Conjugate gradient squared method: Difference between revisions

Revision as of 14:59, 31 December 2023

Revision as of 01:19, 2 January 2024

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{{Short description|Algorithm for solving matrix-vector equations}}

In [[numerical linear algebra]], the '''conjugate gradient squared method (CGS)''' is an [[iterative method|iterative]] algorithm for solving [[systems of linear equations]] of the form <math>A{\bold x} = {\bold b}</math>, particularly in cases where computing the [[transpose]] <math>A^T</math> is impractical.<ref name="mathworld">{{cite web|title=Conjugate Gradient Squared Method|author1=Noel Black|author2=Shirley Moore|publisher=[[MathWorld|Wolfram Mathworld]]|url=https://mathworld.wolfram.com/ConjugateGradientSquaredMethod.html}}</ref> The CGS method was developed as an improvement to the [[Biconjugatebiconjugate gradient method]].<ref name="matlab">{{cite web|title=cgs|author=[[Mathworks]]|website=[[Matlab]] documentation|url=https://au.mathworks.com/help/matlab/ref/cgs.html}}</ref><ref name="vorst03">{{cite book|author=[[Henk van der Vorst]]|title=Iterative Krylov Methods for Large Linear Systems|chapter=Bi-Conjugate Gradients|year=2003|publisher=Cambridge University Press |isbn=0-521-81828-1}}</ref><ref name="SIAM">{{cite journal|title=CGS, A Fast Lanczos-Type Solver for Nonsymmetric Linear systems|author=Peter Sonneveld|journal=SIAM Journal on Scientific and Statistical Computing|volume=10|issue=1|pages=36–52|date=1989|url=https://www.proquest.com/docview/921988114|url-access=limited|doi=10.1137/0910004|id={{ProQuest|921988114}} }}</ref>

== Background ==

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As with the [[conjugate gradient method]], [[biconjugate gradient method]], and similar iterative methods for solving systems of linear equations, the CGS method can be used to find solutions to multi-variable [[optimisation problems]], such as [[power-flow study|power-flow analysis]], [[hyperparameter optimisation]], and [[facial recognition system|facial recognition]].<ref>{{cite web|title=Conjugate gradient methods|author1=Alexandra Roberts|author2=Anye Shi|author3=Yue Sun|access-date=2023-12-26|publisher=[[Cornell University]]|url=https://optimization.cbe.cornell.edu/index.php?title=Conjugate_gradient_methods}}</ref>

== The Algorithm ==

The algorithm is as follows:<ref>{{cite book|author1=R. Barrett|author2=M. Berry|author3=T. F. Chan|author4=J. Demmel|author5=J. Donato|author6=J. Dongarra|author7=V. Eijkhout|author8=R. Pozo|author9=C. Romine|author10=H. Van der Vorst|title=Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, 2nd Edition|publisher=SIAM|year=1994|url=https://netlib.org/linalg/html_templates/Templates.html}}</ref>