libs/numeric/ublas/doc/triangular.htm - platform/external/boost - Git at Google

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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.htm#matrix">Matrix</a> .</p>
 <h4>Type requirements</h4>
 <p>None, except for those imposed by the requirements of <a href=
 "container_concept.htm#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.htm#matrix_expression">Matrix Expression</a>
 .</p>
 <h4>Type requirements</h4>
 <p>None, except for those imposed by the requirements of <a href=
 "expression_concept.htm#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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	<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<T, F1, F2,
	A></code> is the base container adaptor for triangular matrices.
	For a <em>(n x n</em> )-dimensional lower triangular matrix and
	<em>0 <= i < n</em>,<em>0 <= j < n</em> holds
	<em>t</em><sub><em>i, j</em></sub> <em>= 0</em> , if <em>i >
	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 <= i <
	n</em>,<em>0 <= j < n</em> holds <em>t</em><sub><em>i,
	j</em></sub> <em>= 0</em> , if <em>i < 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 <boost/numeric/ublas/triangular.hpp>
	#include <boost/numeric/ublas/io.hpp>

	int main () {
	using namespace boost::numeric::ublas;
	triangular_matrix<double, lower> ml (3, 3);
	for (unsigned i = 0; i < ml.size1 (); ++ i)
	for (unsigned j = 0; j <= i; ++ j)
	ml (i, j) = 3 * i + j;
	std::cout << ml << std::endl;
	triangular_matrix<double, upper> mu (3, 3);
	for (unsigned i = 0; i < mu.size1 (); ++ i)
	for (unsigned j = i; j < mu.size2 (); ++ j)
	mu (i, j) = 3 * i + j;
	std::cout << mu << 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<T></code></td>
	</tr>
	</tbody>
	</table>
	<h4>Model of</h4>
	<p><a href="container_concept.htm#matrix">Matrix</a> .</p>
	<h4>Type requirements</h4>
	<p>None, except for those imposed by the requirements of <a href=
	"container_concept.htm#matrix">Matrix</a> .</p>
	<h4>Public base classes</h4>
	<p><code>matrix_container<triangular_matrix<T, F1, F2, A>
	></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
	&m)</code></td>
	<td>The copy constructor.</td>
	</tr>
	<tr>
	<td><code>template<class AE><br />
	triangular_matrix (const matrix_expression<AE>
	&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 &operator = (const
	triangular_matrix &m)</code></td>
	<td>The assignment operator.</td>
	</tr>
	<tr>
	<td><code>triangular_matrix &assign_temporary
	(triangular_matrix &m)</code></td>
	<td>Assigns a temporary. May change the triangular matrix
	<code>m</code>.</td>
	</tr>
	<tr>
	<td><code>template<class AE><br />
	triangular_matrix &operator = (const
	matrix_expression<AE> &ae)</code></td>
	<td>The extended assignment operator.</td>
	</tr>
	<tr>
	<td><code>template<class AE><br />
	triangular_matrix &assign (const matrix_expression<AE>
	&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<class AE><br />
	triangular_matrix &operator += (const
	matrix_expression<AE> &ae)</code></td>
	<td>A computed assignment operator. Adds the matrix expression to
	the triangular matrix.</td>
	</tr>
	<tr>
	<td><code>template<class AE><br />
	triangular_matrix &plus_assign (const
	matrix_expression<AE> &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<class AE><br />
	triangular_matrix &operator -= (const
	matrix_expression<AE> &ae)</code></td>
	<td>A computed assignment operator. Subtracts the matrix expression
	from the triangular matrix.</td>
	</tr>
	<tr>
	<td><code>template<class AE><br />
	triangular_matrix &minus_assign (const
	matrix_expression<AE> &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<class AT><br />
	triangular_matrix &operator *= (const AT &at)</code></td>
	<td>A computed assignment operator. Multiplies the triangular
	matrix with a scalar.</td>
	</tr>
	<tr>
	<td><code>template<class AT><br />
	triangular_matrix &operator /= (const AT &at)</code></td>
	<td>A computed assignment operator. Divides the triangular matrix
	through a scalar.</td>
	</tr>
	<tr>
	<td><code>void swap (triangular_matrix &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<T></code> ,
	<code>bounded_array<T></code> and
	<code>std::vector<T></code> .</p>
	<h2><a name="triangular_adaptor"></a>Triangular Adaptor</h2>
	<h4>Description</h4>
	<p>The templated class <code>triangular_adaptor<M, F></code>
	is a triangular matrix adaptor for other matrices.</p>
	<h4>Example</h4>
	<pre>
	#include <boost/numeric/ublas/triangular.hpp>
	#include <boost/numeric/ublas/io.hpp>

