Manuals/calci/CHOLESKY

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}}

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

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

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}}}

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 \frac{1}{l_{11}}A_{21}}

Examples

1. =CHOLESKY([[16,32,12],[12, 18, 0],[ -5, 0, 11]])

Result

4	0	0
3	3	0
-1.25	1.25	2.80624

2. =CHOLESKY([[25, 15, -5],[15, 18, 0],[ -5, 0, 11]])

Result

5	0	0
3	3	0
-1	1	3