Computation of Gauss-Kronrod quadrature rules
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- by D. Calvetti, G. H. Golub, W. B. Gragg and L. Reichel;
- Math. Comp. 69 (2000), 1035-1052
- DOI: https://doi.org/10.1090/S0025-5718-00-01174-1
- Published electronically: February 17, 2000
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Abstract:
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.References
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Bibliographic Information
- D. Calvetti
- Affiliation: Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106
- Email: dxc57@po.cwru.edu
- G. H. Golub
- Affiliation: Department of Computer Science, Stanford University, Stanford, California 94305
- Email: golub@chebyshev.stanford.edu
- W. B. Gragg
- Affiliation: Department of Mathematics, Naval Postgraduate School, Monterey, California 93943
- Email: gragg@nps.navy.mil
- L. Reichel
- Affiliation: Department of Mathematics and Computer Science, Kent State University, Kent, Ohio 44242
- Email: reichel@mcs.kent.edu
- 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. - © Copyright 2000 American Mathematical Society
- Journal: Math. Comp. 69 (2000), 1035-1052
- MSC (1991): Primary 65D30, 65D32
- DOI: https://doi.org/10.1090/S0025-5718-00-01174-1
- MathSciNet review: 1677474