Computation of Gauss-Kronrod quadrature rules

Recently Laurie presented a new algorithm for the computation of (2n+1)(2n+1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n+12n+1 from certain mixed moments, and then computes a partial spectral f...

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Bibliographic Details
Published in:Mathematics of computation 2000-07, Vol.69 (231), p.1035-1052
Main Authors: Calvetti, D., Golub, G. H., Gragg, W. B., Reichel, L.
Format: Article
Language:English
Subjects:
Algorithms
Arithmetic
Coefficients
Eigenvalues
Eigenvectors
Exact sciences and technology
Gaussian quadratures
Mathematics
Matrices
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Numerical quadratures
Polynomials
Recursion
Research article
Sciences and techniques of general use
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Summary:Recently Laurie presented a new algorithm for the computation of (2n+1)(2n+1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n+12n+1 from certain mixed moments, and then computes a partial spectral factorization. We describe a new algorithm that does not require the entries of the tridiagonal matrix to be determined, and thereby avoids computations that can be sensitive to perturbations. Our algorithm uses the consolidation phase of a divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. We also discuss how the algorithm can be applied to compute Kronrod extensions of Gauss-Radau and Gauss-Lobatto quadrature rules. Throughout the paper we emphasize how the structure of the algorithm makes efficient implementation on parallel computers possible. Numerical examples illustrate the performance of the algorithm.
ISSN:0025-5718
1088-6842
DOI:10.1090/S0025-5718-00-01174-1