## Return coefficient matrix (lambda) of n-th order Legendre polynomials 

## Arguments
## 	n: order of polynomials
##	gengler: logical value. If TRUE, Genlger's scaling (1999) will be applied. If not specified, TRUE is assumed.

## Literatures
## Mrode, R.A. 2005. Linear Models for the Prediction of Animal Breeding Values. CAB International, Oxon, UK.
## Gengler, N. et. al. 1999. Estimation of (Co)variance Function Coefficients for Test Day Yield with a Expectation-Maximization Restricted Maximum Likelihood Algorithm.  Journal of Dairy Science.  82

## Author: Gota Morota <morota at wisc dot edu>
## Create: 31-Mar-2010
## Last-Modified: 2-Apr-2010
## License: GPLv3 or later

`legendre` <-
function(n, gengler){
	
	if (nargs()==1){
		gengler <- TRUE	
	}
	
	if (gengler != TRUE & gengler != FALSE){
		gengler=TRUE	
	}
		
	N <- n+1
	L <- matrix(0,nrow=N, ncol=N)
	
	for(i in (1:N)){
		if(i==1){
	 		L[i,i] <- 1
		}
		else if(i==2){
			L[i,i] <- 1
		}
		else  {
			tmp <- L[i-1,]
		    tmp2 <- as.numeric()
			tmp2 <- c(0,tmp[1:(N-1)])
			L[i,] <- (1/(i-2+1))*( (2*(i-2) + 1)*tmp2 -(i-2)*L[i-2,] )
		}
	}
	
	# Normalize
	for (j in (1:N)){	
		L[j,] <- (sqrt( (2*(j-1)+1)/2)  )*L[j,]
	}
	
	
	# Gengler (1999)
	if (gengler==TRUE){
		L <- sqrt(2)*L
	}
	
	return(L)
	
}