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- <div class="titlepage"><div><div><h2 class="title" style="clear: both">
- <a name="math_toolkit.real_concepts"></a><a class="link" href="real_concepts.html" title="Conceptual Requirements for Real Number Types">Conceptual Requirements for
- Real Number Types</a>
- </h2></div></div></div>
- <p>
- The functions and statistical distributions in this library can be used with
- any type <span class="emphasis"><em>RealType</em></span> that meets the conceptual requirements
- given below. All the built-in floating-point types like <code class="computeroutput"><span class="keyword">double</span></code>
- will meet these requirements. (Built-in types are also called <a href="http://en.cppreference.com/w/cpp/language/types" target="_top">fundamental
- (built-in) types</a>).
- </p>
- <p>
- User-defined types that meet the conceptual requirements can also be used.
- For example, with <a class="link" href="high_precision/use_ntl.html" title="Using NTL Library">a thin
- wrapper class</a> one of the types provided with <a href="http://shoup.net/ntl/" target="_top">NTL
- (RR)</a> can be used. But now that <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
- library is available, this has become the preferred real-number type, typically
- <a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_dec_float.html" target="_top">cpp_dec_float</a>
- or <a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_bin_float.html" target="_top">cpp_bin_float</a>.
- </p>
- <p>
- Submissions of binding to other extended precision types would also still be
- welcome.
- </p>
- <p>
- The guiding principal behind these requirements is that a <span class="emphasis"><em>RealType</em></span>
- behaves just like a built-in floating-point type.
- </p>
- <h5>
- <a name="math_toolkit.real_concepts.h0"></a>
- <span class="phrase"><a name="math_toolkit.real_concepts.basic_arithmetic_requirements"></a></span><a class="link" href="real_concepts.html#math_toolkit.real_concepts.basic_arithmetic_requirements">Basic Arithmetic
- Requirements</a>
- </h5>
- <p>
- These requirements are common to all of the functions in this library.
- </p>
- <p>
- In the following table <span class="emphasis"><em>r</em></span> is an object of type <code class="computeroutput"><span class="identifier">RealType</span></code>, <span class="emphasis"><em>cr</em></span> and <span class="emphasis"><em>cr2</em></span>
- are objects of type <code class="computeroutput"><span class="keyword">const</span> <span class="identifier">RealType</span></code>,
- and <span class="emphasis"><em>ca</em></span> is an object of type <code class="computeroutput"><span class="keyword">const</span>
- <span class="identifier">arithmetic</span><span class="special">-</span><span class="identifier">type</span></code> (arithmetic types include all the built
- in integers and floating point types).
- </p>
- <div class="informaltable"><table class="table">
- <colgroup>
- <col>
- <col>
- <col>
- </colgroup>
- <thead><tr>
- <th>
- <p>
- Expression
- </p>
- </th>
- <th>
- <p>
- Result Type
- </p>
- </th>
- <th>
- <p>
- Notes
- </p>
- </th>
- </tr></thead>
- <tbody>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">RealType</span><span class="special">(</span><span class="identifier">cr</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- RealType is copy constructible.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">RealType</span><span class="special">(</span><span class="identifier">ca</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- RealType is copy constructible from the arithmetic types.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">r</span> <span class="special">=</span>
- <span class="identifier">cr</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType&
- </p>
- </td>
- <td>
- <p>
- Assignment operator.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">r</span> <span class="special">=</span>
- <span class="identifier">ca</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType&
- </p>
- </td>
- <td>
- <p>
- Assignment operator from the arithmetic types.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">r</span> <span class="special">+=</span>
- <span class="identifier">cr</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType&
- </p>
- </td>
- <td>
- <p>
- Adds cr to r.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">r</span> <span class="special">+=</span>
- <span class="identifier">ca</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType&
- </p>
- </td>
- <td>
- <p>
- Adds ar to r.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">r</span> <span class="special">-=</span>
- <span class="identifier">cr</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType&
- </p>
- </td>
- <td>
- <p>
- Subtracts cr from r.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">r</span> <span class="special">-=</span>
- <span class="identifier">ca</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType&
- </p>
- </td>
- <td>
- <p>
- Subtracts ca from r.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">r</span> <span class="special">*=</span>
- <span class="identifier">cr</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType&
- </p>
- </td>
- <td>
- <p>
- Multiplies r by cr.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">r</span> <span class="special">*=</span>
- <span class="identifier">ca</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType&
- </p>
- </td>
- <td>
- <p>
- Multiplies r by ca.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">r</span> <span class="special">/=</span>
- <span class="identifier">cr</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType&
- </p>
- </td>
- <td>
- <p>
- Divides r by cr.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">r</span> <span class="special">/=</span>
- <span class="identifier">ca</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType&
- </p>
- </td>
- <td>
- <p>
- Divides r by ca.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="special">-</span><span class="identifier">r</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- Unary Negation.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="special">+</span><span class="identifier">r</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType&
- </p>
- </td>
- <td>
- <p>
- Identity Operation.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">+</span>
- <span class="identifier">cr2</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- Binary Addition
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">+</span>
- <span class="identifier">ca</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- Binary Addition
