Press, Teukolsly, Vetterling, Flannery - Numerical Recipes in C, страница 5

modelcount points by quadrants, used by ks2d1squadrant probabilities, used by ks2d1sK–S test in two dimensions, data vs. dataSavitzky-Golay smoothing coefficientsComputer Programs by Chapter and Sectionxxv15.215.315.315.415.415.415.415.415.415.515.515.515.715.7fitfitexychixylfitcovsrtsvdfitsvdvarfpolyflegmrqminmrqcoffgaussmedfitrofuncleast-squares fit data to a straight linefit data to a straight line, errors in both x and yused by fitexy to calculate a χ2general linear least-squares fit by normal equationsrearrange covariance matrix, used by lfitlinear least-squares fit by singular value decompositionvariances from singular value decompositionfit a polynomial using lfit or svdfitfit a Legendre polynomial using lfit or svdfitnonlinear least-squares fit, Marquardt’s methodused by mrqmin to evaluate coefficientsfit a sum of Gaussians using mrqminfit data to a straight line robustly, least absolute deviationfit data robustly, used by medfit16.116.116.216.216.216.316.416.416.416.516.616.616.616.616.6rk4rkdumbrkqsrkckodeintmmidbssteppzextrrzextrstoermstiffjacobnderivssimprstifbsintegrate one step of ODEs, fourth-order Runge-Kuttaintegrate ODEs by fourth-order Runge-Kuttaintegrate one step of ODEs with accuracy monitoringCash-Karp-Runge-Kutta step used by rkqsintegrate ODEs with accuracy monitoringintegrate ODEs by modified midpoint methodintegrate ODEs, Bulirsch-Stoer steppolynomial extrapolation, used by bssteprational function extrapolation, used by bsstepintegrate conservative second-order ODEsintegrate stiff ODEs by fourth-order Rosenbrocksample Jacobian routine for stiffsample derivatives routine for stiffintegrate stiff ODEs by semi-implicit midpoint ruleintegrate stiff ODEs, Bulirsch-Stoer step17.117.217.317.317.317.317.417.417.417.4shootshootfsolvdebksubpinvsredsfroiddifeqsphootsphfptsolve two point boundary value problem by shootingditto, by shooting to a fitting pointtwo point boundary value problem, solve by relaxationbacksubstitution, used by solvdediagonalize a sub-block, used by solvdereduce columns of a matrix, used by solvdespheroidal functions by method of solvdespheroidal matrix coefficients, used by sfroidspheroidal functions by method of shootspheroidal functions by method of shootf18.118.118.218.318.3fred2fredinvoltrawwghtskermomsolve linear Fredholm equations of the second kindinterpolate solutions obtained with fred2linear Volterra equations of the second kindquadrature weights for an arbitrarily singular kernelsample routine for moments of a singular kernelxxviComputer Programs by Chapter and Section18.318.3quadmxfredexsample routine for a quadrature matrixexample of solving a singular Fredholm equation19.519.619.619.619.619.619.619.619.619.619.619.619.619.619.619.619.6sormglinrstrctinterpaddintslvsmlrelaxresidcopyfill0mgfasrelax2slvsm2lopmataddmatsubanorm2elliptic PDE solved by successive overrelaxation methodlinear elliptic PDE solved by multigrid methodhalf-weighting restriction, used by mglin, mgfasbilinear prolongation, used by mglin, mgfasinterpolate and add, used by mglinsolve on coarsest grid, used by mglinGauss-Seidel relaxation, used by mglincalculate residual, used by mglinutility used by mglin, mgfasutility used by mglinnonlinear elliptic PDE solved by multigrid methodGauss-Seidel relaxation, used by mgfassolve on coarsest grid, used by mgfasapplies nonlinear operator, used by mgfasutility used by mgfasutility used by mgfasutility used by mgfas20.120.220.320.320.320.420.420.420.420.520.520.520.620.620.620.620.620.620.6macharigrayicrc1icrcdecchkhufmakhufapphufenchufdecarcmakarcodearcsummpopsmpmulmpinvmpdivmpsqrtmp2dfrmppidiagnose computer’s floating arithmeticGray code and its inversecyclic redundancy checksum, used by icrccyclic redundancy checksumdecimal check digit calculation or verificationconstruct a Huffman codeappend bits to a Huffman code, used by hufmakuse Huffman code to encode and compress a characteruse Huffman code to decode and decompress a characterconstruct an arithmetic codeencode or decode a character using arithmetic codingadd integer to byte string, used by arcodemultiple precision arithmetic, simpler operationsmultiple precision multiply, using FFT methodsmultiple precision reciprocalmultiple precision divide and remaindermultiple precision square rootmultiple precision conversion to decimal basemultiple precision example, compute many digits of πChapter 1.Preliminaries1.0 IntroductionThis book, like its predecessor edition, is supposed to teach you methods ofnumerical computing that are practical, efficient, and (insofar as possible) elegant.We presume throughout this book that you, the reader, have particular tasks that youwant to get done.

