ultraspherical.quadrature: Perform Gauss ultraspherical quadrature

In gaussquad: Collection of Functions for Gaussian Quadrature

Perform Gauss ultraspherical quadrature

Description

This function evaluates the integral of the given function between the lower and upper limits using the weight and abscissa values specified in the rule data frame. The quadrature formula uses the weight function for ultraspherical polynomials.

Usage

ultraspherical.quadrature(functn, rule, alpha = 0, lower = -1, upper = 1, 
weighted = TRUE, ...)

Arguments

functn	an R function which should take a numeric argument x and possibly some parameters. The function returns a numerical vector value for the given argument x.
rule	a data frame containing the order n ultraspherical quadrature rule
alpha	a numeric value for the ultraspherical polynomial parameter
lower	a numeric value for the lower limit of the integral with a default value of -1
upper	a numeric value for the upper limit of the integral with a default value of 1
weighted	a boolean value which if true causes the ultraspherical weight function to be included in the integrand
...	other arguments passed to the give function

Details

The rule argument corresponds to an order n ultraspherical polynomial, weight function and interval ≤ft[ { - 1,1} \right]. The lower and upper limits of the integral must be finite.

Value

The value of definite integral evaluated using Gauss Gegenbauer quadrature

Author(s)

Frederick Novomestky [e-mail]

References

Abramowitz, M. and I. A. Stegun, 1968. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, Inc., New York.

Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, 1992. Numerical Recipes in C, Cambridge University Press, Cambridge, U.K.

Stroud, A. H., and D. Secrest, 1966. Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, NJ.

See Also

ultraspherical.quadrature.rules

Examples

###
### this example evaluates the quadrature function for
### the ultraspherical polynomials.  it computes the integral
### of the product for all pairs of orthogonal polynomials
### from order 0 to order 16.  the results are compared to
### the diagonal matrix of the inner products for the
### polynomials.  it also computes the integral of the product
### of all pairs of orthonormal polynomials from order 0
### to order 16.  the resultant matrix should be an identity matrix
###
###
### set the polynomial parameter
###
alpha <- 0.25
###
### set the value for the maximum polynomial order
###
    n <- 16
###
### maximum order plus 1
###
    np1 <- n + 1
###
### function to construct the polynomial products by column
###
by.column.products <- function( c, p.list, p.p.list )
{
###
### function to construct the polynomial products by row
###
    by.row.products <- function( r, c, p.list )
    {
        row.column.product <- p.list[[r]] * p.list[[c]]
        return (row.column.product )
    }
    np1 <- length( p.list )
    row.list <- lapply( 1:np1, by.row.products, c, p.list )
    return( row.list )
}
###
### function construct the polynomial functions by column
###
by.column.functions <- function( c, p.p.products )
{
###
### function to construct the polynomial functions by row
###
    by.row.functions <- function( r, c, p.p.products )
    {
        row.column.function <- as.function( p.p.products[[r]][[c]] )
        return( row.column.function )
    }
    np1 <- length( p.p.products[[1]] )
    row.list <- lapply( 1:np1, by.row.functions, c, p.p.products )
    return( row.list )
}
###
### function to compute the integral of the polynomials by column
###
by.column.integrals <- function( c, p.p.functions )
{
###
### function to compute the integral of the polynomials by row
###
    by.row.integrals <- function( r, c, p.p.functions )
    {
        row.column.integral <- ultraspherical.quadrature(
            p.p.functions[[r]][[c]], order.np1.rule, alpha )
        return( row.column.integral )
    }
    np1 <- length( p.p.functions[[1]] )
    row.vector <- sapply( 1:np1, by.row.integrals, c, p.p.functions )
    return( row.vector )
}
###
### construct a list of the ultraspherical orthogonal polynomials
###
p.list <- ultraspherical.polynomials( n, alpha )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- ultraspherical.quadrature.rules( np1, alpha )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### construct the diagonal matrix with the inner products
### of the orthogonal polynomials on the diagonal
###
p.p.inner.products <- diag( ultraspherical.inner.products( n,alpha ) )
print( "Integral of cross products for the orthogonal polynomials " )
print( apply( p.p.integrals, 2, round, digits=6 ) )
print( apply( p.p.inner.products, 2, round, digits=6 ) )
###
### construct a list of the ultraspherical orthonormal polynomials
###
p.list <- ultraspherical.polynomials( n, alpha, TRUE )
###
### construct the two dimensional list of pair products
### of polynomials
###
p.p.products <- lapply( 1:np1, by.column.products, p.list )
###
### compute the two dimensional list of functions
### corresponding to the polynomial products in
### the two dimensional list p.p.products
###
p.p.functions <- lapply( 1:np1, by.column.functions, p.p.products )
###
### get the rule table for the order np1 polynomial
###
rules <- gegenbauer.quadrature.rules( np1, alpha, TRUE )
order.np1.rule <- rules[[np1]]
###
### construct the square symmetric matrix containing
### the definite integrals over the default limits
### corresponding to the two dimensional list of
### polynomial functions
###
p.p.integrals <- sapply( 1:np1, by.column.integrals, p.p.functions )
###
### display the matrix of integrals
###
print( "Integral of cross products for the orthonormal polynomials " )
print(apply( p.p.integrals, 2, round, digits=6 ) )

gaussquad documentation built on June 14, 2022, 9:05 a.m.