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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 &lt;cppad/lu_invert.hpp&gt;</font></code>
<code><span style='white-space: nowrap'><br/>
</span></code>
<code><font color="blue"><span style='white-space: nowrap'>LuInvert(</span></font><i><font color="black"><span style='white-space: nowrap'>ip</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>jp</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>LU</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>X</span></font></i><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><i><font color="black"><span style='white-space: nowrap'>A</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>X</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>B</span></font></i></code>
 
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">&#x02026;</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">&#x02026;</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 
<code><i><font color="black"><span style='white-space: nowrap'>ip</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>SizeVector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>ip</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see description for 
<code><i><font color="black"><span style='white-space: nowrap'>SizeVector</span></font></i></code>
 in
<a href="lufactor.xml#SizeVector" target="_top"><span style='white-space: nowrap'>LuFactor</span></a>
 specifications).
The size of 
<code><i><font color="black"><span style='white-space: nowrap'>ip</span></font></i></code>
 is referred to as 
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
 in the
specifications below.
The elements of 
<code><i><font color="black"><span style='white-space: nowrap'>ip</span></font></i></code>
 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 
<code><i><font color="black"><span style='white-space: nowrap'>jp</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>SizeVector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>jp</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see description for 
<code><i><font color="black"><span style='white-space: nowrap'>SizeVector</span></font></i></code>
 in
<a href="lufactor.xml#SizeVector" target="_top"><span style='white-space: nowrap'>LuFactor</span></a>
 specifications).
The size of 
<code><i><font color="black"><span style='white-space: nowrap'>jp</span></font></i></code>
 must be equal to 
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
.
The elements of 
<code><i><font color="black"><span style='white-space: nowrap'>jp</span></font></i></code>
 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 
<code><i><font color="black"><span style='white-space: nowrap'>LU</span></font></i></code>
 has the prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;const&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>FloatVector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>LU</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
and the size of 
<code><i><font color="black"><span style='white-space: nowrap'>LU</span></font></i></code>
 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 
<code><i><font color="black"><span style='white-space: nowrap'>FloatVector</span></font></i></code>
 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 
<code><i><font color="black"><span style='white-space: nowrap'>L</span></font></i></code>
 in terms of 
<code><i><font color="black"><span style='white-space: nowrap'>LU</span></font></i></code>
.
The matrix 
<code><i><font color="black"><span style='white-space: nowrap'>L</span></font></i></code>
 is zero above the diagonal
and the rest of the elements are defined by

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>L</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>i</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>)&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>LU</span></font></i><font color="blue"><span style='white-space: nowrap'>[&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ip</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>i</span></font></i><font color="blue"><span style='white-space: nowrap'>]&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;+&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>jp</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>]&#xA0;]<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">&#x02026;</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">&#x02026;</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 
<code><i><font color="black"><span style='white-space: nowrap'>U</span></font></i></code>
 in terms of 
<code><i><font color="black"><span style='white-space: nowrap'>LU</span></font></i></code>
.
The matrix 
<code><i><font color="black"><span style='white-space: nowrap'>U</span></font></i></code>
 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/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>U</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>i</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>)&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>LU</span></font></i><font color="blue"><span style='white-space: nowrap'>[&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ip</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>i</span></font></i><font color="blue"><span style='white-space: nowrap'>]&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;+&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>jp</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>]&#xA0;]<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">&#x02026;</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">&#x02026;</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 
<code><i><font color="black"><span style='white-space: nowrap'>P</span></font></i></code>
 in terms of 
the matrix 
<code><i><font color="black"><span style='white-space: nowrap'>L</span></font></i></code>
 and the matrix 
<code><i><font color="black"><span style='white-space: nowrap'>U</span></font></i></code>
 
by 
<code><i><font color="black"><span style='white-space: nowrap'>P</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>L</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>U</span></font></i></code>
.

<br/>
<br/>
<b><a name="LU.A" id="LU.A">A</a></b>
<br/>
The matrix 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i></code>
, 
which defines the linear equations that we are solving, is given by

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>P</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>i</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>)&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>A</span></font></i><font color="blue"><span style='white-space: nowrap'>[&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>ip</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>i</span></font></i><font color="blue"><span style='white-space: nowrap'>]&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;+&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>jp</span></font></i><font color="blue"><span style='white-space: nowrap'>[</span></font><i><font color="black"><span style='white-space: nowrap'>j</span></font></i><font color="blue"><span style='white-space: nowrap'>]&#xA0;]<br/>
</span></font></code>
(Hence 

<code><i><font color="black"><span style='white-space: nowrap'>LU</span></font></i></code>
 contains a permuted factorization of the matrix 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i></code>
.)


<br/>
<br/>
<b><big><a name="X" id="X">X</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>X</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>FloatVector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>X</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
(see description for 
<code><i><font color="black"><span style='white-space: nowrap'>FloatVector</span></font></i></code>
 in
<a href="lufactor.xml#FloatVector" target="_top"><span style='white-space: nowrap'>LuFactor</span></a>
 specifications).
The matrix 
<code><i><font color="black"><span style='white-space: nowrap'>X</span></font></i></code>

must have the same number of rows as the matrix 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i></code>
.
The input value of 
<code><i><font color="black"><span style='white-space: nowrap'>X</span></font></i></code>
 is the matrix 
<code><i><font color="black"><span style='white-space: nowrap'>B</span></font></i></code>
 and the 
output value solves the matrix equation 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>X</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>B</span></font></i></code>
.



<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="lu_invert.cpp.xml" target="_top"><span style='white-space: nowrap'>lu_invert.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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