1 Error Analysis 1
1.1 Representation of Numbers 2
1.2 Roundoff Errors and Floating-Point Arithmetic 5
1.3 Error Propagation 9
1.4 Examples 20
1.5 Interval Arithmetic;Statistical Roundoff Estimation 27
Exercises for Chapter 1 33
References for Chapter 1 36
2 Interpolation 37
2.1 Interpolation by Polynomials 38
2.1.1 Theoretical Foundation:The Interpolation Formula of Lagrange 38
2.1.2 Neville's Algorithm 40
2.1.3 Newton's Interpolation Formula:Divided Differences 43
2.1.4 The Error in Polynomial Interpolation 49
2.1.5 Hermite Interpolation 52
2.2 Interpolation by Rational Functions 58
2.2.1 General Properties of Rational Interpolation 58
2.2.2 Inverse and Reciprocal Differences.Thiele's Continued Fraction 63
2.2.3 Algorithms of the Neville Type 67
2.2.4 Comparing Rational and Polynomial Interpolations 71
2.3 Trigonometric Interpolation 72
2.3.1 Basic Facts 72
2.3.2 Fast Fourier Transforms 78
2.3.3 The Algorithms of Goertzel and Reinsch 84
2.3.4 The Calculation of Fourier Coefficients.Attenuation Factors 88
2.4 Interpolation by Spline Functions 93
2.4.1 Theoretical Foundations 93
2.4.2 Determining Interpolating Cubic Spline Functions 97
2.4.3 Convergence Properties of Cubic Spline Functions 102
2.4.4 B-Splines 107
2.4.5 The Computation of B-Splines 110
Exercises for Chapter 2 114
References for Chapter 2 123
3 Topics in Integration 125
3.1 The Integration Formulas of Newton and Cotes 126
3.2 Peano's Error Representation 131
3.3 The Euler-Maclaurin Summation Formula 135
3.4 Integrating by Extrapolation 139
3.5 About Extrapolation Methods 144
3.6 Gaussian Integration Methods 150
3.7 Integrals with Singularities 160
Exercises for Chapter 3 162
References for Chapter 3 166
4 Systems of Linear Equations 167
4.1 Gaussian Elimination.The Triangular Decomposition of a Matrix 167
4.2 The Gauss-Jordan Algorithm 177
4.3 The Cholesky Decomposition 180
4.4 Error Bounds 183
4.5 Roundoff-Error Analysis for Gaussian Elimination 191
4.6 Roundoff Errors in Solving Triangular Systems 196
4.7 Orthogonalization Techniques of Householder and Gram-Schmidt 198
4.8 Data Fitting 205
4.8.1 Linear Least Squares.The Normal Equations 207
4.8.2 The Use of Orthogonalization in Solving Linear Least-Squares Problems 209
4.8.3 The Condition of the Linear Least-Squares Problem 210
4.8.4 Nonlinear Least-Squares Problems 217
4.8.5 The Pseudoinverse of a Matrix 218
4.9 Modification Techniques for Matrix Decompositions 221
4.10 The Simplex Method 230
4.11 Phase One of the Simplex Method 241
Appendix to Chapter 4 245
4.A Elimination Methods for Sparse Matrices 245
Exercises for Chapter 4 253
References for Chapter 4 258
5 Finding Zeros and Minimum Points by Iterative Methods 260
5.1 The Development of Iterative Methods 261
5.2 General Convergence Theorems 264
5.3 The Convergence of Newton's Method in Several Variables 269
5.4 A Modified Newton Method 272
5.4.1 On the Convergence of Minimization Methods 273
5.4.2 Application of the Convergence Criteria to the Modified Newton Method 278
5.4.3 Suggestions for a Practical Implementation of the Modified Newton Method.A Rank-One Method Due to Broyden 282
5.5 Roots of Polynomials.Application of Newton's Method 286
5.6 Sturm Sequences and Bisection Methods 297
5.7 Bairstow's Method 301
5.8 The Sensitivity of Polynomial Roots 303
5.9 Interpolation Methods for Determining Roots 306
5.10 The △2-Method of Aitken 312
5.11 Minimization Problems without Constraints 316
Exercises for Chapter 5 325
References for Chapter 5 328
6 Eigenvalue Problems 330
6.0 Introduction 330
6.1 Basic Facts on Eigenvalues 332
6.2 The Jordan Normal Form of a Matrix 335
6.3 The Frobenius Normal Form of a Matrix 340
6.4 The Schur Normal Form of a Matrix;Hermitian and Normal Matrices;Singular Values of Matrices 345
