Incomplete Cholesky factorization

== Cholesky Factorization ==

When the square matrix '''A''' is symmetric and positive definite then it has an efficient triangular decomposition. ''Symmetric'' means that a<sub>ij</sub> = a<sub>ji</sub> for i,j = 1, ... , N. While ''positive definite'' means that <br>

<math> v \bullet A \bullet v > 0</math>    <math>  \forall v </math> <br>

In cholesky factorization we construct a lower triangular matrix '''L''' whose transpose '''L<sup>T</sup>''' can itself serve as upper triangular part. <br>
In other words we have <br>
'''L <math>\bullet</math>L<sup>T</sup> = A ''' <br>

=== Algorithm for full matrix ''A'' ===

We have by definition 
<math>
L_{ij}^T  = L_{ji} 
</math> <br>
From this we can easily obtain<br>

----
'''for := 1 step 1 until N do''' <br>

<math>
L_{ii}  = \left( {a_{ii}  - \sum\limits_{k = 1}^{i - 1} {L_{ik}^2 } } \right)^{{1 \over 2}} 
</math><br>
and <br>
<math>
L_{ji}  = {1 \over {L_{ii} }}\left( {a_{ij}  - \sum\limits_{k = 1}^{i - 1} {L_{ik} L_{jk} } } \right)
</math> ; where j = i+1, i+2, ..., N <br>

'''end (i-loop)''' <br>
----