Linear Algebra pp 396-403 | Cite as

The Modified LR Algorithm for Complex Hessenberg Matrices

  • R. S. Martin
  • J. H. Wilkinson
Part of the Handbook for Automatic Computation book series (HDBKAUCO, volume 2)

Abstract

The LR algorithm of Rutishauser [3] is based on the observation that if
$$A = LR $$
(1)
where L is unit lower-triangular and R is upper-triangular then B defined by
$$B = {L^{ - 1}}AL = RL$$
(2)
is similar to A. Hence if we write
$${A_s} = {L_s}{R_s},\quad {R_s}{L_s} = {A_{s + 1}}$$
(3)
, a sequence of matrices is obtained each of which is similar to A 1. Rutishauser showed [3, 4, 5] if A 1 has roots of distinct moduli then, in general A s tends to upper triangular form, the diagonal elements tending to the roots arranged in order of decreasing modulus. If A 1 has some roots of equal modulus then A s does not tend to strictly triangular form but rather to block-triangular form. Corresponding to a block of k roots of the same modulus there is a diagonal block of order k which does not tend to a limit, but its roots tend to the k eigenvalues.

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References

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Copyright information

© Springer-Verlag Berlin Heidelberg 1971

Authors and Affiliations

  • R. S. Martin
  • J. H. Wilkinson

There are no affiliations available

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