**UFBA**

# Numerical Recipes in Fortran 77

(Parte **5** de 10)

6.1 beta beta function 6.2 gammp incomplete gamma function

6.2 gammq complement of incomplete gamma function 6.2 gser series used by gammp and gammq 6.2 gcf continued fraction used by gammp and gammq 6.2 erf error function 6.2 erfc complementary error function 6.2 erfcc complementary error function, concise routine

6.3 expint exponential integral En 6.3 ei exponential integral Ei

6.4 betai incomplete beta function 6.4 betacf continued fraction used by betai

6.5 bessj0 Bessel function J0 6.5 bessy0 Bessel function Y0 6.5 bessj1 Bessel function J1

6.5 bessy1 Bessel function Y1 6.5 bessy Bessel function Y of general integer order

6.5 bessj Bessel function J of general integer order

6.6 bessi0 modified Bessel function I0 6.6 bessk0 modified Bessel function K0 6.6 bessi1 modified Bessel function I1

6.6 bessk1 modified Bessel function K1 6.6 bessk modified Bessel function K of integer order

6.6 bessi modified Bessel function I of integer order 6.7 bessjy Bessel functions of fractional order 6.7 beschb Chebyshev expansion used by bessjy 6.7 bessik modified Bessel functions of fractional order 6.7 airy Airy functions

6.7 sphbes spherical Bessel functions jn and yn 6.8 plgndr Legendre polynomials, associated (spherical harmonics)

6.9 frenel Fresnel integrals S(x) and C(x)

6.9 cisi cosine and sine integrals Ci and Si 6.10 dawson Dawson’s integral 6.1 rf Carlson’s elliptic integral of the first kind 6.1 rd Carlson’s elliptic integral of the second kind 6.1 rj Carlson’s elliptic integral of the third kind 6.1 rc Carlson’s degenerate elliptic integral 6.1 ellf Legendre elliptic integral of the first kind 6.1 elle Legendre elliptic integral of the second kind 6.1 ellpi Legendre elliptic integral of the third kind 6.1 sncndn Jacobian elliptic functions 6.12 hypgeo complex hypergeometric function 6.12 hypser complex hypergeometric function, series evaluation 6.12 hypdrv complex hypergeometric function, derivative of

7.1 ran0 random deviate by Park and Miller minimal standard 7.1 ran1 random deviate, minimal standard plus shuffle

Computer Programs by Chapter and Section xxvii

Sample page from NUMERICAL RECIPES IN FORTRAN 7: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X)Copyright (C) 1986-1992 by Cambridge University Press. Programs Copyright (C) 1986-1992 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 machine-readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books, diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to trade@cup.cam.ac.uk (outside North America).

7.1 ran2 random deviate by L’Ecuyer long period plus shuffle 7.1 ran3 random deviate by Knuth subtractive method 7.2 expdev exponential random deviates 7.2 gasdev normally distributed random deviates 7.3 gamdev gamma-law distribution random deviates 7.3 poidev Poisson distributed random deviates 7.3 bnldev binomial distributed random deviates 7.4 irbit1 random bit sequence 7.4 irbit2 random bit sequence 7.5 psdes “pseudo-DES” hashing of 64 bits 7.5 ran4 random deviates from DES-like hashing 7.7 sobseq Sobol’s quasi-random sequence 7.8 vegas adaptive multidimensional Monte Carlo integration 7.8 rebin sample rebinning used by vegas 7.8 miser recursive multidimensional Monte Carlo integration 7.8 ranpt get random point, used by miser

8.1 piksrt sort an array by straight insertion 8.1 piksr2 sort two arrays by straight insertion 8.1 shell sort an array by Shell’s method 8.2 sort sort an array by quicksort method 8.2 sort2 sort two arrays by quicksort method 8.3 hpsort sort an array by heapsort method 8.4 indexx construct an index for an array 8.4 sort3 sort, use an index to sort 3 or more arrays 8.4 rank construct a rank table for an array 8.5 select find the Nth largest in an array 8.5 selip find the Nth largest, without altering an array 8.5 hpsel find M largest values, without altering an array 8.6 eclass determine equivalence classes from list 8.6 eclazz determine equivalence classes from procedure

