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xxiv Computer Programs by Chapter and Section 12.4 fourn fast Fourier transform in multidimensions 12.5 rlft3 FFT of real data in two or three dimensions 12.6 fourfs FFT for huge data sets on external media 12.6 fourew rewind and permute files, used by fourfs 13.1 convlv convolution or deconvolution of data using FFT 13.2 correl correlation or autocorrelation of data using FFT 13.4 spctrm power spectrum estimation using FFT 13.6 memcof evaluate maximum entropy (MEM) coefficients 13.6 fixrts reflect roots of a polynomial into unit circle 13.6 predic linear prediction using MEM coefficients 13.7 evlmem power spectral estimation from MEM coefficients 13.8 period power spectrum of unevenly sampled data 13.8 fasper power spectrum of unevenly sampled larger data sets 13.8 spread extirpolate value into array, used by fasper 13.9 dftcor compute endpoint corrections for Fourier integrals 13.9 dftint high-accuracy Fourier integrals 13.10 wt1 one-dimensional discrete wavelet transform 13.10 daub4 Daubechies 4-coefficient wavelet filter 13.10 pwtset initialize coefficients for pwt 13.10 pwt partial wavelet transform 13.10 wtn multidimensional discrete wavelet transform 14.1 moment calculate moments of a data set 14.2 ttest Student’s t-test for difference of means 14.2 avevar calculate mean and variance of a data set 14.2 tutest Student’s t-test for means, case of unequal variances 14.2 tptest Student’s t-test for means, case of paired data 14.2 ftest F -test for difference of variances 14.3 chsone chi-square test for difference between data and model 14.3 chstwo chi-square test for difference between two data sets 14.3 ksone Kolmogorov-Smirnov test of data against model 14.3 kstwo Kolmogorov-Smirnov test between two data sets 14.3 probks Kolmogorov-Smirnov probability function 14.4 cntab1 contingency table analysis using chi-square 14.4 cntab2 contingency table analysis using entropy measure 14.5 pearsn Pearson’s correlation between two data sets 14.6 spear Spearman’s rank correlation between two data sets 14.6 crank replaces array elements by their rank 14.6 kendl1 correlation between two data sets, Kendall’s tau 14.6 kendl2 contingency table analysis using Kendall’s tau 14.7 ks2d1s K–S test in two dimensions, data vs. model 14.7 quadct count points by quadrants, used by ks2d1s 14.7 quadvl quadrant probabilities, used by ks2d1s 14.7 ks2d2s K–S test in two dimensions, data vs. data 14.8 savgol Savitzky-Golay smoothing coefficients Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).
Computer Programs by Chapter and Section xxv 15.2 fit least-squares fit data to a straight line 15.3 fitexy fit data to a straight line, errors in both x and y 15.3 chixy used by fitexy to calculate a χ 2 15.4 lfit general linear least-squares fit by normal equations 15.4 covsrt rearrange covariance matrix, used by lfit 15.4 svdfit linear least-squares fit by singular value decomposition 15.4 svdvar variances from singular value decomposition 15.4 fpoly fit a polynomial using lfit or svdfit 15.4 fleg fit a Legendre polynomial using lfit or svdfit 15.5 mrqmin nonlinear least-squares fit, Marquardt’s method 15.5 mrqcof used by mrqmin to evaluate coefficients 15.5 fgauss fit a sum of Gaussians using mrqmin 15.7 medfit fit data to a straight line robustly, least absolute deviation 15.7 rofunc fit data robustly, used by medfit 16.1 rk4 integrate one step of ODEs, fourth-order Runge-Kutta 16.1 rkdumb integrate ODEs by fourth-order Runge-Kutta 16.2 rkqs integrate one step of ODEs with accuracy monitoring 16.2 rkck Cash-Karp-Runge-Kutta step used by rkqs 16.2 odeint integrate ODEs with accuracy monitoring 16.3 mmid integrate ODEs by modified midpoint method 16.4 bsstep integrate ODEs, Bulirsch-Stoer step 16.4 pzextr polynomial extrapolation, used by bsstep 16.4 rzextr rational function extrapolation, used by bsstep 16.5 stoerm integrate conservative second-order ODEs 16.6 stiff integrate stiff ODEs by fourth-order Rosenbrock 16.6 jacobn sample Jacobian routine for stiff 16.6 derivs sample derivatives routine for stiff 16.6 simpr integrate stiff ODEs by semi-implicit midpoint rule 16.6 stifbs integrate stiff ODEs, Bulirsch-Stoer step 17.1 shoot solve two point boundary value problem by shooting 17.2 shootf ditto, by shooting to a fitting point 17.3 solvde two point boundary value problem, solve by relaxation 17.3 bksub backsubstitution, used by solvde 17.3 pinvs diagonalize a sub-block, used by solvde 17.3 red reduce columns of a matrix, used by solvde 17.4 sfroid spheroidal functions by method of solvde 17.4 difeq spheroidal matrix coefficients, used by sfroid 17.4 sphoot spheroidal functions by method of shoot 17.4 sphfpt spheroidal functions by method of shootf 18.1 fred2 solve linear Fredholm equations of the second kind 18.1 fredin interpolate solutions obtained with fred2 18.2 voltra linear Volterra equations of the second kind 18.3 wwghts quadrature weights for an arbitrarily singular kernel 18.3 kermom sample routine for moments of a singular kernel Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).
