Mo logo [home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff] german flag
Mathematics-Online lexicon:

Cholesky Factorization

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview
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)

Download:

( .m, 368 ,  24.04.2007)
[Annotations] [Examples] [Links]
  automatically generated 4/24/2007