Numerical Analysis

Preface\0......Page 1
Contents\0......Page 6
1 Mathematical Preliminaries\0......Page 10
1.1 Review of Calculus\0......Page 11
1.2 Roundoff Errors and Computer Arithmetic\0......Page 27
1.3 Algorithms and Convergence\0......Page 40
1.4 Numerical Software\0......Page 49
2 Solutions of Equations in One Variable\0......Page 56
2.1 The Bisection Method\0......Page 57
2.2 Fixed-Point Iteration\0......Page 64
2.3 Newton s Method\0......Page 75
2.4 Error Analysis for Iterative Methods\0......Page 87
2.5 Accelerating Convergence\0......Page 95
2.6 Zeros of Polynomials and Müller s Method\0......Page 100
2.7 Survey of Methods and Software\0......Page 110
3 Interpolation and Polynomial Approximation\0......Page 113
3.1 Interpolation and the Lagrange Polynomial\0......Page 116
3.2 Divided Differences\0......Page 131
3.3 Hermite Interpolation\0......Page 142
3.4 Cubic Spline Interpolation\0......Page 150
3.5 Parametric Curves\0......Page 165
3.6 Survey of Methods and Software\0......Page 172
4 Numerical Differentiation and Integration\0......Page 175
4.1 Numerical Differentiation\0......Page 176
4.2 Richardson s Extrapolation\0......Page 187
4.3 Elements of Numerical Integration\0......Page 195
4.4 Composite Numerical Integration\0......Page 205
4.5 Romberg Integration\0......Page 216
4.6 Adaptive Quadrature Methods\0......Page 222
4.7 Gaussian Quadrature\0......Page 229
4.8 Multiple Integrals\0......Page 236
4.9 Improper Integrals\0......Page 250
4.10 Survey of Methods and software\0......Page 256
5 Initial-Value Problems for Ordinary Differential Equations\0......Page 258
5.1 The Elementary Theory of Initial-Value Problems\0......Page 260
5.2 Euler s Method\0......Page 265
5.3 Higher-Order Taylor Methods\0......Page 275
5.4 Runge-Kutta Methods\0......Page 281
5.5 Error Control and the Runge-Kutta-Fehlberg Method\0......Page 291
5.6 Multistep Methods\0......Page 298
5.7 Variable Step-Size Multistep Methods\0......Page 310
5.8 Extrapolation Methods\0......Page 316
5.9 Higher-Order Equations and Systems of Differential Equations\0......Page 322
5.10 Stability\0......Page 333
5.11 Stiff Differential Equations\0......Page 343
5.12 Survey of Methods and Software\0......Page 351
6 Direct Methods for Solving Linear Systems\0......Page 353
6.1 Linear Systems of Equations\0......Page 354
6.2 Pivoting Strategies\0......Page 368
6.3 Linear Algebra and Matrix Inversion\0......Page 379
6.4 The Determinant of a Matrix\0......Page 392
6.5 Matrix Factorization\0......Page 397
6.6 Special Types of Matrices\0......Page 407
6.7 Survey of Methods and Software\0......Page 422
7 Iterative Techniques in Matrix Algebra\0......Page 426
7.1 Norms of Vectors and Matrices\0......Page 427
7.2 Eigenvalues and Eigenvectors\0......Page 439
7.3 Iterative Techniques for Solving Linear Systems\0......Page 446
7.4 Error Bounds and Iterative Refinement\0......Page 463
7.5 The Conjugate Gradient Method\0......Page 474
7.6 Survey of Methods and Software\0......Page 490
8 Approximation Theory\0......Page 492
8.1 Discrete Least Squares Approximation\0......Page 493
8.2 Orthogonal Polynomials and Least Squares Approximation\0......Page 507
8.3 Chebyshev Polynomials and Economization of power Series\0......Page 516
8.4 Rational Function Approximation\0......Page 526
8.5 Trigonometric Polynomial Approximation\0......Page 538
8.6 Fast Fourier Transforms\0......Page 546
8.7 survey of Methods and software\0......Page 557
9 Approximating Eigenvalues\0......Page 559
9.1 Linear Algebra and Eigenvalues\0......Page 560
9.2 The Power Method\0......Page 569
9.3 Householder s Method\0......Page 586
9.4 The QR Algorithm\0......Page 594
9.5 Survey of Methods and Software\0......Page 606
10 Numerical Solutions of Nonlinear Systems of Equations\0......Page 609
10.1 Fixed Points for Functions of Several Variables\0......Page 611
10.2 Newton s Method\0......Page 620
10.3 Quasi-Newton Methods\0......Page 629
10.4 Steepest Descent Techniques\0......Page 637
10.5 Homotopy and Continuation Methods\0......Page 644
10.6 Survey of Methods and software\0......Page 652
11 Boundary-Value Problems for Ordinary Differential Equations\0......Page 654
11.1 The Linear Shooting Method\0......Page 655
11.2 The Shooting Method for Nonlinear Problems\0......Page 662
11.3 Finite-Difference Methods for Linear Problems\0......Page 669
11.4 Finite-Difference Methods for Nonlinear Problems\0......Page 676
11.5 The Rayleigh-Ritz Method\0......Page 681
11.6 Survey of Methods and Software\0......Page 697
12 Numerical Solutions to partial Differential Equations\0......Page 700
12.1 Elliptic Partial Differential Equations\0......Page 703
12.2 Parabolic Partial Differential Equations\0......Page 713
12.3 Hyperbolic Partial Differential Equations\0......Page 727
12.4 An Introduction to the Finite-Element Method\0......Page 735
12.5 Survey of Methods and Software\0......Page 750
Bibliography\0......Page 752
Answers to Selected Exercises\0......Page 762
Index\0......Page 840