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ublas/doc/triangular.htm
Jörg Walter bbca4f92fa A couple of fixes and improvements
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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&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 upper 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;<br>
#include &lt;boost/numeric/ublas/io.hpp&gt;<br>
<br>
int main () {<br>
using namespace boost::numeric::ublas;<br>
triangular_matrix&lt;double, lower&gt; ml (3, 3);<br>
for (unsigned i = 0; i &lt; ml.size1 (); ++ i)<br>
for (unsigned j = 0; j &lt;= i; ++ j)<br>
ml (i, j) = 3 * i + j;<br>
std::cout &lt;&lt; ml &lt;&lt; std::endl;<br>
triangular_matrix&lt;double, upper&gt; mu (3, 3);<br>
for (unsigned i = 0; i &lt; mu.size1 (); ++ i)<br>
for (unsigned j = i; j &lt; mu.size2 (); ++ j)<br>
mu (i, j) = 3 * i + j;<br>
std::cout &lt;&lt; mu &lt;&lt; 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>&nbsp;</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.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&lt;triangular_matrix&lt;T, F1, F2, A&gt;
&gt;</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
&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)</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 &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>
<h4>Interface</h4>
<pre>
<code>// Array based triangular matrix class<br>
template&lt;class T, class F1, class F2, class A&gt;<br>
class triangular_matrix:<br>
public matrix_expression&lt;triangular_matrix&lt;T, F1, F2, A&gt; &gt; {<br>
public:<br>
typedef std::size_t size_type;<br>
typedef std::ptrdiff_t difference_type;<br>
typedef T value_type;<br>
typedef const T &amp;const_reference;<br>
typedef T &amp;reference;<br>
typedef const T *const_pointer;<br>
typedef T *pointer;<br>
typedef F1 functor1_type;<br>
typedef F2 functor2_type;<br>
typedef A array_type;<br>
typedef const A const_array_type;<br>
typedef const triangular_matrix&lt;T, F1, F2, A&gt; const_self_type;<br>
typedef triangular_matrix&lt;T, F1, F2, A&gt; self_type;<br>
typedef const matrix_const_reference&lt;const_self_type&gt; const_closure_type;<br>
typedef matrix_reference&lt;self_type&gt; closure_type;<br>
typedef packed_tag storage_category;<br>
typedef typename F1::packed_category packed_category;<br>
typedef typename F2::orientation_category orientation_category;<br>
<br>
// Construction and destruction<br>
triangular_matrix ();<br>
triangular_matrix (size_type size1, size_type size2);<br>
triangular_matrix (const triangular_matrix &amp;m);<br>
template&lt;class AE&gt;<br>
triangular_matrix (const matrix_expression&lt;AE&gt; &amp;ae);<br>
<br>
// Accessors<br>
size_type size1 () const;<br>
size_type size2 () const;<br>
const_array_type &amp;data () const;<br>
array_type &amp;data ();<br>
<br>
// Resizing<br>
void resize (size_type size1, size_type size2);<br>
<br>
// Element access<br>
const_reference operator () (size_type i, size_type j) const;<br>
reference operator () (size_type i, size_type j);<br>
<br>
// Assignment<br>
triangular_matrix &amp;operator = (const triangular_matrix &amp;m);<br>
triangular_matrix &amp;assign_temporary (triangular_matrix &amp;m);<br>
template&lt;class AE&gt;<br>
triangular_matrix &amp;operator = (const matrix_expression&lt;AE&gt; &amp;ae);<br>
template&lt;class AE&gt;<br>
triangular_matrix &amp;reset (const matrix_expression&lt;AE&gt; &amp;ae);<br>
template&lt;class AE&gt;<br>