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">ca</span> <span class="special">+</span>
- <span class="identifier">cr</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- Binary Addition
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">-</span>
- <span class="identifier">cr2</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- Binary Subtraction
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">-</span>
- <span class="identifier">ca</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- Binary Subtraction
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">ca</span> <span class="special">-</span>
- <span class="identifier">cr</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- Binary Subtraction
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">*</span>
- <span class="identifier">cr2</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- Binary Multiplication
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">*</span>
- <span class="identifier">ca</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- Binary Multiplication
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">ca</span> <span class="special">*</span>
- <span class="identifier">cr</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- Binary Multiplication
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">/</span>
- <span class="identifier">cr2</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- Binary Subtraction
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">/</span>
- <span class="identifier">ca</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- Binary Subtraction
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">ca</span> <span class="special">/</span>
- <span class="identifier">cr</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- Binary Subtraction
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">==</span>
- <span class="identifier">cr2</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Equality Comparison
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">==</span>
- <span class="identifier">ca</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Equality Comparison
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">ca</span> <span class="special">==</span>
- <span class="identifier">cr</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Equality Comparison
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">!=</span>
- <span class="identifier">cr2</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Inequality Comparison
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">!=</span>
- <span class="identifier">ca</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Inequality Comparison
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">ca</span> <span class="special">!=</span>
- <span class="identifier">cr</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Inequality Comparison
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special"><=</span>
- <span class="identifier">cr2</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Less than equal to.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special"><=</span>
- <span class="identifier">ca</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Less than equal to.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">ca</span> <span class="special"><=</span>
- <span class="identifier">cr</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Less than equal to.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">>=</span>
- <span class="identifier">cr2</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Greater than equal to.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">>=</span>
- <span class="identifier">ca</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Greater than equal to.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">ca</span> <span class="special">>=</span>
- <span class="identifier">cr</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Greater than equal to.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special"><</span>
- <span class="identifier">cr2</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Less than comparison.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special"><</span>
- <span class="identifier">ca</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Less than comparison.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">ca</span> <span class="special"><</span>
- <span class="identifier">cr</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Less than comparison.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">></span>
- <span class="identifier">cr2</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Greater than comparison.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cr</span> <span class="special">></span>
- <span class="identifier">ca</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Greater than comparison.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">ca</span> <span class="special">></span>
- <span class="identifier">cr</span></code>
- </p>
- </td>
- <td>
- <p>
- bool
- </p>
- </td>
- <td>
- <p>
- Greater than comparison.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">digits</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>()</span></code>
- </p>
- </td>
- <td>
- <p>
- int
- </p>
- </td>
- <td>
- <p>
- The number of digits in the significand of RealType.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">max_value</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>()</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- The largest representable number by type RealType.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">min_value</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>()</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- The smallest representable number by type RealType.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">log_max_value</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>()</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- The natural logarithm of the largest representable number by type
- RealType.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">log_min_value</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>()</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- The natural logarithm of the smallest representable number by type
- RealType.
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">epsilon</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>()</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- <td>
- <p>
- The machine epsilon of RealType.
