Numerical Recipes in Pascal

Transcription

1 Numerical Recipes in Pascal The Art of Scientific Computing William H. Press Harvard-Smithsonian Center for Astrophysics Brian P. Flannery EXXON Research and Engineering Company Saul A. Teukolsky Department of Physics, Cornell University William T. Vetterling Polaroid Corporation The right of the University of Cambridge to print and seil all manner of books was granied by Henry Vlll in The University has printed and published continuousiy since CAMBRIDGE UNIVERSITY PRESS Cambridge New York Port ehester Melbourne Sydney

2 Contents Preface to the Pascal Edition Preface List of Computer Programs xi xiü xvii 1 PRELIMINARIES l 1.0 Introduction Program Organization and Control Structures Conventions for Scientific Computing in Pascal Error, Accuracy, and Stability 23 2 SOLUTION OF LINEAR ALGEBRAIC EQUATIONS Introduction Gauss-Jordan Elimination Gaussian Elimination with Backsubstitution LU Decomposition Inverse of a Matrix Determinant of a Matrix Tridiagonal Systems of Equations Iterative Improvement of a Solution to Linear Equations Vandermonde Matrices and Toeplitz Matrices Singular Value Decomposition Sparse Linear Systems Is Matrix Inversion an N 3 Process? 84 3 INTERPOLATION AND EXTRAPOLATION Introduction Polynomial Interpolation and Extrapolation Rational Function Interpolation and Extrapolation Cubic Spline Interpolation How to Search an Ordered Table Coefficients of the Interpolating Polynomial Interpolation in Two or More Dimensions INTEGRATION OF FUNCTIONS 4.0 Introduction Classical Formulas for Equally-Spaced Abscissas 4.2 Elementary Algorithms Romberg Integration Improper Integrals Gaussian Quadratures Multidimensional Integrals v

3 vi Contents 5 EVALUATION OF FUNCTIONS Introduction Series and Their Convergence Evaluation of Continued Fractions Polynomials and Rational Functions Recurrence Relations and Clenshaw's Recurrence Formula Quadratic and Cubic Equations Chebyshev Approximation Derivatives or Integrals of a Chebyshev-approximated Function Polynomial Approximation from Chebyshev Coemcients SPECIAL FUNCTIONS Introduction Gamma Function, Beta Function, Factorials, Binomial Coemcients Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Distribution Incomplete Beta Function, Student's Distribution, F-Distribution, Cumulative Binomial Distribution Bessel Functions of Integer Order Modified Bessel Functions of Integer Order Spherical Harmonics Elliptic Integrals and Jacobian Elliptic Functions RANDOM NUMBERS Introduction Uniform Deviates Transformation Method: Exponential and Normal Deviates Rejection Method: Gamma, Poisson, Binomial Deviates Generation of Random Bits The Data Encryption Standard Monte Carlo Integration SORTING Introduction Straight Insertion and Shell's Method Heapsort Indexing and Ranking Quicksort Determination of Equivalence Classes ROOT FINDING AND NONLINEAR SETS OF EQUATIONS Introduction Bracketing and Bisection Secant Method and False Position Method 279

4 Contents vn 9.3 Van Wijngaarden-Dekker-Brent Method Newton-Raphson Method Using Derivative Roots of Polynomials Newton-Raphson Method for Nonlinear Systems of Equations MINIMIZATION OR MAXIMIZATION OF FUNCTIONS Introduction Golden Section Search in One Dimension Parabolic Interpolation and Brent's Method in One Dimension One-Dimensional Search with First Derivatives Downhill Simplex Method in Multidimensions Direction Set (Powell's) Methods in Multidimensions Conjugate Gradient Methods in Multidimensions Variable Metrie Methods in Multidimensions Linear Programming and the Simplex Method Combinatorial Minimization: Method of Simulated Annealing EIGENSYSTEMS Introduction Jacobi Transformations of a Symmetrie Matrix Reduction of a Symmetrie Matrix to Tridiagonal Form: Givens and Householder Reductions Eigenvalues and Eigenvectors of a Tridiagonal Matrix Hermitian Matrices Reduction of a General Matrix to Hessenberg Form The QR Algorithm for Real Hessenberg Matrices Improving Eigenvalues and/or Finding Eigenvectors by Inverse Iteration FOURIER TRANSFORM SPECTRAL METHODS Introduction Fourier Transform of Discretely Sampled Data Fast Fourier Transform (FFT) FFT of Real Functions, Sine and Cosine Transforms Convolution and Deconvolution Using the FFT Correlation and Autocorrelation Using the FFT Optimal (Wiener) Filtering with the FFT Power Spectrum Estimation Using the FFT Power Spectrum Estimation by the Maximum Entropy (All Poles) Method Digital Filtering in the Time Domain Linear Prediction and Linear Predictive Coding FFT in Two or More Dimensions 493

5 Vlll Contents 13 STATISTICAL DESCRIPTION OF DATA Introduction Moments of a Distribution: Mean, Variance, Skewness, and so forth Efficient Search for the Median Estimation of the Mode for Continuous Data Do Two Distributions Have the Same Means or Variances? Are Two Distributions Different? Contingency Table Analysis of Two Distributions Linear Correlation Nonparametric or Rank Correlation Smoothing of Data MODELING OF DATA Introduction Least Squares as a Maximum Likelihood Estimator Fitting Data to a Straight Line General Linear Least Squares Nonlinear Models Confidence Limits on Estimated Model Parameters Robust Estimation INTEGRATION OF ORDINARY DIFFERENTIAL EQUATIONS Introduction Runge-Kutta Method Adaptive Stepsize Control for Runge-Kutta Modified Midpoint Method Richardson Extrapolation and the Bulirsch-Stoer Method Predictor-Corrector Methods Stiff Sets of Equations TWO POINT BOUNDARY VALUE PROBLEMS Introduction The Shooting Method Shooting to a Fitting Point Relaxation Methods A Worked Example: Spheroidal Harmonics Automated Allocation of Mesh Points Handling Internal Boundary Conditions or Singular Points PARTIAL DIFFERENTIAL EQUATIONS Introduction Flux-Conservative Initial Value Problems Diffusive Initial Value Problems Initial Value Problems in Multidimensions 700

6 Contents ix 17.4 Fourier and Cyclic Reduction Methods for Boundary Value Problems Relaxation Methods for Boundary Value Problems Operator Splitting Methods and ADI 718 APPENDIX A: References 727 APPENDIX B: Table of Program Dependencies 732 Index 737