Numerical Recipes in
Fortran

 Acrobat Edition

Also available is a PostScript edition.

Thanks to special permission from Cambridge University Press, we are able to bring you the complete Numerical Recipes in Fortran 77 book On-Line! To utilize this resource, you will need an Adobe Acrobat viewer linked as a helper program to your WWW browser. Permission is granted by the copyright owners for users of the World Wide Web to make one paper copy of these PostScript 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 Want to download the Numerical Recipes routines? Jump to the Numerical Recipes On-Line Software Store. You'll get instant downloading and can use your credit card to get an immediate, low-cost, single-screen license. You can order Numerical Recipes books, too!


  • 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 Error, Accuracy, and Stability 18
  • 2 Solution of Linear Algebraic Equations

  • 2.0 Introduction 22
  • 2.1 Gauss-Jordan Elimination 27
  • 2.2 Gaussian Elimination with Backsubstitution 33
  • 2.3 LU Decomposition and Its Applications 34
  • 2.4 Tridiagonal and Band Diagonal Systems of Equations 42
  • 2.5 Iterative Improvement of a Solution to Linear Equations 47
  • 2.6 Singular Value Decomposition 51
  • 2.7 Sparse Linear Systems 63
  • 2.8 Vandermonde Matrices and Toeplitz Matrices 82
  • 2.9 Cholesky Decomposition 89
  • 2.10 QR Decomposition 91
  • 2.11 Is Matrix Inversion an $N^3$ Process? 95
  • 3 Interpolation and Extrapolation

  • 3.0 Introduction 99
  • 3.1 Polynomial Interpolation and Extrapolation 102
  • 3.2 Rational Function Interpolation and Extrapolation 104
  • 3.3 Cubic Spline Interpolation 107
  • 3.4 How to Search an Ordered Table 110
  • 3.5 Coefficients of the Interpolating Polynomial 113
  • 3.6 Interpolation in Two or More Dimensions 116
  • 4 Integration of Functions

  • 4.0 Introduction 123
  • 4.1 Classical Formulas for Equally Spaced Abscissas 124
  • 4.2 Elementary Algorithms 130
  • 4.3 Romberg Integration 134
  • 4.4 Improper Integrals 135
  • 4.5 Gaussian Quadratures and Orthogonal Polynomials 140
  • 4.6 Multidimensional Integrals 155
  • 5 Evaluation of Functions

  • 5.0 Introduction 159
  • 5.1 Series and Their Convergence 159
  • 5.2 Evaluation of Continued Fractions 163
  • 5.3 Polynomials and Rational Functions 167
  • 5.4 Complex Arithmetic 171
  • 5.5 Recurrence Relations and Clenshaw's Recurrence Formula 172
  • 5.6 Quadratic and Cubic Equations 178
  • 5.7 Numerical Derivatives 180
  • 5.8 Chebyshev Approximation 184
  • 5.9 Derivatives or Integrals of a Chebyshev-approximated Function 189
  • 5.10 Polynomial Approximation from Chebyshev Coefficients 191
  • 5.11 Economization of Power Series 192
  • 5.12 Pad\'e Approximants 194
  • 5.13 Rational Chebyshev Approximation 197
  • 5.14 Evaluation of Functions by Path Integration 201
  • 6 Special Functions

  • 6.0 Introduction 205
  • 6.1 Gamma Function, Beta Function, Factorials, Binomial Coefficients 206
  • 6.2 Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function 209
  • 6.3 Exponential Integrals 215
  • 6.4 Incomplete Beta Function, Student's Distribution, F-Distribution,Cumulative Binomial Distribution 219
  • 6.5 Bessel Functions of Integer Order 223
  • 6.6 Modified Bessel Functions of Integer Order 229
  • 6.7 Bessel Functions of Fractional Order, Airy Functions, SphericalBessel Functions 234
  • 6.8 Spherical Harmonics 246
  • 6.9 Fresnel Integrals, Cosine and Sine Integrals 248
  • 6.10 Dawson's Integral 252
  • 6.11 Elliptic Integrals and Jacobian Elliptic Functions 254
  • 6.12 Hypergeometric Functions 263
  • 7 Random Numbers