- </p>
- </td>
- </tr>
- </tbody>
- </table></div>
- <p>
- Note that:
- </p>
- <div class="orderedlist"><ol class="orderedlist" type="1">
- <li class="listitem">
- The functions <code class="computeroutput"><span class="identifier">log_max_value</span></code>
- and <code class="computeroutput"><span class="identifier">log_min_value</span></code> can be
- synthesised from the others, and so no explicit specialisation is required.
- </li>
- <li class="listitem">
- The function <code class="computeroutput"><span class="identifier">epsilon</span></code> can
- be synthesised from the others, so no explicit specialisation is required
- provided the precision of RealType does not vary at runtime (see the header
- <a href="../../../../../boost/math/bindings/rr.hpp" target="_top">boost/math/bindings/rr.hpp</a>
- for an example where the precision does vary at runtime).
- </li>
- <li class="listitem">
- The functions <code class="computeroutput"><span class="identifier">digits</span></code>,
- <code class="computeroutput"><span class="identifier">max_value</span></code> and <code class="computeroutput"><span class="identifier">min_value</span></code>, all get synthesised automatically
- from <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span></code>. However, if <code class="computeroutput"><span class="identifier">numeric_limits</span></code> is not specialised for
- type RealType, then you will get a compiler error when code tries to use
- these functions, <span class="emphasis"><em>unless</em></span> you explicitly specialise
- them. For example if the precision of RealType varies at runtime, then
- <code class="computeroutput"><span class="identifier">numeric_limits</span></code> support
- may not be appropriate, see <a href="../../../../../boost/math/bindings/rr.hpp" target="_top">boost/math/bindings/rr.hpp</a>
- for examples.
- </li>
- </ol></div>
- <div class="warning"><table border="0" summary="Warning">
- <tr>
- <td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../doc/src/images/warning.png"></td>
- <th align="left">Warning</th>
- </tr>
- <tr><td align="left" valign="top">
- <p>
- If <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><></span></code>
- is <span class="bold"><strong>not specialized</strong></span> for type <span class="emphasis"><em>RealType</em></span>
- then the default float precision of 6 decimal digits will be used by other
- Boost programs including:
- </p>
- <p>
- Boost.Test: giving misleading error messages like
- </p>
- <p>
- <span class="emphasis"><em>"difference between {9.79796} and {9.79796} exceeds 5.42101e-19%".</em></span>
- </p>
- <p>
- Boost.LexicalCast and Boost.Serialization when converting the number to a
- string, causing potentially serious loss of accuracy on output.
- </p>
- <p>
- Although it might seem obvious that RealType should require <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span></code>
- to be specialized, this is not sensible for <code class="computeroutput"><span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span></code> and
- similar classes where the <span class="bold"><strong>number of digits is a runtime
- parameter</strong></span> (whereas for <code class="computeroutput"><span class="identifier">numeric_limits</span></code>
- everything has to be fixed at compile time).
- </p>
- </td></tr>
- </table></div>
- <h5>
- <a name="math_toolkit.real_concepts.h1"></a>
- <span class="phrase"><a name="math_toolkit.real_concepts.standard_library_support_require"></a></span><a class="link" href="real_concepts.html#math_toolkit.real_concepts.standard_library_support_require">Standard
- Library Support Requirements</a>
- </h5>
- <p>
- Many (though not all) of the functions in this library make calls to standard
- library functions, the following table summarises the requirements. Note that
- most of the functions in this library will only call a small subset of the
- functions listed here, so if in doubt whether a user-defined type has enough
- standard library support to be useable the best advise is to try it and see!
- </p>
- <p>
- In the following table <span class="emphasis"><em>r</em></span> is an object of type <code class="computeroutput"><span class="identifier">RealType</span></code>, <span class="emphasis"><em>cr1</em></span> and <span class="emphasis"><em>cr2</em></span>
- are objects of type <code class="computeroutput"><span class="keyword">const</span> <span class="identifier">RealType</span></code>,
- and <span class="emphasis"><em>i</em></span> is an object of type <code class="computeroutput"><span class="keyword">int</span></code>.
