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1075 lines
29 KiB
HTML
1075 lines
29 KiB
HTML
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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
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"http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
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<html xmlns="http://www.w3.org/1999/xhtml">
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=us-ascii" />
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<link rel="stylesheet" href="../../../../boost.css" type="text/css"/>
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<link rel="stylesheet" href="ublas.css" type="text/css" />
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<script type="text/javascript" src="js/jquery-1.3.2.min.js" async="async" ></script>
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<script type="text/javascript" src="js/jquery.toc-gw.js" async="async" ></script>
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<title>Expression Concepts</title>
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</head>
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<body>
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<h1><img src="../../../../boost.png" align="middle" />Expression Concepts</h1>
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<div class="toc" id="toc"></div>
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<h2><a name="scalar_expression"></a>Scalar Expression</h2>
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<h4>Description</h4>
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<p>A Scalar Expression is an expression convertible to a scalar
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type.</p>
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<h4>Refinement of</h4>
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<p>Default Constructible.</p>
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<h4>Associated types</h4>
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<table border="1" summary="associated types">
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<tbody>
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<tr>
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<td>Public base</td>
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<td>scaler_expression<S></td>
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<td>S must be derived from this public base type.</td>
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</tr>
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<tr>
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<td>Value type</td>
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<td><code>value_type</code></td>
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<td>The type of the scalar expression.</td>
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</tr>
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</tbody>
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</table>
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<h4>Notation</h4>
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<table border="0" summary="notation">
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<tbody>
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<tr>
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<td><code>S</code></td>
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<td>A type that is a model of Scalar Expression</td>
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</tr>
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</tbody>
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</table>
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<h4>Definitions</h4>
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<h4>Valid expressions</h4>
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<p>In addition to the expressions defined in Default Constructible
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the following expressions must be valid.</p>
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<table border="1" summary="expressions">
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<tbody>
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<tr>
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<th>Name</th>
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<th>Expression</th>
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<th>Type requirements</th>
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<th>Return type</th>
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</tr>
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<tr>
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<td>Evaluation</td>
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<td><code>operator value_type () const</code></td>
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<td> </td>
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<td><code>value_type</code></td>
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</tr>
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</tbody>
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</table>
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<h4>Expression semantics</h4>
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<p>Semantics of an expression is defined only where it differs
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from, or is not defined in Default Constructible.</p>
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<table border="1" summary="semantics">
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<tbody>
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<tr>
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<th>Name</th>
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<th>Expression</th>
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<th>Precondition</th>
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<th>Semantics</th>
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<th>Postcondition</th>
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</tr>
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<tr>
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<td>Evaluation</td>
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<td><code>operator value_type () const</code></td>
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<td> </td>
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<td> Evaluates the scalar expression.</td>
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<td> </td>
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</tr>
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</tbody>
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</table>
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<h4>Complexity guarantees</h4>
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<p>The run-time complexity of the evaluation is specific for the
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evaluated scalar expression.</p>
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<h4>Invariants</h4>
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<h4>Models</h4>
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<ul>
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<li><code>vector_scalar_unary</code></li>
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<li><code>vector_scalar_binary</code></li>
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</ul>
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<h2><a name="vector_expression"></a>Vector Expression</h2>
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<h4>Description</h4>
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<p>A Vector Expression is an expression evaluatable to a vector.
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Vector Expression provides an <a href=
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"iterator_concept.html#indexed_bidirectional_iterator">Indexed Bidirectional
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Iterator</a> or an <a href=
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"iterator_concept.html#indexed_random_access_iterator">Indexed Random Access
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Iterator</a> .</p>
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<h4>Refinement of</h4>
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<p>Default Constructible.</p>
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<h4>Associated types</h4>
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<table border="1" summary="associated types">
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<tbody>
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<tr>
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<td>Public base</td>
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<td>vector_expression<V></td>
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<td>V must be derived from this public base type.</td>
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</tr>
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<tr>
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<td>Value type</td>
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<td><code>value_type</code></td>
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<td>
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The element type of the vector expression.
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</td>
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</tr>
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<tr>
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<td>Reference type</td>
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<td><code>reference</code></td>
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<td>
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The return type when accessing an element of a vector expression.
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<br />
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Convertable to a<code>value_type</code>.
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</td>
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</tr>
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<tr>
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<td>Const reference type</td>
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<td><code>const_reference</code></td>
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<td>
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The return type when accessing an element of a constant vector expression.
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<br />
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Convertable to a<code>value_type</code>.
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</td>
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</tr>
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<tr>
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<td>Size type</td>
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<td><code>size_type</code></td>
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<td>
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The index type of the vector expression. Am unsigned integral type used to represent size and index values.
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<br />
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Can represent any nonnegative value of <code>difference_type</code>.
