Eigenvalue Computations Based on IDR

Abstract

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.

Revisiting (k, ℓ)-step methods

“…The representation (4.11) justifies the approximation of some of the eigenvalues of A by the spectrum of the pencil (H M , C m ), where H M denotes the "upper part" of H M ; see [9].…”

An induced dimension reduction algorithm to approximate eigenpairs of large nonsymmetric matrices

“…Update [6] for details). The IDR factorization is a standard Hessenberg relation which can be used to approximate eigenvalues and eigenvectors of a sparse matrix.…”

“…This method has obtained attention and different variants have been proposed to improved its convergence and numerical stability, for example [3,4,5]. Recently, in [6], the IDR(s) method has been adapted to approximate eigenpairs (λ , x) of the matrix A, i.e. Ax = λ x, with λ ∈ C, and x = 0 ∈ C n .…”

On the Convergence Behavior of IDR($s$) and Related Methods

“…The obvious choice is to drop the variant for n = j(s + 1) − 1 and to keep the variant based on the first residual in the new space. See also [13]. The ambiguity does not occur in the variant of IDR(s) described in [30], since in this variant the critical residuals satisfy…”

“…Relations with Bi-CGSTAB are described in [19] and the combination of IDR(s) with BiCGSTAB( ) in [20]. Recent extensions to IDR(s) can be found in [3,29], and IDR(s)-eigenvalue algorithms have been developed [13]. …”