- </p>
- <div class="informaltable"><table class="table">
- <colgroup>
- <col>
- <col>
- </colgroup>
- <thead><tr>
- <th>
- <p>
- Expression
- </p>
- </th>
- <th>
- <p>
- Result Type
- </p>
- </th>
- </tr></thead>
- <tbody>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">fabs</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">abs</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">ceil</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">floor</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">exp</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">pow</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">,</span>
- <span class="identifier">cr2</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">log</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">frexp</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">,</span>
- <span class="special">&</span><span class="identifier">i</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">ldexp</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">,</span>
- <span class="identifier">i</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">cos</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">sin</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">asin</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">tan</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">atan</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">fmod</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">round</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">iround</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- int
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">trunc</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- RealType
- </p>
- </td>
- </tr>
- <tr>
- <td>
- <p>
- <code class="computeroutput"><span class="identifier">itrunc</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code>
- </p>
- </td>
- <td>
- <p>
- int
- </p>
- </td>
- </tr>
- </tbody>
- </table></div>
- <p>
- Note that the table above lists only those standard library functions known
- to be used (or likely to be used in the near future) by this library. The following
- functions: <code class="computeroutput"><span class="identifier">acos</span></code>, <code class="computeroutput"><span class="identifier">atan2</span></code>, <code class="computeroutput"><span class="identifier">fmod</span></code>,
- <code class="computeroutput"><span class="identifier">cosh</span></code>, <code class="computeroutput"><span class="identifier">sinh</span></code>,
- <code class="computeroutput"><span class="identifier">tanh</span></code>, <code class="computeroutput"><span class="identifier">log10</span></code>,
- <code class="computeroutput"><span class="identifier">lround</span></code>, <code class="computeroutput"><span class="identifier">llround</span></code>,
- <code class="computeroutput"><span class="identifier">ltrunc</span></code>, <code class="computeroutput"><span class="identifier">lltrunc</span></code>
- and <code class="computeroutput"><span class="identifier">modf</span></code> are not currently
- used, but may be if further special functions are added.
- </p>
- <p>
- Note that the <code class="computeroutput"><span class="identifier">round</span></code>, <code class="computeroutput"><span class="identifier">trunc</span></code> and <code class="computeroutput"><span class="identifier">modf</span></code>
- functions are not part of the current C++ standard: they are part of the additions
- added to C99 which will likely be in the next C++ standard. There are Boost
- versions of these provided as a backup, and the functions are always called
- unqualified so that argument-dependent-lookup can take place.
- </p>
- <p>
- In addition, for efficient and accurate results, a <a class="link" href="lanczos.html" title="The Lanczos Approximation">Lanczos
- approximation</a> is highly desirable. You may be able to adapt an existing
- approximation from <a href="../../../../../boost/math/special_functions/lanczos.hpp" target="_top">boost/math/special_functions/lanczos.hpp</a>
- or <a href="../../../../../boost/math/bindings/detail/big_lanczos.hpp" target="_top">boost/math/bindings/detail/big_lanczos.hpp</a>:
- in the former case you will need change <code class="computeroutput"><span class="keyword">static_cast</span></code>'s
- to <code class="computeroutput"><span class="identifier">lexical_cast</span></code>'s, and the
- constants to <span class="emphasis"><em>strings</em></span> (in order to ensure the coefficients
- aren't truncated to <code class="computeroutput"><span class="keyword">long</span> <span class="identifier">doubl</span></code>e)
- and then specialise <code class="computeroutput"><span class="identifier">lanczos_traits</span></code>
- for type T. Otherwise you may have to hack <a href="../../../tools/lanczos_generator.cpp" target="_top">libs/math/tools/lanczos_generator.cpp</a>
- to find a suitable approximation for your RealType. The code will still compile
- if you don't do this, but both accuracy and efficiency will be somewhat compromised
- in any function that makes use of the gamma/beta/erf family of functions.
- </p>
- </div>
- <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
- <td align="left"></td>
- <td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
- Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
- Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
- Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
- Daryle Walker and Xiaogang Zhang<p>
- Distributed under 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" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
- </p>
- </div></td>
- </tr></table>
- <hr>
- <div class="spirit-nav">
- <a accesskey="p" href="high_precision/using_test.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../using_udt.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="dist_concept.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a>
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