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Mathematics of Computation

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Inverse iteration on defective matrices

Author: Nai Fu Chen
Journal: Math. Comp. 31 (1977), 726-732
MSC: Primary 65F15
MathSciNet review: 0438682
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Abstract: Very often, inverse iteration is used with shifts to accelerate convergence to an eigenvector. In this paper, it is shown that, if the eigenvalue belongs to a nonlinear elementary divisor, the vector sequences may diverge even when the shift sequences converge to the eigenvalue. The local behavior is discussed through a $ 2 \times 2$ example, and a sufficient condition for the convergence of the vector sequence is given.

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  • [1] N. CHEN, The Rayleigh Quotient Iteration for Non-Normal Matrices, Ph. D. Dissertation, Electronic Research Laboratory Memorandum No. ERL-M548, University of California, Berkeley, 1975.
  • [2] J. M. VARAH, The Computation of Bounds for the Invariant Subspaces of a General Matrix Operator, Ph. D. Dissertation, Stanford University, 1967.
  • [3] J. H. Wilkinson, Inverse iteration in theory and in practice, Symposia Mathematica, Vol. X (Convegno di Analisi Numerica, INDAM, Rome, 1972) Academic Press, London, 1972, pp. 361–379. MR 0366017

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Keywords: Eigenvector, eigenvalue, nonlinear divisors, convergence, inverse iterations, shifts
Article copyright: © Copyright 1977 American Mathematical Society