Generalized eigenvalue methods for Gaussian quadrature rules
Annales Henri Lebesgue, Volume 3 (2020), pp. 1327-1341.

### Abstract

A quadrature rule of a measure $\mu$ on the real line represents a conic combination of finitely many evaluations at points, called nodes, that agrees with integration against $\mu$ for all polynomials up to some fixed degree. In this paper, we present a bivariate polynomial whose roots parametrize the nodes of minimal quadrature rules for measures on the real line. We give two symmetric determinantal formulas for this polynomial, which translate the problem of finding the nodes to solving a generalized eigenvalue problem.

### References

[AK62] Aheizer, Naum I.; Krein, Mark G. Some questions in the theory of moments, Translations of Mathematical Monographs, Volume 2, American Mathematical Society, 1962 (translated from russian by W. Fleming and D. Prill) | MR: 167806

[BDD + 00] Bai, Zhaojun; Demmel, James; Dongarra, Jack; Ruhe, Axel; van der Vorst, Henk Templates for the solution of algebraic eigenvalue problems: a practical guide, Software – Environments – Tools, Volume 11, Society for Industrial and Applied Mathematics, 2000 | Zbl: 0965.65058

[CF91] Curto, Raúl E.; Fialkow, Lawrence A. Recursiveness, positivity, and truncated moment problems, Houston J. Math., Volume 17 (1991) no. 4, pp. 603-635 | MR: 1147276 | Zbl: 0757.44006

[GMV00] Golub, Gene H.; Milanfar, Peyman; Varah, James A stable numerical method for inverting shape from moments, SIAM J. Sci. Comput., Volume 21 (2000) no. 4, pp. 1222-1243 | Article | MR: 1740393 | Zbl: 0956.65030

[HV07] Helton, John William; Vinnikov, Victor Linear matrix inequality representation of sets, Commun. Pure Appl. Math., Volume 60 (2007) no. 5, pp. 654-674 | Article | MR: 2292953 | Zbl: 1116.15016

[Lau09] Laurent, Monique Sums of squares, moment matrices and optimization over polynomials, Emerging applications of algebraic geometry (Putinar, Mihai; Sullivant, Seth, eds.) (The IMA Volumes in Mathematics and its Applications) Volume 149, Springer, 2009, pp. 157-270 | MR: 2500468

[Lau10] Laurent, Monique Sums of squares, moment matrices and optimization over polynomials (2010) (http://homepages.cwi.nl/~monique/files/moment-ima-update-new.pdf) | Zbl: 1163.13021

[Sch17] Schmüdgen, Konrad The moment problem, Graduate Texts in Mathematics, Volume 277, Springer, 2017 | Zbl: 1383.44004

[Sze75] Szegö, Gábor Orthogonal polynomials, Colloquium Publications, Volume 23, American Mathematical Society, 1975 | Zbl: 0305.42011

[Tyr94] Tyrtyshnikov, Evgenij E. How bad are Hankel matrices?, Numer. Math., Volume 67 (1994) no. 2, pp. 261-269 | Article | MR: 1262784 | Zbl: 0797.65039

[Wag11] Wagner, David G. Multivariate stable polynomials: theory and applications, Bull. Am. Math. Soc. (N.S.), Volume 48 (2011) no. 1, pp. 53-84 | Article | MR: 2738906 | Zbl: 1207.32006