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</td>
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</tr>
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<tr>
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<td>Distance type</td>
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<td><code>difference_type</code></td>
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<td>
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A signed integral type used to represent the distance between two of the vector expression's iterators.
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</td>
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</tr>
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<tr>
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<td>Const iterator type</td>
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<td><code>const_iterator</code></td>
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<td>A type of iterator that may be used to examine a vector
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expression's elements.</td>
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</tr>
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<tr>
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<td>Iterator type</td>
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<td><code>iterator</code></td>
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<td>A type of iterator that may be used to modify a vector
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expression's elements.</td>
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</tr>
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<tr>
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<td>Const reverse iterator type</td>
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<td><code>const_reverse_iterator</code></td>
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<td>A Reverse Iterator adaptor whose base iterator type is the
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vector expression's const iterator type.</td>
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</tr>
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<tr>
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<td>Reverse iterator type</td>
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<td><code>reverse_iterator</code></td>
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<td>A Reverse Iterator adaptor whose base iterator type is the
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vector expression's iterator type.</td>
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</tr>
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</tbody>
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</table>
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<h4>Notation</h4>
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<table border="0" summary="notation">
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<tbody>
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<tr>
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<td><code>V</code></td>
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<td>A type that is a model of Vector Expression</td>
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</tr>
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<tr>
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<td><code>v, v1, v2</code></td>
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<td>Object of type <code>V</code></td>
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</tr>
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<tr>
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<td><code>i</code></td>
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<td>Object of a type convertible to <code>size_type</code></td>
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</tr>
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<tr>
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<td><code>t</code></td>
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<td>Object of a type convertible to <code>value_type</code></td>
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</tr>
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</tbody>
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</table>
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<h4>Definitions</h4>
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<h4>Valid expressions</h4>
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<p>In addition to the expressions defined in Default Constructible
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the following expressions must be valid.</p>
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<table border="1" summary="expressions">
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<tbody>
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<tr>
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<th>Name</th>
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<th>Expression</th>
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<th>Type requirements</th>
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<th>Return type</th>
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</tr>
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<tr>
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<td rowspan="2">Beginning of range</td>
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<td><code>v.begin ()</code></td>
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<td> </td>
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<td><code>const_iterator</code></td>
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</tr>
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<tr>
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<td><code>v.begin ()</code></td>
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<td><code>v</code> is mutable.</td>
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<td><code>iterator</code></td>
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</tr>
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<tr>
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<td rowspan="2">End of range</td>
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<td><code>v.end ()</code></td>
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<td> </td>
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<td><code>const_iterator</code></td>
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</tr>
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<tr>
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<td><code>v.end ()</code></td>
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<td><code>v</code> is mutable.</td>
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<td><code>iterator</code></td>
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</tr>
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<tr>
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<td>Size</td>
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<td><code>v.size ()</code></td>
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<td> </td>
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<td><code>size_type</code></td>
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</tr>
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<tr>
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<td>Swap</td>
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<td><code>v1.swap (v2)</code></td>
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<td><code>v1</code> and <code>v2</code> are mutable.</td>
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<td><code>void</code></td>
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</tr>
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<tr>
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<td rowspan="2">Beginning of reverse range</td>
