This textbook is written primarily for undergraduate mathematicians and also appeals to students working at an advanced level in other disciplines. The text begins with a clear motivation for the study of numerical analysis based on real-world problems. The authors then develop the necessary machinery including iteration, interpolation, boundary-value problems and finite elements. Throughout, the authors keep an eye on the analytical basis for the work and add historical notes on the development of the subject. There are numerous exercises for students.
Author(s): Endre Süli, David F. Mayers
Publisher: Cambridge University Press
Year: 2003
Language: English
Commentary: 18055
Pages: 445
Cover......Page 1
Title......Page 3
Copyright......Page 4
Contents......Page 5
Preface......Page 9
1.1 Introduction......Page 13
1.2 Simple iteration......Page 14
1.3 Iterative solution of equations......Page 29
1.4 Relaxation and Newton’s method......Page 31
1.5 The secant method......Page 37
1.6 The bisection method......Page 40
1.7 Global behaviour......Page 41
1.8 Notes......Page 44
Exercises......Page 47
2.1 Introduction......Page 51
2.2 Gaussian elimination......Page 56
2.3 LU factorisation......Page 60
2.4 Pivoting......Page 64
2.5 Solution of systems of equations......Page 67
2.6 Computational work......Page 68
2.7 Norms and condition numbers......Page 70
2.8 Hilbert matrix......Page 84
2.9 Least squares method......Page 86
2.10 Notes......Page 91
Exercises......Page 94
3.2 Symmetric positive definite matrices......Page 99
3.3 Tridiagonal and band matrices......Page 105
3.4 Monotone matrices......Page 110
3.5 Notes......Page 113
Exercises......Page 114
4.1 Introduction......Page 116
4.2 Simultaneous iteration......Page 118
4.3 Relaxation and Newton’s method......Page 128
4.4 Global convergence......Page 135
4.5 Notes......Page 136
Exercises......Page 138
5.1 Introduction......Page 145
5.3 Jacobi’s method......Page 149
5.4 The Gerschgorin theorems......Page 157
5.5 Householder’s method......Page 162
5.6 Eigenvalues of a tridiagonal matrix......Page 168
5.7.1 The QR factorisation revisited......Page 174
5.7.2 The definition of the QR algorithm......Page 176
5.8 Inverse iteration for the eigenvectors......Page 178
5.9 The Rayleigh quotient......Page 182
5.10 Perturbation analysis......Page 184
5.11 Notes......Page 186
Exercises......Page 187
6.1 Introduction......Page 191
6.2 Lagrange interpolation......Page 192
6.3 Convergence......Page 197
6.4 Hermite interpolation......Page 199
6.5 Differentiation......Page 203
6.6 Notes......Page 206
Exercises......Page 207
7.1 Introduction......Page 212
7.2 Newton–Cotes formulae......Page 213
7.3 Error estimates......Page 216
7.4 The Runge phenomenon revisited......Page 220
7.5 Composite formulae......Page 221
7.6 The Euler–Maclaurin expansion......Page 223
7.7 Extrapolation methods......Page 227
7.8 Notes......Page 231
Exercises......Page 232
8.2 Normed linear spaces......Page 236
8.3 Best approximation in the -norm......Page 240
8.4 Chebyshev polynomials......Page 253
8.5 Interpolation......Page 256
8.6 Notes......Page 259
Exercises......Page 260
9.1 Introduction......Page 264
9.2 Inner product spaces......Page 265
9.3 Best approximation in the 2-norm......Page 268
9.4 Orthogonal polynomials......Page 271
9.5 Comparisons......Page 282
9.6 Notes......Page 284
Exercises......Page 285
10.2 Construction of Gauss quadrature rules......Page 289
10.3 Direct construction......Page 292
10.4 Error estimation for Gauss quadrature......Page 294
10.5 Composite Gauss formulae......Page 297
10.6 Radau and Lobatto quadrature......Page 299
Exercises......Page 300
11.1 Introduction......Page 304
11.2 Linear interpolating splines......Page 305
11.3 Basis functions for the linear spline......Page 309
11.4 Cubic splines......Page 310
11.5 Hermite cubic splines......Page 312
11.6 Basis functions for cubic splines......Page 314
11.7 Notes......Page 318
Exercises......Page 319
12.1 Introduction......Page 322
12.2 One-step methods......Page 329
12.3 Consistency and convergence......Page 333
12.4 An implicit one-step method......Page 336
12.5 Runge–Kutta methods......Page 337
12.6 Linear multistep methods......Page 341
12.7 Zero-stability......Page 343
12.8 Consistency......Page 349
12.9 Dahlquist’s theorems......Page 352
12.10 Systems of equations......Page 353
12.11 Stiff systems......Page 355
12.12 Implicit Runge–Kutta methods......Page 361
12.13 Notes......Page 365
Exercises......Page 367
13.2 A model problem......Page 373
13.3 Error analysis......Page 376
13.4 Boundary conditions involving a derivative......Page 379
13.5 The general self-adjoint problem......Page 382
13.6 The Sturm–Liouville eigenvalue problem......Page 385
13.7 The shooting method......Page 387
13.8 Notes......Page 392
Exercises......Page 393
14.1 Introduction: the model problem......Page 397
14.2 Rayleigh–Ritz and Galerkin principles......Page 400
14.3 Formulation of the finite element method......Page 403
14.4 Error analysis of the finite element method......Page 409
14.5 A posteriori error analysis by duality......Page 415
14.6 Notes......Page 424
Exercises......Page 426
Appendix A An overview of results from real analysis......Page 431
Appendix B WWW-resources......Page 435
Bibliography......Page 436
Index......Page 441
