Manuals/calci/CHOLESKYFACTORIZATION

CHOLESKYFACTORIZATION(Matrix)

• $Matrix$ is the array of numeric elements

Description

• This function gives the value of Cholesky factorization.
• It is called Cholesky Decomposition or Cholesky Factorization.
• In $CHOLESKYFACTORIZATION(Matrix)$,$Matrix$ is the set of values.
• The Cholesky Factorization is only defined for symmetric or Hermitian positive definite matrices.
• Every positive definite matrix A can be factored as $A$ = $LL^{T}$

$L$ is lower triangular with positive diagonal elements
$L^{T}$ is is the conjugate transpose value of $L$

• Every Hermitian positive-definite matrix has a unique Cholesky decomposition.
• Here $CHOLESKYFACTORIZATION(Matrix)$, $arr$ is set of values to find the factorization value.
• Partition matrices in $A$ = $LL^{T}$ is

${\begin{bmatrix}a_{11}&A_{21}^{T}\\A_{21}&A_{22}\end{bmatrix}}={\begin{bmatrix}l_{11}&0\\L_{21}&L_{22}\end{bmatrix}}{\begin{bmatrix}l_{11}&L_{21}^{T}\\0&L_{22}^{T}\end{bmatrix}}={\begin{bmatrix}l_{11}^{2}&L_{11}L_{21}^{T}\\L_{11}L_{21}&L_{21}L_{21}^{T}+L_{22}L_{22}^{T}\end{bmatrix}}$

Algorithm

1. Determine $l_{11}$ and $L_{21}$
$l_{11}$ = ${\sqrt {a_{11}}}$ $L_{21}$ = ${\frac {1}{l_{11}}}A_{21}$
2. Compute $L_{22}$ from
$A_{22}-L_{21}L_{21}^{T}$ = $L_{22}L_{22}^{T}$
• this is a Cholesky Factorization of order $n-1$

• If the matrix A is Hermitian and positive semi-definite, then it still has a decomposition of the form A = LL^T if the diagonal entries of L are allowed to be zero.
• Also A can be written as LL^T for some invertible L, lower triangular or otherwise, then A is Hermitian and positive definite.

Examples

1. CHOLESKYFACTORIZATION([[25, 15, -5],[15, 18, 0],[ -5, 0, 11]])

Result

5	0	0
3	3	0
-1	1	3

2. CHOLESKYFACTORIZATION([[8,14],[10,32]])

Result

2.8284271247461903	0
3.5355339059327373	4.415880433163924