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+<title>Matrix Expressions</title>
+</head>
+<body>
+<h1><img src="../../../../boost.png" align="middle" />Matrix Expressions</h1>
+<div class="toc" id="toc"></div>
+<h2><a name="matrix_expression"></a>Matrix Expression</h2>
+<h4>Description</h4>
+<p>The templated class <code>matrix_expression<E></code>
+is required to be a public base of all classes which model the Matrix Expression concept.</p>
+<h4>Definition</h4>
+<p>Defined in the header expression_types.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>E</code></td>
+<td>The type of the matrix expression.</td>
+<td> </td>
+</tr>
+</tbody>
+</table>
+<h4>Model of</h4>
+<p>None. <u>Not a Matrix Expression</u>!
+</p>
+<h4>Type requirements</h4>
+<p>None.</p>
+<h4>Public base classes</h4>
+<p>None.</p>
+<h4>Members</h4>
+<table border="1" summary="members">
+<tbody>
+<tr>
+<th>Member</th>
+<th>Description</th>
+</tr>
+<tr>
+<td><code>const expression_type &operator () ()
+const</code></td>
+<td>Returns a <code>const</code> reference of the expression.</td>
+</tr>
+<tr>
+<td><code>expression_type &operator () ()</code></td>
+<td>Returns a reference of the expression.</td>
+</tr>
+</tbody>
+</table>
+<h4>Notes</h4>
+<p>The <code>operator[]</code>, <code>row</code>, <code>column</code>, <code>range</code>, <code>slice</code> and <code>project</code> functions have been removed. Use the free functions defined in <a href="matrix_proxy.html">matrix proxy</a> instead.</p>
+<h2><a name="matrix_container"></a>Matrix Container</h2>
+<h4>Description</h4>
+<p>The templated class <code>matrix_container<C></code>
+is required to be a public base of all classes which model the Matrix concept.
+This includes the class <code>matrix</code> itself.</p>
+<h4>Definition</h4>
+<p>Defined in the header expression_types.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>E</code></td>
+<td>The type of the matrix expression.</td>
+<td> </td>
+</tr>
+</tbody>
+</table>
+<h4>Model of</h4>
+<p>None. <u>Not a Matrix Expression OR Matrix</u>!
+</p>
+<h4>Type requirements</h4>
+<p>None.</p>
+<h4>Public base classes</h4>
+<p><code>matrix_expression<C></code></p>
+<h4>Members</h4>
+<table border="1" summary="members">
+<tbody>
+<tr>
+<th>Member</th>
+<th>Description</th>
+</tr>
+<tr>
+<td><code>const container_type &operator () ()
+const</code></td>
+<td>Returns a <code>const</code> reference of the container.</td>
+</tr>
+<tr>
+<td><code>container_type &operator () ()</code></td>
+<td>Returns a reference of the container.</td>
+</tr>
+</tbody>
+</table>
+<h2><a name="matrix_references"></a>Matrix References</h2>
+<h3>Reference</h3>
+<h4>Description</h4>
+<p>The templated class <code>matrix_reference<E></code>
+contains a reference to a matrix expression.</p>
+<h4>Definition</h4>
+<p>Defined in the header matrix_expression.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>E</code></td>
+<td>The type of the matrix expression.</td>
+<td> </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<matrix_reference<E>
+></code></p>
+<h4>Members</h4>
+<table border="1" summary="members">
+<tbody>
+<tr>
+<th>Member</th>
+<th>Description</th>
+</tr>
+<tr>
+<td><code>matrix_reference (expression_type &e)</code></td>
+<td>Constructs a constant reference of the expression.</td>
+</tr>
+<tr>
+<td><code>void resize (size_type size1, size2)</code></td>
+<td>Resizes the expression to hold at most <code>size1</code> rows
+of <code>size2</code> elements.</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 the value 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>const_iterator1 begin1 () const</code></td>
+<td>Returns a <code>const_iterator1</code> pointing to the
+beginning of the expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator1 end1 () const</code></td>
+<td>Returns a <code>const_iterator1</code> pointing to the end of
+the expression.</td>
+</tr>
+<tr>
+<td><code>iterator1 begin1 ()</code></td>
+<td>Returns a <code>iterator1</code> pointing to the beginning of
+the expression.</td>
+</tr>
+<tr>
+<td><code>iterator1 end1 ()</code></td>
+<td>Returns a <code>iterator1</code> pointing to the end of the
+expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator2 begin2 () const</code></td>
+<td>Returns a <code>const_iterator2</code> pointing to the
+beginning of the expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator2 end2 () const</code></td>
+<td>Returns a <code>const_iterator2</code> pointing to the end of
+the expression.</td>
+</tr>
+<tr>
+<td><code>iterator2 begin2 ()</code></td>
+<td>Returns a <code>iterator2</code> pointing to the beginning of
+the expression.</td>
+</tr>
+<tr>
+<td><code>iterator2 end2 ()</code></td>
+<td>Returns a <code>iterator2</code> pointing to the end of the
+expression.</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 expression.</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 expression.</td>
