Polminres and Polcg for Solving P(a)x = B Polminres and Polcg for Solving P(a)x = B

We investigate two iterative methods for solving nonsingular linear systems P(A)x = b; () where P(A) = P p i=0 i A i denotes a matrix polynomial and A is hermitian. The key idea of the methods is to choose the approximate solutions x j from the Krylov subspaces with respect to the matrix A instead of P(A). Thus, only one matrix vector multiplication (Gemv) is necessary to extend the current Krylov space, whereas classical methods for solving (), like MINRES or CG, require p Gemvs in each step. Our algorithms are based on short recurrences, and the iterates minimize an appropriate residual norm. The methods are analyzed theoretically, and their eeciency is illustrated for linear systems from lattice Quantum Chromodynamics (lattice QCD) and Tikhonov-Phillips regularization.