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<td><code>v.rbegin ()</code></td>
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<td> </td>
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<td><code>const_reverse_iterator</code></td>
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</tr>
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<tr>
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<td><code>v.rbegin ()</code></td>
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<td><code>v</code> is mutable.</td>
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<td><code>reverse_iterator</code></td>
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</tr>
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<tr>
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<td rowspan="2">End of reverse range</td>
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<td><code>v.rend ()</code></td>
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<td> </td>
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<td><code>const_reverse_iterator</code></td>
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</tr>
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<tr>
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<td><code>v.rend ()</code></td>
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<td><code>v</code> is mutable.</td>
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<td><code>reverse_iterator</code></td>
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</tr>
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<tr>
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<td>Element access</td>
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<td><code>v (i)</code></td>
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<td><code>i</code> is convertible to <code>size_type</code>.</td>
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<td>Convertible to <code>value_type</code>.</td>
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</tr>
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<tr>
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<td rowspan="2">Assignment</td>
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<td><code>v2 = v1</code></td>
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<td><code>v2</code> is mutable and <code>v1</code> is convertible
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to <code>V</code>.</td>
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<td><code>V &</code></td>
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</tr>
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<tr>
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<td><code>v2.assign (v1)</code></td>
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<td><code>v2</code> is mutable and <code>v1</code> is convertible
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to <code>V</code>.</td>
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<td><code>V &</code></td>
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</tr>
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<tr>
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<td rowspan="5">Computed assignment</td>
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<td><code>v2 += v1</code></td>
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<td><code>v2</code> is mutable and <code>v1</code> is convertible
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to <code>V</code>.</td>
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<td><code>V &</code></td>
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</tr>
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<tr>
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<td><code>v2.plus_assign (v1)</code></td>
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<td><code>v2</code> is mutable and <code>v1</code> is convertible
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to <code>V</code>.</td>
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<td><code>V &</code></td>
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</tr>
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<tr>
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<td><code>v2 -= v1</code></td>
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<td><code>v2</code> is mutable and <code>v1</code> is convertible
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to <code>V</code>.</td>
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<td><code>V &</code></td>
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</tr>
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<tr>
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<td><code>v2.minus_assign (v1)</code></td>
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<td><code>v2</code> is mutable and <code>v1</code> is convertible
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to <code>V</code>.</td>
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<td><code>V &</code></td>
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</tr>
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<tr>
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<td><code>v *= t</code></td>
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<td><code>v</code> is mutable and <code>t</code> is convertible to
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<code>value_type</code>.</td>
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<td><code>V &</code></td>
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</tr>
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||
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</tbody>
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||
|
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</table>
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||
|
|
<h4>Expression semantics</h4>
|
||
|
|
<p>Semantics of an expression is defined only where it differs
|
||
|
|
from, or is not defined in Default Constructible.</p>
|
||
|
|
<table border="1" summary="semantics">
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
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<th>Name</th>
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||
|
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<th>Expression</th>
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||
|
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<th>Precondition</th>
|
||
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<th>Semantics</th>
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||
|
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<th>Postcondition</th>
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||
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</tr>
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||
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<tr>
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<td>Beginning of range</td>
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<td><code>v.begin ()</code></td>
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<td> </td>
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||
|
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<td>Returns an iterator pointing to the first element in the vector
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expression.</td>
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|
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<td><code>v.begin ()</code> is either dereferenceable or
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past-the-end. It is past-the-end if and only if <code>v.size () ==
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0</code>.</td>
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</tr>
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<tr>
