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<div class="section">
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<div class="titlepage"><div><div><h2 class="title" style="clear: both">
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<a name="math_toolkit.brent_minima"></a><a class="link" href="brent_minima.html" title="Locating Function Minima using Brent's algorithm">Locating Function Minima using
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Brent's algorithm</a>
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</h2></div></div></div>
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<h5>
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<a name="math_toolkit.brent_minima.h0"></a>
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<span class="phrase"><a name="math_toolkit.brent_minima.synopsis"></a></span><a class="link" href="brent_minima.html#math_toolkit.brent_minima.synopsis">Synopsis</a>
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</h5>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><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">minima</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">brent_find_minima</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">);</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">F</span><span class="special">,</span> <span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">brent_find_minima</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">min</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">max</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span><span class="special">&</span> <span class="identifier">max_iter</span><span class="special">);</span>
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</pre>
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<h5>
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<a name="math_toolkit.brent_minima.h1"></a>
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<span class="phrase"><a name="math_toolkit.brent_minima.description"></a></span><a class="link" href="brent_minima.html#math_toolkit.brent_minima.description">Description</a>
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</h5>
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<p>
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These two functions locate the minima of the continuous function <span class="emphasis"><em>f</em></span>
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using <a href="http://en.wikipedia.org/wiki/Brent%27s_method" target="_top">Brent's method</a>:
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specifically it uses quadratic interpolation to locate the minima, or if that
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fails, falls back to a <a href="http://en.wikipedia.org/wiki/Golden_section_search" target="_top">golden-section
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search</a>.
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</p>
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<p>
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<span class="bold"><strong>Parameters</strong></span>
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</p>
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<div class="variablelist">
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<p class="title"><b></b></p>
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<dl class="variablelist">
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<dt><span class="term">f</span></dt>
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<dd><p>
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The function to minimise: a function object (functor) that should be
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smooth over the range <span class="emphasis"><em>[min, max]</em></span>, with no maxima
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occurring in that interval.
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</p></dd>
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<dt><span class="term">min</span></dt>
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<dd><p>
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The lower endpoint of the range in which to search for the minima.
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</p></dd>
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<dt><span class="term">max</span></dt>
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<dd><p>
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The upper endpoint of the range in which to search for the minima.
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</p></dd>
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<dt><span class="term">bits</span></dt>
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<dd><p>
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The number of bits precision to which the minima should be found.<br>
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Note that in principle, the minima can not be located to greater accuracy
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than the square root of machine epsilon (for 64-bit double, sqrt(1e-16)≅1e-8),
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therefore the value of <span class="emphasis"><em>bits</em></span> will be ignored if it's
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greater than half the number of bits in the mantissa of T.
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</p></dd>
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<dt><span class="term">max_iter</span></dt>
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<dd><p>
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The maximum number of iterations to use in the algorithm, if not provided
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the algorithm will just keep on going until the minima is found.
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</p></dd>
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</dl>
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</div>
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<p>
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<span class="bold"><strong>Returns:</strong></span>
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</p>
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<p>
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A <code class="computeroutput"><span class="identifier">pair</span></code> of type T containing
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the value of the abscissa at the minima and the value of <span class="emphasis"><em>f(x)</em></span>
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at the minima.
