Séance Séminaire

Séminaire ACSIOM

The Jacobi-Davidson method for nonlinear eigenproblems

The Jacobi-Davidson method of Sleijpen and Van der Vorst was proposed initially as a solution method for the standard eigenproblem $Ax = \lambda x$. In essence, the method is based on two known ideas. The first idea is that of computing approximate eigensolutions by projecting the eigenproblem onto a low-dimensional search space. This can be interpreted as the Davidson part of the algorithm. The second idea is that of expanding the search space by solving a correction equation in the orthogonal complement of the current approximate solution. This idea can be traced back to Jacobi. Analysis of the resulting method shows that the Jacobi-Davidson method can be viewed as an inexact-Newton method combined with a search-space acceleration. Consequently, the convergence to an eigenpair is at at least quadratic if the correction equation is solved with sufficient accuracy.