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An equivalence transformationbn → λbn ,an+1 → λan+1(5.2.11)leaves the value of a continued fraction unchanged. By a suitable choice of the scalefactor λ you can often simplify the form of the a’s and the b’s. Of course, youcan carry out successive equivalence transformations, possibly with different λ’s, onsuccessive terms of the continued fraction.The even and odd parts of a continued fraction are continued fractions whosesuccessive convergents are f2n and f2n+1 , respectively.
Their main use is that theyconverge twice as fast as the original continued fraction, and so if their terms are notmuch more complicated than the terms in the original there can be a big savings incomputation. The formula for the even part of (5.2.2) isfeven = d0 +c1c2···d1 + d2 +(5.2.12)where in terms of intermediate variablesα1 =a1b1αn =an,bn bn−1(5.2.13)n≥2we haved 0 = b0 ,cn = −α2n−1 α2n−2,c1 = α 1 ,d1 = 1 + α2dn = 1 + α2n−1 + α2n ,n≥2(5.2.14)You can find the similar formula for the odd part in the review by Blanch [1].
Oftena combination of the transformations (5.2.14) and (5.2.11) is used to get the bestform for numerical work.We will make frequent use of continued fractions in the next chapter.CITED REFERENCES AND FURTHER READING:Abramowitz, M., and Stegun, I.A. 1964, Handbook of Mathematical Functions, Applied Mathematics Series, Volume 55 (Washington: National Bureau of Standards; reprinted 1968 byDover Publications, New York), §3.10.Blanch, G.
1964, SIAM Review, vol. 6, pp. 383–421. [1]Acton, F.S. 1970, Numerical Methods That Work; 1990, corrected edition (Washington: Mathematical Association of America), Chapter 11. [2]Cuyt, A., and Wuytack, L. 1987, Nonlinear Methods in Numerical Analysis (Amsterdam: NorthHolland), Chapter 1.Fike, C.T. 1968, Computer Evaluation of Mathematical Functions (Englewood Cliffs, NJ: PrenticeHall), §§8.2, 10.4, and 10.5.
[3]Wallis, J. 1695, in Opera Mathematica, vol. 1, p. 355, Oxoniae e Theatro Shedoniano. Reprintedby Georg Olms Verlag, Hildeshein, New York (1972). [4]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).an → λan ,5.3 Polynomials and Rational Functions173Thompson, I.J., and Barnett, A.R. 1986, Journal of Computational Physics, vol. 64, pp. 490–509.[5]Lentz, W.J.
1976, Applied Optics, vol. 15, pp. 668–671. [6]Jones, W.B. 1973, in Padé Approximants and Their Applications, P.R. Graves-Morris, ed. (London: Academic Press), p. 125. [7]A polynomial of degree N is represented numerically as a stored array ofcoefficients, c[j] with j= 0, . . . , N . We will always take c[0] to be the constantterm in the polynomial, c[N ] the coefficient of xN ; but of course other conventionsare possible. There are two kinds of manipulations that you can do with a polynomial:numerical manipulations (such as evaluation), where you are given the numericalvalue of its argument, or algebraic manipulations, where you want to transformthe coefficient array in some way without choosing any particular argument.
Let’sstart with the numerical.We assume that you know enough never to evaluate a polynomial this way:p=c[0]+c[1]*x+c[2]*x*x+c[3]*x*x*x+c[4]*x*x*x*x;or (even worse!),p=c[0]+c[1]*x+c[2]*pow(x,2.0)+c[3]*pow(x,3.0)+c[4]*pow(x,4.0);Come the (computer) revolution, all persons found guilty of such criminalbehavior will be summarily executed, and their programs won’t be! It is a matterof taste, however, whether to writep=c[0]+x*(c[1]+x*(c[2]+x*(c[3]+x*c[4])));orp=(((c[4]*x+c[3])*x+c[2])*x+c[1])*x+c[0];If the number of coefficients c[0..n] is large, one writesp=c[n];for(j=n-1;j>=0;j--) p=p*x+c[j];orp=c[j=n];while (j>0) p=p*x+c[--j];Another useful trick is for evaluating a polynomial P (x) and its derivativedP (x)/dx simultaneously:p=c[n];dp=0.0;for(j=n-1;j>=0;j--) {dp=dp*x+p; p=p*x+c[j];}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).5.3 Polynomials and Rational Functions.
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