[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.

Any advice is welcome!

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


Baptiste Auguié

School of Physics
University of Exeter
Stocker Road,
Exeter, Devon,

Phone: +44 1392 264187


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