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 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 ${\Bbb Q}$.