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</p>
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<div class="tip"><table border="0" summary="Tip">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../doc/src/images/tip.png"></td>
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<th align="left">Tip</th>
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</tr>
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<tr><td align="left" valign="top">
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<p>
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Defining BOOST_MATH_INSTRUMENT will show some parameters, for example:
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</p>
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<pre class="programlisting"><span class="identifier">Type</span> <span class="identifier">T</span> <span class="identifier">is</span> <span class="keyword">double</span>
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<span class="identifier">bits</span> <span class="special">=</span> <span class="number">24</span><span class="special">,</span> <span class="identifier">maximum</span> <span class="number">26</span>
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<span class="identifier">tolerance</span> <span class="special">=</span> <span class="number">1.19209289550781e-007</span>
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<span class="identifier">seeking</span> <span class="identifier">minimum</span> <span class="identifier">in</span> <span class="identifier">range</span> <span class="identifier">min</span><span class="special">-</span><span class="number">4</span> <span class="identifier">to</span> <span class="number">1.33333333333333</span>
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<span class="identifier">maximum</span> <span class="identifier">iterations</span> <span class="number">18446744073709551615</span>
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<span class="number">10</span> <span class="identifier">iterations</span><span class="special">.</span>
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</pre>
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</td></tr>
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</table></div>
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<h5>
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<a name="math_toolkit.brent_minima.h2"></a>
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<span class="phrase"><a name="math_toolkit.brent_minima.example"></a></span><a class="link" href="brent_minima.html#math_toolkit.brent_minima.example">Brent
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Minimisation Example</a>
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</h5>
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<p>
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As a demonstration, we replicate this <a href="http://en.wikipedia.org/wiki/Brent%27s_method#Example" target="_top">Wikipedia
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example</a> minimising the function <span class="emphasis"><em>y= (x+3)(x-1)<sup>2</sup></em></span>.
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</p>
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<p>
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It is obvious from the equation and the plot that there is a minimum at exactly
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one and the value of the function at one is exactly zero.
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</p>
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<div class="tip"><table border="0" summary="Tip">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../doc/src/images/tip.png"></td>
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<th align="left">Tip</th>
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</tr>
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<tr><td align="left" valign="top"><p>
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This observation shows that an analytical or <a href="http://en.wikipedia.org/wiki/Closed-form_expression" target="_top">Closed-form
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expression</a> solution always beats brute-force hands-down for both
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speed and precision.
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</p></td></tr>
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</table></div>
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<p>
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<span class="inlinemediaobject"><img src="../../graphs/brent_test_function_1.svg" align="middle"></span>
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</p>
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<p>
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First an include is needed:
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</p>
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<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><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">minima</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
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</pre>
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<p>
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This function is encoded in C++ as function object (functor) using <code class="computeroutput"><span class="keyword">double</span></code> precision thus:
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</p>
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<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">funcdouble</span>
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<span class="special">{</span>
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<span class="keyword">double</span> <span class="keyword">operator</span><span class="special">()(</span><span class="keyword">double</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">x</span><span class="special">)</span>
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<span class="special">{</span> <span class="comment">//</span>
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<span class="keyword">return</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">3</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span> <span class="comment">// (x + 3)(x - 1)^2</span>
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<span class="special">}</span>
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<span class="special">};</span>
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</pre>
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<p>
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The Brent function is conveniently accessed through a <code class="computeroutput"><span class="keyword">using</span></code>
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statement (noting sub-namespace <code class="computeroutput"><span class="special">::</span><span class="identifier">tools</span></code>).
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</p>
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<p>
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The search minimum and maximum are chosen as -4 to 4/3 (as in the Wikipedia
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example).
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</p>
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<div class="tip"><table border="0" summary="Tip">
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<tr>
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../doc/src/images/tip.png"></td>
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<th align="left">Tip</th>
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</tr>
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<tr><td align="left" valign="top"><p>
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S A Stage (reference 6) reports that the Brent algorithm is <span class="emphasis"><em>slow
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to start, but fast to converge</em></span>, so choosing a tight min-max range
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is good.
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</p></td></tr>
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</table></div>
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<p>
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For simplicity, we set the precision parameter <code class="computeroutput"><span class="identifier">bits</span></code>
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to <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits</span></code>, which is effectively the maximum
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possible i.e. <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits</span></code>/2. Nor do we provide a maximum iterations
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parameter <code class="computeroutput"><span class="identifier">max_iter</span></code>, (perhaps
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unwidely), so the function will iterate until it finds a minimum.
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</p>
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<pre class="programlisting"><span class="keyword">int</span> <span class="identifier">bits</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits</span><span class="special">;</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="keyword">double</span><span class="special">></span> <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">brent_find_minima</span><span class="special">(</span><span class="identifier">funcdouble</span><span class="special">(),</span> <span class="special">-</span><span class="number">4.</span><span class="special">,</span> <span class="number">4.</span> <span class="special">/</span> <span class="number">3</span><span class="special">,</span> <span class="identifier">bits</span><span class="special">);</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">);</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"x at minimum = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">", f("</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
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<span class="comment">// x at minimum = 1.00000000112345, f(1.00000000112345) = 5.04852568272458e-018</span>
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</pre>
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<p>
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The resulting <a href="http://en.cppreference.com/w/cpp/utility/pair" target="_top">std::pair</a>
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contains the minimum close to one and the minimum value close to zero.
