A deflation technique for linear systems of equations

Iterative methods for solving linear systems of equations can be very efficient if the structure of the coefficient matrix can be exploited to accelerate the convergence of the iterative process. However, for classes of problems for which suitable preconditioners cannot be found or for which the ite...

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Bibliographic Details
Published inSIAM journal on scientific computing Vol. 19; no. 4; pp. 1245 - 1260
Main Authors BURRAGE, K, ERHEL, J, POHL, B, WILLIAMS, A
Format Journal Article
LanguageEnglish
Published Philadelphia, PA Society for Industrial and Applied Mathematics 01.07.1998
Subjects
Algorithms
Applied mathematics
Deflation
Eigenvalues
Exact sciences and technology
Iterative methods
Mathematics
Methods
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Sciences and techniques of general use
Reverse Gauss Seidel method
Linear system
Jacobi method
Krylov subspace
Iterative method
Jacobi matrix
Deflation method
Algorithm
Gauss Seidel method
Preconditioning
Jacobi coupling
Equation system
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Summary:Iterative methods for solving linear systems of equations can be very efficient if the structure of the coefficient matrix can be exploited to accelerate the convergence of the iterative process. However, for classes of problems for which suitable preconditioners cannot be found or for which the iteration scheme does not converge, iterative techniques may be inappropriate. This paper proposes a technique for deflating the eigenvalues and associated eigenvectors of the iteration matrix which either slow down convergence or cause divergence. This process is completely general and works by approximating the eigenspace ${\Bbb P}$ corresponding to the unstable or slowly converging modes and then applying a coupled iteration scheme on ${\Bbb P}$ and its orthogonal complement ${\Bbb Q}$.
ISSN:1064-8275
1095-7197
DOI:10.1137/S1064827595294721