# Perform Gauss Legendre quadrature 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 Legendre polynomials. ```r legendre.quadrature(functn, rule, 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 $n$ Legendre quadrature rule - `lower`: numeric value for the lower limit of the integral - `upper`: numeric value for the upper limit of the integral - `weighted`: boolean value which if true causes the Legendre weight function to be included in the integrand - ``...``: other arguments passed to the give function ## Details The rule argument corresponds to an order $n$ Legendre polynomial, weight function and interval $\left[ { - 1,1} \right]$. The lower and upper limits of the integral must be finite. ## Returns The value of definite integral evaluated using Gauss Legendre quadrature ## 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. Szego, G., 1939. **Orthogonal Polynomials**, 23, American Mathematical Society Colloquium Publications, Providence, RI. ## Author(s) Frederick Novomestky fnovomes@poly.edu ## See Also `legendre.quadrature.rules` ## Examples ```r ### ### this example evaluates the quadrature function for ### the Legendre polynomials. it computes the integral ### of the product for all pairs of orthogonal polynomials ### from order 0 to order 20. 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 20. the resultant matrix should be an identity matrix ### ### ### set the value for the maximum polynomial order ### n <- 20 ### ### 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 <- legendre.quadrature( p.p.functions[[r]][[c]], order.np1.rule ) 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 Legendre orthogonal polynomials ### p.list <- legendre.polynomials( n ) ### ### 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 <- legendre.quadrature.rules( np1 ) 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( legendre.inner.products( n ) ) print( "Integral of cross products for the orthogonal polynomials " ) print( apply( p.p.integrals, 2, round, digits=5 ) ) print( apply( p.p.inner.products, 2, round, digits=5 ) ) ### ### construct a list of the Legendre orthonormal polynomials ### p.list <- legendre.polynomials( n, 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 <- legendre.quadrature.rules( np1, 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=5 ) ) ```

legendre.quadrature function