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<td>End of range</td>
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<td><code>v.end ()</code></td>
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||
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<td> </td>
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||
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<td>Returns an iterator pointing one past the last element in the
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vector expression.</td>
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||
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<td><code>v.end ()</code> is past-the-end.</td>
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||
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</tr>
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<tr>
|
||
|
|
<td>Size</td>
|
||
|
|
<td><code>v.size ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Returns the size of the vector expression, that is, its number
|
||
|
|
of elements.</td>
|
||
|
|
<td><code>v.size () >= 0</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Swap</td>
|
||
|
|
<td><code>v1.swap (v2)</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Equivalent to <code>swap (v1, v2)</code>.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Beginning of reverse range</td>
|
||
|
|
<td><code>v.rbegin ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Equivalent to <code>reverse_iterator (v.end ())</code>.</td>
|
||
|
|
<td><code>v.rbegin ()</code> is either dereferenceable or
|
||
|
|
past-the-end. It is past-the-end if and only if <code>v.size () ==
|
||
|
|
0</code>.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>End of reverse range</td>
|
||
|
|
<td><code>v.rend ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Equivalent to <code>reverse_iterator (v.begin ())</code>.</td>
|
||
|
|
<td><code>v.rend ()</code> is past-the-end.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Element access</td>
|
||
|
|
<td><code>v (i)</code></td>
|
||
|
|
<td><code>0 <= i < v.size ()</code></td>
|
||
|
|
<td>Returns the <code>i</code>-th element of the vector
|
||
|
|
expression.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="2">Assignment</td>
|
||
|
|
<td><code>v2 = v1</code></td>
|
||
|
|
<td><code>v1.size () == v2.size ()</code></td>
|
||
|
|
<td>Assigns every element of the evaluated vector expression
|
||
|
|
<code>v1</code> to the corresponding element of <code>v2</code>
|
||
|
|
.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>v2.assign (v1)</code></td>
|
||
|
|
<td><code>v1.size () == v2.size ()</code></td>
|
||
|
|
<td>Assigns every element of <code>v1</code> to the corresponding
|
||
|
|
element of <code>v2</code>.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="5">Computed assignment</td>
|
||
|
|
<td><code>v2 += v1</code></td>
|
||
|
|
<td><code>v1.size () == v2.size ()</code></td>
|
||
|
|
<td>Adds every element of the evaluated vector expression
|
||
|
|
<code>v1</code> to the corresponding element of
|
||
|
|
<code>v2</code>.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>v2.plus_assign (v1)</code></td>
|
||
|
|
<td><code>v1.size () == v2.size ()</code></td>
|
||
|
|
<td>Adds every element of <code>v1</code> to the corresponding
|
||
|
|
element of <code>v2</code>.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>v2 -= v1</code></td>
|
||
|
|
<td><code>v1.size () == v2.size ()</code></td>
|
||
|
|
<td>Subtracts every element of the evaluated vector expression
|
||
|
|
<code>v1</code> from the corresponding element of <code>v2</code>
|
||
|
|
.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>v2.minus_assign (v1)</code></td>
|
||
|
|
<td><code>v1.size () == v2.size ()</code></td>
|
||
|
|
<td>Subtracts every element of <code>v1</code> from the
|
||
|
|
corresponding element of <code>v2</code>.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>v *= t</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Multiplies every element of <code>v</code> with <code>t</code>
|
||
|
|
.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
<h4>Complexity guarantees</h4>
|
||
|
|
<p>The run-time complexity of <code>begin ()</code> and <code>end
|
||
|
|
()</code> is specific for the evaluated vector expression,
|
||
|
|
typically amortized constant time.</p>
|
||
|
|
<p>The run-time complexity of <code>size ()</code> is constant
|
||
|
|
time.</p>
|
||
|
|
<p>The run-time complexity of <code>swap ()</code> is specific for
|
||
|
|
the evaluated vector expression, typically constant time.</p>
|
||
|
|
<p>The run-time complexity of <code>rbegin ()</code> and <code>rend
|
||
|
|
()</code> is specific for the evaluated vector expression,
|
||
|
|
typically amortized constant time.</p>
|
||
|
|
<p>The run-time complexity of the element access is specific for
|
||
|
|
the evaluated vector expression, typically amortized constant time
|
||
|
|
for the dense and logarithmic for the sparse case.</p>
|
||
|
|
<p>The run-time complexity of the arithmetic operations is specific
|
||
|
|
for the evaluated vector expressions, typically linear in the size
|
||
|
|
of the expressions.</p>
|
||
|
|
<h4>Invariants</h4>
|
||
|
|
<table border="1" summary="invariants">
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td>Valid range</td>
|
||
|
|
<td>For any vector expression <code>v</code>, <code>[v.begin (),
|
||
|
|
v.end ())</code> is a valid range.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Completeness</td>
|
||
|
|
<td>An algorithm that iterates through the range <code>[v.begin (),
|
||
|
|
v.end ())</code> will pass through every element of <code>v</code>
|
||
|
|
.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Valid reverse range</td>
|
||
|
|
<td><code>[v.rbegin (), v.rend ())</code> is a valid range.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Equivalence of ranges</td>
|
||
|
|
<td>The distance from <code>v.begin ()</code> to <code>v.end
|
||
|
|
()</code> is the same as the distance from <code>v.rbegin ()</code>
|
||
|
|
to <code>v.rend ()</code>.</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
<h4>Models</h4>
|
||
|
|
<ul>
|
||
|
|
<li><code>vector_range;</code></li>
|
||
|
|
<li><code>vector_slice</code></li>
|
||
|
|
<li><code>matrix_row</code></li>
|
||
|
|
<li><code>matrix_column</code></li>
|
||
|
|
<li><code>matrix_vector_range</code></li>
|
||
|
|
<li><code>matrix_vector_slice</code></li>
|
||
|
|
<li><code>vector_unary</code></li>
|
||
|
|
<li><code>vector_binary</code></li>
|
||
|
|
<li><code>vector_binary_scalar1</code></li>
|
||
|
|
<li><code>vector_binary_scalar2</code></li>
|
||
|
|
<li><code>matrix_vector_unary1</code></li>
|
||
|
|
<li><code>matrix_vector_unary2</code></li>
|
||
|
|
<li><code>matrix_vector_binary1</code></li>
|
||
|
|
<li><code>matrix_vector_binary2</code></li>
|
||
|
|
</ul>
|
||
|
|
|
||
|
|
<h2><a name="matrix_expression"></a>Matrix Expression</h2>
|
||
|
|
<h4>Description</h4>
|
||
|
|
<p>A Matrix Expression is an expression evaluatable to a matrix.
|
||
|
|
Matrix Expression provides an <a href=
|
||
|
|
"iterator_concept.html#indexed_bidirectional_cr_iterator">Indexed
|
||
|
|
Bidirectional Column/Row Iterator</a> or an <a href=
|
||
|
|
"iterator_concept.html#indexed_random_access_cr_iterator">Indexed Random
|
||
|
|
Access Column/Row Iterator</a> .</p>
|
||
|
|
<h4>Refinement of</h4>
|
||
|
|
<p>Default Constructible.</p>
|
||
|
|
|
||
|
|
<h4>Associated types</h4>
|
||
|
|
<h5>immutable types</h5>
|
||
|
|
<table border="1" summary="associated immutable types" title="">
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td>Public base</td>
|
||
|
|
<td><code>matrix_expression<M></code></td>
|
||
|
|
<td>M must be derived from this public base type.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Value type</td>
|
||
|
|
<td><code>value_type</code></td>
|
||
|
|
<td>
|
||
|
|
The element type of the matrix expression.
