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<h1><img src="../../../../boost.png" align="middle" />Triangular Matrix</h1>
<div class="toc" id="toc"></div>
<h2><a name="triangular_matrix"></a>Triangular Matrix</h2>
<h4>Description</h4>
<p>The templated class <code>triangular_matrix&lt;T, F1, F2,
A&gt;</code> is the base container adaptor for triangular matrices.
For a <em>(n x n</em> )-dimensional lower triangular matrix and
<em>0 &lt;= i &lt; n</em>,<em>0 &lt;= j &lt; n</em> holds
<em>t</em><sub><em>i, j</em></sub> <em>= 0</em> , if <em>i &gt;
j</em>. If furthermore holds t<sub><em>i, i</em></sub><em>= 1</em>
the matrix is called unit lower triangular. For a <em>(n x n</em>
)-dimensional lower triangular matrix and <em>0 &lt;= i &lt;
n</em>,<em>0 &lt;= j &lt; n</em> holds <em>t</em><sub><em>i,
j</em></sub> <em>= 0</em> , if <em>i &lt; j</em>. If furthermore
holds t<sub><em>i, i</em></sub><em>= 1</em> the matrix is called
unit lower triangular. The storage of triangular matrices is
packed.</p>
<h4>Example</h4>
<pre>
#include &lt;boost/numeric/ublas/triangular.hpp&gt;
#include &lt;boost/numeric/ublas/io.hpp&gt;

int main () {
    using namespace boost::numeric::ublas;
    triangular_matrix&lt;double, lower&gt; ml (3, 3);
    for (unsigned i = 0; i &lt; ml.size1 (); ++ i)
        for (unsigned j = 0; j &lt;= i; ++ j)
            ml (i, j) = 3 * i + j;
    std::cout &lt;&lt; ml &lt;&lt; std::endl;
    triangular_matrix&lt;double, upper&gt; mu (3, 3);
    for (unsigned i = 0; i &lt; mu.size1 (); ++ i)
        for (unsigned j = i; j &lt; mu.size2 (); ++ j)
            mu (i, j) = 3 * i + j;
    std::cout &lt;&lt; mu &lt;&lt; std::endl;
}
</pre>
<p>Please read the <a href="samples/ex_triangular.cpp">full triangular example</a> for more details.</p>

<h4>Definition</h4>
<p>Defined in the header triangular.hpp.</p>
<h4>Template parameters</h4>
<table border="1" summary="parameters">
<tbody>
<tr>
<th>Parameter</th>
<th>Description</th>
<th>Default</th>
</tr>
<tr>
<td><code>T</code></td>
<td>The type of object stored in the matrix.</td>
<td></td>
</tr>
<tr>
<td><code>F1</code></td>
<td>Functor describing the type of the triangular matrix. <a href=
"#triangular_matrix_1">[1]</a></td>
<td><code>lower</code></td>
</tr>
<tr>
<td><code>F2</code></td>
<td>Functor describing the storage organization. <a href=
"#triangular_matrix_2">[2]</a></td>
<td><code>row_major</code></td>
</tr>
<tr>
<td><code>A</code></td>
<td>The type of the adapted array. <a href=
"#triangular_matrix_3">[3]</a></td>
<td><code>unbounded_array&lt;T&gt;</code></td>
</tr>
</tbody>
</table>
<h4>Model of</h4>
<p><a href="container_concept.html#matrix">Matrix</a> .</p>
<h4>Type requirements</h4>
<p>None, except for those imposed by the requirements of <a href=
"container_concept.html#matrix">Matrix</a> .</p>
<h4>Public base classes</h4>
<p><code>matrix_container&lt;triangular_matrix&lt;T, F1, F2, A&gt;
&gt;</code></p>
<h4>Members</h4>
<table border="1" summary="members">
<tbody>
<tr>
<th>Member</th>
<th>Description</th>
</tr>
<tr>
<td><code>triangular_matrix ()</code></td>
<td>Allocates an uninitialized <code>triangular_matrix</code> that
holds zero rows of zero elements.</td>
</tr>
<tr>
<td><code>triangular_matrix (size_type size1, size_type
size2)</code></td>
<td>Allocates an uninitialized <code>triangular_matrix</code> that
holds <code>size1</code> rows of <code>size2</code> elements.</td>
</tr>
<tr>
<td><code>triangular_matrix (const triangular_matrix
&amp;m)</code></td>
<td>The copy constructor.</td>
</tr>
<tr>
<td><code>template&lt;class AE&gt;<br />
triangular_matrix (const matrix_expression&lt;AE&gt;
&amp;ae)</code></td>
<td>The extended copy constructor.</td>
</tr>
<tr>
<td><code>void resize (size_type size1, size_type size2, bool
preserve = true)</code></td>
<td>Reallocates a <code>triangular_matrix</code> to hold
