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Linear Algebra, Optimisation and Numerical Methods / Eigenvalue Computation Algorithms

Inverse Iteration and Shifted Methods

Exercise 2 of the power-iteration note flagged that power iteration only ever finds the dominant eigenvalue. This note supplies the fix: apply power iteration to a shifted, inverted matrix, (A−μI)−1(A-\mu I)^{-1}, chosen so that whichever eigenvalue of AA is closest to the shift μ\mu becomes the dominant eigenvalue of the new matrix — turning power iteration into a tool that can target any eigenvalue, not just the largest.
Learning objectives. By the end of this note you should be able to:
  • explain why (A−μI)−1(A-\mu I)^{-1}'s eigenvalues make the target eigenvalue dominant;
  • run inverse iteration and explain its (much faster) convergence rate;
  • combine inverse iteration with the Rayleigh quotient for a self-improving shift;
  • explain why inverse iteration solves a linear system at each step rather than forming an explicit inverse.

In this lesson

  1. Shifting and inverting
  2. Self-improving shifts: Rayleigh quotient iteration

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