Inverse iteration

Inverse iteration, also called the inverse power method, is an iterative eigenvalue algorithm. It is used to find an approximate eigenvector when an approximation to the corresponding eigenvalue is already known. It is conceptually close to the power method but applies the iteration to the inverse of the matrix shifted by the eigenvalue estimate, so it converges to the eigenvector whose eigenvalue lies closest to the estimate. The closer the estimate, the faster the convergence, and often a single iteration is enough. Each step requires solving a linear system or applying an inverse matrix, which costs on the order of n cubed operations for an unstructured matrix, although reusing the factorisation or the inverse when the estimate stays fixed brings the later steps down to n squared. It appears to have been developed originally to compute resonance frequencies in structural mechanics, and in practice it is typically paired with a method that finds approximate eigenvalues, such as bisection or Rayleigh quotient iteration.