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Const reference type</td>
|
||
|
|
<td><code>const_reference</code></td>
|
||
|
|
<td>
|
||
|
|
The return type when accessing an element of a constant matrix expression.
|
||
|
|
<br />
|
||
|
|
Convertable to a <code>value_type</code>.
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Size type</td>
|
||
|
|
<td><code>size_type</code></td>
|
||
|
|
<td>
|
||
|
|
The index type of the matrix expression. Am unsigned integral type used to represent size and index values.
|
||
|
|
<br />
|
||
|
|
Can represent any nonnegative value of <code>difference_type</code>.
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Distance type</td>
|
||
|
|
<td><code>difference_type</code></td>
|
||
|
|
<td>
|
||
|
|
A signed integral type used to represent the distance between two of the matrix expression's iterators.
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="2">Const iterator types</td>
|
||
|
|
<td><code>const_iterator1</code></td>
|
||
|
|
<td>A type of column iterator that may be used to examine a matrix
|
||
|
|
expression's elements.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>const_iterator2</code></td>
|
||
|
|
<td>A type of row iterator that may be used to examine a matrix
|
||
|
|
expression's elements.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="2">Const reverse iterator types</td>
|
||
|
|
<td><code>const_reverse_iterator1</code></td>
|
||
|
|
<td>A Reverse Iterator adaptor whose base iterator type is the
|
||
|
|
matrix expression's const column iterator type.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>const_reverse_iterator2</code></td>
|
||
|
|
<td>A Reverse Iterator adaptor whose base iterator type is the
|
||
|
|
matrix expression's const row iterator type.</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
|
||
|
|
<h5>mutable types</h5>
|
||
|
|
<table border="1" summary="associated mutable types">
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td>Reference type</td>
|
||
|
|
<td><code>reference</code></td>
|
||
|
|
<td>
|
||
|
|
The return type when accessing an element of a matrix expression.
|
||
|
|
<br />
|
||
|
|
Convertable to a <code>value_type</code>.
|
||
|
|
</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="2">Iterator types</td>
|
||
|
|
<td><code>iterator1</code></td>
|
||
|
|
<td>A type of column iterator that may be used to modify a matrix
|
||
|
|
expression's elements.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>iterator2</code></td>
|
||
|
|
<td>A type of row iterator that may be used to modify a matrix
|
||
|
|
expression's elements.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="2">Reverse iterator types</td>
|
||
|
|
<td><code>reverse_iterator1</code></td>
|
||
|
|
<td>A Reverse Iterator adaptor whose base iterator type is the
|
||
|
|
matrix expression's column iterator type.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>reverse_iterator2</code></td>
|
||
|
|
<td>A Reverse Iterator adaptor whose base iterator type is the
|
||
|
|
matrix expression's row iterator type.</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
|
||
|
|
|
||
|
|
<h4>Notation</h4>
|
||
|
|
<table border="0" summary="notation">
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td><code>M</code></td>
|
||
|
|
<td>A type that is a model of Matrix Expression</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m, m1, m2</code></td>
|
||
|
|
<td>Object of type <code>M</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>i, j</code></td>
|
||
|
|
<td>Objects of a type convertible to <code>size_type</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>t</code></td>
|
||
|
|
<td>Object of a type convertible to <code>value_type</code></td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
<h4>Definitions</h4>
|
||
|
|
<h4>Valid expressions</h4>
|
||
|
|
<p>In addition to the expressions defined in Default Constructible
|
||
|
|
the following expressions must be valid.</p>
|
||
|
|
|
||
|
|
<h5>immutable expressions</h5>
|
||
|
|
<table border="1" summary="expressions">
|
||
|
|
<thead>
|
||
|
|
<tr>
|
||
|
|
<th>Name</th>
|
||
|
|
<th>Expression</th>
|
||
|
|
<th>Type requirements</th>
|
||
|
|
<th>Return type</th>
|
||
|
|
</tr>
|
||
|
|
</thead>
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="2">Size</td>
|
||
|
|
<td><code>m.size1 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td><code>size_type</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.size2 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td><code>size_type</code></td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
|
||
|
|
<h5>possibly mutable expressions</h5>
|
||
|
|
<table border="1" summary="expressions">
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<th>Name</th>
|
||
|
|
<th>Expression</th>
|
||
|
|
<th>Type requirements</th>
|
||
|
|
