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Cholesky Factorization

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A symmetric positive definite matrix $ S$ admits a unique factorization

$\displaystyle S= R^{\operatorname t} R\,,

where $ R$ is an upper triagonal matrix with positive diagonal entries.

The factorization can be computed by solving the equations

$\displaystyle s_{j,k} = r_{1,j} r_{1,k} + \cdots + r_{j,j} r_{j,k} \,, \quad
k \geq j,

successively for $ j= 1, 2, \hdots$. For each $ j$, we first compute

$\displaystyle r_{j,j} = \sqrt{s_{j,j}-\sum\limits_{i<j}r_{i,j}^2}

and then, simultaneously for $ k=j+1 , \hdots , n$,

$\displaystyle r_{j,k}= \left( s_{j,k} - \sum\limits_{i<j} r_{i,j} r_{i,k}
\right)/ r_{j,j}\,,

in both cases using the previously determined values.
(Authors: Höllig/Pfeil/Walter)


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  automatically generated 4/24/2007