<code>size1</code> rows of <code>size2</code> elements. The
existing elements of the <code>triangular_matrix</code> are
preseved when specified.</td>
</tr>
<tr>
<td><code>size_type size1 () const</code></td>
<td>Returns the number of rows.</td>
</tr>
<tr>
<td><code>size_type size2 () const</code></td>
<td>Returns the number of columns.</td>
</tr>
<tr>
<td><code>const_reference operator () (size_type i, size_type j)
const</code></td>
<td>Returns a <code>const</code> reference of the <code>j</code>
-th element in the <code>i</code>-th row.</td>
</tr>
<tr>
<td><code>reference operator () (size_type i, size_type
j)</code></td>
<td>Returns a reference of the <code>j</code>-th element in the
<code>i</code>-th row.</td>
</tr>
<tr>
<td><code>triangular_matrix &amp;operator = (const
triangular_matrix &amp;m)</code></td>
<td>The assignment operator.</td>
</tr>
<tr>
<td><code>triangular_matrix &amp;assign_temporary
(triangular_matrix &amp;m)</code></td>
<td>Assigns a temporary. May change the triangular matrix
<code>m</code>.</td>
</tr>
<tr>
<td><code>template&lt;class AE&gt;<br />
triangular_matrix &amp;operator = (const
matrix_expression&lt;AE&gt; &amp;ae)</code></td>
<td>The extended assignment operator.</td>
</tr>
<tr>
<td><code>template&lt;class AE&gt;<br />
triangular_matrix &amp;assign (const matrix_expression&lt;AE&gt;
&amp;ae)</code></td>
<td>Assigns a matrix expression to the triangular matrix. Left and
right hand side of the assignment should be independent.</td>
</tr>
<tr>
<td><code>template&lt;class AE&gt;<br />
triangular_matrix &amp;operator += (const
matrix_expression&lt;AE&gt; &amp;ae)</code></td>
<td>A computed assignment operator. Adds the matrix expression to
the triangular matrix.</td>
</tr>
<tr>
<td><code>template&lt;class AE&gt;<br />
triangular_matrix &amp;plus_assign (const
matrix_expression&lt;AE&gt; &amp;ae)</code></td>
<td>Adds a matrix expression to the triangular matrix. Left and
right hand side of the assignment should be independent.</td>
</tr>
<tr>
<td><code>template&lt;class AE&gt;<br />
triangular_matrix &amp;operator -= (const
matrix_expression&lt;AE&gt; &amp;ae)</code></td>
<td>A computed assignment operator. Subtracts the matrix expression
from the triangular matrix.</td>
</tr>
<tr>
<td><code>template&lt;class AE&gt;<br />
triangular_matrix &amp;minus_assign (const
matrix_expression&lt;AE&gt; &amp;ae)</code></td>
<td>Subtracts a matrix expression from the triangular matrix. Left
and right hand side of the assignment should be independent.</td>
</tr>
<tr>
<td><code>template&lt;class AT&gt;<br />
triangular_matrix &amp;operator *= (const AT &amp;at)</code></td>
<td>A computed assignment operator. Multiplies the triangular
matrix with a scalar.</td>
</tr>
<tr>
<td><code>template&lt;class AT&gt;<br />
triangular_matrix &amp;operator /= (const AT &amp;at)</code></td>
<td>A computed assignment operator. Divides the triangular matrix
through a scalar.</td>
</tr>
<tr>
<td><code>void swap (triangular_matrix &amp;m)</code></td>
<td>Swaps the contents of the triangular matrices.</td>
</tr>
<tr>
<td><code>void insert (size_type i, size_type j, const_reference
t)</code></td>
<td>Inserts the value <code>t</code> at the <code>j</code>-th
element of the <code>i</code>-th row.</td>
</tr>
<tr>
<td><code>void erase (size_type i, size_type j)</code></td>
<td>Erases the value at the <code>j</code>-th elemenst of the
<code>i</code>-th row.</td>
</tr>
<tr>
<td><code>void clear ()</code></td>
<td>Clears the matrix.</td>
</tr>
<tr>
<td><code>const_iterator1 begin1 () const</code></td>
<td>Returns a <code>const_iterator1</code> pointing to the
beginning of the <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>const_iterator1 end1 () const</code></td>
<td>Returns a <code>const_iterator1</code> pointing to the end of
the <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>iterator1 begin1 ()</code></td>
