A Rational Function Preconditioner For Indefinite Sparse Linear Systems

This paper introduces a rational function preconditioner for linear systems with indefinite sparse matrices $A$. By resorting to rational functions of $A$, the algorithm decomposes the spectrum of $A$ into two disjoint regions and approximates the restriction of $A^{-1}$ on these regions separately. We show a systematic way to construct these rational functions so that they can be applied stably and inexpensively. An attractive feature of the proposed approach is that the construction and application of the preconditioner can exploit two levels of parallelism. Moreover, the proposed preconditioner can be modified at a negligible cost into a preconditioner for a nearby matrix of the form $A-cI$, which can be useful in some applications. The efficiency and robustness of the proposed preconditioner are demonstrated on a few tests with challenging model problems, including problems arising from the Helmholtz equation in three dimensions.

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