Difference between revisions of "Manuals/calci/CHOLESKY"
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*Every Hermitian positive-definite matrix (and thus also every real-valued symmetric positive-definite matrix) has a unique Cholesky decomposition. | *Every Hermitian positive-definite matrix (and thus also every real-valued symmetric 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 as <math>A</math> = <math>LL^{T}<math> (Please take the description from http://www.seas.ucla.edu/~vandenbe/103/lectures/chol.pdf ) | + | *Partition matrices in as <math>A</math> = <math>LL^{T}</math> (Please take the description from http://www.seas.ucla.edu/~vandenbe/103/lectures/chol.pdf ) |
==ZOS Section== | ==ZOS Section== | ||
Revision as of 05:44, 8 April 2015
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 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}}
where
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 (and thus also every real-valued symmetric 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 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}} (Please take the description from http://www.seas.ucla.edu/~vandenbe/103/lectures/chol.pdf )