Abstract
We present an extended Krylov subspace analogue of the two-sided Lanczos method, i.e., a method which, given a nonsingular matrix A and vectors b, c with \(\left \langle {{\mathbf {b}},{\mathbf {c}}}\right \rangle \neq 0\), constructs bi-orthonormal bases of the extended Krylov subspaces \({\mathcal {E}}_{m}(A,{\mathbf {b}})\) and \({\mathcal {E}}_{m}(A^{T}\!,{\mathbf {c}})\) via short recurrences. We investigate the connection of the proposed method to rational moment matching for bilinear forms c T f(A)b, similar to known results connecting the two-sided Lanczos method to moment matching. Numerical experiments demonstrate the quality of the resulting approximations and the numerical behavior of the new extended Krylov subspace method.
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