Page 1 and 2: Numerical Recipes in C The Art of S
Page 3 and 4: Contents Preface to the Second Edit
Page 5 and 6: Contents vii 7.2 Transformation Met
Page 7 and 8: Contents ix 15.6 Confidence Limits
Page 9 and 10: Preface to the Second Edition Our a
Page 11 and 12: Preface to the Second Edition xiii
Page 13 and 14: Preface to the First Edition xv lin
Page 15 and 16: Types of License Offered License In
Page 17 and 18: Computer Programs by Chapter and Se
Page 19 and 20: Computer Programs by Chapter and Se
Page 21: Computer Programs by Chapter and Se
Page 25 and 26: Chapter 1. Preliminaries 1.0 Introd
Page 27 and 28: 1.0 Introduction 3 Tested Machines
Page 29 and 30: 1.1 Program Organization and Contro
Page 33 and 34: true 1.1 Program Organization and C
Page 39 and 40: } 1.2 Some C Conventions for Scient
Page 41 and 42: 1.2 Some C Conventions for Scientif
Page 45 and 46: (a) **m (b) **m 1.2 Some C Conventi
Page 53 and 54: = 0 3 = 0 = 0 = 0 = 0 = 0 1 ⁄ 2 1
Page 55 and 56: 1.3 Error, Accuracy, and Stability
Page 57 and 58: 2.0 Introduction 33 Accumulated rou
Page 59 and 60: 2.0 Introduction 35 between singula
Page 61 and 62: 2.1 Gauss-Jordan Elimination 37 Eli
Page 63 and 64: 2.1 Gauss-Jordan Elimination 39 pre
Page 65 and 66: 2.2 Gaussian Elimination with Backs
Page 67 and 68: 2.3 LU Decomposition and Its Applic
Page 69 and 70: a b c d 2.3 LU Decomposition and It
Page 71 and 72: } 2.3 LU Decomposition and Its Appl
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2.3 LU Decomposition and Its Applic
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2.4 Tridiagonal and Band Diagonal S
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2.4 Tridiagonal and Band Diagonal S
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δx 2.5 Iterative Improvement of a
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2.5 Iterative Improvement of a Solu
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2.6 Singular Value Decomposition 59
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SVD of a Square Matrix 2.6 Singular
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solutions of A ⋅ x = d null space
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} 2.6 Singular Value Decomposition
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2.7 Sparse Linear Systems 2.7 Spars
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2.7 Sparse Linear Systems 73 prescr
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2.7 Sparse Linear Systems 75 Here
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2.7 Sparse Linear Systems 77 Finall
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2.7 Sparse Linear Systems 79 In row
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} 2.7 Sparse Linear Systems 81 jp=i
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2.7 Sparse Linear Systems 83 Matrix
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2.7 Sparse Linear Systems 85 while
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p=dvector(1,n); pp=dvector(1,n); r=
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} 2.7 Sparse Linear Systems 89 void
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2.8 Vandermonde Matrices and Toepli
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2.9 Cholesky Decomposition 97 If yo
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2.10 QR Decomposition 99 The standa
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#include #include "nrutil.h" 2.10
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in terms of which 2.11 Is Matrix In
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Chapter 3. Interpolation and Extrap
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(a) (b) 3.0 Introduction 107 Figure
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3.1 Polynomial Interpolation and Ex
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3.2 Rational Function Interpolation
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3.3 Cubic Spline Interpolation 113
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3.3 Cubic Spline Interpolation 115
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3.4 How to Search an Ordered Table
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} 3.4 How to Search an Ordered Tabl
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#include "nrutil.h" 3.5 Coefficient
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3.6 Interpolation in Two or More Di
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} int j; float *ymtmp; 3.6 Interpol
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3.6 Interpolation in Two or More Di
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Chapter 4. Integration of Functions
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4.1 Classical Formulas for Equally
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#define FUNC(x) ((*func)(x)) 4.2 El
