gauss.m

Produces a [Gauss quadrature rule] for an orthogonal polynomial with a given three term recurrence.

Given a weight function \(w\) encoded by the \(n \times 2\) array ab of the first \(n\) recurrence coefficients for the associated orthogonal polynomials — the first column of ab containing the \(n\) \(\alpha\)-coefficients, and the second column the \(n\) \(\beta\)-coefficients — the call:

xw = GAUSS(n, ab)

generates the nodes and weights \(x, w\) of the \(n\)-point Gauss quadrature rule.

Inputs

• n : The number of weights and nodes required.

• ab : The first column of ab contains the \(n\) \(\alpha\) recurrence coefficients, and the second column the \(n\) contains \(\beta\) recurrence coefficients

Outputs

• xw : A \(n \times 2\) array. The nodes (in increasing order) are stored in the first column, and the \(n\) corresponding weights are stored in the second column.