Rational Function Approximation
Fitting a rational function to data points.
rationalfit(x, y, d1 = 5, d2 = 5)
x |
numeric vector; points on the x-axis; needs to be sorted; at least three points required. |
y |
numeric vector; values of the assumed underlying function;
|
d1, d2 |
maximal degrees of numerator ( |
A rational fit is a rational function of two polynomials p1 and
p2 (of user specified degrees d1 and d2) such that
p1(x)/p2(x) approximates y in a least squares sense.
d1 and d2 must be large enough to get a good fit and usually
d1=d2 gives good results
List with components p1 and p2 for the polynomials in
numerator and denominator of the rational function.
This implementation will later be replaced by a 'barycentric rational interpolation'.
Copyright (c) 2006 by Paul Godfrey for a Matlab version available from the MatlabCentral under BSD license. R re-implementation by Hans W Borchers.
Press, W. H., S. A. Teukolsky, W. T Vetterling, and B. P. Flannery (2007). Numerical Recipes: The Art of Numerical Computing. Third Edition, Cambridge University Press, New York.
## Not run: x <- linspace(0, 15, 151); y <- sin(x)/x rA <- rationalfit(x, y, 10, 10); p1 <- rA$p1; p2 <- rA$p2 ys <- polyval(p1,x) / polyval(p2,x) plot(x, y, type="l", col="blue", ylim=c(-0.5, 1.0)) points(x, Re(ys), col="red") # max(abs(y-ys), na.rm=TRUE) < 1e-6 grid() # Rational approximation of the Zeta function x <- seq(-5, 5, by = 1/16) y <- zeta(x) rA <- rationalfit(x, y, 10, 10); p1 <- rA$p1; p2 <- rA$p2 ys <- polyval(p1,x) / polyval(p2,x) plot(x, y, type="l", col="blue", ylim=c(-5, 5)) points(x, Re(ys), col="red") grid() # Rational approximation to the Gamma function x <- seq(-5, 5, by = 1/32); y <- gamma(x) rA <- rationalfit(x, y, 10, 10); p1 <- rA$p1; p2 <- rA$p2 ys <- polyval(p1,x) / polyval(p2,x) plot(x, y, type="l", col = "blue") points(x, Re(ys), col="red") grid() ## End(Not run)
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