A Comparison of Two Equivalent Real Formulations for Complex-Valued Linear Systems

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Computer Sciences | Physical Sciences and Mathematics


Michael Heroux, Computer Science


Many iterative linear solver packages focus on real-valued systems and do not deal as well with complex-valued systems, even though preconditioned iterative methods typically apply to both real and complex-valued linear systems. Instead, commonly available packages such as PETSc (1) and Aztec (6) tend to focus on the real-valued systems, while the complex-valued systems are seen as a late addition. At the same time, by changing the complex problem into an equivalent real formulation (ERF), a real valued solver can be used. In this paper we consider two ERFs to find a solution for complex-valued linear systems. We investigate the spectral properties of each and show how each can be preconditioned to move eigen values in a cloud around the point (1,0) in the complex plane. Finally, we consider an interleaved formulation, combining each of the approaches, showing that the interleaved form achieves a better outcome than either approach alone. The effectiveness of interleaved ERFs is demonstrated by solving ill-conditioned complex-valued linear systems for a variety of large scale applications.