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

Abstract

For some families of classical orthogonal polynomials defined on appropriate intervals, it is shown that the corresponding Jacobi matrices are totally positive and their bidiagonal factorizations can be accurately computed. By exploiting these facts, an algorithm to compute with high relative accuracy the eigenvalues of those Jacobi matrices, and consequently the nodes of Gaussian quadrature formulae for those families of orthogonal polynomials, is presented. An algorithm is also presented for the computation of the eigenvectors of these Jacobi matrices, and hence the weights of Gaussian quadrature formulae. Although in this case high relative accuracy is not theoretically guaranteed, the numerical experiments with our algorithm provide very accurate results.

The Application of the Bidiagonal Factorization of Totally Positive Matrices in Numerical Linear Algebra

“…An example of this situation (where the Jacobi matrices are totally positive) has been analyzed in [51], and more recently in [52]. Although in [51] Gaussian quadrature formulae were considered, only the nodes were computed in that paper, since the accurate computation of the eigenvectors of the corresponding Jacobi matrices (necessary to compute the weights) could not be achieved with MATLAB.…”

“…While the LAPACK subroutine DLASQ1 (used by the algorithm TNEigenValues) provides high relative accuracy in the computation of the nodes, for the computation of the weights (presented in Section 4 of [52]) the LAPACK subroutine DBDSQR is used to compute the right singular vectors of a bidiagonal matrix. The initial stage is again the accurate computation of the bidiagonal factorization of the corresponding totally positive Jacobi matrices.…”