triangular_matrix &amp;assign (const matrix_expression&lt;AE&gt; &amp;ae);<br>
template&lt;class AE&gt;<br>
triangular_matrix&amp; operator += (const matrix_expression&lt;AE&gt; &amp;ae);<br>
template&lt;class AE&gt;<br>
triangular_matrix &amp;plus_assign (const matrix_expression&lt;AE&gt; &amp;ae);<br>
template&lt;class AE&gt;<br>
triangular_matrix&amp; operator -= (const matrix_expression&lt;AE&gt; &amp;ae);<br>
template&lt;class AE&gt;<br>
triangular_matrix &amp;minus_assign (const matrix_expression&lt;AE&gt; &amp;ae);<br>
template&lt;class AT&gt;<br>
triangular_matrix&amp; operator *= (const AT &amp;at);<br>
template&lt;class AT&gt;<br>
triangular_matrix&amp; operator /= (const AT &amp;at);<br>
<br>
// Swapping<br>
void swap (triangular_matrix &amp;m);<br>
friend void swap (triangular_matrix &amp;m1, triangular_matrix &amp;m2);<br>
<br>
// Element insertion and erasure<br>
void insert (size_type i, size_type j, const_reference t);<br>
void erase (size_type i, size_type j);<br>
void clear ();<br>
<br>
class const_iterator1;<br>
class iterator1;<br>
class const_iterator2;<br>
class iterator2;<br>
typedef reverse_iterator_base1&lt;const_iterator1&gt; const_reverse_iterator1;<br>
typedef reverse_iterator_base1&lt;iterator1&gt; reverse_iterator1;<br>
typedef reverse_iterator_base2&lt;const_iterator2&gt; const_reverse_iterator2;<br>
typedef reverse_iterator_base2&lt;iterator2&gt; reverse_iterator2;<br>
<br>
// Element lookup<br>
const_iterator1 find_first1 (int rank, size_type i, size_type j) const;<br>
iterator1 find_first1 (int rank, size_type i, size_type j);<br>
const_iterator1 find_last1 (int rank, size_type i, size_type j) const;<br>
iterator1 find_last1 (int rank, size_type i, size_type j);<br>
const_iterator2 find_first2 (int rank, size_type i, size_type j) const;<br>
iterator2 find_first2 (int rank, size_type i, size_type j);<br>
const_iterator2 find_last2 (int rank, size_type i, size_type j) const;<br>
iterator2 find_last2 (int rank, size_type i, size_type j);<br>
<br>
// Iterators simply are indices.<br>
<br>
class const_iterator1:<br>
public container_const_reference&lt;triangular_matrix&gt;,<br>
public random_access_iterator_base&lt;const_iterator1, value_type&gt; {<br>
public:<br>
typedef packed_random_access_iterator_tag iterator_category;<br>
typedef typename triangular_matrix::difference_type difference_type;<br>
typedef typename triangular_matrix::value_type value_type;<br>
typedef typename triangular_matrix::const_reference reference;<br>
typedef typename triangular_matrix::const_pointer pointer;<br>
typedef const_iterator2 dual_iterator_type;<br>
typedef const_reverse_iterator2 dual_reverse_iterator_type;<br>
<br>
// Construction and destruction<br>
const_iterator1 ();<br>
const_iterator1 (const triangular_matrix &amp;m, size_type it1, size_type it2);<br>
const_iterator1 (const iterator1 &amp;it);<br>
<br>
// Arithmetic<br>
const_iterator1 &amp;operator ++ ();<br>
const_iterator1 &amp;operator -- ();<br>
const_iterator1 &amp;operator += (difference_type n);<br>
const_iterator1 &amp;operator -= (difference_type n);<br>
difference_type operator - (const const_iterator1 &amp;it) const;<br>
<br>
// Dereference<br>
reference operator * () const;<br>
<br>
const_iterator2 begin () const;<br>
const_iterator2 end () const;<br>
const_reverse_iterator2 rbegin () const;<br>
const_reverse_iterator2 rend () const;<br>
<br>
// Indices<br>
size_type index1 () const;<br>
size_type index2 () const;<br>