<th>Return type</th>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="4">Beginning of range</td>
|
||
|
|
<td><code>m.begin1 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td><code>const_iterator1</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.begin2 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td><code>const_iterator2</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.begin1 ()</code></td>
|
||
|
|
<td><code>m</code> is mutable. </td>
|
||
|
|
<td><code>iterator1</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.begin2 ()</code></td>
|
||
|
|
<td><code>m</code> is mutable.</td>
|
||
|
|
<td><code>iterator2</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="4">End of range</td>
|
||
|
|
<td><code>m.end1 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td><code>const_iterator1</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.end2 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td><code>const_iterator2</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.end1 ()</code></td>
|
||
|
|
<td><code>m</code> is mutable. </td>
|
||
|
|
<td><code>iterator1</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.end2 ()</code></td>
|
||
|
|
<td><code>m</code> is mutable.</td>
|
||
|
|
<td><code>iterator2</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Swap</td>
|
||
|
|
<td><code>m1.swap (m2)</code></td>
|
||
|
|
<td><code>m1</code> and <code>m2</code> are mutable. </td>
|
||
|
|
<td><code>void</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="4">Beginning of reverse range</td>
|
||
|
|
<td><code>m.rbegin1 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td><code>const_reverse_iterator1</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.rbegin2 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td><code>const_reverse_iterator2</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.rbegin1 ()</code></td>
|
||
|
|
<td><code>m</code> is mutable. </td>
|
||
|
|
<td><code>reverse_iterator1</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.rbegin2 ()</code></td>
|
||
|
|
<td><code>m</code> is mutable.</td>
|
||
|
|
<td><code>reverse_iterator2</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="4">End of reverse range</td>
|
||
|
|
<td><code>m.rend1 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td><code>const_reverse_iterator1</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.rend2 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td><code>const_reverse_iterator2</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.rend1 ()</code></td>
|
||
|
|
<td><code>m</code> is mutable.</td>
|
||
|
|
<td><code>reverse_iterator1</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.rend2 ()</code></td>
|
||
|
|
<td><code>m</code> is mutable.</td>
|
||
|
|
<td><code>reverse_iterator2</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Element access</td>
|
||
|
|
<td><code>m (i, j)</code></td>
|
||
|
|
<td><code>i</code> and <code>j</code> are convertible to
|
||
|
|
<code>size_type</code> .</td>
|
||
|
|
<td>Convertible to <code>value_type</code>.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="2">Assignment</td>
|
||
|
|
<td><code>m2 = m1</code></td>
|
||
|
|
<td><code>m2</code> is mutable and <code>m1</code> is convertible
|
||
|
|
to <code>M</code>.</td>
|
||
|
|
<td><code>M &</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m2.assign (m1)</code></td>
|
||
|
|
<td><code>m2</code> is mutable and <code>m1</code> is convertible
|
||
|
|
to <code>M</code>.</td>
|
||
|
|
<td><code>M &</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="5">Computed assignment</td>
|
||
|
|
<td><code>m2 += m1</code></td>
|
||
|
|
<td><code>m2</code> is mutable and <code>m1</code> is convertible
|
||
|
|
to <code>M</code>.</td>
|
||
|
|
<td><code>M &</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m2.plus_assign (m1)</code></td>
|
||
|
|
<td><code>m2</code> is mutable and <code>m1</code> is convertible
|
||
|
|
to <code>M</code>.</td>
|
||
|
|
<td><code>M &</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m2 -= m1</code></td>
|
||
|
|
<td><code>m2</code> is mutable and <code>m1</code> is convertible
|
||
|
|
to <code>M</code>.</td>
|
||
|
|
<td><code>M &</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m2.minus_assign (m1)</code></td>
|
||
|
|
<td><code>m2</code> is mutable and <code>m1</code> is convertible
|
||
|
|
to <code>M</code>.</td>
|
||
|
|
<td><code>M &</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m *= t</code></td>
|
||
|
|
<td><code>m</code> is mutable and <code>t</code> is convertible to
|
||
|
|
<code>value_type</code>.</td>
|
||
|
|
<td><code>M &</code></td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
<h4>Expression semantics</h4>
|
||
|
|
<p>Semantics of an expression is defined only where it differs
|
||
|
|
from, or is not defined in Default Constructible.</p>
|
||
|
|
<table border="1" summary="semantics">
