m4_solvers.hlp

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{smcl}
{* 18jan2005}{...}
{cmd:help m4 solvers}
{hline}
{* index matrix functions}{...}
{* index mathematical functions}{...}
{* index solve AX=B}{...}
{* index inverse matrix}{...}

{title:Title}

{p 4 4 2}
{bf:[M-4] solvers -- Functions to solve AX=B and to obtain A inverse}


{title:Contents}

{col 8} {bf:[M-5]}
{col 5}{bf:Manual entry{col 22}Function{col 35}Purpose}
{col 5}{hline}

{col 5}   {c TLC}{hline 9}{c TRC}
{col 5}{hline 3}{c RT}{it: Solvers }{c LT}{hline}
{col 5}   {c BLC}{hline 9}{c BRC}

{col 8}{bf:{help mf_cholsolve:cholsolve()}}{...}
{col 22}{cmd:cholsolve()}{...}
{col 35}{it:A} positive definite, symmetric or Hermitian

{col 8}{bf:{help mf_lusolve:lusolve()}}{...}
{col 22}{cmd:lusolve()}{...}
{col 35}{it:A} full rank, square, real or complex

{col 8}{bf:{help mf_qrsolve:qrsolve()}}{...}
{col 22}{cmd:qrsolve()}{...}
{col 35}{it:A} general; {it:m x n}, {it:m} >= {it:n}, real or complex;
{col 35}least-squares generalized solution

{col 8}{bf:{help mf_svsolve:svsolve()}}{...}
{col 22}{cmd:svsolve()}{...}
{col 35}generalized; {it:m x n}, real or complex;
{col 35}minimum norm, least-squares solution

{col 5}   {c TLC}{hline 11}{c TRC}
{col 5}{hline 3}{c RT}{it: Inverters }{c LT}{hline}
{col 5}   {c BLC}{hline 11}{c BRC}

{col 8}{bf:{help mf_invsym:invsym()}}{...}
{col 22}{cmd:invsym()}{...}
{col 35}generalized; real symmetric

{col 8}{bf:{help mf_cholinv:cholinv()}}{...}
{col 22}{cmd:cholinv()}{...}
{col 35}positive definite; symmetric or Hermitian

{col 8}{bf:{help mf_luinv:luinv()}}{...}
{col 22}{cmd:luinv()}{...}
{col 35}full rank; square; real or complex

{col 8}{bf:{help mf_qrinv:qrinv()}}{...}
{col 22}{cmd:qrinv()}{...}
{col 35}generalized; {it:m x n}, {it:m} >= {it:n}; real or complex

{col 8}{bf:{help mf_pinv:pinv()}}{...}
{col 22}{cmd:pinv()}{...}
{col 35}generalized; {it:m x n}, real or complex
{col 35}Moore-Penrose pseudoinverse

{col 5}{hline}


{title:Description}

{p 4 4 2}
The above functions solve {it:AX}={it:B} for {it:X}
and solve for {it:A}^(-1).


{title:Remarks}

{p 4 4 2}
Matrix solvers can be used to implement matrix inverters, and so the 
two nearly always come as a pair.

{p 4 4 2}
Solvers solve 
{it:AX}={it:B}
for {it:X}.
One way to obtain {it:A}^(-1) is to solve
{it:AX}={it:I}.  If 
{bind:{it:f}({it:A}, {it:B})} solves {it:AX}={it:B}, 
then 
{bind:{it:f}({it:A}, {cmd:I(rows(}{it:A}{cmd:))}}
solves for the inverse.
Some matrix 
inverters are in fact implemented this way, although usually 
custom code is written because memory savings are possible when it is known 
that {it:B}={it:I}.

{p 4 4 2}
The pairings of inverter and solver are

		inverter         solver
		{hline 37}
		{cmd:invsym()}         (none)
		{cmd:cholinv()}        {bf:{help mf_cholsolve:[M-5] cholsolve()}}
		{cmd:luinv()}          {bf:{help mf_lusolve:[M-5] lusolve()}}
		{cmd:qrinv()}          {bf:{help mf_qrsolve:[M-5] qrsolve()}}
		{cmd:pinv()}           {bf:{help mf_svsolve:[M-5] svsolve()}}
		{hline 37}


{title:Also see}

{p 4 13 2}
Manual:  {hi:[M-4] solvers}

{p 4 13 2}
Online:  help for 
{bf:{help m4_intro:[M-4] intro}};
{bf:{help mata:[M-0] intro}}
{p_end}