Introduction to Numerical Analysis

Transcription

1 J. Stoer R. Bulirsch Introduction to Numerical Analysis Second Edition Translated by R. Bartels, W. Gautschi, and C. Witzgall With 35 Illustrations Springer

2 Contents Preface to the Second Edition Preface to the First Edition v vii 1 Error Analysis Representation of Numbers Roundoff Errors and Floating-Point Arithmetic Error Propagation Examples Interval Arithmetic; Statistical Roundoff Estimation 27 Exercises for Chapter 1 33 References for Chapter Interpolation Interpolation by Polynomials Theoretical Foundation: The Interpolation Formula of Lagrange Neville's Algorithm Newton's Interpolation Formula: Divided Differences The Error in Polynomial Interpolation Hermite Interpolation Interpolation by Rational Functions General Properties of Rational Interpolation Inverse and Reciprocal Differences. Thiele's Continued Fraction Algorithms of the Neville Type Comparing Rational and Polynomial Interpolations Trigonometric Interpolation 72 IX

3 2.3.1 Basic Facts Fast Fourier Transforms The Algorithms of Goertzel and Reinsch The Calculation of Fourier Coefficients. Attenuation Factors Interpolation by Spline Functions Theoretical Foundations Determining Interpolating Cubic Spline Functions Convergence Properties of Cubic Spline Functions B-Splines The Computation of B-Splines 110 Exercises for Chapter References for Chapter Topics in Integration 3.1 The Integration Formulas of Newton and Cotes Peano's Error Representation The Euler-Maclaurin Summation Formula Integrating by Extrapolation About Extrapolation Methods Gaussian Integration Methods Integrals with Singularities 160 Exercises for Chapter References for Chapter Systems of Linear Equations 4.1 Gaussian Elimination. The Triangular Decomposition of a Matrix 4.2 The Gauss-Jordan Algorithm The Cholesky Decomposition Error Bounds Roundoff-Error Analysis for Gaussian Elimination Roundoff Errors in Solving Triangular Systems Orthogonalization Techniques of Householder and Gram-Schmidt 4.8 Data Fitting Linear Least Squares. The Normal Equations The Use of Orthogonalization in Solving Linear Least-Squares Problems The Condition of the Linear Least-Squares Problem Nonlinear Least-Squares Problems The Pseudoinverse of a Matrix Modification Techniques for Matrix Decompositions The Simplex Method Phase One of the Simplex Method 241 Appendix to Chapter A Elimination Methods for Sparse Matrices 245 Exercises for Chapter References for Chapter 4 258

4 Contents XI 5 Finding Zeros and Minimum Points by Iterative Methods The Development of Iterative Methods General Convergence Theorems The Convergence of Newton's Method in Several Variables A Modified Newton Method On the Convergence of Minimization Methods Application of the Convergence Criteria to the Modified Newton Method Suggestions for a Practical Implementation of the Modified Newton Method. A Rank-One Method Due to Broyden Roots of Polynomials. Application of Newton's Method Sturm Sequences and Bisection Methods Bairstow's Method The Sensitivity of Polynomial Roots Interpolation Methods for Determining Roots TheA 2 -MethodofAitken Minimization Problems without Constraints 316 Exercises for Chapter References for Chapter Eigenvalue Problems Introduction Basic Facts on Eigenvalues The Jordan Normal Form of a Matrix The Frobenius Normal Form of a Matrix The Schur Normal Form of a Matrix; Hermitian and Normal Matrices; Singular Values of Matrices Reduction of Matrices to Simpler Form Reduction of a Hermitian Matrix to Tridiagonal Form: The Method of Householder Reduction of a Hermitian Matrix to Tridiagonal or Diagonal Form: The Methods of Givens and Jacobi Reduction of a Hermitian Matrix to Tridiagonal Form: The Method of Lanczos Reduction to Hessenberg Form Methods for Determining the Eigenvalues and Eigenvectors Computation of the Eigenvalues of a Hermitian Tridiagonal Matrix Computation of the Eigenvalues of a Hessenberg Matrix. The Method of Hyman Simple Vector Iteration and Inverse Iteration of Wielandt The LR and QR Methods The Practical Implementation oftheßä Method Computation of the Singular Values of a Matrix Generalized Eigenvalue Problems Estimation of Eigenvalues 406 Exercises for Chapter References for Chapter 6 425

5 7 Ordinary Differential Equations Introduction Some Theorems from the Theory of Ordinary Differential Equations Initial-Value Problems One-Step Methods: Basic Concepts Convergence of One-Step Methods Asymptotic Expansions for the Global Discretization Error of One-Step Methods The Influence of Rounding Errors in One-Step Methods Practical Implementation of One-Step Methods Multistep Methods: Examples General Multistep Methods An Example of Divergence Linear Difference Equations Convergence of Multistep Methods Linear Multistep Methods Asymptotic Expansions of the Global Discretization Error for Linear Multistep Methods Practical Implementation of Multistep Methods Extrapolation Methods for the Solution of the Initial-Value Problem Comparison of Methods for Solving Initial-Value Problems Stiff Differential Equations Implicit Differential Equations. Differential-Algebraic Equations Boundary-Value Problems Introduction The Simple Shooting Method The Simple Shooting Method for Linear Boundary-Value Problems An Existence and Uniqueness Theorem for the Solution of Boundary-Value Problems Difficulties in the Execution of the Simple Shooting Method The Multiple Shooting Method Hints for the Practical Implementation of the Multiple Shooting Method An Example: Optimal Control Program for a Lifting Reentry Space Vehicle The Limiting Case m -* oo of the Multiple Shooting Method (General Newton's Method, Quasilinearization) Difference Methods Variational Methods Comparison of the Methods for Solving Boundary-Value Problems for Ordinary Differential Equations Variational Methods for Partial Differential Equations. The Finite-Element Method 553 Exercises for Chapter References for Chapter 7 566

6 Contents хш 8 Iterative Methods for the Solution of Large Systems of Linear Equations. Some Further Methods Introduction General Procedures for the Construction of Iterative Methods Convergence Theorems Relaxation Methods Applications to Difference Methods An Example Block Iterative Methods The ADI-Method of Peaceman and Rachford The Conjugate-Gradient Method of Hestenes and Stiefel The Algorithm of Buneman for the Solution of the Discretized Poisson Equation Multigrid Methods Comparison of Iterative Methods 632 Exercises for Chapter References for Chapter General Literature on Numerical Methods 646 Index 648