A fast minimal residual algorithm for shifted unitary matrices

Abstract

A new iterative scheme is described for the solution of large linear systems of equations with a matrix of the form A = p U + { I , where p and { are constants, I/ is a unitary matrix and I is the identity matrix. We show that for such matrices a Krylov subspace basis can be generated by recursion formulas with few terms. This leads to a minimal residual algorithm that requires little storage and makes it possible to determine each iterate with fairly little arithmetic work. This algorithm provides a model for iterative methods for non-Hermitian linear systems of equations, in a similar way to the conjugate gradient and conjugate residual algorithms. Our iterative scheme illustrates that results by Faber and Manteuffel [3,4]

Recent computational developments in Krylov subspace methods for linear systems

“…Refined schemes can be obtained when further conditions are imposed to the spectrum location, as described by Fassbender and Ikramov [112]. The special case of normal matrices in the form of scaled and shifted orthogonal matrices, has received distinct attention, and specific implementations for large linear systems have been proposed; see Jagels and Reichel [189], [190] and the references therein.…”

The Arnoldi Process and GMRES for Nearly Symmetric Matrices

“…The latter type of recurrence relations had previously been applied to iterative methods in [11,12]; see also Arnold et al [2] for a recent application to QCD computations. The derivation of our iterative methods for (1.2) differs from the derivation by Barth and Manteuffel [4] of their schemes in that we do not apply properties of orthogonal polynomials on the unit circle.…”

On Optimal Short Recurrences for Generating Orthogonal Krylov Subspace Bases

“…The sufficiency result of Barth and Manteuffel implies that for shifted unitary matrices there exists a (3,1)-term recurrence of the form (4.2). This generalization of the isometric Arnoldi algorithm has been, prior to the work of Barth and Manteuffel, employed by Jagels and Reichel [29,30] for constructing a minimal residual method for solving linear systems with shifted unitary matrices. It is easily seen that (2, 1)-term (resp.…”