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Incomplete Cholesky Factorization

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== Cholesky Factorization ==
 
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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>
 
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<math> v \bullet A \bullet v > 0</math>    <math>  \forall v </math> <br>
 
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In cholesky factorization we construct a lower triangular matrix '''L''' whose transpose '''L<sup>T</sup>''' can itself serve as upper triangular part. <br>
 
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In other words we have <br>
 
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'''L <math>\bullet</math>L<sup>T</sup> = A ''' <br>
 
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=== Algorithm for full matrix ''A'' ===
 
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We have by definition
 
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<math>
 
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L_{ij}^T  = L_{ji}
 
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</math> <br>
 
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From this we can easily obtain<br>
 
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'''for := 1 step 1 until N do''' <br>
 
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<math>
 
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L_{ii}  = \left( {a_{ii}  - \sum\limits_{k = 1}^{i - 1} {L_{ik}^2 } } \right)^{{1 \over 2}}
 
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</math><br>
 
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and <br>
 
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<math>
 
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L_{ji}  = {1 \over {L_{ii} }}\left( {a_{ij}  - \sum\limits_{k = 1}^{i - 1} {L_{ik} L_{jk} } } \right)
 
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</math> ; where j = i+1, i+2, ..., N <br>
 
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'''end (i-loop)''' <br>
 

Latest revision as of 09:47, 17 December 2008

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