Computing Gaussian quadrature rules with high relative accuracy

Abstract

The computation of n-point Gaussian quadrature rules for symmetric weight functions is considered in this paper. It is shown that the nodes and the weights of the Gaussian quadrature rule can be retrieved from the singular value decomposition of a bidiagonal matrix of size n/2. The proposed numerical method allows to compute the nodes with high relative accuracy and a computational complexity of $ \mathcal {O} (n^{2}). $ O ( n 2 ) . We also describe an algorithm for computing the weights of a generic Gaussian quadrature rule with high relative accuracy. Numerical examples show the effectiveness of the proposed approach.

On computing modified moments for half-range Hermite weights

“…, ℓ, be the nodes and the weights of the ℓ-Gauss-Laguerre quadrature rule. They can be computed by solving a symmetric tridiagonal eigenvuale problem [7,8]. Then…”

A Modified Inverse Iteration Method for Computing the Symmetric Tridiagonal Eigenvectors

“…Computing the symmetric tridiagonal (ST) eigenvector is an important task in many research fields, such as the computational quantum physics [1], mathematics [2,3], dynamics [4], computational quantum chemistry [5], etc. The ST eigenvector problem also arises while solving any symmetric eigenproblem because it is a common practice to reduce the generalized symmetric eigenproblems to an ST one.…”

“…A commonly used remedy is to reorthogonalize each approximate eigenvector, by the modified Gram-Schmidt method, against previously computed eigenvectors in the cluster. This remedy increases up to 2n 3 operations if all the eigenvalues cluster, while the time cost for the eigenvectors themselves is only O(n 2 ).…”

Accurate computations with totally positive matrices applied to the computation of Gaussian quadrature formulae

“…In fact, the very recent paper [21] considers the numerical linear algebra approach to compute Gaussian quadrature rules. In that paper, which mainly addresses the special case of symmetric weight functions, the authors indicate that the use of LAPACK subroutine DLASQ1 (which was called by the algorithm TNEigen-Values used in [28]) provides high relative accuracy in the computation of the nodes.…”

“…In that paper, which mainly addresses the special case of symmetric weight functions, the authors indicate that the use of LAPACK subroutine DLASQ1 (which was called by the algorithm TNEigen-Values used in [28]) provides high relative accuracy in the computation of the nodes. As for the computation of the weights, the authors of [21] use an approach which is partly based on the eigenvector computation presented in [32]. Our approach to the computation of the weights, which is presented in detail in Section 4, is based on the computation of the right singular vectors of a bidiagonal matrix by means of the LAPACK routine DBDSQR.…”