Stable solutions of linear systems involving long chain of matrix multiplications

This paper is concerned with solving linear system (In+BL⋯B2B1)x=b(In+BL⋯B2B1)x=b arising from the Green’s function calculation in the quantum Monte Carlo simulation of interacting electrons. The order of the system and integer L   are adjustable. Also adjustable is the conditioning of the coefficient matrix to give rise an extreme ill-conditioned system. Two numerical methods based on the QR decomposition with column pivoting and the singular value decomposition, respectively, are studied in this paper. It is proved that the computed solution x∼ by each of the methods is weakly backward stable   in the sense that the computed x∼ is close to the exact solution of a nearby linear system[In+(BL+ΔBL)⋯(B2+ΔB2)(B1+ΔB1)]x∼=bwith each ΔBiΔBi small in norm relatively to BiBi.