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<title>Lu Factor and Solve with Recorded Pivoting</title>
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<center><b><big><big>Lu Factor and Solve with Recorded Pivoting</big></big></b></center>
<code><span style='white-space: nowrap'><br/>
</span></code><b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>

<br/>

<code><font color="blue"><span style='white-space: nowrap'>int&#xA0;LuVecAD(<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>,<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;size_t&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>m</span></font></i><font color="blue"><span style='white-space: nowrap'>,<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;VecAD&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>double</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>Matrix</span></font></i><font color="blue"><span style='white-space: nowrap'>,<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;VecAD&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>double</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>Rhs</span></font></i><font color="blue"><span style='white-space: nowrap'>,<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;VecAD&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>double</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>Result</span></font></i><font color="blue"><span style='white-space: nowrap'>,<br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;AD&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>double</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>logdet</span></font></i><font color="blue"><span style='white-space: nowrap'>)</span></font></code>



<br/>
<br/>
<b><big><a name="Purpose" id="Purpose">Purpose</a></big></b>
<br/>
Solves the linear equation

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>Matrix</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>Result</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>Rhs</mi>
</mrow></math>

where 
<code><i><font color="black"><span style='white-space: nowrap'>Matrix</span></font></i></code>
 is an 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>n</mi>
</mrow></math>

 matrix,

<code><i><font color="black"><span style='white-space: nowrap'>Rhs</span></font></i></code>
 is an 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
</mrow></math>

 matrix, and

<code><i><font color="black"><span style='white-space: nowrap'>Result</span></font></i></code>
 is an 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
</mrow></math>

 matrix.
<code><span style='white-space: nowrap'><br/>
<br/>
</span></code>The routine <a href="lusolve.xml" target="_top"><span style='white-space: nowrap'>LuSolve</span></a>
 uses an arbitrary vector type,
instead of <a href="vecad.xml" target="_top"><span style='white-space: nowrap'>VecAD</span></a>
,
to hold its elements.
The pivoting operations for a <code><font color="blue">ADFun</font></code> object
corresponding to an <code><font color="blue">LuVecAD</font></code> solution
will change to be optimal for the matrix being factored.
<code><span style='white-space: nowrap'><br/>
<br/>
</span></code>It is often the case that
<code><font color="blue">LuSolve</font></code> is faster than <code><font color="blue">LuVecAD</font></code> when <code><font color="blue">LuSolve</font></code>
uses a simple vector class with 
<a href="simplevector.xml#Elements of Specified Type" target="_top"><span style='white-space: nowrap'>elements&#xA0;of&#xA0;type&#xA0;double</span></a>
,
but the corresponding <a href="adfun.xml" target="_top"><span style='white-space: nowrap'>ADFun</span></a>
 objects have a fixed
set of pivoting operations.

<br/>
<br/>
<b><big><a name="Storage Convention" id="Storage Convention">Storage Convention</a></big></b>
<br/>
The matrices stored in row major order.
To be specific, if 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>A</mi>
</mrow></math>

 contains the vector storage for an

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mi mathvariant='italic'>x</mi>
<mi mathvariant='italic'>m</mi>
</mrow></math>

 matrix,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
</mrow></math>

 is between zero and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>

,
and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
</mrow></math>

 is between zero and 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>m</mi>
<mn>-1</mn>
</mrow></math>

,

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mi mathvariant='italic'>A</mi>
<mrow><mi mathvariant='italic'>i</mi>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>j</mi>
</mrow>
</msub>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>A</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>m</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">]</mo>
</mrow></math>

(The length of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>A</mi>
</mrow></math>

 must be equal to 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>m</mi>
</mrow></math>

.)

<br/>
<br/>
<b><big><a name="n" id="n">n</a></big></b>
<br/>
is the number of rows in 

<code><i><font color="black"><span style='white-space: nowrap'>Matrix</span></font></i></code>
,

<code><i><font color="black"><span style='white-space: nowrap'>Rhs</span></font></i></code>
,
and 
<code><i><font color="black"><span style='white-space: nowrap'>Result</span></font></i></code>
.

<br/>
<br/>
<b><big><a name="m" id="m">m</a></big></b>
<br/>
is the number of columns in 

<code><i><font color="black"><span style='white-space: nowrap'>Rhs</span></font></i></code>

and 
<code><i><font color="black"><span style='white-space: nowrap'>Result</span></font></i></code>
.
It is ok for 
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
 to be zero which is reasonable when
you are only interested in the determinant of 
<code><i><font color="black"><span style='white-space: nowrap'>Matrix</span></font></i></code>
.


<br/>
<br/>
<b><big><a name="Matrix" id="Matrix">Matrix</a></big></b>
<br/>
On input, this is an

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>n</mi>
</mrow></math>

 matrix containing the variable coefficients for 
the equation we wish to solve.
On output, the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>Matrix</span></font></i></code>
 have been overwritten
and are not specified.

<br/>
<br/>
<b><big><a name="Rhs" id="Rhs">Rhs</a></big></b>
<br/>
On input, this is an

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>m</mi>
</mrow></math>

 matrix containing the right hand side
for the equation we wish to solve.
On output, the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>Rhs</span></font></i></code>
 have been overwritten
and are not specified.
If 
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
 is zero, 
<code><i><font color="black"><span style='white-space: nowrap'>Rhs</span></font></i></code>
 is not used.

<br/>
<br/>
<b><big><a name="Result" id="Result">Result</a></big></b>
<br/>
On input, this is an

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">&#x000D7;</mo>
<mi mathvariant='italic'>m</mi>
</mrow></math>

 matrix and the value of its elements do not matter.
On output, the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>Rhs</span></font></i></code>
 contain the solution
of the equation we wish to solve
(unless the value returned by <code><font color="blue">LuVecAD</font></code> is equal to zero).
If 
<code><i><font color="black"><span style='white-space: nowrap'>m</span></font></i></code>
 is zero, 
<code><i><font color="black"><span style='white-space: nowrap'>Result</span></font></i></code>
 is not used.

<br/>
<br/>
<b><big><a name="logdet" id="logdet">logdet</a></big></b>
<br/>
On input, the value of 
<code><i><font color="black"><span style='white-space: nowrap'>logdet</span></font></i></code>
 does not matter.
On output, it has been set to the 
log of the determinant of 
<code><i><font color="black"><span style='white-space: nowrap'>Matrix</span></font></i></code>
 (but not quite).
To be more specific,
if 
<code><i><font color="black"><span style='white-space: nowrap'>signdet</span></font></i></code>
 is the value returned by <code><font color="blue">LuVecAD</font></code>,
the determinant of 
<code><i><font color="black"><span style='white-space: nowrap'>Matrix</span></font></i></code>
 is given by the formula

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<mi mathvariant='italic'>det</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>signdet</mi>
<mi>exp</mi>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>logdet</mi>
<mo stretchy="false">)</mo>
</mrow></math>

This enables <code><font color="blue">LuVecAD</font></code> to use logs of absolute values.


<br/>
<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>

<br/>
The file
<a href="lu_vec_ad_ok.cpp.xml" target="_top"><span style='white-space: nowrap'>lu_vec_ad_ok.cpp</span></a>

contains an example and test of <code><font color="blue">LuVecAD</font></code>.
It returns true if it succeeds and false otherwise.



<hr/>Input File: example/lu_vec_ad.cpp

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