Dear list, This is quite a specific question requiring the package orthopolynom. This package provides a nice implementation of the Legendre polynomials, however I need the associated Legendre polynomial which can be readily expressed in terms of the mth order derivative of the corresponding Legendre polynomial. (For the curious, I'm trying to calculate spherical harmonics [*]). Because legendre.polynomials(l) returns a list of Legendre polynomials of degree 0 to l, I'd like to make use of the whole list of them at a time rather than wasting this information. For a given degree "l" I therefore have a list of l+1 polynomials. For each of these I want to compute l+1 derivatives, from m= 0 to m=l. The last step is to evaluate all of these polynomials with a vector argument and return a list of data.frames. I've come up with the following hack but it's really ugly,
require(orthopolynom)
md <- function(.p, m=2){
test <- list()
if(.p==0) pl.list <- rep(as.polylist(.p), m+1) else {
pl.list <- as.polylist(.p)
for(n in seq(1, m+1)){
pl.list[[n+1]] <- deriv(pl.list[[n]])
}
}
rev(pl.list) # ascending order
}
l <- 3 # example
theta <- seq(0, pi, length= 10) # the variable to evaluate the
polynomials at
Pl <- as.polylist(legendre.polynomials(l))
Plm <- lapply(seq_along(Pl), function(ind) md(Pl[[ind]], ind-1))
Plm.theta <- lapply(seq_along(Plm), function(ind) # treat each order l
sapply(seq_along(Plm[[ind]]), function(ind2) # treat each order m
(-1)^ind2 *(1-cos(theta)^2)^(ind2/2) * as.function(Plm[[ind]]
[[ind2]])( cos(theta)) )) # evaluate the expression in theta
I tried (unsuccessfully) to get inspiration from Recall() but since I want to store the intermediate derivatives it doesn't seem very suitable anyway. Any advice is welcome! [*] http://en.wikipedia.org/wiki/Spherical_harmonic _____________________________ Baptiste Augui? School of Physics University of Exeter Stocker Road, Exeter, Devon, EX4 4QL, UK Phone: +44 1392 264187 http://newton.ex.ac.uk/research/emag