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<center><b><big><big>Invert an LU Factored Equation</big></big></b></center>
<code><span style='white-space: nowrap'><br/>
</span></code><b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>
<code><font color="blue"><br/>
# include <cppad/lu_invert.hpp></font></code>
<code><span style='white-space: nowrap'><br/>
</span></code><code><font color="blue"><span style='white-space: nowrap'>LuInvert(</span></font></code><i><span style='white-space: nowrap'>ip</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>jp</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>LU</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>X</span></i><code><font color="blue"><span style='white-space: nowrap'>)</span></font></code>
<br/>
<br/>
<b><big><a name="Description" id="Description">Description</a></big></b>
<br/>
Solves the matrix equation <code><font color="blue"></font></code><i><span style='white-space: nowrap'>A</span></i><code><font color="blue"><span style='white-space: nowrap'> * </span></font></code><i><span style='white-space: nowrap'>X</span></i><code><font color="blue"><span style='white-space: nowrap'> = </span></font></code><i><span style='white-space: nowrap'>B</span></i>
using an LU factorization computed by <a href="lufactor.xml" target="_top"><span style='white-space: nowrap'>LuFactor</span></a>
.
<br/>
<br/>
<b><big><a name="Include" id="Include">Include</a></big></b>
<br/>
The file <code><font color="blue">cppad/lu_invert.hpp</font></code> is included by <code><font color="blue">cppad/cppad.hpp</font></code>
but it can also be included separately with out the rest of
the <code><font color="blue">CppAD</font></code> routines.
<br/>
<br/>
<b><big><a name="Matrix Storage" id="Matrix Storage">Matrix Storage</a></big></b>
<br/>
All matrices are stored in row major order.
To be specific, if
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Y</mi>
</mrow></math>
is a vector
that contains a
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
</mrow></math>
by
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>q</mi>
</mrow></math>
matrix,
the size of
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>Y</mi>
</mrow></math>
must be equal to
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>q</mi>
</mrow></math>
and for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>p</mi>
<mn>-1</mn>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>q</mi>
<mn>-1</mn>
</mrow></math>
,
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow>
<msub><mi mathvariant='italic'>Y</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'>Y</mi>
<mo stretchy="false">[</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">*</mo>
<mi mathvariant='italic'>q</mi>
<mo stretchy="false">+</mo>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">]</mo>
</mrow></math>
<br/>
<b><big><a name="ip" id="ip">ip</a></big></b>
<br/>
The argument <i>ip</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font></code><i><span style='white-space: nowrap'>SizeVector</span></i><code><font color="blue"><span style='white-space: nowrap'> &</span></font></code><i><span style='white-space: nowrap'>ip</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>(see description for <i>SizeVector</i> in
<a href="lufactor.xml#SizeVector" target="_top"><span style='white-space: nowrap'>LuFactor</span></a>
specifications).
The size of <i>ip</i> is referred to as <i>n</i> in the
specifications below.
The elements of <i>ip</i> determine
the order of the rows in the permuted matrix.
<br/>
<br/>
<b><big><a name="jp" id="jp">jp</a></big></b>
<br/>
The argument <i>jp</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font></code><i><span style='white-space: nowrap'>SizeVector</span></i><code><font color="blue"><span style='white-space: nowrap'> &</span></font></code><i><span style='white-space: nowrap'>jp</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>(see description for <i>SizeVector</i> in
<a href="lufactor.xml#SizeVector" target="_top"><span style='white-space: nowrap'>LuFactor</span></a>
specifications).
The size of <i>jp</i> must be equal to <i>n</i>.
The elements of <i>jp</i> determine
the order of the columns in the permuted matrix.
<br/>
<br/>
<b><big><a name="LU" id="LU">LU</a></big></b>
<br/>
The argument <i>LU</i> has the prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     const </span></font></code><i><span style='white-space: nowrap'>FloatVector</span></i><code><font color="blue"><span style='white-space: nowrap'> &</span></font></code><i><span style='white-space: nowrap'>LU</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>and the size of <i>LU</i> must equal
<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>
(see description for <i>FloatVector</i> in
<a href="lufactor.xml#FloatVector" target="_top"><span style='white-space: nowrap'>LuFactor</span></a>
specifications).
<br/>
<br/>
<b><a name="LU.L" id="LU.L">L</a></b>
<br/>
We define the lower triangular matrix <i>L</i> in terms of <i>LU</i>.
The matrix <i>L</i> is zero above the diagonal
and the rest of the elements are defined by
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font></code><i><span style='white-space: nowrap'>L</span></i><code><font color="blue"><span style='white-space: nowrap'>(</span></font></code><i><span style='white-space: nowrap'>i</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>j</span></i><code><font color="blue"><span style='white-space: nowrap'>) = </span></font></code><i><span style='white-space: nowrap'>LU</span></i><code><font color="blue"><span style='white-space: nowrap'>[ </span></font></code><i><span style='white-space: nowrap'>ip</span></i><code><font color="blue"><span style='white-space: nowrap'>[</span></font></code><i><span style='white-space: nowrap'>i</span></i><code><font color="blue"><span style='white-space: nowrap'>] * </span></font></code><i><span style='white-space: nowrap'>n</span></i><code><font color="blue"><span style='white-space: nowrap'> + </span></font></code><i><span style='white-space: nowrap'>jp</span></i><code><font color="blue"><span style='white-space: nowrap'>[</span></font></code><i><span style='white-space: nowrap'>j</span></i><code><font color="blue"><span style='white-space: nowrap'>] ]<br/>
</span></font></code>for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<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>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>i</mi>
</mrow></math>
.
