The induced dimension reduction (IDR) method, which has been introduced as a transpose-free Krylov space method for solving nonsymmetric linear systems, can also be used to determine approximate eigenvalues of a matrix or operator. The IDR residual polynomials are the products of a residual polynomial constructed by successively appending linear smoothing factors and the residual polynomials of a two-sided (block) Lanczos process with one right-hand side and several left-hand sides. The Hessenberg matrix of the OrthoRes version of this Lanczos process is explicitly obtained in terms of the scalars defining IDR by deflating the smoothing factors. The eigenvalues of this Hessenberg matrix are approximations of eigenvalues of the given matrix or operator.
We generalize an augmented rounding error result that was proven for the symmetric Lanczos process in [SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2347-2359, to the two-sided (usually unsymmetric) Lanczos process for tridiagonalizing a square matrix. We extend the analysis to more general perturbations than rounding errors in order to provide tools for the analysis of inexact and related methods. The aim is to develop a deeper understanding of the behavior of all these methods. Our results take the same form as those for the symmetric Lanczos process, except for the bounds on the backward perturbation terms (the generalizations of backward rounding errors for the augmented system). In general we cannot derive tight a priori bounds for these terms as was done for the symmetric process, but a posteriori bounds are feasible, while bounds related to certain properties of matrices would be theoretically desirable.
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