<br>
// Assignment<br>
const_iterator1 &amp;operator = (const const_iterator1 &amp;it);<br>
<br>
// Comparison<br>
bool operator == (const const_iterator1 &amp;it) const;<br>
bool operator &lt;(const const_iterator1 &amp;it) const;<br>
};<br>
<br>
const_iterator1 begin1 () const;<br>
const_iterator1 end1 () const;<br>
<br>
class iterator1:<br>
public container_reference&lt;triangular_matrix&gt;,<br>
public random_access_iterator_base&lt;iterator1, value_type&gt; {<br>
public:<br>
typedef packed_random_access_iterator_tag iterator_category;<br>
typedef typename triangular_matrix::difference_type difference_type;<br>
typedef typename triangular_matrix::value_type value_type;<br>
typedef typename triangular_matrix::reference reference;<br>
typedef typename triangular_matrix::pointer pointer;<br>
typedef iterator2 dual_iterator_type;<br>
typedef reverse_iterator2 dual_reverse_iterator_type;<br>
<br>
// Construction and destruction<br>
iterator1 ();<br>
iterator1 (triangular_matrix &amp;m, size_type it1, size_type it2);<br>
<br>
// Arithmetic<br>
iterator1 &amp;operator ++ ();<br>
iterator1 &amp;operator -- ();<br>
iterator1 &amp;operator += (difference_type n);<br>
iterator1 &amp;operator -= (difference_type n);<br>
difference_type operator - (const iterator1 &amp;it) const;<br>
<br>
// Dereference<br>
reference operator * () const;<br>
<br>
iterator2 begin () const;<br>
iterator2 end () const;<br>
reverse_iterator2 rbegin () const;<br>
reverse_iterator2 rend () const;<br>
<br>
// Indices<br>
size_type index1 () const;<br>
size_type index2 () const;<br>
<br>
// Assignment<br>
iterator1 &amp;operator = (const iterator1 &amp;it);<br>
<br>
// Comparison<br>
bool operator == (const iterator1 &amp;it) const;<br>
bool operator &lt;(const iterator1 &amp;it) const;<br>
};<br>
<br>
iterator1 begin1 ();<br>
iterator1 end1 ();<br>
<br>
class const_iterator2:<br>
public container_const_reference&lt;triangular_matrix&gt;,<br>
public random_access_iterator_base&lt;const_iterator2, value_type&gt; {<br>
public:<br>
typedef packed_random_access_iterator_tag iterator_category;<br>
typedef typename triangular_matrix::difference_type difference_type;<br>
typedef typename triangular_matrix::value_type value_type;<br>
typedef typename triangular_matrix::const_reference reference;<br>
typedef typename triangular_matrix::const_pointer pointer;<br>
typedef const_iterator1 dual_iterator_type;<br>
typedef const_reverse_iterator1 dual_reverse_iterator_type;<br>
<br>
// Construction and destruction<br>
const_iterator2 ();<br>
const_iterator2 (const triangular_matrix &amp;m, size_type it1, size_type it2);<br>
const_iterator2 (const iterator2 &amp;it);<br>
<br>
// Arithmetic<br>
const_iterator2 &amp;operator ++ ();<br>
const_iterator2 &amp;operator -- ();<br>
const_iterator2 &amp;operator += (difference_type n);<br>
const_iterator2 &amp;operator -= (difference_type n);<br>
difference_type operator - (const const_iterator2 &amp;it) const;<br>
<br>
// Dereference<br>
reference operator * () const;<br>
<br>
const_iterator1 begin () const;<br>
const_iterator1 end () const;<br>
const_reverse_iterator1 rbegin () const;<br>
const_reverse_iterator1 rend () const;<br>
<br>
// Indices<br>
size_type index1 () const;<br>
size_type index2 () const;<br>
<br>
// Assignment<br>
const_iterator2 &amp;operator = (const const_iterator2 &amp;it);<br>
<br>
// Comparison<br>
bool operator == (const const_iterator2 &amp;it) const;<br>
