golub_1969#

G. H. Golub and J. H. Welsch, “Calculation of Gauss Quadrature Rules”, Math. Comp. 23, 221 (1969)

Source record · Catalogue · Review register

Source kind: theory

Recorded technique: Numerical analysis: the nodes and weights of an N-point Gaussian quadrature rule for a weight function are the eigenvalues and the squared first components of the orthonormal eigenvectors (times the zeroth moment) of the symmetric tridiagonal Jacobi matrix of the three-term recurrence of its orthogonal polynomials (section 2, Eqs. 2.1-2.6); a shifted Q-R algorithm that computes only the first components (sections 3-4); and the recurrence from the moments (section 5).

Materials in structured entries: None recorded.

Dataset quantities: None recorded.

Contents: 0 datasets; 0 formulas.

Curation notes#

Identity VALIDATED_WITH_WARNINGS by the library, whose resolver took the received and revised dates (1967, 1968) for the publication year; the DOI and the record were confirmed by the curator (the library’s DOI review). No values are transcribed: the paper is the method of OEP 3’s band rules.

THE RELATION USED (p. 223, Eq. 2.6, checked against the page image): w_j = q_{1,j}^2 / p_0^2(t_j) = q_{1,j}^2 / k_0^2 = q_{1,j}^2 x integral_a^b omega(x) dx = q_{1,j}^2 mu_0, with t_j the eigenvalues of the Jacobi matrix and q_{1,j} the first component of the j-th orthonormal eigenvector. optmat.gas.infrared.gauss_rule applies it to a band’s discrete measure (weights f/E^2 at x = E^2): the Jacobi matrix comes from the Lanczos process on the lines, with full reorthogonalisation, rather than from the moments (section 5), whose recurrence is ill-conditioned; the eigenvalues and vectors from numpy’s symmetric eigensolver, rather than the paper’s Q-R program. The rule is exact for polynomials of degree up to 2N - 1 (p. 222, the Theorem), so it keeps the band’s moments mu_0 to mu_{2N-1}; tested in tests/test_infrared.py.

Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.

Datasets#

No datasets are recorded.

Formulas#

No formulas are recorded.

Corrections#

No dataset errata are recorded in this source record.

Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the review register for the checks and decisions.