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<th>Name</th>
|
||
|
|
<th>Expression</th>
|
||
|
|
<th>Precondition</th>
|
||
|
|
<th>Semantics</th>
|
||
|
|
<th>Postcondition</th>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="2">Beginning of range</td>
|
||
|
|
<td><code>m.begin1 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Returns an iterator pointing to the first element in the first
|
||
|
|
column of a matrix expression.</td>
|
||
|
|
<td><code>m.begin1 ()</code> is either dereferenceable or
|
||
|
|
past-the-end. It is past-the-end if and only if <code>m.size1 () ==
|
||
|
|
0</code>.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.begin2 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Returns an iterator pointing to the first element in the first
|
||
|
|
row of a matrix expression.</td>
|
||
|
|
<td><code>m.begin2 ()</code> is either dereferenceable or
|
||
|
|
past-the-end. It is past-the-end if and only if <code>m.size2 () ==
|
||
|
|
0</code>.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="2">End of range</td>
|
||
|
|
<td><code>m.end1 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Returns an iterator pointing one past the last element in the
|
||
|
|
matrix expression.</td>
|
||
|
|
<td><code>m.end1 ()</code> is past-the-end.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.end2 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Returns an iterator pointing one past the last element in the
|
||
|
|
matrix expression.</td>
|
||
|
|
<td><code>m.end2 ()</code> is past-the-end.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="2">Size</td>
|
||
|
|
<td><code>m.size1 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Returns the number of rows of the matrix expression.</td>
|
||
|
|
<td><code>m.size1 () >= 0</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.size2 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Returns the number of columns of the matrix expression.</td>
|
||
|
|
<td><code>m.size2 () >= 0</code></td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Swap</td>
|
||
|
|
<td><code>m1.swap (m2)</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Equivalent to <code>swap (m1, m2)</code>.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="2">Beginning of reverse range</td>
|
||
|
|
<td><code>m.rbegin1 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Equivalent to <code>reverse_iterator1 (m.end1 ())</code>.</td>
|
||
|
|
<td><code>m.rbegin1 ()</code> is either dereferenceable or
|
||
|
|
past-the-end. It is past-the-end if and only if <code>m.size1 () ==
|
||
|
|
0</code>.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.rbegin2 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Equivalent to <code>reverse_iterator2 (m.end2 ())</code>.</td>
|
||
|
|
<td><code>m.rbegin2 ()</code> is either dereferenceable or
|
||
|
|
past-the-end. It is past-the-end if and only if <code>m.size2 () ==
|
||
|
|
0</code>.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="2">End of reverse range</td>
|
||
|
|
<td><code>m.rend1 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Equivalent to <code>reverse_iterator1 (m.begin1
|
||
|
|
())</code>.</td>
|
||
|
|
<td><code>m.rend1 ()</code> is past-the-end.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m.rend2 ()</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Equivalent to <code>reverse_iterator2 (m.begin2
|
||
|
|
())</code>.</td>
|
||
|
|
<td><code>m.rend2 ()</code> is past-the-end.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Element access</td>
|
||
|
|
<td><code>m (i, j)</code></td>
|
||
|
|
<td><code>0 <= i < m.size1 ()</code> and <code>0 <= j <
|
||
|
|
m.size2 ()</code></td>
|
||
|
|
<td>Returns the <code>j</code>-th element of the <code>i</code>-th
|
||
|
|
row of the matrix expression.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="2">Assignment</td>
|
||
|
|
<td><code>m2 = m1</code></td>
|
||
|
|
<td><code>m1.size1 () == m2.size1 ()</code> and <code><br />
|
||
|
|
m1.size2 () == m2.size2 ()</code></td>
|
||
|
|
<td>Assigns every element of the evaluated matrix expression
|
||
|
|
<code>m1</code> to the corresponding element of <code>m2</code>
|
||
|
|
.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m2.assign (m1)</code></td>
|
||
|
|
<td><code>m1.size1 () == m2.size1 ()</code> and <code><br />
|
||
|
|
m1.size2 () == m2.size2 ()</code></td>
|
||
|
|
<td>Assigns every element of <code>m1</code> to the corresponding
|
||
|
|
element of <code>m2</code>.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td rowspan="5">Computed assignment</td>
|
||
|
|
<td><code>m2 += m1</code></td>
|
||
|
|
<td><code>m1.size1 () == m2.size1 ()</code> and <code><br />
|
||
|
|
m1.size2 () == m2.size2 ()</code></td>
|
||
|
|
<td>Adds every element of the evaluated matrix expression
|
||
|
|
<code>m1</code> to the corresponding element of
