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A deflation technique for linear systems of equations

by K. Burrage and J. Erhel and B. Pohl

(Report number 1994-02)

Abstract
Iterative methods for solving linear systems of equations can be very efficient in a sequential or parallel computing environment 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 are 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 $\Pn$ corresponding to the unstable or slowly converging modes and then applying a coupled iteration scheme on $\Pn$ and its orthogonal complement $\Qn.$

Keywords: linear systems, deflation, preconditioning, iterative %techniques

@Techreport{BEP94_157,
  author = {K. Burrage and J. Erhel and B. Pohl},
  title = {A deflation technique for linear systems of equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1994-02},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1994/1994-02.pdf },
  year = {1994}
}

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