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</p>
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<pre class="programlisting"><span class="identifier">x</span> <span class="identifier">at</span> <span class="identifier">minimum</span> <span class="special">=</span> <span class="number">1.00000000112345</span><span class="special">,</span> <span class="identifier">f</span><span class="special">(</span><span class="number">1.00000000112345</span><span class="special">)</span> <span class="special">=</span> <span class="number">5.04852568272458e-018</span>
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</pre>
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<p>
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The differences from the expected <span class="emphasis"><em>one</em></span> and <span class="emphasis"><em>zero</em></span>
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are less than the uncertainty (for <code class="computeroutput"><span class="keyword">double</span></code>)
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1.5e-008 calculated from <code class="computeroutput"><span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits</span><span class="special">)</span> <span class="special">==</span> <span class="number">53</span></code>.
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</p>
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<p>
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We can use it like this to check that the two values are close-enough to those
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expected,
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</p>
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<pre class="programlisting"> <span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">fpc</span><span class="special">::</span><span class="identifier">is_close_to</span><span class="special">;</span>
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<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">fpc</span><span class="special">::</span><span class="identifier">is_small</span><span class="special">;</span>
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<span class="keyword">double</span> <span class="identifier">uncertainty</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits</span><span class="special">);</span>
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<span class="identifier">is_close_to</span><span class="special">(</span><span class="number">1.</span><span class="special">,</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">,</span> <span class="identifier">uncertainty</span><span class="special">);</span>
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<span class="identifier">is_small</span><span class="special">(</span><span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">,</span> <span class="identifier">uncertainty</span><span class="special">);</span>
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<span class="identifier">x</span> <span class="special">==</span> <span class="number">1</span> <span class="special">(</span><span class="identifier">compared</span> <span class="identifier">to</span> <span class="identifier">uncertainty</span> <span class="number">0.00034527</span><span class="special">)</span> <span class="identifier">is</span> <span class="keyword">true</span>
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<span class="identifier">f</span><span class="special">(</span><span class="identifier">x</span><span class="special">)</span> <span class="special">==</span> <span class="number">0</span> <span class="special">(</span><span class="identifier">compared</span> <span class="identifier">to</span> <span class="identifier">uncertainty</span> <span class="number">0.00034527</span><span class="special">)</span> <span class="identifier">is</span> <span class="keyword">true</span>
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</pre>
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<p>
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It is possible to make this comparison more generally with a templated function,
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returning <code class="computeroutput"><span class="keyword">true</span></code> when this criterion
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is met, for example:
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</p>
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<pre class="programlisting"><span class="comment">//</span>
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<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">></span>
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<span class="keyword">bool</span> <span class="identifier">close</span><span class="special">(</span><span class="identifier">T</span> <span class="identifier">expect</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">got</span><span class="special">,</span> <span class="identifier">T</span> <span class="identifier">tolerance</span><span class="special">)</span>
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<span class="special">{</span>
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<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">fpc</span><span class="special">::</span><span class="identifier">is_close_to</span><span class="special">;</span>
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<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">fpc</span><span class="special">::</span><span class="identifier">is_small</span><span class="special">;</span>
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|
|