<td>Returns a <code>iterator1</code> pointing to the beginning of
the <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>iterator1 end1 ()</code></td>
<td>Returns a <code>iterator1</code> pointing to the end of the
<code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>const_iterator2 begin2 () const</code></td>
<td>Returns a <code>const_iterator2</code> pointing to the
beginning of the <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>const_iterator2 end2 () const</code></td>
<td>Returns a <code>const_iterator2</code> pointing to the end of
the <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>iterator2 begin2 ()</code></td>
<td>Returns a <code>iterator2</code> pointing to the beginning of
the <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>iterator2 end2 ()</code></td>
<td>Returns a <code>iterator2</code> pointing to the end of the
<code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator1 rbegin1 () const</code></td>
<td>Returns a <code>const_reverse_iterator1</code> pointing to the
beginning of the reversed <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator1 rend1 () const</code></td>
<td>Returns a <code>const_reverse_iterator1</code> pointing to the
end of the reversed <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator1 rbegin1 ()</code></td>
<td>Returns a <code>reverse_iterator1</code> pointing to the
beginning of the reversed <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator1 rend1 ()</code></td>
<td>Returns a <code>reverse_iterator1</code> pointing to the end of
the reversed <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator2 rbegin2 () const</code></td>
<td>Returns a <code>const_reverse_iterator2</code> pointing to the
beginning of the reversed <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator2 rend2 () const</code></td>
<td>Returns a <code>const_reverse_iterator2</code> pointing to the
end of the reversed <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator2 rbegin2 ()</code></td>
<td>Returns a <code>reverse_iterator2</code> pointing to the
beginning of the reversed <code>triangular_matrix</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator2 rend2 ()</code></td>
<td>Returns a <code>reverse_iterator2</code> pointing to the end of
the reversed <code>triangular_matrix</code>.</td>
</tr>
</tbody>
</table>
<h4>Notes</h4>
<p><a name="triangular_matrix_1">[1]</a>
Supported parameters for the type of the triangular matrix are
<code>lower</code> , <code>unit_lower</code>, <code>upper</code>
and <code>unit_upper</code> .</p>
<p><a name="triangular_matrix_2">[2]</a>
Supported parameters for the storage organization are
<code>row_major</code> and <code>column_major</code>.</p>
<p><a name="triangular_matrix_3">[3]</a>
Supported parameters for the adapted array are
<code>unbounded_array&lt;T&gt;</code> ,
<code>bounded_array&lt;T&gt;</code> and
<code>std::vector&lt;T&gt;</code> .</p>
<h2><a name="triangular_adaptor"></a>Triangular Adaptor</h2>
<h4>Description</h4>
<p>The templated class <code>triangular_adaptor&lt;M, F&gt;</code>
is a triangular matrix adaptor for other matrices.</p>
<h4>Example</h4>
<pre>
#include &lt;boost/numeric/ublas/triangular.hpp&gt;
#include &lt;boost/numeric/ublas/io.hpp&gt;

int main () {
    using namespace boost::numeric::ublas;
    matrix&lt;double&gt; m (3, 3);
    triangular_adaptor&lt;matrix&lt;double&gt;, lower&gt; tal (m);
    for (unsigned i = 0; i &lt; tal.size1 (); ++ i)
        for (unsigned j = 0; j &lt;= i; ++ j)
            tal (i, j) = 3 * i + j;
    std::cout &lt;&lt; tal &lt;&lt; std::endl;
    triangular_adaptor&lt;matrix&lt;double&gt;, upper&gt; tau (m);
    for (unsigned i = 0; i &lt; tau.size1 (); ++ i)
        for (unsigned j = i; j &lt; tau.size2 (); ++ j)
            tau (i, j) = 3 * i + j;
    std::cout &lt;&lt; tau &lt;&lt; std::endl;
}
</pre>