	int main () {
	using namespace boost::numeric::ublas;
	matrix<double> m (3, 3);
	triangular_adaptor<matrix<double>, lower> tal (m);
	for (unsigned i = 0; i < tal.size1 (); ++ i)
	for (unsigned j = 0; j <= i; ++ j)
	tal (i, j) = 3 * i + j;
	std::cout << tal << std::endl;
	triangular_adaptor<matrix<double>, upper> tau (m);
	for (unsigned i = 0; i < tau.size1 (); ++ i)
	for (unsigned j = i; j < tau.size2 (); ++ j)
	tau (i, j) = 3 * i + j;
	std::cout << tau << 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.htm#matrix_expression">Matrix Expression</a>
	.</p>
	<h4>Type requirements</h4>
	<p>None, except for those imposed by the requirements of <a href=
	"expression_concept.htm#matrix_expression">Matrix Expression</a> .</p>
	<h4>Public base classes</h4>
	<p><code>matrix_expression<triangular_adaptor<M, F>
	></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 &data)</code></td>
	<td>Constructs a <code>triangular_adaptor</code> of a matrix.</td>
	</tr>
	<tr>
	<td><code>triangular_adaptor (const triangular_adaptor
	&m)</code></td>
	<td>The copy constructor.</td>
	</tr>
	<tr>
	<td><code>template<class AE><br />
	triangular_adaptor (const matrix_expression<AE>
	&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 &operator = (const
	triangular_adaptor &m)</code></td>
	<td>The assignment operator.</td>
	</tr>
	<tr>
	<td><code>triangular_adaptor &assign_temporary
	(triangular_adaptor &m)</code></td>
	<td>Assigns a temporary. May change the triangular adaptor
	<code>m</code>.</td>
	</tr>
	<tr>
	<td><code>template<class AE><br />
	triangular_adaptor &operator = (const
	matrix_expression<AE> &ae)</code></td>
	<td>The extended assignment operator.</td>
	</tr>
	<tr>
	<td><code>template<class AE><br />
	triangular_adaptor &assign (const matrix_expression<AE>
	&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<class AE><br />
	triangular_adaptor &operator += (const
	matrix_expression<AE> &ae)</code></td>
	<td>A computed assignment operator. Adds the matrix expression to
	the triangular adaptor.</td>
	</tr>
	<tr>
	<td><code>template<class AE><br />
	triangular_adaptor &plus_assign (const
	matrix_expression<AE> &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<class AE><br />
	triangular_adaptor &operator -= (const
	matrix_expression<AE> &ae)</code></td>
	<td>A computed assignment operator. Subtracts the matrix expression
	from the triangular adaptor.</td>
	</tr>
	<tr>
	<td><code>template<class AE><br />
	triangular_adaptor &minus_assign (const
	matrix_expression<AE> &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<class AT><br />
	triangular_adaptor &operator *= (const AT &at)</code></td>
	<td>A computed assignment operator. Multiplies the triangular
	adaptor with a scalar.</td>
	</tr>
	<tr>
	<td><code>template<class AT><br />
	triangular_adaptor &operator /= (const AT &at)</code></td>
	<td>A computed assignment operator. Divides the triangular adaptor
	through a scalar.</td>
	</tr>
	<tr>
	<td><code>void swap (triangular_adaptor &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 (©) 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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