+</tr>
+<tr>
+<td><code>reverse_iterator1 rbegin1 ()</code></td>
+<td>Returns a <code>reverse_iterator1</code> pointing to the
+beginning of the reversed expression.</td>
+</tr>
+<tr>
+<td><code>reverse_iterator1 rend1 ()</code></td>
+<td>Returns a <code>reverse_iterator1</code> pointing to the end of
+the reversed expression.</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 expression.</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 expression.</td>
+</tr>
+<tr>
+<td><code>reverse_iterator2 rbegin2 ()</code></td>
+<td>Returns a <code>reverse_iterator2</code> pointing to the
+beginning of the reversed expression.</td>
+</tr>
+<tr>
+<td><code>reverse_iterator2 rend2 ()</code></td>
+<td>Returns a <code>reverse_iterator2</code> pointing to the end of
+the reversed expression.</td>
+</tr>
+</tbody>
+</table>
+<h2><a name="matrix_operations"></a>Matrix Operations</h2>
+<h3>Unary Operation Description</h3>
+<h4>Description</h4>
+<p>The templated classes <code>matrix_unary1<E, F></code> and
+<code>matrix_unary2<E, F></code> describe unary matrix
+operations.</p>
+<h4>Definition</h4>
+<p>Defined in the header matrix_expression.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>E</code></td>
+<td>The type of the matrix expression.</td>
+<td> </td>
+</tr>
+<tr>
+<td><code>F</code></td>
+<td>The type of the operation.</td>
+<td> </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<matrix_unary1<E, F> ></code>
+and <code>matrix_expression<matrix_unary2<E, F>
+></code> resp.</p>
+<h4>Members</h4>
+<table border="1" summary="members">
+<tbody>
+<tr>
+<th>Member</th>
+<th>Description</th>
+</tr>
+<tr>
+<td><code>matrix_unary1 (const expression_type &e)</code></td>
+<td>Constructs a description of the expression.</td>
+</tr>
+<tr>
+<td><code>matrix_unary2 (const expression_type &e)</code></td>
+<td>Constructs a description of the expression.</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 the value of the <code>j</code>-th element in the
+<code>i</code>-th row.</td>
+</tr>
+<tr>
+<td><code>const_iterator1 begin1 () const</code></td>
+<td>Returns a <code>const_iterator1</code> pointing to the
+beginning of the expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator1 end1 () const</code></td>
+<td>Returns a <code>const_iterator1</code> pointing to the end of
+the expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator2 begin2 () const</code></td>
+<td>Returns a <code>const_iterator2</code> pointing to the
+beginning of the expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator2 end2 () const</code></td>
+<td>Returns a <code>const_iterator2</code> pointing to the end of
+the expression.</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 expression.</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 expression.</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 expression.</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 expression.</td>
+</tr>
+</tbody>
+</table>
+<h3>Unary Operations</h3>
+<h4>Prototypes</h4>
+<pre>
+<code>template<class E, class F>
+ struct matrix_unary1_traits {
+ typedef matrix_unary1<typename E::const_closure_type, F> expression_type;
+ typedef expression_type result_type;
+ };
+
+ // (- m) [i] [j] = - m [i] [j]
+ template<class E>
+ typename matrix_unary1_traits<E, scalar_negate<typename E::value_type> >::result_type
+ operator - (const matrix_expression<E> &e);
+
+ // (conj m) [i] [j] = conj (m [i] [j])
+ template<class E>
+ typename matrix_unary1_traits<E, scalar_conj<typename E::value_type> >::result_type
+ conj (const matrix_expression<E> &e);
+
+ // (real m) [i] [j] = real (m [i] [j])
+ template<class E>
+ typename matrix_unary1_traits<E, scalar_real<typename E::value_type> >::result_type
+ real (const matrix_expression<E> &e);
+
+ // (imag m) [i] [j] = imag (m [i] [j])
+ template<class E>
+ typename matrix_unary1_traits<E, scalar_imag<typename E::value_type> >::result_type
+ imag (const matrix_expression<E> &e);
+
+ template<class E, class F>
+ struct matrix_unary2_traits {
+ typedef matrix_unary2<typename E::const_closure_type, F> expression_type;
+ typedef expression_type result_type;
+ };
+
+ // (trans m) [i] [j] = m [j] [i]
+ template<class E>
+ typename matrix_unary2_traits<E, scalar_identity<typename E::value_type> >::result_type
+ trans (const matrix_expression<E> &e);
+
+ // (herm m) [i] [j] = conj (m [j] [i])
+ template<class E>
+ typename matrix_unary2_traits<E, scalar_conj<typename E::value_type> >::result_type
+ herm (const matrix_expression<E> &e);</code>
+</pre>
+<h4>Description</h4>
+<p><code>operator -</code> computes the additive inverse of a
+matrix expression. <code>conj</code> computes the complex conjugate
+of a matrix expression. <code>real</code> and <code>imag</code>
+compute the real and imaginary parts of a matrix expression.
+<code>trans</code> computes the transpose of a matrix expression.
