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2008•SIAM Journal on Matrix Analysis and ApplicationsRequires access

The Arnoldi Process and GMRES for Nearly Symmetric Matrices

Bernhard Beckermann, Lothar Reichel

Open publisher page 12 citations

Abstract

Matrices with a skew-symmetric part of low rank arise in many applications, including path following methods and integral equations. This paper explores the properties of the Arnoldi process when applied to such a matrix. We show that an orthogonal Krylov subspace basis can be generated with short recursion formulas and that the Hessenberg matrix generated by the Arnoldi process has a structure, which makes it possible to derive a progressive GMRES method. Eigenvalue computation is also considered.

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What this paper is about

Matrices with a skew-symmetric part of low rank arise in many applications, including path following methods and integral equations. This paper explores the properties of the Arnoldi process when applied to such a matrix. We show that an orthogonal Krylov subspace basis can be generated with short recursion formulas and that the Hessenberg matrix generated by the Arnoldi process has a structure, which makes it possible to derive a progressive GMRES method. Eigenvalue computation is also considered.

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OpenAlex reports 12 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

Matrices with a skew-symmetric part of low rank arise in many applications, including path following methods and integral equations. This paper explores the properties of the Arnoldi process when applied to such a matrix. We show that an orthogonal Krylov subspace basis can be generated with short recursion formulas and that the Hessenberg matrix generated by the Arnoldi process has a structure, which makes it possible to derive a progressive GMRES method. Eigenvalue computation is also considered.

Key concepts: Generalized minimal residual method, Krylov subspace, Arnoldi iteration, Mathematics, Recursion (computer science), Eigenvalues and eigenvectors, Applied mathematics, Orthogonalization

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