iterative eigenvalue algorithm with nonstandard inner product - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T10:16:40Z http://mathoverflow.net/feeds/question/71558 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/71558/iterative-eigenvalue-algorithm-with-nonstandard-inner-product iterative eigenvalue algorithm with nonstandard inner product Victor Liu 2011-07-29T07:46:14Z 2011-10-14T14:19:57Z <p>There exist many iterative algorithms for computing eigenvalues of large sparse matrices, e.g. Power, Lanczos, and Arnoldi iteration. Take the Lanczos algorithm for instance. It assumes that the matrix is Hermitian, in the standard inner product space. Since it only requires matrix-vector multiplies and inner products as basic operations, is it possible to apply the algorithm to a non-Hermitian matrix which <em>is</em> self-adjoint under a non-standard inner product and still expect the algorithm to "work" in some sense? I'm imagining this in the case of discretizing self-adjoint operators with respect to a non-standard bilinear form (not necessarily positive definite).</p> http://mathoverflow.net/questions/71558/iterative-eigenvalue-algorithm-with-nonstandard-inner-product/78138#78138 Answer by 0x2207 for iterative eigenvalue algorithm with nonstandard inner product 0x2207 2011-10-14T14:14:16Z 2011-10-14T14:19:57Z <p>Using inner product $(x,My)$ in Lanczos is equivalent to using $M^{-1}$ preconditioner in its preconditioned version. As far as I know, preconditioned version of Lanczos is used for generalized eigen-problems: $A x = \lambda M x$</p> <p>Preconditioned Lanczos operates with two black-boxes: $Ax$ multiplier and $M^{-1}x$ multiplier.</p>