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5.9 Derivatives or Integrals of a Chebyshev-approximated Function1955.9 Derivatives or Integrals of aChebyshev-approximated FunctionCi =ci−1 − ci+12(i − 1)c0i−1 = c0i+1 + 2(i − 1)ci(i > 1)(5.9.1)(i = m − 1, m − 2, . . . , 2)(5.9.2)Equation (5.9.1) is augmented by an arbitrary choice of C0 , corresponding to anarbitrary constant of integration. Equation (5.9.2), which is a recurrence, is startedwith the values c0m = c0m−1 = 0, corresponding to no information about the m + 1stChebyshev coefficient of the original function f.Here are routines for implementing equations (5.9.1) and (5.9.2).void chder(float a, float b, float c[], float cder[], int n)Given a,b,c[0..n-1], as output from routine chebft §5.8, and given n, the desired degreeof approximation (length of c to be used), this routine returns the array cder[0..n-1], theChebyshev coefficients of the derivative of the function whose coefficients are c.{int j;float con;cder[n-1]=0.0;cder[n-2]=2*(n-1)*c[n-1];for (j=n-3;j>=0;j--)cder[j]=cder[j+2]+2*(j+1)*c[j+1];con=2.0/(b-a);for (j=0;j<n;j++)cder[j] *= con;n-1 and n-2 are special cases.Equation (5.9.2).Normalize to the interval b-a.}void chint(float a, float b, float c[], float cint[], int n)Given a,b,c[0..n-1], as output from routine chebft §5.8, and given n, the desired degreeof approximation (length of c to be used), this routine returns the array cint[0..n-1], theChebyshev coefficients of the integral of the function whose coefficients are c.
The constant ofintegration is set so that the integral vanishes at a.{int j;float sum=0.0,fac=1.0,con;con=0.25*(b-a);for (j=1;j<=n-2;j++) {cint[j]=con*(c[j-1]-c[j+1])/j;Factor that normalizes to the interval b-a.Equation (5.9.1).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 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).If you have obtained the Chebyshev coefficients that approximate a function ina certain range (e.g., from chebft in §5.8), then it is a simple matter to transformthem to Chebyshev coefficients corresponding to the derivative or integral of thefunction.
Having done this, you can evaluate the derivative or integral just as if itwere a function that you had Chebyshev-fitted ab initio.The relevant formulas are these: If ci , i = 0, . . . , m − 1 are the coefficientsthat approximate a function f in equation (5.8.9), Ci are the coefficients thatapproximate the indefinite integral of f, and c0i are the coefficients that approximatethe derivative of f, then196Chapter 5.Evaluation of Functionssum += fac*cint[j];fac = -fac;}cint[n-1]=con*c[n-2]/(n-1);sum += fac*cint[n-1];cint[0]=2.0*sum;Accumulates the constant of integration.Will equal ±1.Special case of (5.9.1) for n-1.Set the constant of integration.}Since a smooth function’s Chebyshev coefficients ci decrease rapidly, generally exponentially, equation (5.9.1) is often quite efficient as the basis for a quadrature scheme.
TheroutinesR x chebft and chint, used in that order, can be followed by repeated calls to chebevif a f (x)dx is required for many different values of x in the range a ≤ x ≤ b.RbIf only the single definite integral a f (x)dx is required, then chint and chebev arereplaced by the simpler formula, derived from equation (5.9.1),Z b1111f (x)dx = (b − a)c1 − c3 −c5 − · · · −c2k+1 − · · ·2315(2k + 1)(2k − 1)a(5.9.3)where the ci ’s are as returned by chebft. The series can be truncated when c2k+1 becomesnegligible, and the first neglected term gives an error estimate.This scheme is known as Clenshaw-Curtis quadrature [1].
