Numerical Mathematics

Preface......Page 7
Contents......Page 10
1.1 Vector Spaces......Page 20
1.2 Matrices......Page 22
1.3 Operations with Matrices......Page 24
1.4 Trace and Determinant of a Matrix......Page 27
1.5 Rank and Kernel of a Matrix......Page 28
1.6 Special Matrices......Page 29
1.7 Eigenvalues and Eigenvectors......Page 31
1.8 Similarity Transformations......Page 33
1.9 The Singular Value Decomposition ( SVD)......Page 35
1.10 Scalar Product and Norms in Vector Spaces......Page 36
1.11 Matrix Norms......Page 40
M- matrices......Page 46
1.13 Exercises......Page 49
Problem......Page 52
2.2 Stability of Numerical Methods......Page 56
2.3 A priori and a posteriori Analysis......Page 60
2.4 Sources of Error in Computational Models......Page 62
2.5 Machine Representation of Numbers......Page 64
2.6 Exercises......Page 73
Direct Methods for the Solution of Linear Systems......Page 76
3.1 Stability Analysis of Linear Systems......Page 77
3.2 Solution of Triangular Systems......Page 84
LU Factorization......Page 87
3.4 Other Types of Factorization......Page 98
= Q......Page 102
3.5 Pivoting......Page 104
3.6 Computing the Inverse of a Matrix......Page 108
3.7 Banded Systems......Page 109
3.8 Block Systems......Page 112
3.9 Sparse Matrices......Page 116
GEM......Page 122
3.11 An Approximate Computation of ( A)......Page 125
3.12 Improving the Accuracy of GEM......Page 128
3.13 Undetermined Systems......Page 131
3.14 Applications......Page 134
3.15 Exercises......Page 140
4.1 On the Convergence of Iterative Methods......Page 142
4.2 Linear Iterative Methods......Page 145
Methods......Page 155
4.4 Methods Based on Krylov Subspace Iterations......Page 178
4.5 The Lanczos Method for Unsymmetric Systems......Page 187
4.6 Stopping Criteria......Page 190
4.7 Applications......Page 193
4.8 Exercises......Page 198
5.1 Geometrical Location of the Eigenvalues......Page 201
5.2 Stability and Conditioning Analysis......Page 204
5. 3 The Power Method......Page 210
5.4 The QR Iteration......Page 218
5.5 The Basic QR Iteration......Page 219
Form......Page 221
5.7 The QR Iteration with Shifting Techniques......Page 233
Matrix......Page 239
5.9 The Generalized Eigenvalue Problem......Page 242
matrices......Page 245
5.11 The Lanczos Method......Page 251
5.12 Applications......Page 253
5.13 Exercises......Page 258
Rootfinding for Nonlinear Equations......Page 262
6.1 Conditioning of a Nonlinear Equation......Page 263
6.2 A Geometric Approach to Root   nding......Page 265
6.3 Fixed- point Iterations for Nonlinear Equations......Page 274
6.4 Zeros of Algebraic Equations......Page 278
6.5 Stopping Criteria......Page 286
Methods......Page 289
6.7 Applications......Page 293
6.8 Exercises......Page 296
Nonlinear Systems and Numerical Optimization......Page 298
7.1 Solution of Systems of Nonlinear Equations......Page 299
7.2 Unconstrained Optimization......Page 311
7.3 Constrained Optimization......Page 328
7.4 Applications......Page 336
7.5 Exercises......Page 342
Polynomial Interpolation......Page 344
8.1 Polynomial Interpolation......Page 345
8.2 Newton Form of the Interpolating Polynomial......Page 350
8.3 Piecewise Lagrange Interpolation......Page 355
8.4 Hermite- Birko   Interpolation......Page 358
8.5 Extension to the Two- Dimensional Case......Page 360
8.6 Approximation by Splines......Page 365
8.7 Splines in Parametric Form......Page 374
8.8 Applications......Page 379
8.9 Exercises......Page 385
9.1 Quadrature Formulae......Page 388
9.2 Interpolatory Quadratures......Page 390
9.3 Newton- Cotes Formulae......Page 395
9.4 Composite Newton- Cotes Formulae......Page 400
9.5 Hermite Quadrature Formulae......Page 403
9.6 Richardson Extrapolation......Page 404
9.7 Automatic Integration......Page 408
( I)......Page 413
9.8 Singular Integrals......Page 415
9.9 Multidimensional Numerical Integration......Page 419
9.10 Applications......Page 425
9.11 Exercises......Page 429
Fourier Series......Page 432
10.2 Gaussian Integration and Interpolation......Page 436
10.3 Chebyshev Integration and Interpolation......Page 441
10.4 Legendre Integration and Interpolation......Page 443
Intervals......Page 445
Quadratures......Page 446
Least- Squares Sense......Page 448
10.8 The Polynomial of Best Approximation......Page 450
10.9 Fourier Trigonometric Polynomials......Page 452
10.10 Approximation of Function Derivatives......Page 459
10.11 Transforms and Their Applications......Page 467
10.12 The Wavelet Transform......Page 475
10.13 Applications......Page 480
10.14 Exercises......Page 484
11.1 The Cauchy Problem......Page 486
11.2 One- Step Numerical Methods......Page 489
11.3 Analysis of One- Step Methods......Page 490
11.4 Di   erence Equations......Page 499
11.5 Multistep Methods......Page 504
11.6 Analysis of Multistep Methods......Page 509
11.7 Predictor- Corrector Methods......Page 519
11.8 Runge- Kutta ( RK) Methods......Page 525
11.9 Systems of ODEs......Page 534
11.10 Sti   Problems......Page 536
11.11 Applications......Page 538
11.12 Exercises......Page 544
12.1 A Model Problem......Page 548
12.2 Finite Di   erence Approximation......Page 550
12.3 The Spectral Collocation Method......Page 559
12.4 The Galerkin Method......Page 561
12.5 Advection- Di   usion Equations......Page 577
12.6 A Quick Glance to the Two- Dimensional Case......Page 589
12.7 Applications......Page 592
12.8 Exercises......Page 595
13.1 The Heat Equation......Page 597
Equation......Page 600
Equation......Page 602
Heat Equation......Page 609
Problem......Page 613
13.6 Systems of Linear Hyperbolic Equations......Page 615
Equations......Page 618
13.8 Analysis of Finite Di   erence Methods......Page 621
13.9 Dissipation and Dispersion......Page 627
Equations......Page 634
13.11 Applications......Page 639
13.12 Exercises......Page 641
References......Page 643
Index of MATLAB Programs......Page 658
Index......Page 661