Sensitivity of the Lanczos recurrence to Gaussian quadrature data: How malignant can small weights be?

Stability of passing from Gaussian quadrature data to the Lanczos recurrence coefficients is considered. Special attention is paid to estimates explicitly expressed in terms of quadrature data and not having weights in denominators. It has been shown that the recent approach, exploiting integral rep...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2010, Vol.233 (5), p.1238-1244
Main Author: Knizhnerman, Leonid
Format: Article
Language:English
Subjects:
Gaussian quadrature formula
Jacobi inverse eigenvalue problem
Lanczos recurrence
Orthogonal polynomials
Small weights
Stability estimates
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Summary:Stability of passing from Gaussian quadrature data to the Lanczos recurrence coefficients is considered. Special attention is paid to estimates explicitly expressed in terms of quadrature data and not having weights in denominators. It has been shown that the recent approach, exploiting integral representation of Hankel determinants, implies quantitative improvement of D. Laurie’s constructive estimate. It has also been demonstrated that a particular implementation on the Hankel determinant approach gives an estimate being unimprovable up to a coefficient; the corresponding example involves quadrature data with a small but not too small weight. It follows that polynomial increase of a general case upper bound in terms of the dimension is unavoidable.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2007.12.028