Contents. Preface to the Third Edition (2007) Preface to the Second Edition (1992) Preface to the First Edition (1985) License and Legal Information

Transcription

1 Contents Preface to the Third Edition (2007) Preface to the Second Edition (1992) Preface to the First Edition (1985) License and Legal Information xi xiv xvii xix 1 Preliminaries Introduction Error, Accuracy, and Stability C Family Syntax Objects, Classes, and Inheritance Vector and Matrix Objects Some Further Conventions and Capabilities Solution of Linear Algebraic Equations Introduction Gauss-Jordan Elimination Gaussian Elimination with Backsubstitution LU Decomposition and Its Applications Tridiagonal and Band-Diagonal Systems of Equations Iterative Improvement of a Solution to Linear Equations Singular Value Decomposition Sparse Linear Systems Vandermonde Matrices and Toeplitz Matrices Cholesky Decomposition QR Decomposition Is Matrix Inversion an N 3 Process? Interpolation and Extrapolation Introduction Preliminaries: Searching an Ordered Table Polynomial Interpolation and Extrapolation Cubic Spline Interpolation Rational Function Interpolation and Extrapolation v

2 vi Contents 3.5 Coefficients of the Interpolating Polynomial Interpolation on a Grid in Multidimensions Interpolation on Scattered Data in Multidimensions Laplace Interpolation Integration of Functions Introduction Classical Formulas for Equally Spaced Abscissas Elementary Algorithms Romberg Integration Improper Integrals Quadrature by Variable Transformation Gaussian Quadratures and Orthogonal Polynomials Adaptive Quadrature Multidimensional Integrals Evaluation of Functions Introduction Polynomials and Rational Functions Evaluation of Continued Fractions Series and Their Convergence Recurrence Relations and Clenshaw s Recurrence Formula Complex Arithmetic Quadratic and Cubic Equations Numerical Derivatives Chebyshev Approximation Derivatives or Integrals of a Chebyshev-Approximated Function Polynomial Approximation from Chebyshev Coefficients Economization of Power Series Padé Approximants Rational Chebyshev Approximation Evaluation of Functions by Path Integration Special Functions Introduction Gamma Function, Beta Function, Factorials, Binomial Coefficients Incomplete Gamma Function and Error Function Exponential Integrals Incomplete Beta Function Bessel Functions of Integer Order Bessel Functions of Fractional Order, Airy Functions, Spherical Bessel Functions Spherical Harmonics Fresnel Integrals, Cosine and Sine Integrals Dawson s Integral Generalized Fermi-Dirac Integrals Inverse of the Function x log.x/ Elliptic Integrals and Jacobian Elliptic Functions

3 Contents vii 6.13 Hypergeometric Functions Statistical Functions Random Numbers Introduction Uniform Deviates Completely Hashing a Large Array Deviates from Other Distributions Multivariate Normal Deviates Linear Feedback Shift Registers Hash Tables and Hash Memories Simple Monte Carlo Integration Quasi- (that is, Sub-) Random Sequences Adaptive and Recursive Monte Carlo Methods Sorting and Selection Introduction Straight Insertion and Shell s Method Quicksort Heapsort Indexing and Ranking Selecting the M th Largest Determination of Equivalence Classes Root Finding and Nonlinear Sets of Equations Introduction Bracketing and Bisection Secant Method, False Position Method, and Ridders Method Van Wijngaarden-Dekker-Brent Method Newton-Raphson Method Using Derivative Roots of Polynomials Newton-Raphson Method for Nonlinear Systems of Equations Globally Convergent Methods for Nonlinear Systems of Equations Minimization or Maximization of Functions Introduction Initially Bracketing a Minimum 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 Line Methods in Multidimensions Direction Set (Powell s) Methods in Multidimensions Conjugate Gradient Methods in Multidimensions Quasi-Newton or Variable Metric Methods in Multidimensions Linear Programming: The Simplex Method Linear Programming: Interior-Point Methods Simulated Annealing Methods Dynamic Programming

4 viii Contents 11 Eigensystems Introduction Jacobi Transformations of a Symmetric Matrix Real Symmetric Matrices Reduction of a Symmetric Matrix to Tridiagonal Form: Givens and Householder Reductions Eigenvalues and Eigenvectors of a Tridiagonal Matrix Hermitian Matrices Real Nonsymmetric Matrices The QR Algorithm for Real Hessenberg Matrices Improving Eigenvalues and/or Finding Eigenvectors by Inverse Iteration Fast Fourier Transform Introduction Fourier Transform of Discretely Sampled Data Fast Fourier Transform (FFT) FFT of Real Functions Fast Sine and Cosine Transforms FFT in Two or More Dimensions Fourier Transforms of Real Data in Two and Three Dimensions External Storage or Memory-Local FFTs Fourier and Spectral Applications Introduction Convolution and Deconvolution Using the FFT Correlation and Autocorrelation Using the FFT Optimal (Wiener) Filtering with the FFT Power Spectrum Estimation Using the FFT Digital Filtering in the Time Domain Linear Prediction and Linear Predictive Coding Power Spectrum Estimation by the Maximum Entropy (All-Poles) Method Spectral Analysis of Unevenly Sampled Data Computing Fourier Integrals Using the FFT Wavelet Transforms Numerical Use of the Sampling Theorem Statistical Description of Data Introduction Moments of a Distribution: Mean, Variance, Skewness, and So Forth 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 Information-Theoretic Properties of Distributions Do Two-Dimensional Distributions Differ?

5 Contents ix 14.9 Savitzky-Golay Smoothing Filters Modeling of Data Introduction Least Squares as a Maximum Likelihood Estimator Fitting Data to a Straight Line Straight-Line Data with Errors in Both Coordinates General Linear Least Squares Nonlinear Models Confidence Limits on Estimated Model Parameters Robust Estimation Markov Chain Monte Carlo Gaussian Process Regression Classification and Inference Introduction Gaussian Mixture Models and k-means Clustering Viterbi Decoding Markov Models and Hidden Markov Modeling Hierarchical Clustering by Phylogenetic Trees Support Vector Machines Integration of Ordinary Differential Equations Introduction Runge-Kutta Method Adaptive Stepsize Control for Runge-Kutta Richardson Extrapolation and the Bulirsch-Stoer Method Second-Order Conservative Equations Stiff Sets of Equations Multistep, Multivalue, and Predictor-Corrector Methods Stochastic Simulation of Chemical Reaction Networks 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 Integral Equations and Inverse Theory Introduction Fredholm Equations of the Second Kind Volterra Equations Integral Equations with Singular Kernels Inverse Problems and the Use of A Priori Information Linear Regularization Methods Backus-Gilbert Method

6 x Contents 19.7 Maximum Entropy Image Restoration Partial Differential Equations Introduction Flux-Conservative Initial Value Problems Diffusive Initial Value Problems Initial Value Problems in Multidimensions Fourier and Cyclic Reduction Methods for Boundary Value Problems Relaxation Methods for Boundary Value Problems Multigrid Methods for Boundary Value Problems Spectral Methods Computational Geometry Introduction Points and Boxes KD Trees and Nearest-Neighbor Finding Triangles in Two and Three Dimensions Lines, Line Segments, and Polygons Spheres and Rotations Triangulation and Delaunay Triangulation Applications of Delaunay Triangulation Quadtrees and Octrees: Storing Geometrical Objects Less-Numerical Algorithms Introduction Plotting Simple Graphs Diagnosing Machine Parameters Gray Codes Cyclic Redundancy and Other Checksums Huffman Coding and Compression of Data Arithmetic Coding Arithmetic at Arbitrary Precision Index 1195