pade: Pade Approximation

Description

A Pade approximation is a rational function (of a specified order) whose power series expansion agrees with a given function and its derivatives to the highest possible order.

Usage

pade(p1, p2 = c(1), d1 = 5, d2 = 5)

Arguments

polynomial representing or approximating the function, preferably the Taylor series of the function around some point.

if present, the function is given as p1/p2.

the degree of the numerator of the rational function.

the degree of the denominator of the rational function.

Value

List with components r1 and r2 for the numerator and denominator polynomials, i.e. r1/r2 is the rational approximation sought.

Details

The relationship between the coefficients of p1 (and p2) and r1 and r2 is determined by a system of linear equations. The system is then solved by applying the pseudo-inverse pinv for for the left-hand matrix.

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.

Examples

Run this code

##  Exponential function
p1 <- c(1/24, 1/6, 1/2, 1.0, 1.0)  # Taylor series of exp(x) at x=0
R  <- pade(p1); r1 <- R$r1; r2 <- R$r2
f1 <- function(x) polyval(r1, x) / polyval(r2, x)
xs <- seq(-1, 1, length.out=51); ys1 <- exp(xs); ys2 <- f1(xs)
plot(xs, ys1, type = "l", col="blue")
lines(xs, ys2, col = "red")
grid()

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