+<code>herm</code> computes the hermitian, i.e. the complex
+conjugate of the transpose of a matrix expression.</p>
+<h4>Definition</h4>
+<p>Defined in the header matrix_expression.hpp.</p>
+<h4>Type requirements</h4>
+<ul>
+<li><code>E</code> is a model of <a href=
+"expression_concept.html#matrix_expression">Matrix Expression</a> .</li>
+</ul>
+<h4>Preconditions</h4>
+<p>None.</p>
+<h4>Complexity</h4>
+<p>Quadratic depending from the size of the matrix expression.</p>
+<h4>Examples</h4>
+<pre>
+#include <boost/numeric/ublas/matrix.hpp>
+#include <boost/numeric/ublas/io.hpp>
+
+int main () {
+ using namespace boost::numeric::ublas;
+ matrix<std::complex<double> > m (3, 3);
+ for (unsigned i = 0; i < m.size1 (); ++ i)
+ for (unsigned j = 0; j < m.size2 (); ++ j)
+ m (i, j) = std::complex<double> (3 * i + j, 3 * i + j);
+
+ std::cout << - m << std::endl;
+ std::cout << conj (m) << std::endl;
+ std::cout << real (m) << std::endl;
+ std::cout << imag (m) << std::endl;
+ std::cout << trans (m) << std::endl;
+ std::cout << herm (m) << std::endl;
+}
+</pre>
+<h3>Binary Operation Description</h3>
+<h4>Description</h4>
+<p>The templated class <code>matrix_binary<E1, E2, F></code>
+describes a binary matrix operation.</p>
+<h4>Definition</h4>
+<p>Defined in the header matrix_expression.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>E1</code></td>
+<td>The type of the first matrix expression.</td>
+<td></td>
+</tr>
+<tr>
+<td><code>E2</code></td>
+<td>The type of the second matrix expression.</td>
+<td></td>
+</tr>
+<tr>
+<td><code>F</code></td>
+<td>The type of the operation.</td>
+<td></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<matrix_binary<E1, E2, F>
+></code>.</p>
+<h4>Members</h4>
+<table border="1" summary="members">
+<tbody>
+<tr>
+<th>Member</th>
+<th>Description</th>
+</tr>
+<tr>
+<td><code>matrix_binary (const expression1_type &e1, const
+expression2_type &e2)</code></td>
+<td>Constructs a description of the expression.</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 the value of the <code>j</code>-th element in the
+<code>i</code>-th row.</td>
+</tr>
+<tr>
+<td><code>const_iterator1 begin1 () const</code></td>
+<td>Returns a <code>const_iterator1</code> pointing to the
+beginning of the expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator1 end1 () const</code></td>
+<td>Returns a <code>const_iterator1</code> pointing to the end of
+the expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator2 begin2 () const</code></td>
+<td>Returns a <code>const_iterator2</code> pointing to the
+beginning of the expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator2 end2 () const</code></td>
+<td>Returns a <code>const_iterator2</code> pointing to the end of
+the expression.</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 expression.</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 expression.</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 expression.</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 expression.</td>
+</tr>
+</tbody>
+</table>
+<h3>Binary Operations</h3>
+<h4>Prototypes</h4>
+<pre>
+<code>template<class E1, class E2, class F>
+ struct matrix_binary_traits {
+ typedef matrix_binary<typename E1::const_closure_type,
+ typename E2::const_closure_type, F> expression_type;
+ typedef expression_type result_type;
+ };
+
+ // (m1 + m2) [i] [j] = m1 [i] [j] + m2 [i] [j]
+ template<class E1, class E2>
+ typename matrix_binary_traits<E1, E2, scalar_plus<typename E1::value_type,
+ typename E2::value_type> >::result_type
+ operator + (const matrix_expression<E1> &e1,
+ const matrix_expression<E2> &e2);
+
+ // (m1 - m2) [i] [j] = m1 [i] [j] - m2 [i] [j]
+ template<class E1, class E2>
+ typename matrix_binary_traits<E1, E2, scalar_minus<typename E1::value_type,
+ typename E2::value_type> >::result_type
+ operator - (const matrix_expression<E1> &e1,
+ const matrix_expression<E2> &e2);</code>
+</pre>
+<h4>Description</h4>
+<p><code>operator +</code> computes the sum of two matrix
+expressions. <code>operator -</code> computes the difference of two
+matrix expressions.</p>
+<h4>Definition</h4>
+<p>Defined in the header matrix_expression.hpp.</p>
+<h4>Type requirements</h4>
+<ul>
+<li><code>E1</code> is a model of <a href=
+"expression_concept.html#matrix_expression">Matrix Expression</a> .</li>
+<li><code>E2</code> is a model of <a href=
+"expression_concept.html#matrix_expression">Matrix Expression</a> .</li>
+</ul>
+<h4>Preconditions</h4>
+<ul>
+<li><code>e1 ().size1 () == e2 ().size1 ()</code></li>
+<li><code>e1 ().size2 () == e2 ().size2 ()</code></li>
+</ul>
+<h4>Complexity</h4>