bool operator &lt;(const const_iterator2 &amp;it) const;<br>
};<br>
<br>
const_iterator2 begin2 () const;<br>
const_iterator2 end2 () const;<br>
<br>
class iterator2:<br>
public container_reference&lt;triangular_matrix&gt;,<br>
public random_access_iterator_base&lt;iterator2, value_type&gt; {<br>
public:<br>
typedef packed_random_access_iterator_tag iterator_category;<br>
typedef typename triangular_matrix::difference_type difference_type;<br>
typedef typename triangular_matrix::value_type value_type;<br>
typedef typename triangular_matrix::reference reference;<br>
typedef typename triangular_matrix::pointer pointer;<br>
typedef iterator1 dual_iterator_type;<br>
typedef reverse_iterator1 dual_reverse_iterator_type;<br>
<br>
// Construction and destruction<br>
iterator2 ();<br>
iterator2 (triangular_matrix &amp;m, size_type it1, size_type it2);<br>
<br>
// Arithmetic<br>
iterator2 &amp;operator ++ ();<br>
iterator2 &amp;operator -- ();<br>
iterator2 &amp;operator += (difference_type n);<br>
iterator2 &amp;operator -= (difference_type n);<br>
difference_type operator - (const iterator2 &amp;it) const;<br>
<br>
// Dereference<br>
reference operator * () const;<br>
<br>
iterator1 begin () const;<br>
iterator1 end () const;<br>
reverse_iterator1 rbegin () const;<br>
reverse_iterator1 rend () const;<br>
<br>
// Indices<br>
size_type index1 () const;<br>
size_type index2 () const;<br>
<br>
// Assignment<br>
iterator2 &amp;operator = (const iterator2 &amp;it);<br>
<br>
// Comparison<br>
bool operator == (const iterator2 &amp;it) const;<br>
bool operator &lt;(const iterator2 &amp;it) const;<br>
};<br>
<br>
iterator2 begin2 ();<br>
iterator2 end2 ();<br>
<br>
// Reverse iterators<br>
<br>
const_reverse_iterator1 rbegin1 () const;<br>
const_reverse_iterator1 rend1 () const;<br>
<br>
reverse_iterator1 rbegin1 ();<br>
reverse_iterator1 rend1 ();<br>
<br>
const_reverse_iterator2 rbegin2 () const;<br>
const_reverse_iterator2 rend2 () const;<br>
<br>
reverse_iterator2 rbegin2 ();<br>
reverse_iterator2 rend2 ();<br>
};</code>
</pre>
<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;<br>
#include &lt;boost/numeric/ublas/io.hpp&gt;<br>
<br>
int main () {<br>
using namespace boost::numeric::ublas;<br>
matrix&lt;double&gt; m (3, 3);<br>
triangular_adaptor&lt;matrix&lt;double&gt;, lower&gt; tal (m);<br>
for (unsigned i = 0; i &lt; tal.size1 (); ++ i)<br>
for (unsigned j = 0; j &lt;= i; ++ j)<br>
tal (i, j) = 3 * i + j;<br>
std::cout &lt;&lt; tal &lt;&lt; std::endl;<br>
triangular_adaptor&lt;matrix&lt;double&gt;, upper&gt; tau (m);<br>
for (unsigned i = 0; i &lt; tau.size1 (); ++ i)<br>
for (unsigned j = i; j &lt; tau.size2 (); ++ j)<br>
tau (i, j) = 3 * i + j;<br>
std::cout &lt;&lt; tau &lt;&lt; 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>&nbsp;</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&lt;triangular_adaptor&lt;M, F&gt;
&gt;</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 &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>
<h4>Interface</h4>
<pre>
<code>// Triangular matrix adaptor class<br>
template&lt;class M, class F&gt;<br>
class triangular_adaptor:<br>
public matrix_expression&lt;triangular_adaptor&lt;M, F&gt; &gt; {<br>
public:<br>
typedef const M const_matrix_type;<br>
typedef M matrix_type;<br>
typedef F functor_type;<br>
typedef typename M::size_type size_type;<br>
typedef typename M::difference_type difference_type;<br>
typedef typename M::value_type value_type;<br>
typedef typename M::const_reference const_reference;<br>
