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<h1><img src="c++boost.gif" alt="c++boost.gif" align="Center">
Triangular Matrix</h1>
<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 upper 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><br>
#include <boost/numeric/ublas/io.hpp><br>
<br>
int main () {<br>
using namespace boost::numeric::ublas;<br>
triangular_matrix<double, lower> ml (3, 3);<br>
for (unsigned i = 0; i < ml.size1 (); ++ i)<br>
for (unsigned j = 0; j <= i; ++ j)<br>
ml (i, j) = 3 * i + j;<br>
std::cout << ml << std::endl;<br>
triangular_matrix<double, upper> mu (3, 3);<br>
for (unsigned i = 0; i < mu.size1 (); ++ i)<br>
for (unsigned j = i; j < mu.size2 (); ++ j)<br>
mu (i, j) = 3 * i + j;<br>
std::cout << mu << std::endl;<br>
}<br>
</pre>
<h4>Definition</h4>
<p>Defined in the header triangular.hpp.</p>
<h4>Template parameters</h4>
<table border="1">
<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.htm#matrix">Matrix</a> .</p>
<h4>Type requirements</h4>
<p>None, except for those imposed by the requirements of <a href=
"container.htm#matrix">Matrix</a> .</p>
<h4>Public base classes</h4>
<p><code>matrix_expression<triangular_matrix<T, F1, F2, A>
></code></p>
<h4>Members</h4>
<table border="1">
<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)</code></td>
<td>Reallocates a <code>triangular_matrix</code> to hold
<code>size1</code> rows of <code>size2</code> elements. The content
of the <code>triangular_matrix</code> is not preserved.</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><br>
#include <boost/numeric/ublas/io.hpp><br>
<br>
int main () {<br>
using namespace boost::numeric::ublas;<br>
matrix<double> m (3, 3);<br>
triangular_adaptor<matrix<double>, lower> tal (m);<br>
for (unsigned i = 0; i < tal.size1 (); ++ i)<br>
for (unsigned j = 0; j <= i; ++ j)<br>
tal (i, j) = 3 * i + j;<br>
std::cout << tal << std::endl;<br>
triangular_adaptor<matrix<double>, upper> tau (m);<br>
for (unsigned i = 0; i < tau.size1 (); ++ i)<br>
for (unsigned j = i; j < tau.size2 (); ++ j)<br>
tau (i, j) = 3 * i + j;<br>
std::cout << tau << std::endl;<br>
}<br>
</pre>
<h4>Definition</h4>
<p>Defined in the header triangular.hpp.</p>
<h4>Template parameters</h4>
<table border="1">
<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.htm#matrix_expression">Matrix Expression</a>
.</p>
<h4>Type requirements</h4>
<p>None, except for those imposed by the requirements of <a href=
"expression.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">
<tbody>
<tr>
<th>Member</th>
<th>Description</th>
</tr>
<tr>
<td><code>triangular_adaptor ()</code> </td>
<td>Constructs a <code>triangular_adaptor</code> that holds zero
rows of zero elements.</td>
</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>
Permission to copy, use, modify, sell and distribute this document
is granted provided this copyright notice appears in all copies.
This document is provided ``as is'' without express or implied
warranty, and with no claim as to its suitability for any
purpose.</p>
<p>Last revised: 1/15/2003</p>
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