+<p>Quadratic depending from the size of the matrix expressions.</p>
+<h4>Examples</h4>
+<pre>
+#include <boost/numeric/ublas/matrix.hpp>
+#include <boost/numeric/ublas/io.hpp>
+
+int main () {
+ using namespace boost::numeric::ublas;
+ matrix<double> m1 (3, 3), m2 (3, 3);
+ for (unsigned i = 0; i < std::min (m1.size1 (), m2.size1 ()); ++ i)
+ for (unsigned j = 0; j < std::min (m1.size2 (), m2.size2 ()); ++ j)
+ m1 (i, j) = m2 (i, j) = 3 * i + j;
+
+ std::cout << m1 + m2 << std::endl;
+ std::cout << m1 - m2 << std::endl;
+}
+</pre>
+<h3>Scalar Matrix Operation Description</h3>
+<h4>Description</h4>
+<p>The templated classes <code>matrix_binary_scalar1<E1, E2,
+F></code> and <code>matrix_binary_scalar2<E1, E2,
+F></code> describe binary operations between a scalar and a
+matrix.</p>
+<h4>Definition</h4>
+<p>Defined in the header matrix_expression.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>E1/E2</code></td>
+<td>The type of the scalar expression.</td>
+<td></td>
+</tr>
+<tr>
+<td><code>E2/E1</code></td>
+<td>The type of the matrix expression.</td>
+<td></td>
+</tr>
+<tr>
+<td><code>F</code></td>
+<td>The type of the operation.</td>
+<td></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<matrix_binary_scalar1<E1, E2,
+F> ></code> and
+<code>matrix_expression<matrix_binary_scalar2<E1, E2, F>
+></code> resp.</p>
+<h4>Members</h4>
+<table border="1" summary="members">
+<tbody>
+<tr>
+<th>Member</th>
+<th>Description</th>
+</tr>
+<tr>
+<td><code>matrix_binary_scalar1 (const expression1_type &e1,
+const expression2_type &e2)</code></td>
+<td>Constructs a description of the expression.</td>
+</tr>
+<tr>
+<td><code>matrix_binary_scalar1 (const expression1_type &e1,
+const expression2_type &e2)</code></td>
+<td>Constructs a description of the expression.</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 the value of the <code>j</code>-th element in the
+<code>i</code>-th row.</td>
+</tr>
+<tr>
+<td><code>const_iterator1 begin1 () const</code></td>
+<td>Returns a <code>const_iterator1</code> pointing to the
+beginning of the expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator1 end1 () const</code></td>
+<td>Returns a <code>const_iterator1</code> pointing to the end of
+the expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator2 begin2 () const</code></td>
+<td>Returns a <code>const_iterator2</code> pointing to the
+beginning of the expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator2 end2 () const</code></td>
+<td>Returns a <code>const_iterator2</code> pointing to the end of
+the expression.</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 expression.</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 expression.</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 expression.</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 expression.</td>
+</tr>
+</tbody>
+</table>
+<h3>Scalar Matrix Operations</h3>
+<h4>Prototypes</h4>
+<pre>
+<code>template<class T1, class E2, class F>
+ struct matrix_binary_scalar1_traits {
+ typedef matrix_binary_scalar1<scalar_const_reference<T1>,
+ typename E2::const_closure_type, F> expression_type;
+ typedef expression_type result_type;
+ };
+
+ // (t * m) [i] [j] = t * m [i] [j]
+ template<class T1, class E2>
+ typename matrix_binary_scalar1_traits<T1, E2, scalar_multiplies<T1, typename E2::value_type> >::result_type
+ operator * (const T1 &e1,
+ const matrix_expression<E2> &e2);
+
+ template<class E1, class T2, class F>
+ struct matrix_binary_scalar2_traits {
+ typedef matrix_binary_scalar2<typename E1::const_closure_type,
+ scalar_const_reference<T2>, F> expression_type;
+ typedef expression_type result_type;
+ };
+
+ // (m * t) [i] [j] = m [i] [j] * t
+ template<class E1, class T2>
+ typename matrix_binary_scalar2_traits<E1, T2, scalar_multiplies<typename E1::value_type, T2> >::result_type
+ operator * (const matrix_expression<E1> &e1,
+ const T2 &e2);
+
+ // (m / t) [i] [j] = m [i] [j] / t
+ template<class E1, class T2>
+ typename matrix_binary_scalar2_traits<E1, T2, scalar_divides<typename E1::value_type, T2> >::result_type
+ operator / (const matrix_expression<E1> &e1,
+ const T2 &e2);</code>
+</pre>
+<h4>Description</h4>
+<p><code>operator *</code> computes the product of a scalar and a
+matrix expression. <code>operator /</code> multiplies the matrix
+with the reciprocal of the scalar.</p>
+<h4>Definition</h4>
+<p>Defined in the header matrix_expression.hpp.</p>
+<h4>Type requirements</h4>
+<ul>
+<li><code>T1/T2</code> is a model of <a href=
+"expression_concept.html#scalar_expression">Scalar Expression</a> .</li>
+<li><code>E2/E1</code> is a model of <a href=
+"expression_concept.html#matrix_expression">Matrix Expression</a> .</li>