<span class="keyword">if</span> <span class="special">(</span><span class="identifier">is_small</span><span class="special"><</span><span class="identifier">T</span><span class="special">>(</span><span class="identifier">expect</span><span class="special">,</span> <span class="identifier">tolerance</span><span class="special">))</span>
|
|
<span class="special">{</span>
|
|
<span class="keyword">return</span> <span class="identifier">is_small</span><span class="special"><</span><span class="identifier">T</span><span class="special">>(</span><span class="identifier">got</span><span class="special">,</span> <span class="identifier">tolerance</span><span class="special">);</span>
|
|
<span class="special">}</span>
|
|
<span class="keyword">else</span>
|
|
<span class="special">{</span>
|
|
<span class="keyword">return</span> <span class="identifier">is_close_to</span><span class="special"><</span><span class="identifier">T</span><span class="special">>(</span><span class="identifier">expect</span><span class="special">,</span> <span class="identifier">got</span><span class="special">,</span> <span class="identifier">tolerance</span><span class="special">);</span>
|
|
<span class="special">}</span>
|
|
<span class="special">}</span>
|
|
</pre>
|
|
<p>
|
|
In practical applications, we might want to know how many iterations, and maybe
|
|
to limit iterations and perhaps to trade some loss of precision for speed,
|
|
for example:
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">maxit</span> <span class="special">=</span> <span class="number">20</span><span class="special">;</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">it</span> <span class="special">=</span> <span class="identifier">maxit</span><span class="special">;</span>
|
|
<span class="identifier">r</span> <span class="special">=</span> <span class="identifier">brent_find_minima</span><span class="special">(</span><span class="identifier">funcdouble</span><span class="special">(),</span> <span class="special">-</span><span class="number">4.</span><span class="special">,</span> <span class="number">4.</span> <span class="special">/</span> <span class="number">3</span><span class="special">,</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">it</span><span class="special">);</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"x at minimum = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">", f("</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span>
|
|
<span class="special"><<</span> <span class="string">" after "</span> <span class="special"><<</span> <span class="identifier">it</span> <span class="special"><<</span> <span class="string">" iterations. "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
</pre>
|
|
<p>
|
|
limits to a maximum of 20 iterations (a reasonable estimate for this application,
|
|
even for higher precision shown later).
|
|
</p>
|
|
<p>
|
|
The parameter <code class="computeroutput"><span class="identifier">it</span></code> is updated
|
|
to return the actual number of iterations (so it may be useful to also keep
|
|
a record of the limit in <code class="computeroutput"><span class="identifier">maxit</span></code>).
|
|
</p>
|
|
<p>
|
|
It is neat to avoid showing insignificant digits by computing the number of
|
|
decimal digits to display.
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">streamsize</span> <span class="identifier">prec</span> <span class="special">=</span> <span class="keyword">static_cast</span><span class="special"><</span><span class="keyword">int</span><span class="special">>(</span><span class="number">2</span> <span class="special">+</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">bits</span><span class="special">));</span> <span class="comment">// Number of significant decimal digits.</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Showing "</span> <span class="special"><<</span> <span class="identifier">bits</span> <span class="special"><<</span> <span class="string">" bits precision with "</span> <span class="special"><<</span> <span class="identifier">prec</span>
|
|
<span class="special"><<</span> <span class="string">" decimal digits from tolerance "</span> <span class="special"><<</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">())</span>
|
|
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">streamsize</span> <span class="identifier">precision</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">prec</span><span class="special">);</span> <span class="comment">// Save.</span>
|
|
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"x at minimum = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">", f("</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span>
|
|
<span class="special"><<</span> <span class="string">" after "</span> <span class="special"><<</span> <span class="identifier">it</span> <span class="special"><<</span> <span class="string">" iterations. "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
</pre>
|
|
<pre class="programlisting"><span class="identifier">Showing</span> <span class="number">53</span> <span class="identifier">bits</span> <span class="identifier">precision</span> <span class="identifier">with</span> <span class="number">9</span> <span class="identifier">decimal</span> <span class="identifier">digits</span> <span class="identifier">from</span> <span class="identifier">tolerance</span> <span class="number">1.49011611938477e-008</span>
|
|
<span class="identifier">x</span> <span class="identifier">at</span> <span class="identifier">minimum</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">f</span><span class="special">(</span><span class="number">1</span><span class="special">)</span> <span class="special">=</span> <span class="number">5.04852568e-018</span>
|
|
</pre>
|
|
<p>
|
|
We can also half the number of precision bits from 52 to 26.
