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Solution of a Symmetric Positive Linear System

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A linear system $ Sx=b$ with symmetric positive definite matrix $ S$ can be solved with the aid of the Cholesky factorization $ S = R^{\operatorname t} R$:

$\displaystyle R^{\operatorname t} \underbrace{Rx}_y =b \; \Longleftrightarrow \; R^{\operatorname t} y=b \; \wedge \; Rx=y \,. $

The solutions $ y$ and $ x$ of the two triangular systems are determined via forward and backward substitution, respectively.

In particular, the Cholesky factorization can be used for solving the normal equations

$\displaystyle \underbrace{A^{\operatorname t}A}_S x = \underbrace{A^{\operatorname t}c}_b $

if the $ m\times n$ matrix $ A$ has maximal rank $ n$.
(Authors: Höllig/Pfeil/Walter)
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  automatically generated 4/24/2007