Alternating-direction implicit method: Difference between revisions

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The equation <math> AX-XB=C</math> has a unique solution if and only if <math> \sigma(A) \cap \sigma(B) = \emptyset</math>, where <math> \sigma(M) </math> is the [[Spectrum of a matrix|spectrum]] of <math>M</math>.<ref name=":1" /> However, the ADI method performs especially well when <math>\sigma(A)</math> and <math>\sigma(B)</math> are well-separated, and <math>A</math> and <math>B</math> are [[Normal matrix|normal matrices]]. These assumptions are met, for example, by the Lyapunov equation <math>AX + XA^* = C</math> when <math>A</math> is [[Positive-definite matrix|positive definite]]. Under these assumptions, near-optimal shift parameters are known for several choices of <math>A</math> and <math>B</math>.<ref name=":4" /><ref name=":5" /> Additionally, a priori error bounds can be computed, thereby eliminating the need to monitor the residual error in implementation.
 
The ADI method can still be applied when the above assumptions are not met. The use of suboptimal shift parameters may adversely affect convergence,<ref name=":1" /> and convergence is also affected by the non-normality of <math>A</math> or <math>B</math> (sometimes advantageously).<ref name=":6">{{Cite thesis|last=Sabino|first=J|date=2007|title=Solution of large-scale Lyapunov equations via the block modified Smith method|journal=PHDPhD Diss., Rice Univ.|hdl=1911/20641|type=Thesis}}</ref> [[Krylov subspace]] methods, such as the Rational Krylov Subspace Method,<ref>{{Cite journal|last1=Druskin|first1=V.|last2=Simoncini|first2=V.|date=2011|title=Adaptive rational Krylov subspaces for large-scale dynamical systems|journal=Systems & Control Letters|volume=60|issue=8|pages=546–560|doi=10.1016/j.sysconle.2011.04.013|issn=0167-6911}}</ref> are observed to typically converge more rapidly than ADI in this setting,<ref name=":1" /><ref name=":3" /> and this has led to the development of hybrid ADI-projection methods.<ref name=":3" />
 
=== Shift-parameter selection and the ADI error equation ===