golub_1969 ========== G\. H\. Golub and J\. H\. Welsch, "Calculation of Gauss Quadrature Rules", Math\. Comp\. 23, 221 \(1969\) `Source record `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1090/s0025\-5718\-69\-99647\-1 `__ **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 :doc:`review register ` for the checks and decisions.