+</ul>
+<h4>Preconditions</h4>
+<p>None.</p>
+<h4>Complexity</h4>
+<p>Quadratic depending from the size of the matrix expression.</p>
+<h4>Examples</h4>
+<pre>
+#include <boost/numeric/ublas/matrix.hpp>
+#include <boost/numeric/ublas/io.hpp>
+
+int main () {
+ using namespace boost::numeric::ublas;
+ matrix<double> m (3, 3);
+ for (unsigned i = 0; i < m.size1 (); ++ i)
+ for (unsigned j = 0; j < m.size2 (); ++ j)
+ m (i, j) = 3 * i + j;
+
+ std::cout << 2.0 * m << std::endl;
+ std::cout << m * 2.0 << std::endl;
+}
+</pre>
+<h2><a name="matrix_vector_operations"></a>Matrix Vector Operations</h2>
+<h3>Binary Operation Description</h3>
+<h4>Description</h4>
+<p>The templated classes <code>matrix_vector_binary1<E1, E2,
+F></code> and <code>matrix_vector_binary2<E1, E2,
+F></code> describe binary matrix vector operations.</p>
+<h4>Definition</h4>
+<p>Defined in the header matrix_expression.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>E1</code></td>
+<td>The type of the matrix or vector expression.</td>
+<td></td>
+</tr>
+<tr>
+<td><code>E2</code></td>
+<td>The type of the vector or matrix expression.</td>
+<td></td>
+</tr>
+<tr>
+<td><code>F</code></td>
+<td>The type of the operation.</td>
+<td></td>
+</tr>
+</tbody>
+</table>
+<h4>Model of</h4>
+<p><a href="expression_concept.html#vector_expression">Vector Expression</a>
+.</p>
+<h4>Type requirements</h4>
+<p>None, except for those imposed by the requirements of <a href=
+"expression_concept.html#vector_expression">Vector Expression</a> .</p>
+<h4>Public base classes</h4>
+<p><code>vector_expression<matrix_vector_binary1<E1, E2,
+F> ></code> and
+<code>vector_expression<matrix_vector_binary2<E1, E2, F>
+></code> resp.</p>
+<h4>Members</h4>
+<table border="1" summary="members">
+<tbody>
+<tr>
+<th>Member</th>
+<th>Description</th>
+</tr>
+<tr>
+<td><code>matrix_vector_binary1 (const expression1_type &e1,
+const expression2_type &e2)</code></td>
+<td>Constructs a description of the expression.</td>
+</tr>
+<tr>
+<td><code>matrix_vector_binary2 (const expression1_type &e1,
+const expression2_type &e2)</code></td>
+<td>Constructs a description of the expression.</td>
+</tr>
+<tr>
+<td><code>size_type size () const</code></td>
+<td>Returns the size of the expression.</td>
+</tr>
+<tr>
+<td><code>const_reference operator () (size_type i)
+const</code></td>
+<td>Returns the value of the <code>i</code>-th element.</td>
+</tr>
+<tr>
+<td><code>const_iterator begin () const</code></td>
+<td>Returns a <code>const_iterator</code> pointing to the beginning
+of the expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator end () const</code></td>
+<td>Returns a <code>const_iterator</code> pointing to the end of
+the expression.</td>
+</tr>
+<tr>
+<td><code>const_reverse_iterator rbegin () const</code></td>
+<td>Returns a <code>const_reverse_iterator</code> pointing to the
+beginning of the reversed expression.</td>
+</tr>
+<tr>
+<td><code>const_reverse_iterator rend () const</code></td>
+<td>Returns a <code>const_reverse_iterator</code> pointing to the
+end of the reversed expression.</td>
+</tr>
+</tbody>
+</table>
+<h3>Binary Operations</h3>
+<h4>Prototypes</h4>
+<pre>
+<code>template<class T1, class E1, class T2, class E2>
+ struct matrix_vector_binary1_traits {
+ typedef row_major_tag dispatch_category;
+ typedef typename promote_traits<T1, T2>::promote_type promote_type;
+ typedef matrix_vector_binary1<typename E1::const_closure_type,
+ typename E2::const_closure_type,
+ matrix_vector_prod1<T1, T2, promote_type> > expression_type;
+ typedef expression_type result_type;
+ };
+
+ template<class E1, class E2>
+ typename matrix_vector_binary1_traits<typename E1::value_type, E1,
+ typename E2::value_type, E2>::result_type
+ prod (const matrix_expression<E1> &e1,
+ const vector_expression<E2> &e2,
+ row_major_tag);
+
+ // Dispatcher
+ template<class E1, class E2>
+ typename matrix_vector_binary1_traits<typename E1::value_type, E1,
+ typename E2::value_type, E2>::result_type
+ prod (const matrix_expression<E1> &e1,
+ const vector_expression<E2> &e2);
+
+ template<class E1, class E2>
+ typename matrix_vector_binary1_traits<typename type_traits<typename E1::value_type>::precision_type, E1,
+ typename type_traits<typename E2::value_type>::precision_type, E2>::result_type
+ prec_prod (const matrix_expression<E1> &e1,
+ const vector_expression<E2> &e2,
+ row_major_tag);
+
+ // Dispatcher
+ template<class E1, class E2>
+ typename matrix_vector_binary1_traits<typename type_traits<typename E1::value_type>::precision_type, E1,
+ typename type_traits<typename E2::value_type>::precision_type, E2>::result_type
