Front Matter, Contents, and Prefaces xi 
 Legal Matters xvi 
 Computer Programs by Chapter and Section xix 
1 Preliminaries 
 1.0 Introduction 1 
 1.1 Program Organization and Control Structures 5 
 1.2 Some C Conventions for Scientific Computing 15 
 1.3 Error, Accuracy, and Stability 15 
2 Solution of Linear Algebraic Equations 
 2.0 Introduction 32 
 2.1 Gauss-Jordan Elimination 36 
 2.2 Gaussian Elimination with Backsubstitution 41 
 2.3 LU Decomposition and Its Applications 43 
 2.4 Tridiagonal and Band Diagonal Systems of Equations 50 
 2.5 Iterative Improvement of a Solution to Linear Equations 55 
 2.6 Singular Value Decomposition 59 
 2.7 Sparse Linear Systems 71 
 2.8 Vandermonde Matrices and Toeplitz Matrices 90 
 2.9 Cholesky Decomposition 96 
 2.10 QR Decomposition 98 
 2.11 Is Matrix Inversion an $N^3$ Process? 102 
3 Interpolation and Extrapolation 
 3.0 Introduction 105 
 3.1 Polynomial Interpolation and Extrapolation 108 
 3.2 Rational Function Interpolation and Extrapolation 111 
 3.3 Cubic Spline Interpolation 113 
 3.4 How to Search an Ordered Table 117 
 3.5 Coefficients of the Interpolating Polynomial 120 
 3.6 Interpolation in Two or More Dimensions 123 
4 Integration of Functions 
 4.0 Introduction 129 
 4.1 Classical Formulas for Equally Spaced Abscissas 130 
 4.2 Elementary Algorithms 136 
 4.3 Romberg Integration 140 
 4.4 Improper Integrals 141 
 4.5 Gaussian Quadratures and Orthogonal Polynomials 147 
 4.6 Multidimensional Integrals 161 
5 Evaluation of Functions 
 5.0 Introduction 165 
 5.1 Series and Their Convergence 165 
 5.2 Evaluation of Continued Fractions 169 
 5.3 Polynomials and Rational Functions 173 
 5.4 Complex Arithmetic 176 
 5.5 Recurrence Relations and Clenshaw's Recurrence Formula 178 
 5.6 Quadratic and Cubic Equations 183 
 5.7 Numerical Derivatives 186 
 5.8 Chebyshev Approximation 190 
 5.9 Derivatives or Integrals of a Chebyshev-approximated Function 195 
 5.10 Polynomial Approximation from Chebyshev Coefficients 197 
 5.11 Economization of Power Series 198 
 5.12 Pad\'e Approximants 200 
 5.13 Rational Chebyshev Approximation 204 
 5.14 Evaluation of Functions by Path Integration 208 
6 Special Functions 
 6.0 Introduction 212 
 6.1 Gamma Function, Beta Function, Factorials, Binomial Coefficients 213 
 6.2 Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function 216 
 6.3 Exponential Integrals 222 
 6.4 Incomplete Beta Function, Student's Distribution, F-Distribution,Cumulative Binomial Distribution 226 
 6.5 Bessel Functions of Integer Order 230 
 6.6 Modified Bessel Functions of Integer Order 236 
 6.7 Bessel Functions of Fractional Order, Airy Functions, SphericalBessel Functions 240 
 6.8 Spherical Harmonics 252 
 6.9 Fresnel Integrals, Cosine and Sine Integrals 255 
 6.10 Dawson's Integral 259 
 6.11 Elliptic Integrals and Jacobian Elliptic Functions 261 
 6.12 Hypergeometric Functions 271 
7 Random Numbers
 7.0 Introduction 274 
 7.1 Uniform Deviates 275 
 7.2 Transformation Method: Exponential and Normal Deviates 287 
 7.3 Rejection Method: Gamma, Poisson, Binomial Deviates 290 
 7.4 Generation of Random Bits 296 
 7.5 Random Sequences Based on Data Encryption 300 
 7.6 Simple Monte Carlo Integration 304 
 7.7 Quasi- (that is, Sub-) Random Sequences 309 
 7.8 Adaptive and Recursive Monte Carlo Methods 316 
8 Sorting 
 8.0 Introduction 329 
 8.1 Straight Insertion and Shell's Method 330 
 8.2 Quicksort 332 
 8.3 Heapsort 336 
 8.4 Indexing and Ranking 338 
 8.5 Selecting the $M$th Largest 341 
 8.6 Determination of Equivalence Classes 345 
9 Root Finding and Nonlinear Sets of Equations
 9.0 Introduction 347 
 9.1 Bracketing and Bisection 350 
 9.2 Secant Method, False Position Method, and Ridders' Method 354 
 9.3 Van Wijngaarden--Dekker--Brent Method 359 
 9.4 Newton-Raphson Method Using Derivative 362 
 9.5 Roots of Polynomials 369 
 9.6 Newton-Raphson Method for Nonlinear Systems of Equations 379 
 9.7 Globally Convergent Methods for Nonlinear Systems of Equations 383 
10 Minimization or Maximization of Functions 
 10.0 Introduction 394 
 10.1 Golden Section Search in One Dimension 397 
 10.2 Parabolic Interpolation and Brent's Method in One Dimension 402 
 10.3 One-Dimensional Search with First Derivatives 305 
 10.4 Downhill Simplex Method in Multidimensions 408 
 10.5 Direction Set (Powell's) Methods in Multidimensions 412 
