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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>
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Revision as of 07:20, 14 September 2005
