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<center><b><big><big>LU Factorization of A Square Matrix</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_factor.hpp></font></code>
<code><span style='white-space: nowrap'><br/>
</span></code><code><font color="blue"></font></code><i><span style='white-space: nowrap'>sign</span></i><code><font color="blue"><span style='white-space: nowrap'> = LuFactor(</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>
<br/>
<br/>
<b><big><a name="Description" id="Description">Description</a></big></b>
<br/>
Computes an LU factorization of the matrix <i>A</i>
where <i>A</i> is a square matrix.
<br/>
<br/>
<b><big><a name="Include" id="Include">Include</a></big></b>
<br/>
The file <code><font color="blue">cppad/lu_factor.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="sign" id="sign">sign</a></big></b>
<br/>
The return value <i>sign</i> has prototype
<code><font color="blue"><span style='white-space: nowrap'><br/>
     int </span></font></code><i><span style='white-space: nowrap'>sign</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>If <i>A</i> is invertible, <i>sign</i> is plus or minus one
and is the sign of the permutation corresponding to the row ordering
<i>ip</i> and column ordering <i>jp</i>.
If <i>A</i> is not invertible, <i>sign</i> is zero.
<br/>
<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/>
     </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 of <a href="lufactor.xml#SizeVector" target="_top"><span style='white-space: nowrap'>SizeVector</span></a>
below).
The size of <i>ip</i> is referred to as <i>n</i> in the
specifications below.
The input value of the elements of <i>ip</i> does not matter.
The output value of 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/>
     </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 of <a href="lufactor.xml#SizeVector" target="_top"><span style='white-space: nowrap'>SizeVector</span></a>
below).
The size of <i>jp</i> must be equal to <i>n</i>.
The input value of the elements of <i>jp</i> does not matter.
The output value of 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/>
     </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 of <a href="lufactor.xml#FloatVector" target="_top"><span style='white-space: nowrap'>FloatVector</span></a>
below).
<br/>
<br/>
<b><a name="LU.A" id="LU.A">A</a></b>
<br/>
We define <i>A</i> as the matrix corresponding to the input
value of <i>LU</i>.
<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 <i>A</i> 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><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 the
output value 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 the
output value 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.Factor" id="LU.Factor">Factor</a></b>
<br/>
If the return value <i>sign</i> is non-zero,
<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'>U</span></i><code><font color="blue"><span style='white-space: nowrap'> = </span></font></code><i><span style='white-space: nowrap'>P</span></i><code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>If the return value of <i>sign</i> is zero,
the contents of <i>L</i> and <i>U</i> are not defined.
<br/>
<br/>
<b><a name="LU.Determinant" id="LU.Determinant">Determinant</a></b>
<br/>
If the return value <i>sign</i> is zero,
the determinant of <i>A</i> is zero.
If <i>sign</i> is non-zero,
using the output value of <i>LU</i>
the determinant of the matrix <i>A</i> is equal to
<code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code><i><span style='white-space: nowrap'>sign</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'>[0], </span></font></code><i><span style='white-space: nowrap'>jp</span></i><code><font color="blue"><span style='white-space: nowrap'>[0]] * </span></font></code><i><span style='white-space: nowrap'>...</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'>n</span></i><code><font color="blue"><span style='white-space: nowrap'>-1], </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'>n</span></i><code><font color="blue"><span style='white-space: nowrap'>-1]] <br/>
</span></font></code><br/>
<b><big><a name="SizeVector" id="SizeVector">SizeVector</a></big></b>
<br/>
The type <i>SizeVector</i> must be a <a href="simplevector.xml" target="_top"><span style='white-space: nowrap'>SimpleVector</span></a>
class with
<a href="simplevector.xml#Elements of Specified Type" target="_top"><span style='white-space: nowrap'>elements of type size_t</span></a>
.
The routine <a href="checksimplevector.xml" target="_top"><span style='white-space: nowrap'>CheckSimpleVector</span></a>
will generate an error message
if this is not the case.
<br/>
<br/>
<b><big><a name="FloatVector" id="FloatVector">FloatVector</a></big></b>
<br/>
The type <i>FloatVector</i> must be a
<a href="simplevector.xml" target="_top"><span style='white-space: nowrap'>simple vector class</span></a>
.
The routine <a href="checksimplevector.xml" target="_top"><span style='white-space: nowrap'>CheckSimpleVector</span></a>
will generate an error message
if this is not the case.
<br/>
<br/>
<b><big><a name="Float" id="Float">Float</a></big></b>
<br/>
This notation is used to denote the type corresponding
to the elements of a <i>FloatVector</i>.
