BIT Numerical Mathematics

, Volume 51, Issue 4, pp 937–957 | Cite as

Analyzing the convergence factor of residual inverse iteration

  • Elias Jarlebring
  • Wim Michiels


We will establish here a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well-known inverse iteration. The formula for the convergence factor is explicit and involves quantities associated with the eigenvalue to which the iteration converges, in particular the eigenvalue and eigenvector. Residual inverse iteration allows for some freedom in the choice of a vector w k and we can use the formula for the convergence factor to analyze how it depends on the choice of w k . We also use the formula to illustrate the convergence when the shift is close to the eigenvalue. Finally, we explain the slow convergence for double eigenvalues by showing that under generic conditions, the convergence factor is one, unless the eigenvalue is semisimple. If the eigenvalue is semisimple, it turns out that we can expect convergence similar to the simple case.


Nonlinear eigenvalue problems Residual inverse iteration Convergence factors 

Mathematics Subject Classification (2000)

65F15 65H17 15A18 


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Copyright information

© Springer Science + Business Media B.V. 2011

Authors and Affiliations

  1. 1.K.U. LeuvenHeverleeBelgium

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