typedef typename M::reference reference;<br>
typedef typename M::const_pointer const_pointer;<br>
typedef typename M::pointer pointer;<br>
typedef const triangular_adaptor&lt;M, F&gt; const_self_type;<br>
typedef triangular_adaptor&lt;M, F&gt; self_type;<br>
typedef const matrix_const_reference&lt;const_self_type&gt; const_closure_type;<br>
typedef matrix_reference&lt;self_type&gt; closure_type;<br>
typedef typename storage_restrict_traits&lt;typename M::storage_category,<br>
packed_proxy_tag&gt;::storage_category storage_category;<br>
typedef typename F::packed_category packed_category;<br>
typedef typename M::orientation_category orientation_category;<br>
<br>
// Construction and destruction<br>
triangular_adaptor ();<br>
triangular_adaptor (matrix_type &amp;data);<br>
triangular_adaptor (const triangular_adaptor &amp;m);<br>
<br>
// Accessors<br>
size_type size1 () const;<br>
size_type size2 () const;<br>
const_matrix_type &amp;data () const;<br>
matrix_type &amp;data ();<br>
<br>
<br>
// Element access<br>
const_reference operator () (size_type i, size_type j) const;<br>
reference operator () (size_type i, size_type j);<br>
<br>
// Assignment<br>
triangular_adaptor &amp;operator = (const triangular_adaptor &amp;m);<br>
triangular_adaptor &amp;assign_temporary (triangular_adaptor &amp;m);<br>
template&lt;class AE&gt;<br>
triangular_adaptor &amp;operator = (const matrix_expression&lt;AE&gt; &amp;ae);<br>
template&lt;class AE&gt;<br>
triangular_adaptor &amp;assign (const matrix_expression&lt;AE&gt; &amp;ae);<br>
template&lt;class AE&gt;<br>
triangular_adaptor&amp; operator += (const matrix_expression&lt;AE&gt; &amp;ae);<br>
template&lt;class AE&gt;<br>
triangular_adaptor &amp;plus_assign (const matrix_expression&lt;AE&gt; &amp;ae);<br>
template&lt;class AE&gt;<br>
triangular_adaptor&amp; operator -= (const matrix_expression&lt;AE&gt; &amp;ae);<br>
template&lt;class AE&gt;<br>
triangular_adaptor &amp;minus_assign (const matrix_expression&lt;AE&gt; &amp;ae);<br>
template&lt;class AT&gt;<br>
triangular_adaptor&amp; operator *= (const AT &amp;at);<br>
template&lt;class AT&gt;<br>
triangular_adaptor&amp; operator /= (const AT &amp;at);<br>
<br>
// Swapping<br>
void swap (triangular_adaptor &amp;m);<br>
friend void swap (triangular_adaptor &amp;m1, triangular_adaptor &amp;m2);<br>
<br>
class const_iterator1;<br>
class iterator1;<br>
class const_iterator2;<br>
class iterator2;<br>
typedef reverse_iterator_base1&lt;const_iterator1&gt; const_reverse_iterator1;<br>
typedef reverse_iterator_base1&lt;iterator1&gt; reverse_iterator1;<br>
typedef reverse_iterator_base2&lt;const_iterator2&gt; const_reverse_iterator2;<br>
typedef reverse_iterator_base2&lt;iterator2&gt; reverse_iterator2;<br>
<br>
// Element lookup<br>
const_iterator1 find_first1 (int rank, size_type i, size_type j) const;<br>
iterator1 find_first1 (int rank, size_type i, size_type j);<br>
const_iterator1 find_last1 (int rank, size_type i, size_type j) const;<br>
iterator1 find_last1 (int rank, size_type i, size_type j);<br>
const_iterator2 find_first2 (int rank, size_type i, size_type j) const;<br>
iterator2 find_first2 (int rank, size_type i, size_type j);<br>
const_iterator2 find_last2 (int rank, size_type i, size_type j) const;<br>
iterator2 find_last2 (int rank, size_type i, size_type j);<br>
<br>
// Iterators simply are indices.<br>
<br>
class const_iterator1:<br>
public container_const_reference&lt;triangular_adaptor&gt;,<br>
