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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).will cancel out the leading order error term. But there is no error term of order1/N 3 , by (4.2.1). The surviving error is of order 1/N 4 , the same as Simpson’s rule.In fact, it should not take long for you to see that (4.2.4) is exactly Simpson’s rule(4.1.13), alternating 2/3’s, 4/3’s, and all. This is the preferred method for evaluatingthat rule, and we can write it as a routine exactly analogous to qtrap above:140Chapter 4.Integration of Functions4.3 Romberg Integration#include <math.h>#define EPS 1.0e-6#define JMAX 20#define JMAXP (JMAX+1)#define K 5Here EPS is the fractional accuracy desired, as determined by the extrapolation error estimate;JMAX limits the total number of steps; K is the number of points used in the extrapolation.float qromb(float (*func)(float), float a, float b)Returns the integral of the function func from a to b.

Integration is performed by Romberg’smethod of order 2K, where, e.g., K=2 is Simpson’s rule.{void polint(float xa[], float ya[], int n, float x, float *y, float *dy);float trapzd(float (*func)(float), float a, float b, int n);void nrerror(char error_text[]);float ss,dss;float s[JMAXP],h[JMAXP+1];These store the successive trapezoidal approxiint j;mations and their relative stepsizes.h[1]=1.0;for (j=1;j<=JMAX;j++) {s[j]=trapzd(func,a,b,j);if (j >= K) {polint(&h[j-K],&s[j-K],K,0.0,&ss,&dss);if (fabs(dss) <= EPS*fabs(ss)) return ss;}h[j+1]=0.25*h[j];This is a key step: The factor is 0.25 even though the stepsize is decreased by only0.5. This makes the extrapolation a polynomial in h2 as allowed by equation (4.2.1),not just a polynomial in h.}nrerror("Too many steps in routine qromb");return 0.0;Never get here.}The routine qromb, along with its required trapzd and polint, is quitepowerful for sufficiently smooth (e.g., analytic) integrands, integrated over intervalsSample 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 servercomputer, 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).We can view Romberg’s method as the natural generalization of the routineqsimp in the last section to integration schemes that are of higher order thanSimpson’s rule. The basic idea is to use the results from k successive refinementsof the extended trapezoidal rule (implemented in trapzd) to remove all terms inthe error series up to but not including O(1/N 2k ).

The routine qsimp is the caseof k = 2. This is one example of a very general idea that goes by the name ofRichardson’s deferred approach to the limit: Perform some numerical algorithm forvarious values of a parameter h, and then extrapolate the result to the continuumlimit h = 0.Equation (4.2.4), which subtracts off the leading error term, is a special case ofpolynomial extrapolation. In the more general Romberg case, we can use Neville’salgorithm (see §3.1) to extrapolate the successive refinements to zero stepsize.Neville’s algorithm can in fact be coded very concisely within a Romberg integrationroutine.

For clarity of the program, however, it seems better to do the extrapolationby function call to polint, already given in §3.1..

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