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4.2 Elementary Algorithms 139 trapz
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4.4 Improper Integrals 141 which co
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4.4 Improper Integrals 143 The rout
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4.4 Improper Integrals 145 If you n
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4.5 Gaussian Quadratures and Orthog
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Gauss-Legendre: Gauss-Chebyshev: Ga
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} 4.5 Gaussian Quadratures and Orth
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4.6 Multidimensional Integrals 161
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y 4.6 Multidimensional Integrals 16
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Chapter 5. Evaluation of Functions
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5.1 Series and Their Convergence 16
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5.2 Evaluation of Continued Fractio
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5.2 Evaluation of Continued Fractio
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5.3 Polynomials and Rational Functi
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5.3 Polynomials and Rational Functi
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5.4 Complex Arithmetic 177 Actually
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5.5 Recurrence Relations and Clensh
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5.5 Recurrence Relations and Clensh
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5.6 Quadratic and Cubic Equations 1
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5.6 Quadratic and Cubic Equations 1
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5.7 Numerical Derivatives 187 With
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} 5.7 Numerical Derivatives 189 The
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Chebyshev polynomials 1 .5 0 −.5
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} 5.8 Chebyshev Approximation 193 f
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5.9 Derivatives or Integrals of a C
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5.10 Polynomial Approximation from
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5.11 Economization of Power Series
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5.12 Padé Approximants 201 then R(
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5.12 Padé Approximants 203 cof[2*n
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5.13 Rational Chebyshev Approximati
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5.13 Rational Chebyshev Approximati
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5.14 Evaluation of Functions by Pat
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5.14 Evaluation of Functions by Pat
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6.1 Gamma, Beta, and Related Functi
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6.1 Gamma, Beta, and Related Functi
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incomplete gamma function P(a,x) 1.
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} void nrerror(char error_text[]);
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#include 6.2 Incomplete Gamma Func
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6.3 Exponential Integrals 223 where
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6.3 Exponential Integrals 225 #incl
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6.4 Incomplete Beta Function 227 Th
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6.4 Incomplete Beta Function 229 So
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Bessel functions 1 .5 0 −.5 −1
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} 6.5 Bessel Functions of Integer O
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6.5 Bessel Functions of Integer Ord
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modified Bessel functions 4 3 2 1 6
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6.6 Modified Bessel Functions of In
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6.7 Bessel Functions of Fractional
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} 6.7 Bessel Functions of Fractiona
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6.8 Spherical Harmonics 253 The fir
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6.9 Fresnel Integrals, Cosine and S
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} 6.9 Fresnel Integrals, Cosine and
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} 6.10 Dawson’s Integral 259 } if
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6.11 Elliptic Integrals and Jacobia
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where 6.11 Elliptic Integrals and J
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#define TINY 1.0e-25 #define BIG 4.