public random_access_iterator_base&lt;const_iterator1, value_type&gt; {<br>
public:<br>
typedef packed_random_access_iterator_tag iterator_category;<br>
typedef typename triangular_adaptor::difference_type difference_type;<br>
typedef typename triangular_adaptor::value_type value_type;<br>
typedef typename triangular_adaptor::const_reference reference;<br>
typedef typename triangular_adaptor::const_pointer pointer;<br>
typedef const_iterator2 dual_iterator_type;<br>
typedef const_reverse_iterator2 dual_reverse_iterator_type;<br>
<br>
// Construction and destruction<br>
const_iterator1 ();<br>
const_iterator1 (const triangular_adaptor &amp;m, size_type it1, size_type it2);<br>
const_iterator1 (const iterator1 &amp;it);<br>
<br>
// Arithmetic<br>
const_iterator1 &amp;operator ++ ();<br>
const_iterator1 &amp;operator -- ();<br>
const_iterator1 &amp;operator += (difference_type n);<br>
const_iterator1 &amp;operator -= (difference_type n);<br>
difference_type operator - (const const_iterator1 &amp;it) const;<br>
<br>
// Dereference<br>
reference operator * () const;<br>
<br>
const_iterator2 begin () const;<br>
const_iterator2 end () const;<br>
const_reverse_iterator2 rbegin () const;<br>
const_reverse_iterator2 rend () const;<br>
<br>
// Indices<br>
size_type index1 () const;<br>
size_type index2 () const;<br>
<br>
// Assignment<br>
const_iterator1 &amp;operator = (const const_iterator1 &amp;it);<br>
<br>
// Comparison<br>
bool operator == (const const_iterator1 &amp;it) const;<br>
bool operator &lt;(const const_iterator1 &amp;it) const;<br>
};<br>
<br>
const_iterator1 begin1 () const;<br>
const_iterator1 end1 () const;<br>
<br>
class iterator1:<br>
public container_reference&lt;triangular_adaptor&gt;,<br>
public random_access_iterator_base&lt;iterator1, value_type&gt; {<br>
public:<br>
typedef packed_random_access_iterator_tag iterator_category;<br>
typedef typename triangular_adaptor::difference_type difference_type;<br>
typedef typename triangular_adaptor::value_type value_type;<br>
typedef typename triangular_adaptor::reference reference;<br>
typedef typename triangular_adaptor::pointer pointer;<br>
typedef iterator2 dual_iterator_type;<br>
typedef reverse_iterator2 dual_reverse_iterator_type;<br>
<br>
// Construction and destruction<br>
iterator1 ();<br>
iterator1 (triangular_adaptor &amp;m, size_type it1, size_type it2);<br>
<br>
// Arithmetic<br>
iterator1 &amp;operator ++ ();<br>
iterator1 &amp;operator -- ();<br>
iterator1 &amp;operator += (difference_type n);<br>
iterator1 &amp;operator -= (difference_type n);<br>
difference_type operator - (const iterator1 &amp;it) const;<br>
<br>
// Dereference<br>
reference operator * () const;<br>
<br>
iterator2 begin () const;<br>
iterator2 end () const;<br>
reverse_iterator2 rbegin () const;<br>
reverse_iterator2 rend () const;<br>
<br>
// Indices<br>
size_type index1 () const;<br>
size_type index2 () const;<br>
<br>
// Assignment<br>
iterator1 &amp;operator = (const iterator1 &amp;it);<br>
<br>
// Comparison<br>
bool operator == (const iterator1 &amp;it) const;<br>
bool operator &lt;(const iterator1 &amp;it) const;<br>
};<br>
<br>
iterator1 begin1 ();<br>
iterator1 end1 ();<br>
<br>
class const_iterator2:<br>
public container_const_reference&lt;triangular_adaptor&gt;,<br>
public random_access_iterator_base&lt;const_iterator2, value_type&gt; {<br>
public:<br>
typedef packed_random_access_iterator_tag iterator_category;<br>
typedef typename triangular_adaptor::difference_type difference_type;<br>