  • 7.0 Introduction 266
  • 7.1 Uniform Deviates 267
  • 7.2 Transformation Method: Exponential and Normal Deviates 277
  • 7.3 Rejection Method: Gamma, Poisson, Binomial Deviates 281
  • 7.4 Generation of Random Bits 287
  • 7.5 Random Sequences Based on Data Encryption 290
  • 7.6 Simple Monte Carlo Integration 295
  • 7.7 Quasi- (that is, Sub-) Random Sequences 299
  • 7.8 Adaptive and Recursive Monte Carlo Methods 306
  • 8 Sorting

  • 8.0 Introduction 320
  • 8.1 Straight Insertion and Shell's Method 321
  • 8.2 Quicksort 323
  • 8.3 Heapsort 327
  • 8.4 Indexing and Ranking 329
  • 8.5 Selecting the $M$th Largest 333
  • 8.6 Determination of Equivalence Classes 337
  • 9 Root Finding and Nonlinear Sets of Equations

  • 9.0 Introduction 340
  • 9.1 Bracketing and Bisection 343
  • 9.2 Secant Method, False Position Method, and Ridders' Method 347
  • 9.3 Van Wijngaarden--Dekker--Brent Method 352
  • 9.4 Newton-Raphson Method Using Derivative 355
  • 9.5 Roots of Polynomials 362
  • 9.6 Newton-Raphson Method for Nonlinear Systems of Equations 372
  • 9.7 Globally Convergent Methods for Nonlinear Systems of Equations 376
  • 10 Minimization or Maximization of Functions

  • 10.0 Introduction 387
  • 10.1 Golden Section Search in One Dimension 390
  • 10.2 Parabolic Interpolation and Brent's Method in One Dimension 395
  • 10.3 One-Dimensional Search with First Derivatives 399
  • 10.4 Downhill Simplex Method in Multidimensions 402
  • 10.5 Direction Set (Powell's) Methods in Multidimensions 406
  • 10.6 Conjugate Gradient Methods in Multidimensions 413
  • 10.7 Variable Metric Methods in Multidimensions 418
  • 10.8 Linear Programming and the Simplex Method 423
  • 10.9 Simulated Annealing Methods 436
  • 11 Eigensystems

  • 11.0 Introduction 449
  • 11.1 Jacobi Transformations of a Symmetric Matrix 456
  • 11.2 Reduction of a Symmetric Matrix to Tridiagonal Form: Givens and Householder Reductions 462
  • 11.3 Eigenvalues and Eigenvectors of a Tridiagonal Matrix 469
  • 11.4 Hermitian Matrices 475
  • 11.5 Reduction of a General Matrix to Hessenberg Form 476
  • 11.6 The QR Algorithm for Real Hessenberg Matrices 480
  • 11.7 Improving Eigenvalues and/or Finding Eigenvectors by Inverse Iteration 487
  • 12 Fast Fourier Transform

  • 12.0 Introduction 490
  • 12.1 Fourier Transform of Discretely Sampled Data 494
  • 12.2 Fast Fourier Transform (FFT) 498
  • 12.3 FFT of Real Functions, Sine and Cosine Transforms 504
  • 12.4 FFT in Two or More Dimensions 515
  • 12.5 Fourier Transforms of Real Data in Two and Three Dimensions 519
  • 12.6 External Storage or Memory-Local FFTs 525
  • 13 Fourier and Spectral Applications

