Principles and procedures of numerical analysis

Title Page......Page 1
Copyright Page......Page 2
Preface......Page 3
Contents......Page 7
1.1. Number Systems and Representations of Numbers......Page 11
1.2. Error Analysis......Page 19
1.2.1. Upper Bounds in Arithmetic......Page 20
1.2.2. Probabilistic Error Analysis......Page 28
1.2.3. Propagation of Errors......Page 29
1.3. Supplementary Notes and Discussion......Page 33
2. Approximation and Interpolation of Functions......Page 35
2.1.1. Lagrange Interpolating Polynomials......Page 39
2.1.2. Error Bounds for Interpolating Polynomials......Page 41
2.1.3. Differences......Page 45
2.1.4. The Fraser Diagram......Page 48
2.1.5. Aitken's Method and Computational Requirements of Interpolation......Page 56
2.1.6. Hermite Interpolation......Page 58
2.2. Uniform Approximations......Page 59
2.3. Least Squares Approximation......Page 63
2.4. Spline Functions......Page 69
2.5. Asymptotic Properties of Polynomial Approximations......Page 73
2.6. Supplementary Notes and Discussion......Page 79
3. Numerical Differentiation and Integration......Page 83
3.1. Numerical Differentiation......Page 85
3.2.1. Interpolatory Quadrature Formulas......Page 89
3.2.2. Error Analysis and Richardson Extrapolation......Page 91
3.2.3. Gaussian Quadrature......Page 98
3.2.4. The Euler-Maclaurin Formula......Page 107
3.2.5. Romberg Integration......Page 115
3.3. Supplementary Notes and Discussion......Page 117
4. General Theory for Iteration Methods......Page 121
4.1. Metric Spaces......Page 122
4.2. Examples of Metric Spaces......Page 124
4.3. Operators on Metric Spaces......Page 127
4.4. Examples of Bounded Operators......Page 128
4.5. Iterations of Operators......Page 131
4.6. Fixed-Point Theorems......Page 134
4.7. Systems of Operator Equations......Page 141
4.8. Norms of Vectors and Matrices......Page 144
4.9. The Order of Convergence of an Iteration Process......Page 147
4.10. Inner Products......Page 148
4.11. Supplementary Notes and Discussion......Page 149
5.1.1. The Bisection Method......Page 151
5.1.2. The Method of False Position (Regula Falsi)......Page 153
5.1.3. The Secant Method......Page 157
5.1.4. Newton's Method......Page 159
5.1.5. Application of Fixed-Point Theory......Page 164
5.1.6. Acceleration of Convergence, and Aitken's GSM-Method......Page 168
5.2. Solution of Polynomial Equations......Page 170
5.2.1. Sturm Sequences......Page 171
5.2.2. The Lehmer-Schur Method......Page 173
5.2.3. Bairstow's Method......Page 175
5.2.4. The Effect of Coefficient Errors on the Roots......Page 177
5.3. Systems of Nonlinear Equations and Nonlinear Programming......Page 179
5.3.1. Iterative Methods for Solution of Systems of Equations......Page 180
5.3.2. The Gradient Method and Related Techniques......Page 184
5.4. Supplementary Notes and Discussion......Page 188
6. The Solution of Simultaneous Linear Equations......Page 189
6.1.1. Gaussian Elimination......Page 190
6.1.2. Variants of Gaussian Elimination......Page 196
6.1.3. Inversion by Partitioning......Page 202
6.2. Iteration Methods......Page 205
6.2.1. Stationary Iteration Processes......Page 206
6.2.2. Iteration Processes Based on the Minimization of Quadratic Forms......Page 211
6.2.3. Application of the Gradient Method......Page 216
6.2.4. The Conjugate Gradient Method......Page 217
6.3.1. Bounds for Errors of Perturbed Linear Equations......Page 222
6.3.2. Error Bounds for Rounding in Gaussian Elimination......Page 226
6.4. Supplementary Notes and Discussion......Page 229
7. The Solution of Matrix Eigenvalue Problems......Page 231
7.1.1. Some Matrix Algebra Background......Page 232
7.1.2. The Householder Transformation and Reduction to Hessenberg Form......Page 240
7.2. Some Basic Eigenvalue Approximation Methods......Page 244
7.2.1. The Power Method......Page 246
7.2.2. The Inverse Power Method......Page 249
7.2.3. The Rayleigh Quotient Iteration Method......Page 250
7.2.4. Jacobi-Type Methods......Page 255
7.3.1. Principles and Convergence Rates......Page 258
7.3.2. Implementation of the QR Algorithm......Page 260
7.4. Eigenproblem Error Analysis......Page 262
7.5. Supplementary Notes and Discussion......Page 265
8. The Numerical Solution of Ordinary Differential Equations......Page 267
8.1.1. Picard's Method of Successive Approximation......Page 269
8.1.2. The Power Series Method......Page 270
8.1.3. Methods of the Runge-Kutta Type......Page 273
8.1.4. Linear Multistep Methods......Page 282
8.1.5. Step Size and Its Adaptive Selection......Page 288
8.1.6. The Method of Quasi I inearization......Page 290
8.2.1. Reduction to Initial-Value Problems......Page 294
8.2.2. The Method of Undetermined Coefficients......Page 295
8.2.3. The Difference Method......Page 297
8.2.4. The Method of Quasilinearization......Page 299
8.3. The Solution of Eigenvalue Problems......Page 300
8.4. Supplementary Notes and Discussion......Page 302
9. The Numerical Solution of Partial Differential Equations......Page 305
9.1. The Difference Method......Page 306
9.2. The Method of Quasilinearization......Page 313
9.3.1. The Ritz Method......Page 314
9.3.2. The Galerkin Method......Page 320
9.3.3. The Finite-Element Method......Page 321
9.4. Supplementary Notes and Discussion......Page 328
REFERENCES......Page 331
INDEX......Page 337