Twofold deflation preconditioning of linear algebraic systems. I. Theory

In this paper, preconditioning of linear algebraic systems with symmetric positive-definite coefficient matrices by deflation is considered. The twofold deflation technique for simultaneously deflating largest $s$ and smallest $s$ eigenvalues using an appropriate deflating subspace of dimension $s$ is suggested. The possibility of using the extreme Ritz vectors of the coefficient matrix for deflation is analyzed. Bibliography: 15 titles.