<br/>
<br/>
<b><a name="LU.U" id="LU.U">U</a></b>
<br/>
We define the upper triangular matrix <i>U</i> in terms of <i>LU</i>.
The matrix <i>U</i> is zero below the diagonal,
one on the diagonal,
and the rest of the elements are defined by
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font></code><i><span style='white-space: nowrap'>U</span></i><code><font color="blue"><span style='white-space: nowrap'>(</span></font></code><i><span style='white-space: nowrap'>i</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>j</span></i><code><font color="blue"><span style='white-space: nowrap'>) = </span></font></code><i><span style='white-space: nowrap'>LU</span></i><code><font color="blue"><span style='white-space: nowrap'>[ </span></font></code><i><span style='white-space: nowrap'>ip</span></i><code><font color="blue"><span style='white-space: nowrap'>[</span></font></code><i><span style='white-space: nowrap'>i</span></i><code><font color="blue"><span style='white-space: nowrap'>] * </span></font></code><i><span style='white-space: nowrap'>n</span></i><code><font color="blue"><span style='white-space: nowrap'> + </span></font></code><i><span style='white-space: nowrap'>jp</span></i><code><font color="blue"><span style='white-space: nowrap'>[</span></font></code><i><span style='white-space: nowrap'>j</span></i><code><font color="blue"><span style='white-space: nowrap'>] ]<br/>
</span></font></code>for
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">=</mo>
<mn>0</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-2</mn>
</mrow></math>
and
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>j</mi>
<mo stretchy="false">=</mo>
<mi mathvariant='italic'>i</mi>
<mo stretchy="false">+</mo>
<mn>1</mn>
<mo stretchy="false">,</mo>
<mo stretchy="false">…</mo>
<mo stretchy="false">,</mo>
<mi mathvariant='italic'>n</mi>
<mn>-1</mn>
</mrow></math>
.
<br/>
<br/>
<b><a name="LU.P" id="LU.P">P</a></b>
<br/>
We define the permuted matrix <i>P</i> in terms of
the matrix <i>L</i> and the matrix <i>U</i>
by <code><font color="blue"></font></code><i><span style='white-space: nowrap'>P</span></i><code><font color="blue"><span style='white-space: nowrap'> = </span></font></code><i><span style='white-space: nowrap'>L</span></i><code><font color="blue"><span style='white-space: nowrap'> * </span></font></code><i><span style='white-space: nowrap'>U</span></i>.
<br/>
<br/>
<b><a name="LU.A" id="LU.A">A</a></b>
<br/>
The matrix <i>A</i>,
which defines the linear equations that we are solving, is given by
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font></code><i><span style='white-space: nowrap'>P</span></i><code><font color="blue"><span style='white-space: nowrap'>(</span></font></code><i><span style='white-space: nowrap'>i</span></i><code><font color="blue"><span style='white-space: nowrap'>, </span></font></code><i><span style='white-space: nowrap'>j</span></i><code><font color="blue"><span style='white-space: nowrap'>) = </span></font></code><i><span style='white-space: nowrap'>A</span></i><code><font color="blue"><span style='white-space: nowrap'>[ </span></font></code><i><span style='white-space: nowrap'>ip</span></i><code><font color="blue"><span style='white-space: nowrap'>[</span></font></code><i><span style='white-space: nowrap'>i</span></i><code><font color="blue"><span style='white-space: nowrap'>] * </span></font></code><i><span style='white-space: nowrap'>n</span></i><code><font color="blue"><span style='white-space: nowrap'> + </span></font></code><i><span style='white-space: nowrap'>jp</span></i><code><font color="blue"><span style='white-space: nowrap'>[</span></font></code><i><span style='white-space: nowrap'>j</span></i><code><font color="blue"><span style='white-space: nowrap'>] ]<br/>
</span></font></code>(Hence
<i>LU</i> contains a permuted factorization of the matrix <i>A</i>.)
<br/>
<br/>
<b><big><a name="X" id="X">X</a></big></b>
<br/>
The argument <i>X</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     </span></font></code><i><span style='white-space: nowrap'>FloatVector</span></i><code><font color="blue"><span style='white-space: nowrap'> &</span></font></code><i><span style='white-space: nowrap'>X</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>(see description for <i>FloatVector</i> in
<a href="lufactor.xml#FloatVector" target="_top"><span style='white-space: nowrap'>LuFactor</span></a>
specifications).
The matrix <i>X</i>
must have the same number of rows as the matrix <i>A</i>.
The input value of <i>X</i> is the matrix <i>B</i> and the
output value solves the matrix equation <code><font color="blue"></font></code><i><span style='white-space: nowrap'>A</span></i><code><font color="blue"><span style='white-space: nowrap'> * </span></font></code><i><span style='white-space: nowrap'>X</span></i><code><font color="blue"><span style='white-space: nowrap'> = </span></font></code><i><span style='white-space: nowrap'>B</span></i>.
<br/>
<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>
<br/>
The file <a href="lu_solve.hpp.xml" target="_top"><span style='white-space: nowrap'>lu_solve.hpp</span></a>
is a good example usage of
<code><font color="blue">LuFactor</font></code> with <code><font color="blue">LuInvert</font></code>.
The file
<a href="luinvert.cpp.xml" target="_top"><span style='white-space: nowrap'>LuInvert.cpp</span></a>
contains an example and test of using <code><font color="blue">LuInvert</font></code> by itself.
It returns true if it succeeds and false otherwise.
<br/>
<br/>
<b><big><a name="Source" id="Source">Source</a></big></b>
<br/>
The file <a href="lu_invert.hpp.xml" target="_top"><span style='white-space: nowrap'>lu_invert.hpp</span></a>
contains the
current source code that implements these specifications.
<hr/>Input File: cppad/lu_invert.hpp
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