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">bits</span> <span class="special">/=</span> <span class="number">2</span><span class="special">;</span> <span class="comment">// Half digits precision (effective maximum).</span>
|
|
<span class="keyword">double</span> <span class="identifier">epsilon_2</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">pow</span><span class="special"><-(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits</span><span class="special">/</span><span class="number">2</span> <span class="special">-</span> <span class="number">1</span><span class="special">),</span> <span class="keyword">double</span><span class="special">>(</span><span class="number">2</span><span class="special">);</span>
|
|
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Showing "</span> <span class="special"><<</span> <span class="identifier">bits</span> <span class="special"><<</span> <span class="string">" bits precision with "</span> <span class="special"><<</span> <span class="identifier">prec</span>
|
|
<span class="special"><<</span> <span class="string">" decimal digits from tolerance "</span> <span class="special"><<</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">epsilon_2</span><span class="special">)</span>
|
|
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">streamsize</span> <span class="identifier">precision</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">prec</span><span class="special">);</span> <span class="comment">// Save.</span>
|
|
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">it</span> <span class="special">=</span> <span class="identifier">maxit</span><span class="special">;</span>
|
|
<span class="identifier">r</span> <span class="special">=</span> <span class="identifier">brent_find_minima</span><span class="special">(</span><span class="identifier">funcdouble</span><span class="special">(),</span> <span class="special">-</span><span class="number">4.</span><span class="special">,</span> <span class="number">4.</span> <span class="special">/</span> <span class="number">3</span><span class="special">,</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">it</span><span class="special">);</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"x at minimum = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">", f("</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">it</span> <span class="special"><<</span> <span class="string">" iterations. "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
</pre>
|
|
<p>
|
|
showing no change in the result and no change in the number of iterations,
|
|
as expected.
|
|
</p>
|
|
<p>
|
|
It is only if we reduce the precision to a quarter, specifying only 13 precision
|
|
bits
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">bits</span> <span class="special">/=</span> <span class="number">2</span><span class="special">;</span> <span class="comment">// Quarter precision.</span>
|
|
<span class="keyword">double</span> <span class="identifier">epsilon_4</span> <span class="special">=</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">pow</span><span class="special"><-(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits</span> <span class="special">/</span> <span class="number">4</span> <span class="special">-</span> <span class="number">1</span><span class="special">),</span> <span class="keyword">double</span><span class="special">>(</span><span class="number">2</span><span class="special">);</span>
|
|
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"Showing "</span> <span class="special"><<</span> <span class="identifier">bits</span> <span class="special"><<</span> <span class="string">" bits precision with "</span> <span class="special"><<</span> <span class="identifier">prec</span>
|
|
<span class="special"><<</span> <span class="string">" decimal digits from tolerance "</span> <span class="special"><<</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">epsilon_4</span><span class="special">)</span>
|
|
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">streamsize</span> <span class="identifier">precision</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">prec</span><span class="special">);</span> <span class="comment">// Save.</span>
|
|
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">it</span> <span class="special">=</span> <span class="identifier">maxit</span><span class="special">;</span>
|
|
<span class="identifier">r</span> <span class="special">=</span> <span class="identifier">brent_find_minima</span><span class="special">(</span><span class="identifier">funcdouble</span><span class="special">(),</span> <span class="special">-</span><span class="number">4.</span><span class="special">,</span> <span class="number">4.</span> <span class="special">/</span> <span class="number">3</span><span class="special">,</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">it</span><span class="special">);</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"x at minimum = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">", f("</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span>
|
|
<span class="special"><<</span> <span class="string">", after "</span> <span class="special"><<</span> <span class="identifier">it</span> <span class="special"><<</span> <span class="string">" iterations. "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
</pre>
|
|
<p>
|
|
that we reduce the number of iterations from 10 to 7 and the result significantly
|
|
differing from <span class="emphasis"><em>one</em></span> and <span class="emphasis"><em>zero</em></span>.