+ prec_prod (const matrix_expression<E1> &e1,
+ const vector_expression<E2> &e2);
+
+ template<class V, class E1, class E2>
+ V
+ prod (const matrix_expression<E1> &e1,
+ const vector_expression<E2> &e2);
+
+ template<class V, class E1, class E2>
+ V
+ prec_prod (const matrix_expression<E1> &e1,
+ const vector_expression<E2> &e2);
+
+ template<class T1, class E1, class T2, class E2>
+ struct matrix_vector_binary2_traits {
+ typedef column_major_tag dispatch_category;
+ typedef typename promote_traits<T1, T2>::promote_type promote_type;
+ typedef matrix_vector_binary2<typename E1::const_closure_type,
+ typename E2::const_closure_type,
+ matrix_vector_prod2<T1, T2, promote_type> > expression_type;
+ typedef expression_type result_type;
+ };
+
+ template<class E1, class E2>
+ typename matrix_vector_binary2_traits<typename E1::value_type, E1,
+ typename E2::value_type, E2>::result_type
+ prod (const vector_expression<E1> &e1,
+ const matrix_expression<E2> &e2,
+ column_major_tag);
+
+ // Dispatcher
+ template<class E1, class E2>
+ typename matrix_vector_binary2_traits<typename E1::value_type, E1,
+ typename E2::value_type, E2>::result_type
+ prod (const vector_expression<E1> &e1,
+ const matrix_expression<E2> &e2);
+
+ template<class E1, class E2>
+ typename matrix_vector_binary2_traits<typename type_traits<typename E1::value_type>::precision_type, E1,
+ typename type_traits<typename E2::value_type>::precision_type, E2>::result_type
+ prec_prod (const vector_expression<E1> &e1,
+ const matrix_expression<E2> &e2,
+ column_major_tag);
+
+ // Dispatcher
+ template<class E1, class E2>
+ typename matrix_vector_binary2_traits<typename type_traits<typename E1::value_type>::precision_type, E1,
+ typename type_traits<typename E2::value_type>::precision_type, E2>::result_type
+ prec_prod (const vector_expression<E1> &e1,
+ const matrix_expression<E2> &e2);
+
+ template<class V, class E1, class E2>
+ V
+ prod (const vector_expression<E1> &e1,
+ const matrix_expression<E2> &e2);
+
+ template<class V, class E1, class E2>
+ V
+ prec_prod (const vector_expression<E1> &e1,
+ const matrix_expression<E2> &e2);</code>
+</pre>
+<h4>Description</h4>
+<p><code>prod</code> computes the product of the matrix and the
+vector expression. <code>prec_prod</code> computes the double
+precision product of the matrix and the vector expression.</p>
+<h4>Definition</h4>
+<p>Defined in the header matrix_expression.hpp.</p>
+<h4>Type requirements</h4>
+<ul>
+<li><code>E1</code> is a model of <a href=
+"expression_concept.html#matrix_expression">Matrix Expression</a> or
+<a href="expression_concept.html#vector_expression">Vector Expression</a>
+.</li>
+<li><code>E2</code> is a model of <a href=
+"expression_concept.html#vector_expression">Vector Expression</a> or
+<a href="expression_concept.html#matrix_expression">Matrix Expression</a>
+.</li>
+</ul>
+<h4>Preconditions</h4>
+<ul>
+<li><code>e1 ().size2 () == e2 ().size ()</code></li>
+<li><code>e1 ().size () == e2 ().size1 ()</code></li>
+</ul>
+<h4>Complexity</h4>
+<p>Quadratic depending from the size of the matrix expression.</p>
+<h4>Examples</h4>
+<pre>
+#include <boost/numeric/ublas/matrix.hpp>
+#include <boost/numeric/ublas/io.hpp>
+
+int main () {
+ using namespace boost::numeric::ublas;
+ matrix<double> m (3, 3);
+ vector<double> v (3);
+ for (unsigned i = 0; i < std::min (m.size1 (), v.size ()); ++ i) {
+ for (unsigned j = 0; j < m.size2 (); ++ j)
+ m (i, j) = 3 * i + j;
+ v (i) = i;
+ }
+
+ std::cout << prod (m, v) << std::endl;
+ std::cout << prod (v, m) << std::endl;
+}
+</pre>
+<h3>Triangular Solver</h3>
+<h4>Prototypes</h4>
+<pre>
+<code>template<class E1, class E2>
+ struct matrix_vector_solve_traits {
+ typedef typename promote_traits<typename E1::value_type, typename E2::value_type>::promote_type promote_type;
+ typedef vector<promote_type> result_type;
+ };
+
+ template<class E1, class E2>
+ void inplace_solve (const matrix_expression<E1> &e1,
+ E2 &e2,
+ lower_tag,
+ vector_tag);
+ template<class E1, class E2>
+ void inplace_solve (const matrix_expression<E1> &e1,
+ E2 &e2,
+ upper_tag,
+ vector_tag);
+ template<class E1, class E2>
+ void inplace_solve (const matrix_expression<E1> &e1,
+ E2 &e2,
+ unit_lower_tag,
+ vector_tag);
+ template<class E1, class E2>
+ void inplace_solve (const matrix_expression<E1> &e1,
+ E2 &e2,
+ unit_upper_tag,
+ vector_tag);
+
+ template<class E1, class E2, class C>
+ typename matrix_vector_solve_traits<E1, E2>::result_type
+ solve (const matrix_expression<E1> &e1,
+ const vector_expression<E2> &e2,
+ C);
+
+ template<class E1, class E2>
+ void inplace_solve (E1 &e1,
+ const matrix_expression<E2> &e2,