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6.11 Elliptic Integrals and Jacobia
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6.12 Hypergeometric Functions 271 C
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#include "complex.h" #define ONE Co
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7.1 Uniform Deviates 275 As for ref
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7.1 Uniform Deviates 277 This corre
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#define IA 16807 #define IM 2147483
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RAN 3 1 7.1 Uniform Deviates 281 iy
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7.1 Uniform Deviates 283 from the w
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7.1 Uniform Deviates 285 Constants
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7.2 Transformation Method: Exponent
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7.2 Transformation Method: Exponent
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first random deviate in 7.3 Rejecti
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} 7.3 Rejection Method: Gamma, Pois
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} 7.3 Rejection Method: Gamma, Pois
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7.4 Generation of Random Bits 297 s
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} } *iseed
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7.5 Random Sequences Based on Data
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} } 7.5 Random Sequences Based on D
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7.6 Simple Monte Carlo Integration
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7.6 Simple Monte Carlo Integration
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7.7 Quasi- (that is, Sub-) Random S
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} 7.7 Quasi- (that is, Sub-) Random
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7.8 Adaptive and Recursive Monte Ca
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} } 7.8 Adaptive and Recursive Mont
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} 7.8 Adaptive and Recursive Monte
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Chapter 8. Sorting 8.0 Introduction
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8.1 Straight Insertion and Shell’
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8.2 Quicksort 333 question of the b
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8.2 Quicksort 335 istack=lvector(1,
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a 4 a 2 8.3 Heapsort 337 a 5 a 1 a
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original array 1 2 3 4 5 6 14 8 32
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8.5 Selecting the Mth Largest 341 w
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8.5 Selecting the Mth Largest 343 r
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} } } } 8.6 Determination of Equiva
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Chapter 9. Root Finding and Nonline
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9.0 Introduction 349 good first gue
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(a) (b) (c) (d) 9.1 Bracketing and
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Bisection Method 9.1 Bracketing and
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9.2 Secant Method, False Position M
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} 9.2 Secant Method, False Position
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} 9.3 Van Wijngaarden-Dekker-Brent
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9.3 Van Wijngaarden-Dekker-Brent Me
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9.4 Newton-Raphson Method Using Der
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9.4 Newton-Raphson Method Using Der
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} 9.4 Newton-Raphson Method Using D
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9.5 Roots of Polynomials 9.5 Roots
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9.5 Roots of Polynomials 371 tentat
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9.5 Roots of Polynomials 373 The me
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Eigenvalue Methods 9.5 Roots of Pol
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9.5 Roots of Polynomials 377 of goi
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9.6 Newton-Raphson Method for Nonli
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9.6 Newton-Raphson Method for Nonli
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9.7 Globally Convergent Methods for
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X1 A B C 10.0 Introduction 395 D Fi
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10.1 Golden Section Search in One D
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10.1 Golden Section Search in One D
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} } 10.1 Golden Section Search in O
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10.2 Parabolic Interpolation and Br
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} 10.3 One-Dimensional Search with
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10.3 One-Dimensional Search with Fi
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(a) (b) (c) (d) 10.4 Downhill Simpl
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10.4 Downhill Simplex Method in Mul
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10.5 Direction Set (Powell’s) Met
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(a) 10.6 Conjugate Gradient Methods
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10.6 Conjugate Gradient Methods in
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} 10.7 Variable Metric Methods in M
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10.7 Variable Metric Methods in Mul
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} 10.7 Variable Metric Methods in M
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10.8 Linear Programming and the Sim
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} 10.8 Linear Programming and the S
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10.9 Simulated Annealing Methods 44
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(a) (b) (c) 1 .5 0 1 .5 0 1 .5 0 10
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} } 10.9 Simulated Annealing Method
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} free_ivector(jorder,1,ncity); #in
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Definitions and Basic Facts 11.0 In
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11.0 Introduction 459 In both the d
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11.0 Introduction 461 “Eigenpacka
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11.1 Jacobi Transformations of a Sy
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} } 11.2 Reduction of a Symmetric M
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11.2 Reduction of a Symmetric Matri
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11.2 Reduction of a Symmetric Matri
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11.3 Eigenvalues and Eigenvectors o