6.5 Reduction of Matrices to Simpler Form 351
6.5.1 Reduction of a Hermitian Matrix to Tridiagonal Form:The Method of Householder 353
6.5.2 Reduction of a Hermitian Matrix to Tridiagonal or Diagonal Form:The Methods of Givens and Jacobi 358
6.5.3 Reduction of a Hermitian Matrix to Tridiagonal Form:The Method of Lanczos 362
6.5.4 Reduction to Hessenberg Form 366
6.6 Methods for Determining the Eigenvalues and Eigenvectors 370
6.6.1 Computation of the Eigenvalues of a Hermitian Tridiagonal Matrix 370
6.6.2 Computation of the Eigenvalues of a Hessenberg Matrix.The Method of Hyman 372
6.6.3 Simple Vector Iteration and Inverse Iteration of Wielandt 373
6.6.4 The LR and QR Methods 380
6.6.5 The Practical Implementation of the QR Method 389
6.7 Computation of the Singular Values of a Matrix 400
6.8 Generalized Eigenvalue Problems 405
6.9 Estimation of Eigenvalues 406
Exercises for Chapter 6 419
References for Chapter 6 425
7 Ordinary Differential Equations 428
7.0 Introduction 428
7.1 Some Theorems from the Theory of Ordinary Differential Equations 430
7.2 Initial-Value Problems 434
7.2.1 One-Step Methods:Basic Concepts 434
7.2.2 Convergence of One-Step Methods 439
7.2.3 Asymptotic Expansions for the Global Discretization Error of One-Step Methods 443
7.2.4 The Influence of Rounding Errors in One-Step Methods 445
7.2.5 Practical Implementation of One-Step Methods 448
7.2.6 Multistep Methods:Examples 455
7.2.7 General Multistep Methods 458
7.2.8 An Example of Divergence 461
7.2.9 Linear Difference Equations 464
7.2.10 Convergence of Multistep Methods 467
7.2.11 Linear Multistep Methods 471
7.2.12 Asymptotic Expansions of the Global Discretization Error for Linear Multistep Methods 476
7.2.13 Practical Implementation of Multistep Methods 481
7.2.14 ExtrapolationMethodsfortheSolutionoftheInitial-ValueProblem 484
7.2.15 Comparison of Methods for Solving Initial-Value Problems 487
7.2.16 Stiff Differential Equations 488
7.2.17 Implicit Differential Equations.Differential-Algebraic Equations 494
7.3 Boundary-Value Problems 499
7.3.0 Introduction 499
7.3.1 The Simple Shooting Method 502
7.3.2 The Simple Shooting Method for Linear Boundary-Value Problems 507
7.3.3 An Existence and Uniqueness Theorem for the Solution of Boundary-Value Problems 509
7.3.4 Difficulties in the Execution of the Simple Shooting Method 511
7.3.5 The Multiple Shooting Method 516
7.3.6 Hints for the Practical Implementation of the Multiple Shooting Method 520
7.3.7 An Example:Optimal Control Program for a Lifting Reentry Space Vehicle 524
7.3.8 The Limiting Case m→∞ of the Multiple Shooting Method(General Newton's Method,Quasilinearization) 531
7.4 Difference Methods 535
7.5 Variational Methods 540
7.6 Comparison of the Methods for Solving Boundary-Value Problems for Ordinary Differential Equations 549
7.7 Variational Methods for Partial Differential Equations.The Finite-Element Method 553
Exercises for Chapter 7 560
References for Chapter 7 566
8 Iterative Methods for the Solution of Large Systems of Linear Equations.Some Further Methods 570
8.0 Introduction 570
8.1 General Procedures for the Construction of Iterative Methods 571
8.2 Convergence Theorems 574
8.3 Relaxation Methods 579
8.4 Applications to Difference Methods—An Example 588
8.5 Block Iterative Methods 594
8.6 The ADI-Method of Peaceman and Rachford 597
8.7 The Conjugate-Gradient Method of Hestenes and Stiefel 606
8.8 The Algorithm of Buneman for the Solution of the Discretized Poisson Equation 614
8.9 Multigrid Methods 622
8.10 Comparison of Iterative Methods 632
Exercises for Chapter 8 636
References for Chapter 8 643
General Literature on Numerical Methods 646
Index 648