The type <i>Float</i> must satisfy the conditions
for a <a href="numerictype.xml" target="_top"><span style='white-space: nowrap'>NumericType</span></a>
type.
The routine <a href="checknumerictype.xml" target="_top"><span style='white-space: nowrap'>CheckNumericType</span></a>
will generate an error message
if this is not the case.
In addition, the following operations must be defined for any pair
of <i>Float</i> objects <i>x</i> and <i>y</i>:
<table><tr><td align='left' valign='top'>
<b>Operation</b> </td><td align='left' valign='top'>
<b>Description</b> </td></tr><tr><td align='left' valign='top'>
<code><font color="blue"><span style='white-space: nowrap'>log(</span></font></code><i><span style='white-space: nowrap'>x</span></i><code><font color="blue"><span style='white-space: nowrap'>)</span></font></code> </td><td align='left' valign='top'>
returns the logarithm of <i>x</i> as a <i>Float</i> object
</td></tr>
</table>
<br/>
<b><big><a name="AbsGeq" id="AbsGeq">AbsGeq</a></big></b>
<br/>
Including the file <code><font color="blue">lu_factor.hpp</font></code> defines the template function
<code><font color="blue"><span style='white-space: nowrap'><br/>
     template <typename </span></font></code><i><span style='white-space: nowrap'>Float</span></i><code><font color="blue"><span style='white-space: nowrap'>><br/>
     bool AbsGeq<</span></font></code><i><span style='white-space: nowrap'>Float</span></i><code><font color="blue"><span style='white-space: nowrap'>>(const </span></font></code><i><span style='white-space: nowrap'>Float</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'>, const </span></font></code><i><span style='white-space: nowrap'>Float</span></i><code><font color="blue"><span style='white-space: nowrap'> &</span></font></code><i><span style='white-space: nowrap'>y</span></i><code><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>in the <code><font color="blue">CppAD</font></code> namespace.
This function returns true if the absolute value of
<i>x</i> is greater than or equal the absolute value of <i>y</i>.
It is used by <code><font color="blue">LuFactor</font></code> to choose the pivot elements.
This template function definition uses the operator
<code><font color="blue"><=</font></code> to obtain the absolute value for <i>Float</i> objects.
If this operator is not defined for your use of <i>Float</i>,
you will need to specialize this template so that it works for your
use of <code><font color="blue">LuFactor</font></code>.
<code><span style='white-space: nowrap'><br/>
<br/>
</span></code>Complex numbers do not have the operation <code><font color="blue"><=</font></code> defined.
The specializations
<code><font color="blue"><span style='white-space: nowrap'><br/>
bool AbsGeq< std::complex<float> > <br/>
     (const std::complex<float> &</span></font></code><i><span style='white-space: nowrap'>x</span></i><code><font color="blue"><span style='white-space: nowrap'>, const std::complex<float> &</span></font></code><i><span style='white-space: nowrap'>y</span></i><code><font color="blue"><span style='white-space: nowrap'>)<br/>
bool AbsGeq< std::complex<double> > <br/>
     (const std::complex<double> &</span></font></code><i><span style='white-space: nowrap'>x</span></i><code><font color="blue"><span style='white-space: nowrap'>, const std::complex<double> &</span></font></code><i><span style='white-space: nowrap'>y</span></i><code><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>are define by including <code><font color="blue">lu_factor.hpp</font></code>
These return true if the sum of the square of the real and imaginary parts
of <i>x</i> is greater than or equal the
sum of the square of the real and imaginary parts of <i>y</i>.
<br/>
<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>
<br/>
The file
<a href="lufactor.cpp.xml" target="_top"><span style='white-space: nowrap'>LuFactor.cpp</span></a>
contains an example and test of using <code><font color="blue">LuFactor</font></code> by itself.
It returns true if it succeeds and false otherwise.
<code><span style='white-space: nowrap'><br/>
<br/>
</span></code>The file <a href="lu_solve.hpp.xml" target="_top"><span style='white-space: nowrap'>lu_solve.hpp</span></a>
provides a useful example usage of
<code><font color="blue">LuFactor</font></code> with <code><font color="blue">LuInvert</font></code>.
<br/>
<br/>
<b><big><a name="Source" id="Source">Source</a></big></b>
<br/>
The file <a href="lu_factor.hpp.xml" target="_top"><span style='white-space: nowrap'>lu_factor.hpp</span></a>
contains the
current source code that implements these specifications.
<hr/>Input File: cppad/lu_factor.hpp
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