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**Solving complex-valued linear systems via equivalent real formulations.**
*(English)*
Zbl 0992.65020

Summary: Most preconditioned iterative methods apply to both real- and complex-valued linear systems. At the same time, most iterative linear solver packages available today focus exclusively on real-valued systems or deal with complex-valued systems as an afterthought. By recasting the complex problem in a real formulation, a real-valued solver can be applied to the equivalent real system.

On one hand, real formulations have been dismissed due to their unfavorable spectral properties. On the other hand, using an equivalent preconditioned real formulation can be very effective. We give theoretical and experimental evidence that an equivalent real formulation is useful in a number of practical situations. Furthermore, we show how to use the advanced features of modern solver packages to formulate equivalent real preconditioners that are computationally efficient and mathematically identical to their complex counterparts.

The effectiveness of equivalent real formulations is demonstrated by solving ill-conditioned complex-valued linear systems for a variety of large scale applications. Moreover, the circumstances under which certain equivalent real formulations are competitive is more clearly delineated.

On one hand, real formulations have been dismissed due to their unfavorable spectral properties. On the other hand, using an equivalent preconditioned real formulation can be very effective. We give theoretical and experimental evidence that an equivalent real formulation is useful in a number of practical situations. Furthermore, we show how to use the advanced features of modern solver packages to formulate equivalent real preconditioners that are computationally efficient and mathematically identical to their complex counterparts.

The effectiveness of equivalent real formulations is demonstrated by solving ill-conditioned complex-valued linear systems for a variety of large scale applications. Moreover, the circumstances under which certain equivalent real formulations are competitive is more clearly delineated.

### MSC:

65F10 | Iterative numerical methods for linear systems |

65E05 | General theory of numerical methods in complex analysis (potential theory, etc.) |

65F35 | Numerical computation of matrix norms, conditioning, scaling |

65F50 | Computational methods for sparse matrices |

65Y15 | Packaged methods for numerical algorithms |