Author:

Abstract:

It will be shown that extended Krylov subspaces —under some assumptions— can be computed approximately without any explicit inversion or system solves involved. Instead we do the necessary computations in an implicit way using the information from an enlarged standard Krylov subspace. For Krylov spaces the matrices capturing the recurrence coefficients of the orthogonal basis can be retrieved by projecting the original matrix on a particular orthogonal basis of the associated Krylov space. It is also well-known that for (extended) Krylov spaces of full dimension, this matrix can be obtained directly via similarity transformations on the original matrix. In this talk the iterative and the direct similarity approach are combined. First, an orthogonal basis of a large standard Krylov subspace is constructed iteratively. Second, cleverly chosen similarity transformations are executed to alter the matrix of recurrences, thereby also changing the orthogonal basis vectors spanning the large Krylov space. Finally, only a few of the new basis vectors are retained resulting in an orthogonal basis approximately spanning a chosen extended Krylov subspace K_{l,p}(A,v) = span {A^{-p+1}v, ... , A^{-1}v, v, Av, A^{2}v, ... , A^{l-1}v}. Numerical experiments will reveal advantages and disadvantages of this approach.