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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
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+<html xmlns="http://www.w3.org/1999/xhtml">
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+<meta name="generator" content=
+"HTML Tidy for Linux/x86 (vers 1st March 2004), see www.w3.org" />
+<meta http-equiv="Content-Type" content=
+"text/html; charset=us-ascii" />
+<link rel="stylesheet" href="../../../../boost.css" type="text/css"/>
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+<title>Triangular Matrix</title>
+</head>
+<body>
+<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.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<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.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<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>
+<script type="text/javascript">
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+ $('#toc').toc();
+})(jQuery);
+</script>
+</body>
+</html>