Rational Function Approximation

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Description

Fitting a rational function to data points.

Usage

1
rationalfit(x, y, d1 = 5, d2 = 5)

Arguments

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; x and y must be of the same length.

d1, d2

maximal degrees of numerator (d1) and denominator (d1) of the requested rational function.

Details

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

Value

List with components p1 and p2 for the polynomials in numerator and denominator of the rational function.

Note

This implementation will later be replaced by a 'barycentric rational interpolation'.

Author(s)

Copyright (c) 2006 by Paul Godfrey for a Matlab version available from the MatlabCentral under BSD license. R re-implementation by Hans W Borchers.

References

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.

See Also

ratinterp

Examples

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## 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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