Solving complex-valued linear systems via equivalent real formulations

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Most algorithms used in preconditioned iterative methods are generally applicable to complex valued linear systems, with real valued linear systems simply being a special case. However, most iterative solver packages available today focus exclusively on real valued systems, or deal with complex valued systems as an afterthought. One obvious approach to addressing this problem is to recast the complex problem into one of a several equivalent real forms and then use a real valued solver to solve the related system. However, well-known theoretical results showing unfavorable spectral properties for the equivalent real forms have diminished enthusiasm for this approach. At ... continued below

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13 p.

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DAY,DAVID M. & HEROUX,MICHAEL A. May 22, 2000.

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This article is part of the collection entitled: Office of Scientific & Technical Information Technical Reports and was provided by UNT Libraries Government Documents Department to Digital Library, a digital repository hosted by the UNT Libraries. It has been viewed 15 times . More information about this article can be viewed below.

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  • Sandia National Laboratories
    Publisher Info: Sandia National Labs., Albuquerque, NM, and Livermore, CA (United States)
    Place of Publication: Albuquerque, New Mexico

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Most algorithms used in preconditioned iterative methods are generally applicable to complex valued linear systems, with real valued linear systems simply being a special case. However, most iterative solver packages available today focus exclusively on real valued systems, or deal with complex valued systems as an afterthought. One obvious approach to addressing this problem is to recast the complex problem into one of a several equivalent real forms and then use a real valued solver to solve the related system. However, well-known theoretical results showing unfavorable spectral properties for the equivalent real forms have diminished enthusiasm for this approach. At the same time, experience has shown that there are situations where using an equivalent real form can be very effective. In this paper, the authors explore this approach, giving both theoretical and experimental evidence that an equivalent real form can be useful for a number of practical situations. Furthermore, they show that by making good use of some of the advance features of modem solver packages, they can easily generate equivalent real form preconditioners that are computationally efficient and mathematically identical to their complex counterparts. Using their techniques, they are able to solve very ill-conditioned complex valued linear systems for a variety of large scale applications. However, more importantly, they shed more light on the effectiveness of equivalent real forms and more clearly delineate how and when they should be used.

Physical Description

13 p.

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OSTI as DE00756121

Medium: P; Size: 13 pages

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  • Journal Name: SIAM Journal of Scientific Computing; Other Information: Submitted to SIAM Journal of Scientific Computing

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  • Report No.: SAND2000-1274J
  • Grant Number: AC04-94AL85000
  • Office of Scientific & Technical Information Report Number: 756121
  • Archival Resource Key: ark:/67531/metadc710032

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  • May 22, 2000

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  • Sept. 12, 2015, 6:31 a.m.

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  • April 7, 2017, 1:13 p.m.

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DAY,DAVID M. & HEROUX,MICHAEL A. Solving complex-valued linear systems via equivalent real formulations, article, May 22, 2000; Albuquerque, New Mexico. (digital.library.unt.edu/ark:/67531/metadc710032/: accessed September 26, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.