Julia package to compute points and weights for Gauss quadrature rules using the Golub-Welsch algorithm.
Handles the classical Legendre, Chebyshev, Jacobi, Laguerre and Hermite weights, as well as a logarithmic weight function. In fact, the Gauss rule is available for any custom weight function such that the coefficients are known for the three-term recurrence relation satisfied by the associated orthogonal polynomials. The modified Chebyshev algorithm is provided to determine these coefficients from the modified moments of the weight function.
The Lobatto and Radau variants of all these rules are also provided by
appropriate choice of the endpt argument: neither (the default),
both, left or right.
For example, to obtain a plain 5-point Gauss-Legendre rule with weight
function w(x)=1 on the interval -1 < x < -1 do
julia> using GaussQuadrature
julia> x, w = legendre(5)
whereas for the Lobatto version do
julia> x, w = legendre(5, both)
The package supports BigFloats; for example,
julia> x, w = legendre(BigFloat, 5)
gives a plain 5-point Gauss-Legendre rule accurate to about 75 significant figures.
Read the initial comments in the src/GaussQuadrature.jl module
for full details, or read the help documentation for the individual
functions called legendre, chebyshev, jacobi, laguerre, hermite,
logweight and custom_gauss_rule.