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<?xml version='1.0'?>
<html xmlns='http://www.w3.org/1999/xhtml'
      xmlns:math='http://www.w3.org/1998/Math/MathML'
>
<head>
<title>Determinant Using Expansion by Minors</title>
<meta http-equiv='Content-Type' content='text/html' charset='utf-8'/>
<meta name="description" id="description" content="Determinant Using Expansion by Minors"/>
<meta name="keywords" id="keywords" content=" determinant using expansion by minors minor syntax inclusion constructor scalar n det a d vector example source code "/>
<style type='text/css'>
body { color : black }
body { background-color : white }
A:link { color : blue }
A:visited { color : purple }
A:active { color : purple }
</style>
<script type='text/javascript' language='JavaScript' src='_det_by_minor_xml.js'>
</script>
</head>
<body>
<table><tr>
<td>
<a href="http://www.coin-or.org/CppAD/" target="_top"><img border="0" src="_image.gif"/></a>
</td>
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</td><td>
<select onchange='choose_across0(this)'>
<option>Index-&gt;</option>
<option>contents</option>
<option>reference</option>
<option>index</option>
<option>search</option>
<option>external</option>
</select>
</td>
<td>
<select onchange='choose_up0(this)'>
<option>Up-&gt;</option>
<option>CppAD</option>
<option>speed</option>
<option>speed_utility</option>
<option>det_by_minor</option>
</select>
</td>
<td>
<select onchange='choose_down3(this)'>
<option>CppAD-&gt;</option>
<option>Install</option>
<option>Introduction</option>
<option>AD</option>
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<option>Example</option>
<option>speed</option>
<option>Appendix</option>
</select>
</td>
<td>
<select onchange='choose_down2(this)'>
<option>speed-&gt;</option>
<option>speed_main</option>
<option>speed_utility</option>
<option>speed_double</option>
<option>speed_adolc</option>
<option>speed_cppad</option>
<option>speed_fadbad</option>
<option>speed_sacado</option>
</select>
</td>
<td>
<select onchange='choose_down1(this)'>
<option>speed_utility-&gt;</option>
<option>det_by_lu</option>
<option>det_of_minor</option>
<option>det_by_minor</option>
<option>det_33</option>
<option>det_grad_33</option>
<option>mat_sum_sq</option>
<option>ode_evaluate</option>
<option>sparse_jac_fun</option>
<option>sparse_hes_fun</option>
<option>uniform_01</option>
</select>
</td>
<td>
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<option>det_by_minor-&gt;</option>
<option>det_by_minor.cpp</option>
<option>det_by_minor.hpp</option>
</select>
</td>
<td>
<select onchange='choose_current0(this)'>
<option>Headings-&gt;</option>
<option>Syntax</option>
<option>Inclusion</option>
<option>Constructor</option>
<option>Scalar</option>
<option>n</option>
<option>det</option>
<option>---..a</option>
<option>---..d</option>
<option>Vector</option>
<option>Example</option>
<option>Source Code</option>
</select>
</td>
</tr></table><br/>



<center><b><big><big>Determinant Using Expansion by Minors</big></big></b></center>
<br/>
<b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>

<br/>

<code><font color="blue"><span style='white-space: nowrap'>#&#xA0;include&#xA0;&lt;cppad/speed/det_by_minor.hpp&gt;<br/>
</span></font></code>

<code><font color="blue"><span style='white-space: nowrap'>det_by_minor&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Scalar</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>det</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>

<code><i><font color="black"><span style='white-space: nowrap'>d</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>det</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>a</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
<br/>
<b><big><a name="Inclusion" id="Inclusion">Inclusion</a></big></b>
<br/>
The template class <code><font color="blue">det_by_minor</font></code> is defined in the <code><font color="blue">CppAD</font></code>
namespace by including 
the file <code><font color="blue">cppad/speed/det_by_minor.hpp</font></code>
(relative to the CppAD distribution directory).
It is only intended for example and testing purposes, 
so it is not automatically included by
<a href="cppad.xml" target="_top"><span style='white-space: nowrap'>cppad.hpp</span></a>
.

<br/>
<br/>
<b><big><a name="Constructor" id="Constructor">Constructor</a></big></b>
<br/>
The syntax

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;det_by_minor&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Scalar</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>det</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
constructs the object 
<code><i><font color="black"><span style='white-space: nowrap'>det</span></font></i></code>
 which can be used for 
evaluating the determinant of 
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
 by 
<code><i><font color="black"><span style='white-space: nowrap'>n</span></font></i></code>
 matrices
using expansion by minors.

<br/>
<br/>
<b><big><a name="Scalar" id="Scalar">Scalar</a></big></b>
<br/>
The type 
<code><i><font color="black"><span style='white-space: nowrap'>Scalar</span></font></i></code>
 must satisfy the same conditions
as in the function <a href="det_of_minor.xml#Scalar" target="_top"><span style='white-space: nowrap'>det_of_minor</span></a>
.

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

<code><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'>n</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
<br/>
<b><big><a name="det" id="det">det</a></big></b>
<br/>
The syntax

<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'>d</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>det</span></font></i><font color="blue"><span style='white-space: nowrap'>(</span></font><i><font color="black"><span style='white-space: nowrap'>a</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
returns the determinant of the matrix 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i></code>
 using expansion by minors.

<br/>
<br/>
<b><a name="det.a" id="det.a">a</a></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>a</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'>Vector</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>a</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It must be a 
<code><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i></code>
 with length 
<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>

 and with
elements of type 
<code><i><font color="black"><span style='white-space: nowrap'>Scalar</span></font></i></code>
.
The elements of the 
<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 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>A</mi>
</mrow></math>

 are defined,
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'>n</mi>
<mn>-1</mn>
</mrow></math>

, by

<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>

<br/>
<b><a name="det.d" id="det.d">d</a></b>
<br/>
The return value 
<code><i><font color="black"><span style='white-space: nowrap'>d</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'>Scalar</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>d</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
It is equal to the determinant of 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>A</mi>
</mrow></math>

.

<br/>
<br/>
<b><big><a name="Vector" id="Vector">Vector</a></big></b>
<br/>
If 
<code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i></code>
 is a 
<code><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i></code>
 object, 
it must support the syntax

<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'>y</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'>]<br/>
</span></font></code>
where 
<code><i><font color="black"><span style='white-space: nowrap'>i</span></font></i></code>
 has type <code><font color="blue">size_t</font></code> with value less than 
<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>

.
This must return a 
<code><i><font color="black"><span style='white-space: nowrap'>Scalar</span></font></i></code>
 value corresponding to the <code><i>i</i></code>-th
element of the vector 
<code><i><font color="black"><span style='white-space: nowrap'>y</span></font></i></code>
.
This is the only requirement of the type 
<code><i><font color="black"><span style='white-space: nowrap'>Vector</span></font></i></code>
.




<br/>
<br/>
<b><big><a name="Example" id="Example">Example</a></big></b>
<br/>
The file
<a href="det_by_minor.cpp.xml" target="_top"><span style='white-space: nowrap'>det_by_minor.cpp</span></a>
 
contains an example and test of <code><font color="blue">det_by_minor.hpp</font></code>.
It returns true if it succeeds and false otherwise.

<br/>
<br/>
<b><big><a name="Source Code" id="Source Code">Source Code</a></big></b>
<br/>
The file
<a href="det_by_minor.hpp.xml" target="_top"><span style='white-space: nowrap'>det_by_minor.hpp</span></a>
 
contains the source for this template function.



<hr/>Input File: cppad/speed/det_by_minor.hpp

</body>
</html>