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Mathematics of Computation

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Computation of Gauss-Kronrod quadrature rules

Authors: D. Calvetti, G. H. Golub, W. B. Gragg and L. Reichel
Journal: Math. Comp. 69 (2000), 1035-1052
MSC (1991): Primary 65D30, 65D32
Published electronically: February 17, 2000
MathSciNet review: 1677474
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Recently Laurie presented a new algorithm for the computation of $(2n+1)$-point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order $2n+1$ 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.

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Additional Information

D. Calvetti
Affiliation: Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106

G. H. Golub
Affiliation: Department of Computer Science, Stanford University, Stanford, California 94305

W. B. Gragg
Affiliation: Department of Mathematics, Naval Postgraduate School, Monterey, California 93943

L. Reichel
Affiliation: Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242

Keywords: Jacobi matrix, inverse eigenvalue problem, divide-and-conquer algorithm, generalized Gauss-Kronrod rule
Received by editor(s): May 12, 1998
Published electronically: February 17, 2000
Additional Notes: Research of the first author was supported in part by NSF grants DMS-9404692 and DMS-9896073.
Research of the second author was supported in part by NSF grant CCR-9505393.
Research of the fourth author was supported in part by NSF grant DMS-9404706.
Article copyright: © Copyright 2000 American Mathematical Society