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HSL_MI02: Symmetric possibly-indefinite system: SYMMBK method

This routine uses the SYMMBK method to solve the \(n \times n\) symmetric but possibly indefinite linear system \(\mathbf{Ax} = \mathbf{b}\), optionally using preconditioning. If \(\mathbf{P} \mathbf{P} ^T\) is the preconditioning matrix, the routine actually solves the preconditioned system

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

with \(\bar{\mathbf{A}} = \mathbf{P} \mathbf{A}\mathbf{P} ^T\) and \(\bar{\mathbf{b}} = \mathbf{P} \mathbf{b}\) and recovers the solution \(\mathbf{x} = \mathbf{P} ^T \bar{\mathbf{x}}\). Reverse communication is used for preconditioning operations and matrix-vector products of the form \(\mathbf{Az}\).