How to make the Lanczos algorithm converge slowly

Author:
D. S. Scott

Journal:
Math. Comp. **33** (1979), 239-247

MSC:
Primary 65F15

DOI:
https://doi.org/10.1090/S0025-5718-1979-0514821-5

MathSciNet review:
514821

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Abstract | References | Similar Articles | Additional Information

Abstract: The Paige style Lanczos algorithm is an iterative method for finding a few eigenvalues of large sparse symmetric matrices. Some beautiful relationships among the elements of the eigenvectors of a symmetric tridiagonal matrix are used to derive a perverse starting vector which delays convergence as long as possible. Why such slow convergence is never seen in practice is also examined.

**[1]**Shmuel Kaniel,*Estimates for some computational techniques in linear algebra*, Math. Comp.**20**(1966), 369–378. MR**0234618**, https://doi.org/10.1090/S0025-5718-1966-0234618-4**[2]**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****[3]**C. C. PAIGE,*The Computation of Eigenvalues and Eigenvectors of Very Large Sparse Matrices*, Ph.D. Thesis, University of London, 1971.**[4]**R. C. Thompson and P. McEnteggert,*Principal submatrices. II. The upper and lower quadratic inequalities.*, Linear Algebra and Appl.**1**(1968), 211–243. MR**0237532****[5]**D. BOLEY & G. H. BOLEY,*Inverse Eigenvalue Problems for Banded Matrices*, Technical Report STAN-CS-77-623, Computer Science Department, Stanford University, 1977.**[6]**C. de Boor and G. H. Golub,*The numerically stable reconstruction of a Jacobi matrix from spectral data*, Linear Algebra Appl.**21**(1978), no. 3, 245–260. MR**504044**, https://doi.org/10.1016/0024-3795(78)90086-1

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DOI:
https://doi.org/10.1090/S0025-5718-1979-0514821-5

Keywords:
Eigenvalues,
Lanczos algorithm,
sparse symmetric matrices

Article copyright:
© Copyright 1979
American Mathematical Society