Analyzing the convergence factor of residual inverse iteration (Q657887)

A formula for the convergence factor of the method called \textit{residual inverse iteration} for nonlinear eigenvalue problems and generalization of the well-known \textit{inverse iteration} is established. This formula is explicit and involves quantities associated with the eigenvalue to which the iteration converges, in particular the eigenvalue and eigenvector. Residual inverse iteration allows the choice of a vector \(w_k\) and the formula may be used for the convergence factor so as to analyze the dependence on the choice of \(w_k\). The formula is also used to illustrate the convergence when the shift is close to the eigenvalue. The slow convergence for double eigenvalues is explained by showing that under generic conditions the convergence factor is one, unless the eigenvalue is semisimple. Convergence similar to the simple case is expected when the eigenvalue is semisimple.