<p>Please read the <a href="samples/ex_triangular.cpp">full triangular example</a> for more details.</p>

<h4>Definition</h4>
<p>Defined in the header triangular.hpp.</p>
<h4>Template parameters</h4>
<table border="1" summary="parameters">
<tbody>
<tr>
<th>Parameter</th>
<th>Description</th>
<th>Default</th>
</tr>
<tr>
<td><code>M</code></td>
<td>The type of the adapted matrix.</td>
<td></td>
</tr>
<tr>
<td><code>F</code></td>
<td>Functor describing the type of the triangular adaptor. <a href=
"#triangular_adaptor_1">[1]</a></td>
<td><code>lower</code></td>
</tr>
</tbody>
</table>
<h4>Model of</h4>
<p><a href="expression_concept.html#matrix_expression">Matrix Expression</a>
.</p>
<h4>Type requirements</h4>
<p>None, except for those imposed by the requirements of <a href=
"expression_concept.html#matrix_expression">Matrix Expression</a> .</p>
<h4>Public base classes</h4>
<p><code>matrix_expression&lt;triangular_adaptor&lt;M, F&gt;
&gt;</code></p>
<h4>Members</h4>
<table border="1" summary="members">
<tbody>
<tr>
<th>Member</th>
<th>Description</th>
</tr>
<tr>
<td><code>triangular_adaptor (matrix_type &amp;data)</code></td>
<td>Constructs a <code>triangular_adaptor</code> of a matrix.</td>
</tr>
<tr>
<td><code>triangular_adaptor (const triangular_adaptor
&amp;m)</code></td>
<td>The copy constructor.</td>
</tr>
<tr>
<td><code>template&lt;class AE&gt;<br />
triangular_adaptor (const matrix_expression&lt;AE&gt;
&amp;ae)</code></td>
<td>The extended copy constructor.</td>
</tr>
<tr>
<td><code>size_type size1 () const</code></td>
<td>Returns the number of rows.</td>
</tr>
<tr>
<td><code>size_type size2 () const</code></td>
<td>Returns the number of columns.</td>
</tr>
<tr>
<td><code>const_reference operator () (size_type i, size_type j)
const</code></td>
<td>Returns a <code>const</code> reference of the <code>j</code>
-th element in the <code>i</code>-th row.</td>
</tr>
<tr>
<td><code>reference operator () (size_type i, size_type
j)</code></td>
<td>Returns a reference of the <code>j</code>-th element in the
<code>i</code>-th row.</td>
</tr>
<tr>
<td><code>triangular_adaptor &amp;operator = (const
triangular_adaptor &amp;m)</code></td>
<td>The assignment operator.</td>
</tr>
<tr>
<td><code>triangular_adaptor &amp;assign_temporary
(triangular_adaptor &amp;m)</code></td>
<td>Assigns a temporary. May change the triangular adaptor
<code>m</code>.</td>
</tr>
<tr>
<td><code>template&lt;class AE&gt;<br />
triangular_adaptor &amp;operator = (const
matrix_expression&lt;AE&gt; &amp;ae)</code></td>
<td>The extended assignment operator.</td>
</tr>
<tr>
<td><code>template&lt;class AE&gt;<br />
triangular_adaptor &amp;assign (const matrix_expression&lt;AE&gt;
&amp;ae)</code></td>
<td>Assigns a matrix expression to the triangular adaptor. Left and
right hand side of the assignment should be independent.</td>
</tr>
<tr>
<td><code>template&lt;class AE&gt;<br />
triangular_adaptor &amp;operator += (const
matrix_expression&lt;AE&gt; &amp;ae)</code></td>
<td>A computed assignment operator. Adds the matrix expression to
the triangular adaptor.</td>
</tr>
<tr>
<td><code>template&lt;class AE&gt;<br />
triangular_adaptor &amp;plus_assign (const
matrix_expression&lt;AE&gt; &amp;ae)</code></td>
<td>Adds a matrix expression to the triangular adaptor. Left and
right hand side of the assignment should be independent.</td>
</tr>
<tr>
<td><code>template&lt;class AE&gt;<br />
triangular_adaptor &amp;operator -= (const
matrix_expression&lt;AE&gt; &amp;ae)</code></td>
<td>A computed assignment operator. Subtracts the matrix expression
from the triangular adaptor.</td>
</tr>
<tr>
<td><code>template&lt;class AE&gt;<br />
triangular_adaptor &amp;minus_assign (const
matrix_expression&lt;AE&gt; &amp;ae)</code></td>
<td>Subtracts a matrix expression from the triangular adaptor. Left
and right hand side of the assignment should be independent.</td>
</tr>
<tr>
