We are concerned with the minimal residual method combined with polynomial preconditioning for solving large linear systems Ax=b with indefinite Hermitian coefficient matrices A. The standard approach for choosing the polynomial preconditioner leads to preconditioned systems which are positive definite. Here, we investigate a different strategy which leaves the preconditioned coefficient matrix indefinite. More precisely, the polynomial preconditioner is designed to cluster the positive (negative) eigenvalues of A around 1 (around some negative constant). In particular, it is shown that such indefinite polynomial preconditioners can be obtained as the optimal solutions of a certain two-parameter family of Chebyshev approximation problems. The problem of selecting the parameters so that the resulting indefinite polynomial preconditioner speeds up the convergence of the minimal residual method optimally is also addressed. For this task, we propose an approach based on the concept of asymptotic convergence factors. Finally, some numerical examples of indefinite polynomial preconditioners are given. © 1991.
Freund, R. (1991). On polynomial preconditioning and asymptotic convergence factors for indefinite Hermitian matrices. Linear Algebra and Its Applications, 154–156(C), 259–288. https://doi.org/10.1016/0024-3795(91)90380-F