1937- A practical guide to splines Applied mathematical sciences Spline theory Spline-Approximation (DE-588)4182394-1 gnd Spline-Interpolation (DE-588)4182396-5 gnd Spline-Funktion (DE-588)4056332-7 gnd |
| subject_GND | (DE-588)4182394-1 (DE-588)4182396-5 (DE-588)4056332-7 |
| title | A practical guide to splines |
| title_auth | A practical guide to splines |
| title_exact_search | A practical guide to splines |
| title_full | A practical guide to splines Carl de Boor |
| title_fullStr | A practical guide to splines Carl de Boor |
| title_full_unstemmed | A practical guide to splines Carl de Boor |
| title_short | A practical guide to splines |
| title_sort | a practical guide to splines |
| topic | Spline theory Spline-Approximation (DE-588)4182394-1 gnd Spline-Interpolation (DE-588)4182396-5 gnd Spline-Funktion (DE-588)4056332-7 gnd |
| topic_facet | Spline theory Spline-Approximation Spline-Interpolation Spline-Funktion |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009646183&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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