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<title>LU Factorization of A Square Matrix and Stability Calculation</title>
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<option>jp</option>
<option>LU</option>
<option>---..A</option>
<option>---..P</option>
<option>---..L</option>
<option>---..U</option>
<option>---..Factor</option>
<option>---..Determinant</option>
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<center><b><big><big>LU Factorization of A Square Matrix and Stability Calculation</big></big></b></center>
<br/>
<b><big><a name="Syntax" id="Syntax">Syntax</a></big></b>

<code><font color="blue"><br/>
# include &lt;cppad/cppad.hpp&gt;</font></code>
<code><span style='white-space: nowrap'><br/>
</span></code>
<code><i><font color="black"><span style='white-space: nowrap'>sign</span></font></i><font color="blue"><span style='white-space: nowrap'>&#xA0;=&#xA0;LuRatio(</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'>ratio</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/>
Computes an LU factorization of the matrix 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i></code>
 
where 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i></code>
 is a square matrix.
A measure of the numerical stability called 
<code><i><font color="black"><span style='white-space: nowrap'>ratio</span></font></i></code>
 is calculated.
This ratio is useful when the results of <code><font color="blue">LuRatio</font></code> are
used as part of an <a href="adfun.xml" target="_top"><span style='white-space: nowrap'>ADFun</span></a>
 object.

<br/>
<br/>
<b><big><a name="Include" id="Include">Include</a></big></b>
<br/>
This routine is designed to be used with AD objects and
requires the <code><font color="blue">cppad/cppad.hpp</font></code> file to be included.

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

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;int&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>sign</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
If 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i></code>
 is invertible, 
<code><i><font color="black"><span style='white-space: nowrap'>sign</span></font></i></code>
 is plus or minus one
and is the sign of the permutation corresponding to the row ordering

<code><i><font color="black"><span style='white-space: nowrap'>ip</span></font></i></code>
 and column ordering 
<code><i><font color="black"><span style='white-space: nowrap'>jp</span></font></i></code>
.
If 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i></code>
 is not invertible, 
<code><i><font color="black"><span style='white-space: nowrap'>sign</span></font></i></code>
 is zero.

<br/>
<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;</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 of <a href="lufactor.xml#SizeVector" target="_top"><span style='white-space: nowrap'>SizeVector</span></a>
 below).
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 input value of the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>ip</span></font></i></code>
 does not matter.
The output value of 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;</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 of <a href="lufactor.xml#SizeVector" target="_top"><span style='white-space: nowrap'>SizeVector</span></a>
 below).
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 input value of the elements of 
<code><i><font color="black"><span style='white-space: nowrap'>jp</span></font></i></code>
 does not matter.
The output value of 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;</span></font><i><font color="black"><span style='white-space: nowrap'>ADvector</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 of <a href="luratio.xml#ADvector" target="_top"><span style='white-space: nowrap'>ADvector</span></a>
 below).

<br/>
<br/>
<b><a name="LU.A" id="LU.A">A</a></b>
<br/>
We define 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i></code>
 as the matrix corresponding to the input 
value of 
<code><i><font color="black"><span style='white-space: nowrap'>LU</span></font></i></code>
.

<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 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i></code>
 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>
<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 the 
output value 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 the
output value 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.Factor" id="LU.Factor">Factor</a></b>
<br/>
If the return value 
<code><i><font color="black"><span style='white-space: nowrap'>sign</span></font></i></code>
 is non-zero,

<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'>&#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'>&#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'><br/>
</span></font></code>
If the return value of 
<code><i><font color="black"><span style='white-space: nowrap'>sign</span></font></i></code>
 is zero,
the contents of 
<code><i><font color="black"><span style='white-space: nowrap'>L</span></font></i></code>
 and 
<code><i><font color="black"><span style='white-space: nowrap'>U</span></font></i></code>
 are not defined. 

<br/>
<br/>
<b><a name="LU.Determinant" id="LU.Determinant">Determinant</a></b>

<br/>
If the return value 
<code><i><font color="black"><span style='white-space: nowrap'>sign</span></font></i></code>
 is zero,
the determinant of 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i></code>
 is zero.
If 
<code><i><font color="black"><span style='white-space: nowrap'>sign</span></font></i></code>
 is non-zero,
using the output value of 
<code><i><font color="black"><span style='white-space: nowrap'>LU</span></font></i></code>

the determinant of the matrix 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i></code>
 is equal to