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} } 11.4 Hermitian Matrices 481 for
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11.5 Reduction of a General Matrix
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11.5 Reduction of a General Matrix
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11.6 The QR Algorithm for Real Hess
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11.7 Eigenvalues or Eigenvectors by
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11.7 Eigenvalues or Eigenvectors by
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and equation (12.0.1) looks like th
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⏐h(t)⏐ 2 (a) Ph( f ) (one-sided
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12.1 Fourier Transform of Discretel
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12.1 Fourier Transform of Discretel
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12.2 Fast Fourier Transform (FFT) 5
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12.2 Fast Fourier Transform (FFT) 5
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Other FFT Algorithms 12.2 Fast Four
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12.3 FFT of Real Functions, Sine an
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12.3 FFT of Real Functions, Sine an
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(a) (b) (c) +1 0 −1 +1 0 −1 +1
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#include 12.3 FFT of Real Function
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} 12.3 FFT of Real Functions, Sine
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} } } 12.4 FFT in Two or More Dimen
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data [1] Re Im 12.4 FFT in Two or M
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12.5 Fourier Transforms of Real Dat
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12.6 External Storage or Memory-Loc
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} } 12.6 External Storage or Memory
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Chapter 13. Fourier and Spectral Ap
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(t) r*s(t) 13.1 Convolution and Dec
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13.1 Convolution and Deconvolution
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13.1 Convolution and Deconvolution
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13.2 Correlation and Autocorrelatio
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13.3 Optimal (Wiener) Filtering wit
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log scale 13.4 Power Spectrum Estim
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13.4 Power Spectrum Estimation Usin
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amplitude 1 .8 .6 .4 .2 0 13.4 Powe
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13.5 Digital Filtering in the Time
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IIR (Recursive) Filters 13.5 Digita
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(a) (b) 13.5 Digital Filtering in t
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13.6 Linear Prediction and Linear P
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} 13.6 Linear Prediction and Linear
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13.7 Maximum Entropy (All Poles) Me
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13.8 Spectral Analysis of Unevenly
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} 13.8 Spectral Analysis of Unevenl
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} 13.8 Spectral Analysis of Unevenl
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13.9 Computing Fourier Integrals Us
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Cubic order: 13.9 Computing Fourier
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13.9 Computing Fourier Integrals Us
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13.10 Wavelet Transforms 591 The sp
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13.10 Wavelet Transforms 593 p.”
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13.10 Wavelet Transforms 595 comple
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} 13.10 Wavelet Transforms 597 wfil
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.2 0 −.2 .2 0 −.2 DAUB4 e10 + e
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wavelet amplitude 1.5 1 .5 .0001 10
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} nprev=nnew; } free_vector(wksp,1,
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13.10 Wavelet Transforms 605 Figure
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13.11 Numerical Use of the Sampling
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Chapter 14. Statistical Description
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14.1 Moments of a Distribution: Mea
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14.1 Moments of a Distribution: Mea
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14.2 Do Two Distributions Have the
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14.3 Are Two Distributions Differen
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1. male 2. female . 14.4 Contingenc
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14.4 Contingency Table Analysis of
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14.5 Linear Correlation 637 two dis
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14.6 Nonparametric or Rank Correlat
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} } 14.7 Do Two-Dimensional Distrib
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14.7 Do Two-Dimensional Distributio
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14.8 Savitzky-Golay Smoothing Filte
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8 6 4 2 0 8 6 4 2 0 8 6 4 2 0 befor
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14.8 Savitzky-Golay Smoothing Filte
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15.1 Least Squares as a Maximum Lik
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15.1 Least Squares as a Maximum Lik
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15.2 Fitting Data to a Straight Lin
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15.2 Fitting Data to a Straight Lin
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15.3 Straight-Line Data with Errors
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15.3 Straight-Line Data with Errors
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15.4 General Linear Least Squares 6
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} 15.4 General Linear Least Squares
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15.5 Nonlinear Models 681 Lawson, C
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15.5 Nonlinear Models 683 Note that
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15.5 Nonlinear Models 685 pivots, b
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#include "nrutil.h" 15.5 Nonlinear
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15.6 Confidence Limits on Estimated
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actual data set 15.6 Confidence Lim
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15.7 Robust Estimation 699 M� 1 C
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15.7 Robust Estimation 701 where th