+ vector_tag,
+ lower_tag);
+ template<class E1, class E2>
+ void inplace_solve (E1 &e1,
+ const matrix_expression<E2> &e2,
+ vector_tag,
+ upper_tag);
+ template<class E1, class E2>
+ void inplace_solve (E1 &e1,
+ const matrix_expression<E2> &e2,
+ vector_tag,
+ unit_lower_tag);
+ template<class E1, class E2>
+ void inplace_solve (E1 &e1,
+ const matrix_expression<E2> &e2,
+ vector_tag,
+ unit_upper_tag);
+
+ template<class E1, class E2, class C>
+ typename matrix_vector_solve_traits<E1, E2>::result_type
+ solve (const vector_expression<E1> &e1,
+ const matrix_expression<E2> &e2,
+ C);</code>
+</pre>
+<h4>Description</h4>
+<p><code>solve</code> solves a linear equation for lower or upper
+(unit) triangular matrices.</p>
+<h4>Definition</h4>
+<p>Defined in the header triangular.hpp.</p>
+<h4>Type requirements</h4>
+<ul>
+<li><code>E1</code> is a model of <a href=
+"expression_concept.html#matrix_expression">Matrix Expression</a> or
+<a href="expression_concept.html#vector_expression">Vector Expression</a>
+.</li>
+<li><code>E2</code> is a model of <a href=
+"expression_concept.html#vector_expression">Vector Expression</a> or
+<a href="expression_concept.html#matrix_expression">Matrix Expression</a>
+.</li>
+</ul>
+<h4>Preconditions</h4>
+<ul>
+<li><code>e1 ().size1 () == e1 ().size2 ()</code></li>
+<li><code>e1 ().size2 () == e2 ().size ()</code></li>
+<li><code>e1 ().size () == e2 ().size1 ()</code></li>
+<li><code>e2 ().size1 () == e2 ().size2 ()</code></li>
+</ul>
+<h4>Complexity</h4>
+<p>Quadratic depending from the size of the matrix expression.</p>
+<h4>Examples</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);
+ vector<double> v (3);
+ for (unsigned i = 0; i < std::min (m.size1 (), v.size ()); ++ i) {
+ for (unsigned j = 0; j <= i; ++ j)
+ m (i, j) = 3 * i + j + 1;
+ v (i) = i;
+ }
+
+ std::cout << solve (m, v, lower_tag ()) << std::endl;
+ std::cout << solve (v, m, lower_tag ()) << std::endl;
+}
+</pre>
+<h2><a name="matrix_matrix_operations"></a>Matrix Matrix Operations</h2>
+<h3>Binary Operation Description</h3>
+<h4>Description</h4>
+<p>The templated class <code>matrix_matrix_binary<E1, E2,
+F></code> describes a binary matrix operation.</p>
+<h4>Definition</h4>
+<p>Defined in the header matrix_expression.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>E1</code></td>
+<td>The type of the first matrix expression.</td>
+<td></td>
+</tr>
+<tr>
+<td><code>E2</code></td>
+<td>The type of the second matrix expression.</td>
+<td></td>
+</tr>
+<tr>
+<td><code>F</code></td>
+<td>The type of the operation.</td>
+<td></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<matrix_matrix_binary<E1, E2, F>
+></code> .</p>
+<h4>Members</h4>
+<table border="1" summary="members">
+<tbody>
+<tr>
+<th>Member</th>
+<th>Description</th>
+</tr>
+<tr>
+<td><code>matrix_matrix_binary (const expression1_type &e1,
+const expression2_type &e2)</code></td>
+<td>Constructs a description of the expression.</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 the value of the <code>j</code>-th element in the
+<code>i</code>-th row.</td>
+</tr>
+<tr>
+<td><code>const_iterator1 begin1 () const</code></td>
+<td>Returns a <code>const_iterator1</code> pointing to the
+beginning of the expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator1 end1 () const</code></td>
+<td>Returns a <code>const_iterator1</code> pointing to the end of
+the expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator2 begin2 () const</code></td>
+<td>Returns a <code>const_iterator2</code> pointing to the
+beginning of the expression.</td>
+</tr>
+<tr>
+<td><code>const_iterator2 end2 () const</code></td>
+<td>Returns a <code>const_iterator2</code> pointing to the end of
+the expression.</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 expression.</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 expression.</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 expression.</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 expression.</td>
+</tr>
+</tbody>
+</table>
+<h3>Binary Operations</h3>
+<h4>Prototypes</h4>
+<pre>
+<code>template<class T1, class E1, class T2, class E2>
+ struct matrix_matrix_binary_traits {
+ typedef unknown_orientation_tag dispatch_category;
+ typedef typename promote_traits<T1, T2>::promote_type promote_type;
+ typedef matrix_matrix_binary<typename E1::const_closure_type,
+ typename E2::const_closure_type,
+ matrix_matrix_prod<T1, T2, promote_type> > expression_type;
+ typedef expression_type result_type;
+ };
+
+ template<class E1, class E2>
+ typename matrix_matrix_binary_traits<typename E1::value_type, E1,
+ typename E2::value_type, E2>::result_type
