A generalized isometric Arnoldi algorithm

Abstract

This paper describes a generalization of the isometric Arnoldi algorithm and shows that it can be interpreted as a structured form of modified Gram–Schmidt. Given an isometry A, the algorithm efficiently orthogonalizes the columns of a sequence of matrices M j for j ⩾ 0 (with M −1 = 0) for which the columns of M j − AM j−1 are in a fixed finite dimensional subspace for each j ⩾ 0. The dimension of the subspace is analogous to displacement rank in the generalized Schur algorithm. The algorithm is described in terms of projections and inner products. This is in contrast to orthogonalization methods based on the generalized Schur algorithm, for which Cholesky factorization is central to the computation. Numerical experiments suggest that, relative to a generalized Schur algorithm, the new algorithm improves the numerical orthogonality of the computed orthonormal sequence.