We view our job as educating you on how to proceed. Occasionallywe may try to reroute you briefly onto a particularly beautiful side road; but by andlarge, we will guide you along main highways that lead to practical destinations.Throughout this book, you will find us fearlessly editorializing, telling youwhat you should and shouldn’t do. This prescriptive tone results from a consciousdecision on our part, and we hope that you will not find it irritating.

We do notclaim that our advice is infallible! Rather, we are reacting against a tendency, inthe textbook literature of computation, to discuss every possible method that hasever been invented, without ever offering a practical judgment on relative merit. Wedo, therefore, offer you our practical judgments whenever we can. As you gainexperience, you will form your own opinion of how reliable our advice is.We presume that you are able to read computer programs in C, that beingthe language of this version of Numerical Recipes (Second Edition). The bookNumerical Recipes in FORTRAN (Second Edition) is separately available, if youprefer to program in that language. Earlier editions of Numerical Recipes in Pascaland Numerical Recipes Routines and Examples in BASIC are also available; whilenot containing the additional material of the Second Edition versions in C andFORTRAN, these versions are perfectly serviceable if Pascal or BASIC is yourlanguage of choice.When we include programs in the text, they look like this:#include <math.h>#define RAD (3.14159265/180.0)void flmoon(int n, int nph, long *jd, float *frac)Our programs begin with an introductory comment summarizing their purpose and explainingtheir calling sequence.

This routine calculates the phases of the moon. Given an integer n anda code nph for the phase desired (nph = 0 for new moon, 1 for first quarter, 2 for full, 3 for lastquarter), the routine returns the Julian Day Number jd, and the fractional part of a day fracto be added to it, of the nth such phase since January, 1900. Greenwich Mean Time is assumed.{void nrerror(char error_text[]);int i;float am,as,c,t,t2,xtra;This is how we comment an individualline.c=n+nph/4.0;12Chapter 1.Preliminariest=c/1236.85;t2=t*t;as=359.2242+29.105356*c;You aren’t really intended to understandam=306.0253+385.816918*c+0.010730*t2;this algorithm, but it does work!*jd=2415020+28L*n+7L*nph;xtra=0.75933+1.53058868*c+((1.178e-4)-(1.55e-7)*t)*t2;if (nph == 0 || nph == 2)xtra += (0.1734-3.93e-4*t)*sin(RAD*as)-0.4068*sin(RAD*am);else if (nph == 1 || nph == 3)xtra += (0.1721-4.0e-4*t)*sin(RAD*as)-0.6280*sin(RAD*am);else nrerror("nph is unknown in flmoon");This is how we will indicate errori=(int)(xtra >= 0.0 ? floor(xtra) : ceil(xtra-1.0));conditions.*jd += i;*frac=xtra-i;}If the syntax of the function definition above looks strange to you, then you areprobably used to the older Kernighan and Ritchie (“K&R”) syntax, rather than that ofthe newer ANSI C.

In this edition, we adopt ANSI C as our standard. You might wantto look ahead to §1.2 where ANSI C function prototypes are discussed in more detail.Note our convention of handling all errors and exceptional cases with a statementlike nrerror("some error message");. The function nrerror() is part of asmall file of utility programs, nrutil.c, listed in Appendix B at the back of thebook. This Appendix includes a number of other utilities that we will describe later inthis chapter. Function nrerror() prints the indicated error message to your stderrdevice (usually your terminal screen), and then invokes the function exit(), whichterminates execution.

The function exit() is in every C library we know of; but ifyou find it missing, you can modify nrerror() so that it does anything else that willhalt execution. For example, you can have it pause for input from the keyboard, andthen manually interrupt execution. In some applications, you will want to modifynrerror() to do more sophisticated error handling, for example to transfer controlsomewhere else, with an error flag or error code set.We will have more to say about the C programming language, its conventionsand style, in §1.1 and §1.2.Computational Environment and Program ValidationOur goal is that the programs in this book be as portable as possible, acrossdifferent platforms (models of computer), across different operating systems, andacross different C compilers.