Spline functions - computations methods.

Contents......Page 5
Preface......Page 9
 1.1 Introduction......Page 13
 1.2 B-splines......Page 14
 1.3 Computing with the B-form......Page 16
 1.4 Interpolation with univariate splines......Page 20
 1.5 Hermite interpolation......Page 25
 1.6 Interpolation with shape control......Page 28
 1.7 Quasi-interpolation......Page 32
 1.8 Discrete least-squares approximation......Page 34
 1.9 The effect of noise......Page 37
 1.10 Penalized least-squares approximation......Page 38
 1.11 Choosing optimal knots for least-squares splines......Page 41
 1.12 Two-point boundary-value problems......Page 44
 1.13 Error bounds......Page 51
 1.14 Remarks......Page 57
 1.15 Historical notes......Page 61
 2.1 Introduction......Page 63
 2.3 Computing with the tensor B-form......Page 64
 2.4 Lagrange interpolation on a grid......Page 69
 2.5 Hermite interpolation......Page 72
 2.6 Tensor-product quasi-interpolation......Page 76
 2.7 Tensor-product discrete least-squares......Page 78
 2.8 The effect of noise......Page 80
 2.9 Penalized least-squares approximation......Page 81
 2.10 Error bounds......Page 84
 2.11 Remarks......Page 85
 2.12 Historical notes......Page 88
 3.2 Triangulations with hanging vertices......Page 91
 3.3 Storing and plotting triangulations......Page 92
 3.4 Additional data structure for triangulations......Page 93
 3.6 Barycentric coordinates......Page 96
 3.7 Finding the triangle containing a point......Page 97
 3.8 Constructing a triangulation with given vertices......Page 98
 3.10 Triangulations of gridded data on a rectangle......Page 101
 3.11 Refining a triangulation......Page 102
 3.12 Adjusting triangulations by edge swapping......Page 103
 3.13 Euler relations......Page 104
 3.14 Remarks......Page 106
 3.15 Historical notes......Page 107
 4.1 Bernstein basis polynomials......Page 109
 4.2 Domain points on a triangle......Page 110
 4.3 Evaluating polynomials in B-form......Page 111
 4.4 Derivatives of polynomials in B-form......Page 112
 4.5 Integrals and inner products of polynomials......Page 113
 4.6 Integrals of general functions over triangles......Page 114
 4.7 Plotting a polynomial patch......Page 116
 4.8 Domain points on a triangulation......Page 117
 4.9 The space......Page 118
 4.10 Evaluating splines and their derivatives......Page 121
 4.12 Rendering a spline surface......Page 122
 4.13 Spaces of smooth splines......Page 125
 4.14 Some useful macro-element spaces......Page 132
 4.15 Degree raising......Page 142
 4.16 Subdivision......Page 143
 4.17 Remarks......Page 144
 4.18 Historical notes......Page 145
 5.1 Introduction......Page 147
 5.2 The C1 Powell–Sabin interpolant......Page 149
 5.3 The C1 Powell–Sabin-12 interpolant......Page 153
 5.4 The C1 Clough–Tocher interpolant......Page 156
 5.5 The C1 quintic Argyris interpolant......Page 159
 5.6 The C2 Wang interpolant......Page 161
 5.7 A comparison of the methods......Page 162
 5.8 Rate of convergence for less smooth functions......Page 163
 5.9 Interpolation on domains with holes......Page 165
 5.10 Interpolation at badly spaced points......Page 166
 5.11 Scale invariance......Page 167
 5.12 The role of the minimal angle in error bounds......Page 169
 5.13 Remarks......Page 170
 5.14 Historical notes......Page 172
 6.1 Introduction......Page 173
 6.2 Interpolation with......Page 174
 6.3 Minimal energy interpolating splines......Page 178
 6.4 A local C0 cubic spline method......Page 185
 6.5 A local C1 Clough–Tocher method......Page 187
 6.6 Estimating derivatives from scattered data......Page 190
 6.7 Local two-stage methods based on derivative estimation......Page 197
 6.8 A comparison of the methods......Page 202
 6.9 Scale invariance......Page 204
 6.10 Nonrectangular domains and holes......Page 207
 6.11 Interpolating at a large number of data points......Page 210
 6.12 The error as a function of mesh size......Page 214
 6.13 Non-Delaunay triangulations......Page 215
 6.14 Scattered data interpolation with radial basis functions......Page 216
 6.15 Almost interpolation with tensor-product splines......Page 220
 6.16 Remarks......Page 225
 6.17 Historical notes......Page 228
 7.1 Least-squares fitting......Page 231
 7.2 Least-squares fitting with C0 splines......Page 235
 7.3 Least-squares fitting with C1 splines......Page 238
 7.4 Least-squares fitting with C2 splines......Page 243
 7.5 Scale invariance of least-squares fitting......Page 245
 7.6 A comparison of the methods......Page 246
 7.8 Penalized least-squares fitting......Page 248
 7.9 Penalized least-squares fitting with macro-element spline spaces......Page 251
 7.10 Nonscale invariance of penalized least-squares fitting......Page 255
 7.11 Remarks......Page 256
 7.12 Historical notes......Page 258
 8.1 Nonnegative interpolation......Page 259
 8.2 Nonnegative least-squares......Page 263
 8.3 Monotone interpolation......Page 266
 8.4 Monotone least-squares......Page 280
 8.5 Convex interpolation......Page 282
 8.6 Convex least-squares......Page 295
 8.7 Remarks......Page 297
 8.8 Historical notes......Page 299
 9.1 Examples of boundary-value problems......Page 301
 9.2 The Ritz–Galerkin method......Page 302
 9.3 Solving second order boundary-value problems with......Page 306
 9.4 Solving second order boundary-value problems using macro-element spaces......Page 319
 9.5 Solving second order boundary-value problems using the space......Page 320
 9.6 Solving the biharmonic equation with C1 quintic splines......Page 328
 9.7 Remarks......Page 334
 9.8 Historical notes......Page 336
 10.1 Spherical triangulations......Page 337
 10.2 Spherical polynomials......Page 345
 on a spherical triangulation......Page 352
 10.4 Spaces of smooth spherical splines......Page 356
 10.5 Spherical macro-element spaces......Page 360
 10.6 Remarks......Page 365
 10.7 Historical notes......Page 367
 11.1 Interpolation with......Page 369
 11.2 Hermite interpolation with macro-element spaces......Page 373
 11.3 Scattered data interpolation on the sphere......Page 381
 11.4 Least-squares fitting with spherical splines......Page 396
 11.5 The effect of noise......Page 403
 11.6 Spherical penalized least-squares......Page 405
 11.7 PDE’s on the sphere......Page 408
 11.8 Remarks......Page 411
 11.9 Historical notes......Page 414
Bibliography......Page 415
Script Index......Page 419
Function Index......Page 421
Subject Index......Page 423