Nothing
quadrature.rules <- function( recurrences, inner.products )
{
###
### This function returns a list with n elements
### containing the order k quadrature rule data frames
### for orders k=1,2,...n. An order k quadrature data frame
### contains the roots and abscissa values for the degree k polynomial
###
### Parameters
### recurrences = a data frame with recurrence parameters c, d, e and f
### inner.products = a vector of norm squared values
###
np1 <- nrow( recurrences )
n <- np1 - 1
rules <- as.list( rep( NULL, n ) )
monic.recurrences <- monic.polynomial.recurrences( recurrences )
matrices <- jacobi.matrices( monic.recurrences )
matrix.eigens <- lapply( matrices, eigen )
roots <- polynomial.roots( monic.recurrences )
h.0 <- inner.products[1]
for ( k in 1:n ) {
values <- matrix.eigens[[k]]$values
vectors <- matrix.eigens[[k]]$vectors
x <- values
w <- rep( 0, k )
for ( j in 1:k ) {
v.j <- vectors[1,j]
w[j] <- h.0 * v.j * v.j
}
rule <- data.frame( cbind( x, w ) )
names( rule ) <- c( "x", "w" )
rules[[k]] <- rule
}
return( rules )
}
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