Difference between revisions of "Manuals/calci/CHOLESKY"
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</math> | </math> | ||
| − | Algorithm | + | :Algorithm |
| − | 1 | + | #1 Determine <math>l_{11}</math> and <math>L_{21}</math>\\ |
<math>l_{11}</math> = <math>\sqrt{a_{11}}</math> | <math>l_{11}</math> = <math>\sqrt{a_{11}}</math> | ||
Revision as of 05:41, 10 April 2015
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle arr} is the array of numeric elements
Description
- 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle LL^{T}}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L}
is lower triangular with positive diagonal elements
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^{T}}
is is the conjugate transpose value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L}
- Every Hermitian positive-definite matrix has a unique Cholesky decomposition.
- Here Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle CHOLESKY(arr)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle arr} is set of values to find the factorization value.
- Partition matrices in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle LL^{T}} is
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{11}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_{21}} \\
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_{11}} = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{a_{11}}}
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 |