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]**S. KANIEL, "Estimates for some computational techniques in linear algebra,"*Math. Comp.*, v. 20, 1966, pp. 369-378. MR**0234618 (38:2934)****[2]**C. LANCZOS, "An iteration method for the solution of the eigenvalue problem of linear differential and integral operators,"*J. Res. Nat. Bur. Standards*, v. 45, 1950, pp. 255-282. MR**0042791 (13:163d)****[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 & P. McENTEGGERT, "Principal submatrices. II: The upper and lower quadratic inequalities,"*Linear Algebra Appl.*, v. 1, 1968, pp. 211-243. MR**0237532 (38:5813)****[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. deBOOR & G. H. GOLUB, "The numerically stable reconstruction of a Jacobi matrix from spectral data,"*Linear Algebra Appl.*, v. 21, 1978, pp. 245-260. MR**504044 (80i:15007)**

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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