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15.7 Robust Estimation 703 algorith
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} *a=aa; *b=bb; *abdev=abdevt/ndata
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Chapter 16. Integration of Ordinary
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16.0 Introduction 709 problem where
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y(x) 16.1 Runge-Kutta Method 711 1
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} 16.1 Runge-Kutta Method 713 yt=ve
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16.2 Adaptive Stepsize Control for
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16.3 Modified Midpoint Method 723 H
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16.4 Richardson Extrapolation and t
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16.5 Second-Order Conservative Equa
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y 16.6 Stiff Sets of Equations 735
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16.6 Stiff Sets of Equations 737 If
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16.6 Stiff Sets of Equations 739 od
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} 16.6 Stiff Sets of Equations 741
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16.6 Stiff Sets of Equations 743 Co
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16.6 Stiff Sets of Equations 745 Se
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} 16.7 Multistep, Multivalue, and P
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16.7 Multistep, Multivalue, and Pre
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16.7 Multistep, Multivalue, and Pre
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Chapter 17. Two Point Boundary Valu
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equired boundary value 17.0 Introdu
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17.1 The Shooting Method 17.1 The S
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} 17.1 The Shooting Method 759 floa
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17.2 Shooting to a Fitting Point 76
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17.3 Relaxation Methods 763 depende
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X X X X X X X X X X X X X X X X X X
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17.3 Relaxation Methods 767 Next co
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17.3 Relaxation Methods 769 calcula
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} 17.3 Relaxation Methods 771 for (
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17.4 A Worked Example: Spheroidal H
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Page 801 and 802:
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} 17.4 A Worked Example: Spheroidal
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17.5 Automated Allocation of Mesh P
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17.6 Handling Internal Boundary Con
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17.6 Handling Internal Boundary Con
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18.0 Introduction 789 is called a F
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18.1 Fredholm Equations of the Seco
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#include "nrutil.h" 18.1 Fredholm E
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18.2 Volterra Equations 795 in §18
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18.3 Integral Equations with Singul
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f (x) 1 .5 0 −.5 0 18.3 Integral
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18.4 Inverse Problems and the Use o
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18.4 Inverse Problems and the Use o
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18.5 Linear Regularization Methods
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Page 837 and 838:
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18.6 Backus-Gilbert Method 815 nece
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18.6 Backus-Gilbert Method 817 Subs
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18.7 Maximum Entropy Image Restorat
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Page 849 and 850:
Page 851 and 852:
Chapter 19. Partial Differential Eq
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19.0 Introduction 829 possible stab
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y L ∆ y1 19.0 Introduction 831 A
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19.0 Introduction 833 So-called rap
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19.1 Flux-Conservative Initial Valu
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Page 867 and 868:
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t or n 19.1 Flux-Conservative Initi
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19.2 Diffusive Initial Value Proble
Page 873 and 874:
19.2 Diffusive Initial Value Proble
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and the heuristic stability criteri
Page 877 and 878:
19.3 Initial Value Problems in Mult
Page 879 and 880:
19.3 Initial Value Problems in Mult
Page 881 and 882:
19.4 Fourier and Cyclic Reduction M
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Page 885 and 886:
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FACR Method 19.5 Relaxation Methods
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19.5 Relaxation Methods for Boundar
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19.6 Multigrid Methods for Boundary
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S S S E γ = 1 19.6 Multigrid Metho
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} 19.6 Multigrid Methods for Bounda
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#include 19.6 Multigrid Methods fo
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Chapter 20. Less-Numerical Algorith
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20.1 Diagnosing Machine Parameters
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20.1 Diagnosing Machine Parameters
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(a) G(i) (b) MSB 4 3 2 1 0 LSB i MS
Page 921 and 922:
20.3 Cyclic Redundancy and Other Ch
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Page 925 and 926:
Page 927 and 928:
} 20.4 Huffman Coding and Compressi
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20.4 Huffman Coding and Compression
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} 20.4 Huffman Coding and Compressi
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} } } 20.4 Huffman Coding and Compr
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1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
Page 937 and 938:
20.5 Arithmetic Coding 913 Individu
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20.6 Arithmetic at Arbitrary Precis
Page 941 and 942:
} for (j=n;j>=1;j--) { ireg=u[j]+HI
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#include "nrutil.h" #define RX 256.
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} 20.6 Arithmetic at Arbitrary Prec
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Page 949:
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