  • 13.0 Introduction 530
  • 13.1 Convolution and Deconvolution Using the FFT 531
  • 13.2 Correlation and Autocorrelation Using the FFT 538
  • 13.3 Optimal (Wiener) Filtering with the FFT 539
  • 13.4 Power Spectrum Estimation Using the FFT 542
  • 13.5 Digital Filtering in the Time Domain 551
  • 13.6 Linear Prediction and Linear Predictive Coding 557
  • 13.7 Power Spectrum Estimation by the Maximum Entropy (All Poles) Method 565
  • 13.8 Spectral Analysis of Unevenly Sampled Data 569
  • 13.9 Computing Fourier Integrals Using the FFT 577
  • 13.10 Wavelet Transforms 584
  • 13.11 Numerical Use of the Sampling Theorem 600
  • 14 Statistical Description of Data

  • 14.0 Introduction 603
  • 14.1 Moments of a Distribution: Mean, Variance, Skewness, and So Forth 604
  • 14.2 Do Two Distributions Have the Same Means or Variances? 609
  • 14.3 Are Two Distributions Different? 614
  • 14.4 Contingency Table Analysis of Two Distributions 622
  • 14.5 Linear Correlation 630
  • 14.6 Nonparametric or Rank Correlation 633
  • 14.7 Do Two-Dimensional Distributions Differ? 640
  • 14.8 Savitzky-Golay Smoothing Filters 644
  • 15 Modeling of Data

  • 15.0 Introduction 650
  • 15.1 Least Squares as a Maximum Likelihood Estimator 651
  • 15.2 Fitting Data to a Straight Line 655
  • 15.3 Straight-Line Data with Errors in Both Coordinates 660
  • 15.4 General Linear Least Squares 665
  • 15.5 Nonlinear Models 675
  • 15.6 Confidence Limits on Estimated Model Parameters 684
  • 15.7 Robust Estimation 694
  • 16 Integration of Ordinary Differential Equations

  • 16.0 Introduction 701
  • 16.1 Runge-Kutta Method 704
  • 16.2 Adaptive Stepsize Control for Runge-Kutta 708
  • 16.3 Modified Midpoint Method 716
  • 16.4 Richardson Extrapolation and the Bulirsch-Stoer Method 718
  • 16.5 Second-Order Conservative Equations 726
  • 16.6 Stiff Sets of Equations 727
  • 16.7 Multistep, Multivalue, and Predictor-Corrector Methods 740
  • 17 Two Point Boundary Value Problems

  • 17.0 Introduction 745
  • 17.1 The Shooting Method 749
  • 17.2 Shooting to a Fitting Point 751
  • 17.3 Relaxation Methods 753
  • 17.4 A Worked Example: Spheroidal Harmonics 764
  • 17.5 Automated Allocation of Mesh Points 774
  • 17.6 Handling Internal Boundary Conditions or Singular Points 775
  • 18 Integral Equations and Inverse Theory

  • 18.0 Introduction 779
  • 18.1 Fredholm Equations of the Second Kind 782
  • 18.2 Volterra Equations 786
  • 18.3 Integral Equations with Singular Kernels 788
  • 18.4 Inverse Problems and the Use of A Priori Information 795
  • 18.5 Linear Regularization Methods 799
  • 18.6 Backus-Gilbert Method 806
  • 18.7 Maximum Entropy Image Restoration 809
  • 19 Partial Differential Equations

  • 19.0 Introduction 818
  • 19.1 Flux-Conservative Initial Value Problems 825
  • 19.2 Diffusive Initial Value Problems 838
  • 19.3 Initial Value Problems in Multidimensions 844
  • 19.4 Fourier and Cyclic Reduction Methods for Boundary Value Problems 848
  • 19.5 Relaxation Methods for Boundary Value Problems 854
  • 19.6 Multigrid Methods for Boundary Value Problems 862
  • 20 Less-Numerical Algorithms

  • 20.0 Introduction 881
  • 20.1 Diagnosing Machine Parameters 881
  • 20.2 Gray Codes 886
  • 20.3 Cyclic Redundancy and Other Checksums 888
  • 20.4 Huffman Coding and Compression of Data 896
  • 20.5 Arithmetic Coding 902
  • 20.6 Arithmetic at Arbitrary Precision 906
  • References and Program Dependencies 916
  • General Index 935