Pade Approximation

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.

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

Arguments

p1

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

p2

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

d1

the degree of the numerator of the rational function.

d2

the degree of the denominator of the rational function.

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.

Value

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

Note

In general, errors for Pade approximations are smallest when the degrees of numerator and denominator are the same or when the degree of the numerator is one larger than that of the denominator.

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

taylor, ratInterp

Examples

##  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)
if (FALSE) { # \dontrun{
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()} # }