|
||
|
|
<code>m2</code>.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m2.plus_assign (m1)</code></td>
|
||
|
|
<td><code>m1.size1 () == m2.size1 ()</code> and <code><br />
|
||
|
|
m1.size2 () == m2.size2 ()</code></td>
|
||
|
|
<td>Adds every element of <code>m1</code> to the corresponding
|
||
|
|
element of <code>m2</code>.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m2 -= m1</code></td>
|
||
|
|
<td><code>m1.size1 () == m2.size1 ()</code> and <code><br />
|
||
|
|
m1.size2 () == m2.size2 ()</code></td>
|
||
|
|
<td>Subtracts every element of the evaluated matrix expression
|
||
|
|
<code>m1</code> from the corresponding element of <code>m2</code>
|
||
|
|
.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m2.minus_assign (m1)</code></td>
|
||
|
|
<td><code>m1.size1 () == m2.size1 ()</code> and <code><br />
|
||
|
|
m1.size2 () == m2.size2 ()</code></td>
|
||
|
|
<td>Subtracts every element of <code>m1</code> from the
|
||
|
|
corresponding element of <code>m2</code>.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td><code>m *= t</code></td>
|
||
|
|
<td> </td>
|
||
|
|
<td>Multiplies every element of <code>m</code> with <code>t</code>
|
||
|
|
.</td>
|
||
|
|
<td> </td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
|
||
|
|
<h4>Complexity guarantees</h4>
|
||
|
|
<p>The run-time complexity of <code>begin1 ()</code>, <code>begin2
|
||
|
|
()</code> , <code>end1 ()</code> and <code>end2 ()</code> is
|
||
|
|
specific for the evaluated matrix expression.</p>
|
||
|
|
<p>The run-time complexity of <code>size1 ()</code> and <code>size2
|
||
|
|
()</code> is constant time.</p>
|
||
|
|
<p>The run-time complexity of <code>swap ()</code> is specific for
|
||
|
|
the evaluated matrix expression, typically constant time.</p>
|
||
|
|
<p>The run-time complexity of <code>rbegin1 ()</code>,
|
||
|
|
<code>rbegin2 ()</code> , <code>rend1 ()</code> and <code>rend2
|
||
|
|
()</code> is specific for the evaluated matrix expression.</p>
|
||
|
|
<p>The run-time complexity of the element access is specific for
|
||
|
|
the evaluated matrix expression, typically amortized constant time
|
||
|
|
for the dense and logarithmic for the sparse case.</p>
|
||
|
|
<p>The run-time complexity of the arithmetic operations is specific
|
||
|
|
for the evaluated matrix expressions, typically quadratic in the
|
||
|
|
size of the proxies.</p>
|
||
|
|
<h4>Invariants</h4>
|
||
|
|
<table border="1" summary="invariants">
|
||
|
|
<tbody>
|
||
|
|
<tr>
|
||
|
|
<td>Valid range</td>
|
||
|
|
<td>For any matrix expression <code>m</code>, <code>[m.begin1 (),
|
||
|
|
m.end1 ())</code> and <code>[m.begin2 (), m.end2 ())</code> are
|
||
|
|
valid ranges.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Completeness</td>
|
||
|
|
<td>An algorithm that iterates through the range <code>[m.begin1
|
||
|
|
(), m.end1 ())</code> will pass through every row of <code>m</code>
|
||
|
|
, an algorithm that iterates through the range <code>[m.begin2 (),
|
||
|
|
m.end2 ())</code> will pass through every column of <code>m</code>
|
||
|
|
.</td>
|
||
|
|
</tr>
|
||
|
|
<tr>
|
||
|
|
<td>Valid reverse range</td>
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<td><code>[m.rbegin1 (), m.rend1 ())</code> and <code>[m.rbegin2
|
||
|
|
(), m.rend2 ())</code> are valid ranges.</td>
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||
|
|
</tr>
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||
|
|
<tr>
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||
|
|
<td>Equivalence of ranges</td>
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||
|
|
<td>The distance from <code>m.begin1 ()</code> to <code>m.end1
|
||
|
|
()</code> is the same as the distance from <code>m.rbegin1
|
||
|
|
()</code> to <code>m.rend1 ()</code> and the distance from
|
||
|
|
<code>m.begin2 ()</code> to <code>m.end2 ()</code> is the same as
|
||
|
|
the distance from <code>m.rbegin2 ()</code> to <code>m.rend2
|
||
|
|
()</code>.</td>
|
||
|
|
</tr>
|
||
|
|
</tbody>
|
||
|
|
</table>
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||
|
|
<h4>Models</h4>
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||
|
|
<ul>
|
||
|
|
<li><code>matrix_range</code></li>
|
||
|
|
<li><code>matrix_slice;</code></li>
|
||
|
|
<li><code>triangular_adaptor</code></li>
|
||
|
|
<li><code>symmetric_adaptor</code></li>
|
||
|
|
<li><code>banded_adaptor</code></li>
|
||
|
|
<li><code>vector_matrix_binary</code></li>
|
||
|
|
<li><code>matrix_unary1</code></li>
|
||
|
|
<li><code>matrix_unary2</code></li>
|
||
|
|
<li><code>matrix_binary</code></li>
|
||
|
|
<li><code>matrix_binary_scalar1</code></li>
|
||
|
|
<li><code>matrix_binary_scalar2</code></li>
|
||
|
|
<li><code>matrix_matrix_binary</code></li>
|
||
|
|
</ul>
|
||
|
|
<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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