The Jacobi–Davidson method

The Jacobi–Davidson method is a popular technique to compute a few eigenpairs of large sparse matrices. Its introduction, about a decade ago, was motivated by the fact that standard eigensolvers often require an expensive factorization of the matrix to compute interior eigenvalues. Such a factorization may be infeasible for large matrices as arise in today’s large-scale simulations. In the Jacobi–Davidson method, one still needs to solve “inner” linear systems, but a factorization is avoided because the method is designed so as to favor the efficient use of modern iterative solution techniques, based on preconditioning and Krylov subspace acceleration. Here we review the Jacobi–Davidson method, with the emphasis on recent developments that are important in practical use.