<code><font color="blue"><span style='white-space: nowrap'><br/>
</span></font><i><font color="black"><span style='white-space: nowrap'>sign</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'>[</span></font><i><font color="black"><span style='white-space: nowrap'>ip</span></font></i><font color="blue"><span style='white-space: nowrap'>[0],&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>jp</span></font></i><font color="blue"><span style='white-space: nowrap'>[0]]&#xA0;*&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>...</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'>[</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'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>-1],&#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'>n</span></font></i><font color="blue"><span style='white-space: nowrap'>-1]]&#xA0;<br/>
</span></font></code>
<br/>
<b><big><a name="ratio" id="ratio">ratio</a></big></b>
<br/>
The argument 
<code><i><font color="black"><span style='white-space: nowrap'>ratio</span></font></i></code>
 has prototype

<code><font color="blue"><span style='white-space: nowrap'><br/>
&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;&#xA0;AD&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;&#xA0;&amp;</span></font><i><font color="black"><span style='white-space: nowrap'>ratio</span></font></i><font color="blue"><span style='white-space: nowrap'><br/>
</span></font></code>
On input, the value of 
<code><i><font color="black"><span style='white-space: nowrap'>ratio</span></font></i></code>
 does not matter.
On output it is a measure of how good the choice of pivots is.
For 
<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mi mathvariant='italic'>p</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>

, 
the <i>p</i>-th pivot element is the element of maximum absolute value of a 

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">&#x000D7;</mo>
<mo stretchy="false">(</mo>
<mi mathvariant='italic'>n</mi>
<mo stretchy="false">-</mo>
<mi mathvariant='italic'>p</mi>
<mo stretchy="false">)</mo>
</mrow></math>

 sub-matrix.
The ratio of each element of sub-matrix divided by the pivot element
is computed.
The return value of 
<code><i><font color="black"><span style='white-space: nowrap'>ratio</span></font></i></code>
 is the maximum absolute value of
such ratios over with respect to all elements and all the pivots.

<br/>
<br/>
<b><a name="ratio.Purpose" id="ratio.Purpose">Purpose</a></b>
<br/>
Suppose that the execution of a call to <code><font color="blue">LuRatio</font></code> 
is recorded in the 
<code><font color="blue"><span style='white-space: nowrap'>ADFun&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;</span></font></code>
 object 
<code><i><font color="black"><span style='white-space: nowrap'>F</span></font></i></code>
.
Then a call to <a href="forward.xml" target="_top"><span style='white-space: nowrap'>Forward</span></a>
 of the form

<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'>F</span></font></i><font color="blue"><span style='white-space: nowrap'>.Forward(</span></font><i><font color="black"><span style='white-space: nowrap'>k</span></font></i><font color="blue"><span style='white-space: nowrap'>,&#xA0;</span></font><i><font color="black"><span style='white-space: nowrap'>xk</span></font></i><font color="blue"><span style='white-space: nowrap'>)<br/>
</span></font></code>
with 
<code><i><font color="black"><span style='white-space: nowrap'>k</span></font></i></code>
 equal to zero will revaluate this Lu factorization
with the same pivots and a new value for 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i></code>
.
In this case, the resulting 
<code><i><font color="black"><span style='white-space: nowrap'>ratio</span></font></i></code>
 may not be one.
If 
<code><i><font color="black"><span style='white-space: nowrap'>ratio</span></font></i></code>
 is too large (the meaning of too large is up to you), 
the current pivots do not yield a stable LU factorization of 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i></code>
.
A better choice for the pivots (for this value of 
<code><i><font color="black"><span style='white-space: nowrap'>A</span></font></i></code>
)
will be made if you recreate the <code><font color="blue">ADFun</font></code> object
starting with the <a href="independent.xml" target="_top"><span style='white-space: nowrap'>Independent</span></a>
 variable values
that correspond to the vector 
<code><i><font color="black"><span style='white-space: nowrap'>xk</span></font></i></code>
.

<br/>
<br/>
<b><big><a name="SizeVector" id="SizeVector">SizeVector</a></big></b>
<br/>
The type 
<code><i><font color="black"><span style='white-space: nowrap'>SizeVector</span></font></i></code>
 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&#xA0;of&#xA0;type&#xA0;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="ADvector" id="ADvector">ADvector</a></big></b>
<br/>
The type 
<code><i><font color="black"><span style='white-space: nowrap'>ADvector</span></font></i></code>
 must be a 
<a href="simplevector.xml" target="_top"><span style='white-space: nowrap'>simple&#xA0;vector&#xA0;class</span></a>
 with elements of type

<code><font color="blue"><span style='white-space: nowrap'>AD&lt;</span></font><i><font color="black"><span style='white-space: nowrap'>Base</span></font></i><font color="blue"><span style='white-space: nowrap'>&gt;</span></font></code>
.
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="Example" id="Example">Example</a></big></b>

<br/>
The file <a href="lu_ratio.cpp.xml" target="_top"><span style='white-space: nowrap'>lu_ratio.cpp</span></a>

contains an example and test of using <code><font color="blue">LuRatio</font></code>.
It returns true if it succeeds and false otherwise.


<hr/>Input File: cppad/local/lu_ratio.hpp

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