Editing Cholesky decomposition (section)

{{Short description|Matrix decomposition method}}
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In [[linear algebra]], the '''Cholesky decomposition''' or '''Cholesky factorization''' (pronounced {{IPAc-en|ʃ|ə|ˈ|l|ɛ|s|k|i}} {{respell|shə|LES|kee}}) is a [[matrix decomposition|decomposition]] of a [[Hermitian matrix|Hermitian]], [[positive-definite matrix]] into the product of a [[lower triangular matrix]] and its [[conjugate transpose]], which is useful for efficient numerical solutions, e.g., [[Monte Carlo simulation]]s. It was discovered by [[André-Louis Cholesky]] for real matrices, and posthumously published in 1924.<ref name="Bulletin">{{Cite journal|last=Benoit|date=1924|title=Note sur une méthode de résolution des équations normales provenant de l'application de la méthode des moindres carrés à un système d'équations linéaires en nombre inférieur à celui des inconnues (Procédé du Commandant Cholesky)|journal=[[Bulletin Géodésique]]|language=fr|volume=2|pages=66–67| doi=10.1007/BF03031308}}</ref>
When it is applicable, the Cholesky decomposition is roughly twice as efficient as the [[LU decomposition]] for solving [[System of linear equations|systems of linear equations]].<ref name="NR">{{cite book|last=Press|first=William H.|author2=Saul A. Teukolsky|author3=William T. Vetterling|author4=Brian P. Flannery|title=Numerical Recipes in C: The Art of Scientific Computing|edition=second|publisher=Cambridge University England EPress|year=1992|page=[https://archive.org/details/numericalrecipes0865unse/page/994 994]|url=https://archive.org/details/numericalrecipes0865unse/page/994|isbn=0-521-43108-5|access-date=2009-01-28}}</ref>