A look-ahead Lánczos algorithm for unsymmetric matrices

Authors:
Beresford N. Parlett, Derek R. Taylor and Zhishun A. Liu

Journal:
Math. Comp. **44** (1985), 105-124

MSC:
Primary 65F15

DOI:
https://doi.org/10.1090/S0025-5718-1985-0771034-2

MathSciNet review:
771034

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Abstract | References | Similar Articles | Additional Information

Abstract: The two-sided Lanczos algorithm sometimes suffers from serious breakdowns. These occur when the associated moment matrix does not permit triangular factorization. We modify the algorithm slightly so that it corresponds to using a pivot in triangular factorization whenever a pivot would be dangerous. The likelihood of breakdown is greatly reduced. The price paid is that the tridiagonal matrix produced by the algorithm now has bumps whenever a pivot is used. Experiments with several versions of the algorithm on a variety of matrices are described, including some large problems arising in the study of plasma instability.

**[1]**Chandler Davis and W. M. Kahan,*The rotation of eigenvectors by a perturbation. III*, SIAM J. Numer. Anal.**7**(1970), 1–46. MR**0264450**, https://doi.org/10.1137/0707001**[2]**W. Gragg, Notes from a "Kentucky Workshop" on the moment problem and indefinite metrics.**[3]**Alston S. Householder,*The theory of matrices in numerical analysis*, Blaisdell Publishing Co. Ginn and Co. New York-Toronto-London, 1964. MR**0175290****[4]**W. Kahan, B. N. Parlett, and E. Jiang,*Residual bounds on approximate eigensystems of nonnormal matrices*, SIAM J. Numer. Anal.**19**(1982), no. 3, 470–484. MR**656463**, https://doi.org/10.1137/0719030**[5]**Cornelius Lanczos,*An iteration method for the solution of the eigenvalue problem of linear differential and integral operators*, J. Research Nat. Bur. Standards**45**(1950), 255–282. MR**0042791****[6]**Beresford Parlett,*Laguerre’s method applied to the matrix eigenvalue problem*, Math. Comp.**18**(1964), 464–485. MR**0165668**, https://doi.org/10.1090/S0025-5718-1964-0165668-2**[7]**Beresford N. Parlett,*The symmetric eigenvalue problem*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980. Prentice-Hall Series in Computational Mathematics. MR**570116****[8]**J. R. Bunch and B. N. Parlett,*Direct methods for solving symmetric indefinite systems of linear equations*, SIAM J. Numer. Anal.**8**(1971), 639–655. MR**0305564**, https://doi.org/10.1137/0708060**[9]**Y. Saad,*Variations on Arnoldi’s method for computing eigenelements of large unsymmetric matrices*, Linear Algebra Appl.**34**(1980), 269–295. MR**591435**, https://doi.org/10.1016/0024-3795(80)90169-X**[10]**Y. Saad,*The Lanczos biorthogonalization algorithm and other oblique projection methods for solving large unsymmetric systems*, SIAM J. Numer. Anal.**19**(1982), no. 3, 485–506. MR**656464**, https://doi.org/10.1137/0719031**[11]**D. R. Taylor,*Analysis of the Look Ahead Lanczos Algorithm*, Ph.D. thesis, University of California, Berkeley, 1982.**[12]**J. H. Wilkinson,*The algebraic eigenvalue problem*, Clarendon Press, Oxford, 1965. MR**0184422**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0771034-2

Article copyright:
© Copyright 1985
American Mathematical Society