mf_solve_tol.hlp

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{smcl}
{* 28mar2005}{...}
{cmd:help mata solve_tol()}
{hline}
{* index solve_tol()}{...}
{* index tolerances}{...}
{* index solve AX=B}{...}
{* index inverse matrix}{...}
{* index _solvetolerance tt}{...}

{title:Title}

{p 4 8 2}
{bf:[M-5] solve_tol() -- Tolerance used by solvers and inverters}


{title:Syntax}

{p 8 12 2}
{it:real scalar}
{cmd:solve_tol(}{it:numeric matrix Z}{cmd:,}
{it:real scalar usertol}{cmd:)}


{title:Description}

{p 4 4 2}
{cmd:solve_tol(}{it:Z}{cmd:,} {it:usertol}{cmd:)} 
returns the tolerance used by many 
Mata solvers to solve {it:AX}={it:B} and by many Mata inverters to obtain 
{it:A}^(-1).
{it:usertol} is the tolerance specified by the 
user or is missing value if the user did not specify a tolerance.


{title:Remarks}

{p 4 4 2}
The tolerance used by many Mata solvers to solve {it:AX}={it:B} and 
by many Mata inverters to obtain {it:A}^(-1) is

		{it:eta} = {it:s} * trace(abs({it:Z}))/{it:n}{right:(1)    }

{p 4 4 2}
where {it:s}=1e-13 or a value specified by the user, {it:n} is the
min(rows({it:Z}), cols({it:Z})), and {it:Z} is a matrix related to 
{it:A}, usually by some form of decomposition, but could be {it:A} 
itself (for instance, if {it:A} were triangular).
See, for instance, 
{bf:{help mf_solvelower:[M-5] solvelower()}} and 
{bf:{help mf_cholsolve:[M-5] cholsolve()}}.


{p 4 4 2}
When {it:usertol}>0 and {it:usertol}<{cmd:.} is specified, 
{cmd:solvetol()} returns {it:eta} calculated with {it:s}={it:usertol}.

{p 4 4 2}
When {it:usertol}<=0 is specified, 
{cmd:solvetol()} returns -{it:usertol}.  

{p 4 4 2}
When {it:usertol}>={cmd:.} is specified, 
{cmd:solvetol()} returns a default result, calculated as

{p 8 12 2}
1.  If external real scalar {cmd:_solvetolerance} does not exist, as 
    is usually the case, the value of {it:eta} is returned using 
    {it:s}=1e-13.

{p 8 12 2}
2.  If external real scalar {cmd:_solvetolerance} does exist, 

{p 12 16 2}
a.  If {cmd:_solvetolerance}>0, the value of {it:eta} is returned 
    using {it:s}={cmd:solvetolerance}.

{p 12 16 2}
b.  If {cmd:_solvetolerance}<=0, -{cmd:_solvetolerance} is returned.



{title:Conformability}

    {cmd:solve_tol(}{it:Z}{cmd:,} {it:usertol}{cmd:)}:
		{it:Z}:  {it:r x c}
	  {it:usertol}:  1 {it:x} 1
	   {it:result}:  1 {it:x} 1


{title:Diagnostics}

{p 4 4 2}
{cmd:solve_tol(}{it:Z}{cmd:,} {it:usertol}{cmd:)} 
skips over missing values in {it:Z} in calculating (1); {it:n} is 
defined as the number of nonmissing elements on the diagonal.


{title:Source code}

{p 4 4 2}
{view solve_tol.mata, adopath asis:solve_tol.mata}


{title:Also see}

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

{p 4 13 2}
Online:  help for 
{bf:{help m4_utility:[M-4] utility}}
{p_end}