Numerical Recipes in C

 Acrobat ® Edition

Thanks to special permission from Cambridge University Press, we are able to bring you the complete Numerical Recipes in C book On-Line! To utilize this resource, you will need an Adobe Acrobat viewer linked as a helper program to your web browser. Permission is granted by the copyright owners for users of this resource to make one paper copy of these Acrobat files for their own personal use. Further reproduction, or the extraction of, or copying of, machine readable files to any server computer, is strictly prohibited. This on-line resource is not intended as a substitute for purchasing the book, or for obtaining a license for the use of Numerical Recipes source code.

Numerical Recipes On-Line Software Store Downloads of the Numerical Recipes source code in machine-readable format are not available as part of this free resource. For information on downloads, please go to the Numerical Recipes On-Line Software Store.

  • 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