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In [[numerical linear algebra]], the '''Alternating Direction Implicit (ADI) method''' is an iterative method used to solve [[Sylvester equation|Sylvester]] matrix equations. It is a popular method for solving the large matrix equations that arise in [[systems theory]] and [[Control theory|control]],<ref name=":1">{{Cite journal|last=Simoncini|first=V.|date=2016|title=Computational Methods for Linear Matrix Equations|journal=SIAM Review|language=en|volume=58|issue=3|pages=377–441|doi=10.1137/130912839|issn=0036-1445|url=https://semanticscholar.org/paper/bf31e4aaa0f2f0cc1318d1742c60a409d68f12ce}}</ref> and can be formulated to construct solutions in a memory-efficient, factored form.<ref name=":2">{{Cite journal|last=Li|first=Jing-Rebecca|author1-link=Jing-Rebecca Li|last2=White|first2=Jacob|date=2002|title=Low Rank Solution of Lyapunov Equations|journal=SIAM Journal on Matrix Analysis and Applications|language=en|volume=24|issue=1|pages=260–280|doi=10.1137/s0895479801384937|issn=0895-4798}}</ref><ref name=":3">{{Cite journal|last=Benner|first=Peter|last2=Li|first2=Ren-Cang|last3=Truhar|first3=Ninoslav|date=2009|title=On the ADI method for Sylvester equations|journal=Journal of Computational and Applied Mathematics|volume=233|issue=4|pages=1035–1045|doi=10.1016/j.cam.2009.08.108|issn=0377-0427|bibcode=2009JCoAM.233.1035B}}</ref> It is also used to numerically solve [[Parabolic partial differential equation|parabolic]] and [[Elliptic partial differential equation|elliptic]] partial differential equations, and is a classic method used for modeling [[heat conduction]] and solving the [[diffusion equation]] in two or more dimensions.<ref name=":0">{{Citation|title=The numerical solution of parabolic and elliptic differential equations|year=1955|last1=Peaceman|last2=Rachford Jr.|first1=D. W.|first2=H. H.|journal=Journal of the Society for Industrial and Applied Mathematics|volume=3|issue=1|pages=28–41|doi=10.1137/0103003|mr=0071874|hdl=10338.dmlcz/135399|hdl-access=free}}.</ref> It is an example of an [[operator splitting]] method.<ref>*{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | ___location=New York | isbn=978-0-521-88068-8 | chapter=Section 20.3.3. Operator Splitting Methods Generally | chapter-url=http://apps.nrbook.com/empanel/index.html#pg=1052}}
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</math>
where the infimum is taken over all rational functions of degree <math>(K, K)</math>.<ref name=":5" /> This approximation problem is related to several results in [[potential theory]],<ref>{{Cite book|title=Logarithmic potentials with external fields|last=1944-|first=Saff, E. B.|others=Totik, V.|isbn=9783662033296|___location=Berlin|oclc=883382758|date = 2013-11-11}}</ref><ref>{{Cite journal|last=Gonchar|first=A.A.|date=1969|title=Zolotarev problems connected with rational functions|url=|journal=Mat. Sb. (N.S.)|volume=78 (120):4|issue=4|pages=640–654|bibcode=1969SbMat...7..623G|doi=10.1070/SM1969v007n04ABEH001107}}</ref> and was solved by [[Yegor Ivanovich Zolotarev|Zolotarev]] in 1877 for <math>E</math> = [a, b] and <math> F=-E.</math><ref>{{Cite journal|last=Zolotarev|first=D.I.|date=1877|title=Application of elliptic functions to questions of functions deviating least and most from zero|url=|journal=Zap. Imp. Akad. Nauk. St. Petersburg|volume=30|pages=1–59|via=}}</ref> The solution is also known when <math> E</math> and <math> F</math> are disjoint disks in the complex plane.<ref>{{Cite journal|last=Starke|first=Gerhard|date=July 1992|title=Near-circularity for the rational Zolotarev problem in the complex plane|journal=Journal of Approximation Theory|volume=70|issue=1|pages=115–130|doi=10.1016/0021-9045(92)90059-w|issn=0021-9045|doi-access=free}}</ref>
==== Heuristic shift parameter strategies ====
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