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We do this in such a way as to take the most “straightline” route through the tableau to its apex, updating ns accordingly to keep track ofwhere we are. This route keeps the partial approximations centered (insofar as possible)on the target x. The last dy added is thus the error indication.}free_vector(d,1,n);free_vector(c,1,n);3.2 Rational Function Interpolation and Extrapolation1113.2 Rational Function Interpolation andExtrapolationRi(i+1)...(i+m) =p 0 + p1 x + · · · + pµ x µPµ (x)=Qν (x)q 0 + q 1 x + · · · + q ν xν(3.2.1)Since there are µ + ν + 1 unknown p’s and q’s (q0 being arbitrary), we must havem+1 = µ+ν +1(3.2.2)In specifying a rational function interpolating function, you must give the desiredorder of both the numerator and the denominator.Rational functions are sometimes superior to polynomials, roughly speaking,because of their ability to model functions with poles, that is, zeros of the denominatorof equation (3.2.1). These poles might occur for real values of x, if the functionto be interpolated itself has poles.
More often, the function f(x) is finite for allfinite real x, but has an analytic continuation with poles in the complex x-plane.Such poles can themselves ruin a polynomial approximation, even one restricted toreal values of x, just as they can ruin the convergence of an infinite power seriesin x. If you draw a circle in the complex plane around your m tabulated points,then you should not expect polynomial interpolation to be good unless the nearestpole is rather far outside the circle. A rational function approximation, by contrast,will stay “good” as long as it has enough powers of x in its denominator to accountfor (cancel) any nearby poles.For the interpolation problem, a rational function is constructed so as to gothrough a chosen set of tabulated functional values.
However, we should alsomention in passing that rational function approximations can be used in analyticwork. One sometimes constructs a rational function approximation by the criterionthat the rational function of equation (3.2.1) itself have a power series expansionthat agrees with the first m + 1 terms of the power series expansion of the desiredfunction f(x). This is called P adé approximation, and is discussed in §5.12.Bulirsch and Stoer found an algorithm of the Neville type which performsrational function extrapolation on tabulated data.
A tableau like that of equation(3.1.2) is constructed column by column, leading to a result and an error estimate.The Bulirsch-Stoer algorithm produces the so-called diagonal rational function, withthe degrees of numerator and denominator equal (if m is even) or with the degreeof the denominator larger by one (if m is odd, cf. equation 3.2.2 above). For thederivation of the algorithm, refer to [1]. The algorithm is summarized by a recurrenceSample 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).Some functions are not well approximated by polynomials, but are wellapproximated by rational functions, that is quotients of polynomials.
We denote by Ri(i+1)...(i+m) a rational function passing through the m + 1 points(xi , yi ) . . . (xi+m , yi+m ). More explicitly, suppose.
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