 10.6 Conjugate Gradient Methods in Multidimensions 420 
 10.7 Variable Metric Methods in Multidimensions 425 
 10.8 Linear Programming and the Simplex Method 430 
 10.9 Simulated Annealing Methods 444 
11 Eigensystems
 11.0 Introduction 456 
 11.1 Jacobi Transformations of a Symmetric Matrix 463 
 11.2 Reduction of a Symmetric Matrix to Tridiagonal Form: Givens and Householder Reductions 469 
 11.3 Eigenvalues and Eigenvectors of a Tridiagonal Matrix 475 
 11.4 Hermitian Matrices 481 
 11.5 Reduction of a General Matrix to Hessenberg Form 482 
 11.6 The QR Algorithm for Real Hessenberg Matrices 486 
 11.7 Improving Eigenvalues and/or Finding Eigenvectors by Inverse Iteration 493 
12 Fast Fourier Transform
 12.0 Introduction 496 
 12.1 Fourier Transform of Discretely Sampled Data 500 
 12.2 Fast Fourier Transform (FFT) 504 
 12.3 FFT of Real Functions, Sine and Cosine Transforms 510 
 12.4 FFT in Two or More Dimensions 521 
 12.5 Fourier Transforms of Real Data in Two and Three Dimensions 525 
 12.6 External Storage or Memory-Local FFTs 532 
13 Fourier and Spectral Applications
 13.0 Introduction 537 
 13.1 Convolution and Deconvolution Using the FFT 538 
 13.2 Correlation and Autocorrelation Using the FFT 545 
 13.3 Optimal (Wiener) Filtering with the FFT 547 
 13.4 Power Spectrum Estimation Using the FFT 549 
 13.5 Digital Filtering in the Time Domain 558 
 13.6 Linear Prediction and Linear Predictive Coding 564 
 13.7 Power Spectrum Estimation by the Maximum Entropy (All Poles) Method 572 
 13.8 Spectral Analysis of Unevenly Sampled Data 575 
 13.9 Computing Fourier Integrals Using the FFT 584 
 13.10 Wavelet Transforms 591 
 13.11 Numerical Use of the Sampling Theorem 606 
14 Statistical Description of Data 
 14.0 Introduction 609 
 14.1 Moments of a Distribution: Mean, Variance, Skewness, and So Forth 610 
 14.2 Do Two Distributions Have the Same Means or Variances? 615 
 14.3 Are Two Distributions Different? 620 
 14.4 Contingency Table Analysis of Two Distributions 628 
 14.5 Linear Correlation 636 
 14.6 Nonparametric or Rank Correlation 639 
 14.7 Do Two-Dimensional Distributions Differ? 645 
 14.8 Savitzky-Golay Smoothing Filters 650 
15 Modeling of Data 
 15.0 Introduction 656 
 15.1 Least Squares as a Maximum Likelihood Estimator 657 
 15.2 Fitting Data to a Straight Line 661 
 15.3 Straight-Line Data with Errors in Both Coordinates 666 
 15.4 General Linear Least Squares 671 
 15.5 Nonlinear Models 681 
 15.6 Confidence Limits on Estimated Model Parameters 689 
 15.7 Robust Estimation 699 
16 Integration of Ordinary Differential Equations 
 16.0 Introduction 707 
 16.1 Runge-Kutta Method 710 
 16.2 Adaptive Stepsize Control for Runge-Kutta 714 
 16.3 Modified Midpoint Method 722 
 16.4 Richardson Extrapolation and the Bulirsch-Stoer Method 724 
 16.5 Second-Order Conservative Equations 732 
 16.6 Stiff Sets of Equations 734 
 16.7 Multistep, Multivalue, and Predictor-Corrector Methods 747 
17 Two Point Boundary Value Problems 
 17.0 Introduction 753 
 17.1 The Shooting Method 757 
 17.2 Shooting to a Fitting Point 760 
 17.3 Relaxation Methods 762 
 17.4 A Worked Example: Spheroidal Harmonics 772 
 17.5 Automated Allocation of Mesh Points 783 
 17.6 Handling Internal Boundary Conditions or Singular Points 784 
18 Integral Equations and Inverse Theory 
 18.0 Introduction 788 
 18.1 Fredholm Equations of the Second Kind 791 
 18.2 Volterra Equations 794 
 18.3 Integral Equations with Singular Kernels 797 
 18.4 Inverse Problems and the Use of A Priori Information 804 
 18.5 Linear Regularization Methods 808 
 18.6 Backus-Gilbert Method 815 
 18.7 Maximum Entropy Image Restoration 818 
19 Partial Differential Equations 
 19.0 Introduction 827 
 19.1 Flux-Conservative Initial Value Problems 834 
 19.2 Diffusive Initial Value Problems 847 
 19.3 Initial Value Problems in Multidimensions 853 
 19.4 Fourier and Cyclic Reduction Methods for Boundary Value Problems 857 
 19.5 Relaxation Methods for Boundary Value Problems 863 
 19.6 Multigrid Methods for Boundary Value Problems 871 
20 Less-Numerical Algorithms 
 20.0 Introduction 889 
 20.1 Diagnosing Machine Parameters 889 
 20.2 Gray Codes 894 
 20.3 Cyclic Redundancy and Other Checksums 896 
 20.4 Huffman Coding and Compression of Data 903 
 20.5 Arithmetic Coding 910 
 20.6 Arithmetic at Arbitrary Precision 915 
 References and Program Dependencies 926 
 General Index 965