Difference between revisions of "Manuals/calci/CHOLESKY"
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*The Cholesky Factorization is only defined for symmetric or Hermitian positive definite matrices. | *The Cholesky Factorization is only defined for symmetric or Hermitian positive definite matrices. | ||
*Every positive definite matrix A can be factored as <math>A</math> = <math>LL^{T}</math> | *Every positive definite matrix A can be factored as <math>A</math> = <math>LL^{T}</math> | ||
| − | |||
<math>L</math> is lower triangular with positive diagonal elements | <math>L</math> is lower triangular with positive diagonal elements | ||
<math>L^{T}</math> is is the conjugate transpose value of <math>L</math> | <math>L^{T}</math> is is the conjugate transpose value of <math>L</math> | ||
*Every Hermitian positive-definite matrix has a unique Cholesky decomposition. | *Every Hermitian positive-definite matrix has a unique Cholesky decomposition. | ||
*Here <math>CHOLESKY(arr)</math>, <math>arr</math> is set of values to find the factorization value. | *Here <math>CHOLESKY(arr)</math>, <math>arr</math> is set of values to find the factorization value. | ||
| − | *Partition matrices in | + | *Partition matrices in <math>A</math> = <math>LL^{T}</math> is |
| + | <math> | ||
| + | \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} | ||
| + | </math> | ||
==ZOS Section== | ==ZOS Section== | ||
Revision as of 05:20, 10 April 2015
CHOLESKY(arr)
- is the array of numeric elements
is the array of numeric elementsDescription
- This function gives the value of Cholesky factorization.
- It is called Cholesky Decomposition or Cholesky Factorization.
- The Cholesky Factorization is only defined for symmetric or Hermitian positive definite matrices.
- Every positive definite matrix A can be factored as =
= 
is lower triangular with positive diagonal elements is is the conjugate transpose value of
is lower triangular with positive diagonal elements
is is the conjugate transpose value of - Every Hermitian positive-definite matrix has a unique Cholesky decomposition.
- Here , is set of values to find the factorization value.
- Partition matrices in = is
,
ZOS Section
Examples
1. =CHOLESKY([[16,32,12],[12, 18, 0],[ -5, 0, 11]])
| 4 | 0 | 0 |
| 3 | 3 | 0 |
| -1.25 | 1.25 | 2.80624 |
2. =CHOLESKY([[25, 15, -5],[15, 18, 0],[ -5, 0, 11]])
| 5 | 0 | 0 |
| 3 | 3 | 0 |
| -1 | 1 | 3 |