typedef typename triangular_adaptor::value_type value_type;<br>
typedef typename triangular_adaptor::const_reference reference;<br>
typedef typename triangular_adaptor::const_pointer pointer;<br>
typedef const_iterator1 dual_iterator_type;<br>
typedef const_reverse_iterator1 dual_reverse_iterator_type;<br>
<br>
// Construction and destruction<br>
const_iterator2 ();<br>
const_iterator2 (const triangular_adaptor &amp;m, size_type it1, size_type it2);<br>
const_iterator2 (const iterator2 &amp;it);<br>
<br>
// Arithmetic<br>
const_iterator2 &amp;operator ++ ();<br>
const_iterator2 &amp;operator -- ();<br>
const_iterator2 &amp;operator += (difference_type n);<br>
const_iterator2 &amp;operator -= (difference_type n);<br>
difference_type operator - (const const_iterator2 &amp;it) const;<br>
<br>
// Dereference<br>
reference operator * () const;<br>
<br>
const_iterator1 begin () const;<br>
const_iterator1 end () const;<br>
const_reverse_iterator1 rbegin () const;<br>
const_reverse_iterator1 rend () const;<br>
<br>
// Indices<br>
size_type index1 () const;<br>
size_type index2 () const;<br>
<br>
// Assignment<br>
const_iterator2 &amp;operator = (const const_iterator2 &amp;it);<br>
<br>
// Comparison<br>
bool operator == (const const_iterator2 &amp;it) const;<br>
bool operator &lt;(const const_iterator2 &amp;it) const;<br>
};<br>
<br>
const_iterator2 begin2 () const;<br>
const_iterator2 end2 () const;<br>
<br>
class iterator2:<br>
public container_reference&lt;triangular_adaptor&gt;,<br>
public random_access_iterator_base&lt;iterator2, value_type&gt; {<br>
public:<br>
typedef packed_random_access_iterator_tag iterator_category;<br>
typedef typename triangular_adaptor::difference_type difference_type;<br>
typedef typename triangular_adaptor::value_type value_type;<br>
typedef typename triangular_adaptor::reference reference;<br>
typedef typename triangular_adaptor::pointer pointer;<br>
typedef iterator1 dual_iterator_type;<br>
typedef reverse_iterator1 dual_reverse_iterator_type;<br>
<br>
// Construction and destruction<br>
iterator2 ();<br>
iterator2 (triangular_adaptor &amp;m, size_type it1, size_type it2);<br>
<br>
// Arithmetic<br>
iterator2 &amp;operator ++ ();<br>
iterator2 &amp;operator -- ();<br>
iterator2 &amp;operator += (difference_type n);<br>
iterator2 &amp;operator -= (difference_type n);<br>
difference_type operator - (const iterator2 &amp;it) const;<br>
<br>
// Dereference<br>
reference operator * () const;<br>
<br>
iterator1 begin () const;<br>
iterator1 end () const;<br>
reverse_iterator1 rbegin () const;<br>
reverse_iterator1 rend () const;<br>
<br>
// Indices<br>
size_type index1 () const;<br>
size_type index2 () const;<br>
<br>
// Assignment<br>
iterator2 &amp;operator = (const iterator2 &amp;it);<br>
<br>
// Comparison<br>
bool operator == (const iterator2 &amp;it) const;<br>
bool operator &lt;(const iterator2 &amp;it) const;<br>
};<br>
<br>
iterator2 begin2 ();<br>
iterator2 end2 ();<br>
<br>
// Reverse iterators<br>
<br>
const_reverse_iterator1 rbegin1 () const;<br>
const_reverse_iterator1 rend1 () const;<br>
<br>
reverse_iterator1 rbegin1 ();<br>
reverse_iterator1 rend1 ();<br>
<br>
const_reverse_iterator2 rbegin2 () const;<br>
const_reverse_iterator2 rend2 () const;<br>
<br>
reverse_iterator2 rbegin2 ();<br>
reverse_iterator2 rend2 ();<br>
};</code>
</pre>
<hr>
<p>Copyright (&copy;) 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>
</body>
</html>