BIT Numerical Mathematics

, Volume 40, Issue 4, pp 671-684

First online:

Inexact Inverse Iteration for Generalized Eigenvalue Problems

  • Gene H. GolubAffiliated withDepartment of Computer Science, Stanford University
  • , Qiang YeAffiliated withDepartment of Mathematics, University of Kentucky

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In this paper, we study an inexact inverse iteration with inner-outer iterations for solving the generalized eigenvalu problem Ax = λBx, and analyze how the accuracy in the inner iterations affects the convergence of the outer iterations. By considering a special stopping criterion depending on a threshold parameter, we show that the outer iteration converges linearly with the inner threshold parameter as the convergence rate. We also discuss the total amount of work and asymptotic equivalence between this stopping criterion and a more standard one. Numerical examples are given to illustrate the theoretical results.

Inverse iteration shift-and-invert inner-outer iterations