Gregory_weights {GregoryQuadrature}

Calculate the Gregory quadrature weights for equispaced integration. If f is a row vector containing the function values, the integral is approximated by the statement f %*% t(w) where w are the returned weights. Translated from https://www.colorado.edu/amath/sites/default/files/attached-files/gregory.pdf.

Description

Calculate the Gregory quadrature weights for equispaced integration. If f is a row vector containing the function values, the integral is approximated by the statement f %*% t(w) where w are the returned weights. Translated from https://www.colorado.edu/amath/sites/default/files/attached-files/gregory.pdf.

Usage

Gregory_weights(n_nodes, h, order)

Arguments

n_nodes

Total number of nodes

h

Step size

order

Order of accuracy desired. 2, 3, 4, ... (with 2 giving the trapezoidal rule). The value must satisfy 2 <= order <= n_nodes

Value

The weights to be used for the successive function values

Examples

n_nodes = 11
order = 8
h = 2/(n_nodes-1)
x = pracma::linspace(-1, 1, n_nodes)
f = exp(x)

w = GregoryQuadrature::Gregory_weights(n_nodes, h, order)
int = f %*% w
# Exact value for integral
exact = exp(1) - exp(-1)

error = int - exact