A fast minimal residual algorithm for shifted unitary matrices

✍ Scribed by Carl F. Jagels; Lothar Reichel

• Publisher: John Wiley and Sons
• Year: 1994
• Tongue: English
• Weight: 649 KB
• Volume: 1
• Category: Article
• ISSN: 1070-5325
• DOI: 10.1002/nla.1680010604

✦ Synopsis

Abstract

A new iterative scheme is described for the solution of large linear systems of equations with a matrix of the form A = ρ__U__ + ζ__I__, where ρ and ζ are constants, U 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] on the existence of conjugate gradient algorithms with short recurrence relations, and related results by Joubert and Young [13], can be extended.