Biorthogonal rational Krylov subspace methods

Abstract

Extended Krylov subspace methods generalize classical Krylov, since also products with the inverse of the matrix under consideration are allowed. Recent advances have shown how to efficiently construct an orthogonal basis for the extended subspace, as well as how to build the highly structured projected matrix, named an extended Hessenberg matrix. It was shown that this alternative can lead to faster convergence for particular applications.In this text biorthogonal extended Krylov subspace methods are considered for general nonsymmetric matrices. We will show that the data resulting from the oblique projection can be expressed as a nonsymmetric tridiagonal matrix-pair. This is a direct generalization of the classical biorthogonal Krylov subspace method where the projection becomes a single nonsymmetric tridiagonal matrix. To obtain this result we first need to revisit the classical extended Krylov subspace algorithm and prove that the projected matrix can be written efficiently as a structured matrix-pair, where the structure can take several forms, such as, e.g., Hessenberg or inverse Hessenberg. Based on the compact storage of a tridiagonal matrix-pair in the biorthogonal setting, we can develop short recurrences.The numerical experiments confirm the validity of the approach.

Generation of orthogonal rational functions by procedures for structured matrices

“…The Hessenberg pencil IEP is solved by applying rational Arnoldi iteration [23,24] in Section 3.1 and in our setting this iteration cannot break down. The solution of the tridiagonal pencil IEP is discussed in Section 3.2 and relies on a rational Lanczos iteration [28]. Due to its biorthogonal nature, this iteration can break down an therefore we must rely on our assumption.…”

“…Thus a tridiagonal pencil (T, S), where both the subdiagonal and superdiagonal elements should satisfy some restrictions on their ratios, must be constructed. These ratio restrictions guarantee that the corresponding rational Krylov subspaces have the appropriate poles [28]. Solving this IEP corresponds to computing recurrence coefficients of one of two sequences of biorthogonal rational functions, orthogonal with respect to a discrete bilinear form.…”

“…The rational Lanczos iteration [28] constructs a pair of biorthogonal bases V, W for rational Krylov subspaces K m (A, v; Ξ) and K m (A H , w; Ψ), respectively. Here…”

“…Section 3 uses the relation of structured matrices to rational Krylov subspace methods, to provide iterations to solve the main problem. These iterations are the rational Arnoldi iteration [24] and the rational Lanczos iteration [28]. Solution procedures based on updating the IEP are introduced in Section 4 which, in the orthogonal case, are numerically stable.…”

Non-unitary CMV-decomposition

“…Note that the structures of the matrices of recurrence coe cients appearing in Theorem 2 and Theorem 4 are known [9]. The contribution of this manuscript is the procedure to compute these matrices in an e cient manner.…”