It is often combined with anadaptive choice of N , the number of Chebyshev coefficients calculated via equation (5.8.7),which is also the number of function evaluations of f (x). If a modest choice of N doesnot give a sufficiently small c2k+1 in equation (5.9.3), then a larger value is tried. In thisadaptive case, it is even better to replace equation (5.8.7) by the so-called “trapezoidal” orGauss-Lobatto (§4.5) variant,cj = Nπkπ(j − 1)k2 X00f coscosNNNj = 1, . . .
, N(5.9.4)k=0where (N.B.!) the two primes signify that the first and last terms in the sum are to bemultiplied by 1/2. If N is doubled in equation (5.9.4), then half of the new functionevaluation points are identical to the old ones, allowing the previous function evaluations to bereused. This feature, plus the analytic weights and abscissas (cosine functions in 5.9.4), giveClenshaw-Curtis quadrature an edge over high-order adaptive Gaussian quadrature (cf. §4.5),which the method otherwise resembles.If your problem forces you to large values of N , you should be aware that equation (5.9.4)can be evaluated rapidly, and simultaneously for all the values of j, by a fast cosine transform.(See §12.3, especially equation 12.3.17.) (We already remarked that the nontrapezoidal form(5.8.7) can also be done by fast cosine methods, cf.
equation 12.3.22.)CITED REFERENCES AND FURTHER READING:Goodwin, E.T. (ed.) 1961, Modern Computing Methods, 2nd ed. (New York: Philosophical Library), pp. 78–79.Clenshaw, C.W., and Curtis, A.R. 1960, Numerische Mathematik, vol. 2, pp. 197–205. [1]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 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).Clenshaw-Curtis Quadrature5.10 Polynomial Approximation from Chebyshev Coefficients1975.10 Polynomial Approximation fromChebyshev Coefficientsf(x) ≈m−1Xgk xk(5.10.1)k=0Yes, you can do this (and we will give you the algorithm to do it), but wecaution you against it: Evaluating equation (5.10.1), where the coefficient g’s reflectan underlying Chebyshev approximation, usually requires more significant figuresthan evaluation of the Chebyshev sum directly (as by chebev).
This is becausethe Chebyshev polynomials themselves exhibit a rather delicate cancellation: Theleading coefficient of Tn (x), for example, is 2n−1 ; other coefficients of Tn (x) areeven bigger; yet they all manage to combine into a polynomial that lies between ±1.Only when m is no larger than 7 or 8 should you contemplate writing a Chebyshevfit as a direct polynomial, and even in those cases you should be willing to toleratetwo or so significant figures less accuracy than the roundoff limit of your machine.You get the g’s in equation (5.10.1) from the c’s output from chebft (suitablytruncated at a modest value of m) by calling in sequence the following two procedures:#include "nrutil.h"void chebpc(float c[], float d[], int n)Chebyshev polynomial coefficients. Given a coefficient array c[0..n-1], this routine generatesPPn-1ka coefficient array d[0..n-1] such that n-1k=0 dk y =k=0 ck Tk (y) − c0 /2.
The methodis Clenshaw’s recurrence (5.8.11), but now applied algebraically rather than arithmetically.{int k,j;float sv,*dd;dd=vector(0,n-1);for (j=0;j<n;j++) d[j]=dd[j]=0.0;d[0]=c[n-1];for (j=n-2;j>=1;j--) {for (k=n-j;k>=1;k--) {sv=d[k];d[k]=2.0*d[k-1]-dd[k];dd[k]=sv;}sv=d[0];d[0] = -dd[0]+c[j];dd[0]=sv;}for (j=n-1;j>=1;j--)d[j]=d[j-1]-dd[j];d[0] = -dd[0]+0.5*c[0];free_vector(dd,0,n-1);}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 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).You may well ask after reading the preceding two sections, “Must I store andevaluate my Chebyshev approximation as an array of Chebyshev coefficients for atransformed variable y? Can’t I convert the ck ’s into actual polynomial coefficientsin the original variable x and have an approximation of the following form?”.
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