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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: 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.


References [Enhancements On Off] (What's this?)

  • [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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Additional Information

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

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