# [R] recursive derivative a list of polynomials

baptiste auguie ba208 at exeter.ac.uk
Sun Feb 8 13:15:39 CET 2009

```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.

[*] http://en.wikipedia.org/wiki/Spherical_harmonic

_____________________________

Baptiste Auguié

School of Physics
University of Exeter