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polynomial.powers <- function( polynomials )
{
###
### This function returns a list of vectors that define linear combinations
### of the given polynomials equal to powers of the polynomial argument.
###
### Parameters:
### polynomials = a list of polynomial objects in increase polynomial order
###
coefficients <- polynomial.coefficients( polynomials )
np1 <- length( polynomials )
powers <- as.list( rep( NULL, np1 ) )
lambda.matrix <- matrix( 0, nrow=np1, ncol=np1 )
for ( j in 1:np1 ) {
lambda <- coefficients[[j]]
for ( k in 1:j ) {
lambda.matrix[j,k] <- lambda[k]
}
}
alpha.matrix <- solve( lambda.matrix )
for ( j in 1:np1 ) {
alpha <- rep( 0, j )
for ( k in 1:j ) {
alpha[k] <- alpha.matrix[j,k]
}
powers[[j]] <- alpha
}
return( powers )
}
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