<td><code>template&lt;class AT&gt;<br />
triangular_adaptor &amp;operator *= (const AT &amp;at)</code></td>
<td>A computed assignment operator. Multiplies the triangular
adaptor with a scalar.</td>
</tr>
<tr>
<td><code>template&lt;class AT&gt;<br />
triangular_adaptor &amp;operator /= (const AT &amp;at)</code></td>
<td>A computed assignment operator. Divides the triangular adaptor
through a scalar.</td>
</tr>
<tr>
<td><code>void swap (triangular_adaptor &amp;m)</code></td>
<td>Swaps the contents of the triangular adaptors.</td>
</tr>
<tr>
<td><code>const_iterator1 begin1 () const</code></td>
<td>Returns a <code>const_iterator1</code> pointing to the
beginning of the <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>const_iterator1 end1 () const</code></td>
<td>Returns a <code>const_iterator1</code> pointing to the end of
the <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>iterator1 begin1 ()</code></td>
<td>Returns a <code>iterator1</code> pointing to the beginning of
the <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>iterator1 end1 ()</code></td>
<td>Returns a <code>iterator1</code> pointing to the end of the
<code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>const_iterator2 begin2 () const</code></td>
<td>Returns a <code>const_iterator2</code> pointing to the
beginning of the <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>const_iterator2 end2 () const</code></td>
<td>Returns a <code>const_iterator2</code> pointing to the end of
the <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>iterator2 begin2 ()</code></td>
<td>Returns a <code>iterator2</code> pointing to the beginning of
the <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>iterator2 end2 ()</code></td>
<td>Returns a <code>iterator2</code> pointing to the end of the
<code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator1 rbegin1 () const</code></td>
<td>Returns a <code>const_reverse_iterator1</code> pointing to the
beginning of the reversed <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator1 rend1 () const</code></td>
<td>Returns a <code>const_reverse_iterator1</code> pointing to the
end of the reversed <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator1 rbegin1 ()</code></td>
<td>Returns a <code>reverse_iterator1</code> pointing to the
beginning of the reversed <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator1 rend1 ()</code></td>
<td>Returns a <code>reverse_iterator1</code> pointing to the end of
the reversed <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator2 rbegin2 () const</code></td>
<td>Returns a <code>const_reverse_iterator2</code> pointing to the
beginning of the reversed <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>const_reverse_iterator2 rend2 () const</code></td>
<td>Returns a <code>const_reverse_iterator2</code> pointing to the
end of the reversed <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator2 rbegin2 ()</code></td>
<td>Returns a <code>reverse_iterator2</code> pointing to the
beginning of the reversed <code>triangular_adaptor</code>.</td>
</tr>
<tr>
<td><code>reverse_iterator2 rend2 ()</code></td>
<td>Returns a <code>reverse_iterator2</code> pointing to the end of
the reversed <code>triangular_adaptor</code>.</td>
</tr>
</tbody>
</table>
<h4>Notes</h4>
<p><a name="triangular_adaptor_1">[1]</a>
Supported parameters for the type of the triangular adaptor are
<code>lower</code> , <code>unit_lower</code>, <code>upper</code>
and <code>unit_upper</code> .</p>
<hr />
<p>Copyright (&copy;) 2000-2002 Joerg Walter, Mathias Koch<br />
   Use, modification and distribution are subject to the
   Boost Software License, Version 1.0.
   (See accompanying file LICENSE_1_0.txt
   or copy at <a href="http://www.boost.org/LICENSE_1_0.txt">
      http://www.boost.org/LICENSE_1_0.txt
   </a>).
</p>
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