Jacobi-Davidson Methods for Symmetric Eigenproblems

1 Why and how The Lanczos method is quite eeective if the desired eigenvalue is either max or min and if this eigenvalue is relatively well separated from the remaining spectrum, or when the method is applied with (A ? I) ?1 , for some reasonable guess for an eigenvalue. If none of these conditions is fulllled, for instance the computation of a vector (A ? I) ?1 y for given y may be computationally unattractive, then variants of the Jacobi-Davidson method 8] ooer an attractive alternative. The Jacobi-Davidson method is based on a combination of two basic principles. The rst one is to apply a Ritz-Galerkin approach for the eigenproblem Ax = x, with respect to some given subspace spanned by v 1 , : : :, v k. The usage of other than Krylov subspaces was suggested by Davidson 2] who also suggested speciic choices, diierent from the ones that we will take, for the construction of orthonormal basis vectors v j. The Ritz-Galerkin condition is which leads to the reduced system V k AV k s ? s = 0: (1) This equation has k solutions ((k) j ; s (k) j). The k pairs ((k) j ; z (k) j V k s (k) j) are called the Ritz values and Ritz vectors, respectively, of A with respect to the subspace spanned by the v j. These Ritz pairs are approximations for eigenpairs of A, and our goal is to get better approximations by a well-chosen expansion of the subspace. At this point the other principle behind the Jacobi-Davidson approach comes into the play. The idea goes back to Jacobi 4]. Suppose that we have an eigenvector approximation z (k) j for an eigenvector corresponding to given eigenvalue. Then Jacobi suggested (in the original paper for strongly diagonally dominant matrices) to compute the orthogonal correction t for z (k) j so that A(z (k) j + t) = (z (k) j + t): Since t ? z (k) j , we can restrict ourselves to the subspace orthogonal to z (k) j. The operator A restricted to that subspace is given by (I ? z (k) j z (k) j)A(I ? z (k) j z (k) j); and we nd that t satisses the equation (I ? z (k) j z (k) j)(A ? I)(I ? z (k) j z (k) j)t = ?(A ? I)z …