9.0 scrsho graph a function to search for roots 9.1 zbrac outward search for brackets on roots 9.1 zbrak inward search for brackets on roots 9.1 rtbis find root of a function by bisection 9.2 rtflsp find root of a function by false-position 9.2 rtsec find root of a function by secant method 9.2 zriddr find root of a function by Ridders’ method 9.3 zbrent find root of a function by Brent’s method 9.4 rtnewt find root of a function by Newton-Raphson 9.4 rtsafe find root of a function by Newton-Raphson and bisection 9.5 laguer find a root of a polynomial by Laguerre’s method 9.5 zroots roots of a polynomial by Laguerre’s method with deflation 9.5 zrhqr roots of a polynomial by eigenvalue methods 9.5 qroot complex or double root of a polynomial, Bairstow xxviii Computer Programs by Chapter and Section

Sample page from NUMERICAL RECIPES IN FORTRAN 7: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X)Copyright (C) 1986-1992 by Cambridge University Press. Programs Copyright (C) 1986-1992 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 machine-readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books, diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to trade@cup.cam.ac.uk (outside North America).

9.6 mnewt Newton’s method for systems of equations 9.7 lnsrch search along a line, used by newt 9.7 newt globally convergent multi-dimensionalNewton’s method 9.7 fdjac finite-difference Jacobian, used by newt 9.7 fmin norm of a vector function, used by newt 9.7 broydn secant method for systems of equations

10.1 mnbrak bracket the minimum of a function 10.1 golden find minimum of a function by golden section search 10.2 brent find minimum of a function by Brent’s method 10.3 dbrent find minimum of a function using derivative information 10.4 amoeba minimize in N-dimensions by downhill simplex method 10.4 amotry evaluate a trial point, used by amoeba 10.5 powell minimize in N-dimensions by Powell’s method 10.5 linmin minimum of a function along a ray in N-dimensions 10.5 f1dim function used by linmin 10.6 frprmn minimize in N-dimensions by conjugate gradient 10.6 df1dim alternative function used by linmin 10.7 dfpmin minimize in N-dimensions by variable metric method 10.8 simplx linear programming maximization of a linear function 10.8 simp1 linear programming, used by simplx 10.8 simp2 linear programming, used by simplx 10.8 simp3 linear programming, used by simplx 10.9 anneal traveling salesman problem by simulated annealing 10.9 revcst cost of a reversal, used by anneal 10.9 revers do a reversal, used by anneal 10.9 trncst cost of a transposition, used by anneal 10.9 trnspt do a transposition, used by anneal 10.9 metrop Metropolis algorithm, used by anneal 10.9 amebsa simulated annealing in continuous spaces 10.9 amotsa evaluate a trial point, used by amebsa

1.1 jacobi eigenvalues and eigenvectors of a symmetric matrix 1.1 eigsrt eigenvectors, sorts into order by eigenvalue 1.2 tred2 Householder reduction of a real, symmetric matrix 1.3 tqli eigensolution of a symmetric tridiagonal matrix 1.5 balanc balance a nonsymmetric matrix 1.5 elmhes reduce a general matrix to Hessenberg form 1.6 hqr eigenvalues of a Hessenberg matrix

12.2 four1 fast Fourier transform (FFT) in one dimension 12.3 twofft fast Fourier transform of two real functions 12.3 realft fast Fourier transform of a single real function

12.3 sinft fast sine transform 12.3 cosft1 fast cosine transform with endpoints

12.3 cosft2 “staggered” fast cosine transform 12.4 fourn fast Fourier transform in multidimensions

Computer Programs by Chapter and Section xxix

Sample page from NUMERICAL RECIPES IN FORTRAN 7: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43064-X)Copyright (C) 1986-1992 by Cambridge University Press. Programs Copyright (C) 1986-1992 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 machine-readable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books, diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to trade@cup.cam.ac.uk (outside North America).

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

15.2 fit least-squares fit data to a straight line x Computer Programs by Chapter and Section

15.3 fitexy fit data to a straight line, errors in both x and y 15.3 chixy used by fitexy to calculate a 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 18.3 quadmx sample routine for a quadrature matrix

Computer Programs by Chapter and Section xxxi

(Parte **5** de 10)