mf_cholesky.hlp

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{smcl}
{* 11mar2005}{...}
{cmd:help mata cholesky()}
{hline}
{* index LAPACK}{...}
{* index cholesky()}{...}
{* index _cholesky()}{...}
{* index Cholesky decomposition}{...}
{* index square root}{...}
{* index decomposition}{...}

{title:Title}

{p 4 4 2}
{bf:[M-5] cholesky() -- Cholesky square-root decomposition}


{title:Syntax}

{p 8 12 2}
{it:numeric matrix}
{cmd:cholesky(}{it:numeric matrix A}{cmd:)}

{p 8 12 2}
{it:void}{bind:         }
{cmd:_cholesky(}{it:numeric matrix A}{cmd:)}


{title:Description}

{p 4 4 2}
{cmd:cholesky(}{it:A}{cmd:)} returns the Cholesky decomposition {it:G} of
symmetric (Hermitian), positive-definite matrix {it:A}.  
{cmd:cholesky()} returns a lower-triangular matrix of missing values if
{it:A} is not positive definite.

{p 4 4 2}
{cmd:_cholesky(}{it:A}{cmd:)} does the same thing, except that it overwrites
{it:A} with the Cholesky result.


{title:Remarks}

{p 4 4 2}
The Cholesky decomposition {it:G} of a symmetric, positive-definite matrix
{it:A} is 

		{it:A} = {it:G}*{it:G}{bf:'}

{p 4 4 2}
where {it:G} is lower triangular.
When {it:A} is complex, {it:A} must be Hermitian, and {it:G}{bf:'}, of course,
is the conjugate transpose of {it:G}.

{p 4 4 2}
Decomposition is performed via {bf:{help m1_lapack:[M-1] LAPACK}}.


{title:Conformability}

    {cmd:cholesky(}{it:A}{cmd:)}
		{it:A}:  {it:n x n} 
	   {it:result}:  {it:n x n} 

    {cmd:_cholesky(}{it:A}{cmd:)}
	{it:input:}
		{it:A}:  {it:n x n} 
	{it:output:}
		{it:A}:  {it:n x n} 


{title:Diagnostics}

{p 4 4 2}
{cmd:cholesky(}{cmd:)} returns a lower-triangular matrix of missing values
if {it:A} contains missing values or if {it:A} is not positive definite.

{p 4 4 2}
{cmd:_cholesky(}{it:A}{cmd:)} overwrites {it:A} with a lower-triangular
matrix of missing values if {it:A} contains missing values or if {it:A} is
not positive definite.

{p 4 4 2}
Both functions use the elements from the lower triangle of {it:A} without
checking whether {it:A} is symmetric or, in the complex case, Hermitian.  


{title:Source code}

{p 4 4 2}
{view cholesky.mata, adopath asis:cholesky.mata};
{cmd:_cholesky()} is built-in.


{title:Also see}

{p 4 13 2}
Manual:  {hi:[M-5] cholesky()}

{p 4 13 2}
Online:  help for 
{bf:{help mf_lud:[M-5] lud()}}
{bf:{help m4_matrix:[M-4] matrix}}
{p_end}