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Version 1.3.1

2nd December 2015

MI24: Unsymmetric system: GMRES (generalized minimal residual) method

This routine uses the Generalized Minimal Residual method with restarts every \(m\) iterations, GMRES(\(m\)), to solve the \(n \times n\) unsymmetric linear system \(\mathbf{Ax} = \mathbf{b}\), optionally using preconditioning. If \(\mathbf{P} _L\), \(\mathbf{P} _R\) are left and right preconditioning matrices, the routine actually solves the preconditioned system

\[\bar{\mathbf{A}} \bar{\mathbf{x}} = \bar{\mathbf{b}} ,\]

with \(\bar{\mathbf{A}} = \mathbf{P} _L\mathbf{A}\mathbf{P} _R\) and \(\bar{\mathbf{b}} = \mathbf{P} _L \mathbf{b}\). The solution may be recovered as \(\mathbf{x} = \mathbf{P} _R\bar{\mathbf{x}}\). If \(\mathbf{P} _L = \mathbf{I}\), preconditioning is said to be from the right, if \(\mathbf{P} _R = \mathbf{I}\), it is said to be from the left, and otherwise it is from both sides. Reverse communication is used for preconditioning operations and matrix-vector products of the form \(\mathbf{Az}\).