+ prod (const matrix_expression<E1> &e1,
+ const matrix_expression<E2> &e2,
+ unknown_orientation_tag);
+
+ // Dispatcher
+ template<class E1, class E2>
+ typename matrix_matrix_binary_traits<typename E1::value_type, E1,
+ typename E2::value_type, E2>::result_type
+ prod (const matrix_expression<E1> &e1,
+ const matrix_expression<E2> &e2);
+
+ template<class E1, class E2>
+ typename matrix_matrix_binary_traits<typename type_traits<typename E1::value_type>::precision_type, E1,
+ typename type_traits<typename E2::value_type>::precision_type, E2>::result_type
+ prec_prod (const matrix_expression<E1> &e1,
+ const matrix_expression<E2> &e2,
+ unknown_orientation_tag);
+
+ // Dispatcher
+ template<class E1, class E2>
+ typename matrix_matrix_binary_traits<typename type_traits<typename E1::value_type>::precision_type, E1,
+ typename type_traits<typename E2::value_type>::precision_type, E2>::result_type
+ prec_prod (const matrix_expression<E1> &e1,
+ const matrix_expression<E2> &e2);
+
+ template<class M, class E1, class E2>
+ M
+ prod (const matrix_expression<E1> &e1,
+ const matrix_expression<E2> &e2);
+
+ template<class M, class E1, class E2>
+ M
+ prec_prod (const matrix_expression<E1> &e1,
+ const matrix_expression<E2> &e2);</code>
+</pre>
+<h4>Description</h4>
+<p><code>prod</code> computes the product of the matrix
+expressions. <code>prec_prod</code> computes the double precision
+product of the matrix expressions.</p>
+<h4>Definition</h4>
+<p>Defined in the header matrix_expression.hpp.</p>
+<h4>Type requirements</h4>
+<ul>
+<li><code>E1</code> is a model of <a href=
+"expression_concept.html#matrix_expression">Matrix Expression</a> .</li>
+<li><code>E2</code> is a model of <a href=
+"expression_concept.html#matrix_expression">Matrix Expression</a> .</li>
+</ul>
+<h4>Preconditions</h4>
+<ul>
+<li><code>e1 ().size2 () == e2 ().size1 ()</code></li>
+</ul>
+<h4>Complexity</h4>
+<p>Cubic depending from the size of the matrix expression.</p>
+<h4>Examples</h4>
+<pre>
+#include <boost/numeric/ublas/matrix.hpp>
+#include <boost/numeric/ublas/io.hpp>
+
+int main () {
+ using namespace boost::numeric::ublas;
+ matrix<double> m1 (3, 3), m2 (3, 3);
+ for (unsigned i = 0; i < std::min (m1.size1 (), m2.size1 ()); ++ i)
+ for (unsigned j = 0; j < std::min (m1.size2 (), m2.size2 ()); ++ j)
+ m1 (i, j) = m2 (i, j) = 3 * i + j;
+
+ std::cout << prod (m1, m2) << std::endl;
+}
+</pre>
+<h3>Triangular Solvers</h3>
+<h4>Prototypes</h4>
+<pre>
+<code>template<class E1, class E2>
+ struct matrix_matrix_solve_traits {
+ typedef typename promote_traits<typename E1::value_type, typename E2::value_type>::promote_type promote_type;
+ typedef matrix<promote_type> result_type;
+ };
+
+ template<class E1, class E2>
+ void inplace_solve (const matrix_expression<E1> &e1,
+ E2 &e2,
+ lower_tag,
+ matrix_tag);
+ template<class E1, class E2>
+ void inplace_solve (const matrix_expression<E1> &e1,
+ E2 &e2,
+ upper_tag,
+ matrix_tag);
+ template<class E1, class E2>
+ void inplace_solve (const matrix_expression<E1> &e1,
+ E2 &e2,
+ unit_lower_tag,
+ matrix_tag);
+ template<class E1, class E2>
+ void inplace_solve (const matrix_expression<E1> &e1,
+ E2 &e2,
+ unit_upper_tag,
+ matrix_tag);
+
+ template<class E1, class E2, class C>
+ typename matrix_matrix_solve_traits<E1, E2>::result_type
+ solve (const matrix_expression<E1> &e1,
+ const matrix_expression<E2> &e2,
+ C);</code>
+</pre>
+<h4>Description</h4>
+<p><code>solve</code> solves a linear equation for lower or upper
+(unit) triangular matrices.</p>
+<h4>Definition</h4>
+<p>Defined in the header triangular.hpp.</p>
+<h4>Type requirements</h4>
+<ul>
+<li><code>E1</code> is a model of <a href=
+"expression_concept.html#matrix_expression">Matrix Expression</a> .</li>
+<li><code>E2</code> is a model of <a href=
+"expression_concept.html#matrix_expression">Matrix Expression</a> .</li>
+</ul>
+<h4>Preconditions</h4>
+<ul>
+<li><code>e1 ().size1 () == e1 ().size2 ()</code></li>
+<li><code>e1 ().size2 () == e2 ().size1 ()</code></li>
+</ul>
+<h4>Complexity</h4>
+<p>Cubic depending from the size of the matrix expressions.</p>
+<h4>Examples</h4>
+<pre>
+#include <boost/numeric/ublas/triangular.hpp>
+#include <boost/numeric/ublas/io.hpp>
+
+int main () {
+ using namespace boost::numeric::ublas;
+ matrix<double> m1 (3, 3), m2 (3, 3);
+ for (unsigned i = 0; i < std::min (m1.size1 (), m2.size1 ()); ++ i)
+ for (unsigned j = 0; j <= i; ++ j)
+ m1 (i, j) = m2 (i, j) = 3 * i + j + 1;
+
+ std::cout << solve (m1, m2, lower_tag ()) << std::endl;
+}
+</pre>
+<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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