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<option>Result</option>
<option>logdet</option>
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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 LuVecAD(<br/>
     size_t </span></font></code><i><span style='white-space: nowrap'>n</span></i><code><font color="blue"><span style='white-space: nowrap'>,<br/>
     size_t </span></font></code><i><span style='white-space: nowrap'>m</span></i><code><font color="blue"><span style='white-space: nowrap'>,<br/>
     VecAD<</span></font></code><i><span style='white-space: nowrap'>double</span></i><code><font color="blue"><span style='white-space: nowrap'>> &</span></font></code><i><span style='white-space: nowrap'>Matrix</span></i><code><font color="blue"><span style='white-space: nowrap'>,<br/>
     VecAD<</span></font></code><i><span style='white-space: nowrap'>double</span></i><code><font color="blue"><span style='white-space: nowrap'>> &</span></font></code><i><span style='white-space: nowrap'>Rhs</span></i><code><font color="blue"><span style='white-space: nowrap'>,<br/>
     VecAD<</span></font></code><i><span style='white-space: nowrap'>double</span></i><code><font color="blue"><span style='white-space: nowrap'>> &</span></font></code><i><span style='white-space: nowrap'>Result</span></i><code><font color="blue"><span style='white-space: nowrap'>,<br/>
     AD<</span></font></code><i><span style='white-space: nowrap'>double</span></i><code><font color="blue"><span style='white-space: nowrap'>> &</span></font></code><i><span style='white-space: nowrap'>logdet</span></i><code><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 <i>Matrix</i> is an
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">×</mo>
<mi mathvariant='italic'>n</mi>
</mrow></math>
matrix,
<i>Rhs</i> 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
<i>Result</i> 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 of type 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
<i>Matrix</i>,
<i>Rhs</i>,
and <i>Result</i>.
<br/>
<br/>
<b><big><a name="m" id="m">m</a></big></b>
<br/>
is the number of columns in
<i>Rhs</i>
and <i>Result</i>.
It is ok for <i>m</i> to be zero which is reasonable when
you are only interested in the determinant of <i>Matrix</i>.
<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">×</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 <i>Matrix</i> 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">×</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 <i>Rhs</i> have been overwritten
and are not specified.
If <i>m</i> is zero, <i>Rhs</i> 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">×</mo>
<mi mathvariant='italic'>m</mi>
</mrow></math>
matrix and the value of its elements do not matter.
On output, the elements of <i>Rhs</i> 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 <i>m</i> is zero, <i>Result</i> is not used.
<br/>
<br/>
<b><big><a name="logdet" id="logdet">logdet</a></big></b>
<br/>
On input, the value of <i>logdet</i> does not matter.
On output, it has been set to the
log of the determinant of <i>Matrix</i> (but not quite).
To be more specific,
if <i>signdet</i> is the value returned by <code><font color="blue">LuVecAD</font></code>,
the determinant of <i>Matrix</i> 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="luvecadok.cpp.xml" target="_top"><span style='white-space: nowrap'>LuVecADOk.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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