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">Showing</span> <span class="number">13</span> <span class="identifier">bits</span> <span class="identifier">precision</span> <span class="identifier">with</span> <span class="number">9</span> <span class="identifier">decimal</span> <span class="identifier">digits</span> <span class="identifier">from</span> <span class="identifier">tolerance</span> <span class="number">0.015625</span>
|
|
<span class="identifier">x</span> <span class="identifier">at</span> <span class="identifier">minimum</span> <span class="special">=</span> <span class="number">0.9999776</span><span class="special">,</span> <span class="identifier">f</span><span class="special">(</span><span class="number">0.9999776</span><span class="special">)</span> <span class="special">=</span> <span class="number">2.0069572e-009</span> <span class="identifier">after</span> <span class="number">7</span> <span class="identifier">iterations</span><span class="special">.</span>
|
|
</pre>
|
|
<h6>
|
|
<a name="math_toolkit.brent_minima.h3"></a>
|
|
<span class="phrase"><a name="math_toolkit.brent_minima.template"></a></span><a class="link" href="brent_minima.html#math_toolkit.brent_minima.template">Templating
|
|
on floating-point type</a>
|
|
</h6>
|
|
<p>
|
|
If we want to switch the floating-point type, then the functor must be revised.
|
|
Since the functor is stateless, the easiest option is to simply make <code class="computeroutput"><span class="keyword">operator</span><span class="special">()</span></code> a
|
|
template member function:
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">struct</span> <span class="identifier">func</span>
|
|
<span class="special">{</span>
|
|
<span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
|
<span class="identifier">T</span> <span class="keyword">operator</span><span class="special">()(</span><span class="identifier">T</span> <span class="keyword">const</span><span class="special">&</span> <span class="identifier">x</span><span class="special">)</span>
|
|
<span class="special">{</span> <span class="comment">//</span>
|
|
<span class="keyword">return</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">3</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">-</span> <span class="number">1</span><span class="special">);</span> <span class="comment">//</span>
|
|
<span class="special">}</span>
|
|
<span class="special">};</span>
|
|
</pre>
|
|
<p>
|
|
The <code class="computeroutput"><span class="identifier">brent_find_minima</span></code> function
|
|
can now be used in template form.
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">);</span>
|
|
<span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">bracket_min</span> <span class="special">=</span> <span class="special">-</span><span class="number">4.</span><span class="special">;</span>
|
|
<span class="keyword">long</span> <span class="keyword">double</span> <span class="identifier">bracket_max</span> <span class="special">=</span> <span class="number">4.</span> <span class="special">/</span> <span class="number">3</span><span class="special">;</span>
|
|
<span class="keyword">int</span> <span class="identifier">bits</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">>::</span><span class="identifier">digits</span><span class="special">;</span>
|
|
<span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">maxit</span> <span class="special">=</span> <span class="number">20</span><span class="special">;</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">it</span> <span class="special">=</span> <span class="identifier">maxit</span><span class="special">;</span>
|
|
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span><span class="special">></span> <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">brent_find_minima</span><span class="special">(</span><span class="identifier">func</span><span class="special">(),</span> <span class="identifier">bracket_min</span><span class="special">,</span> <span class="identifier">bracket_max</span><span class="special">,</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">it</span><span class="special">);</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"x at minimum = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">", f("</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span>
|
|
<span class="special"><<</span> <span class="string">", after "</span> <span class="special"><<</span> <span class="identifier">it</span> <span class="special"><<</span> <span class="string">" iterations. "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
</pre>
|
|
<p>
|
|
The form shown uses the floating-point type <code class="computeroutput"><span class="keyword">long</span>
|
|
<span class="keyword">double</span></code> by deduction, but it is also
|
|
possible to be more explicit, for example:
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span><span class="special">></span> <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">brent_find_minima</span><span class="special"><</span><span class="identifier">func</span><span class="special">,</span> <span class="keyword">long</span> <span class="keyword">double</span><span class="special">></span>
|
|
<span class="special">(</span><span class="identifier">func</span><span class="special">(),</span> <span class="identifier">bracket_min</span><span class="special">,</span> <span class="identifier">bracket_max</span><span class="special">,</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">it</span><span class="special">);</span>
|
|
</pre>
|
|
<p>
|
|
In order to show the use of multiprecision below, it may be convenient to write
|
|
a templated function to use this.
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">T</span><span class="special">></span>
|
|
<span class="keyword">void</span> <span class="identifier">show_minima</span><span class="special">()</span>
|
|
<span class="special">{</span>
|
|
<span class="keyword">using</span> <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">brent_find_minima</span><span class="special">;</span>
|
|
<span class="keyword">try</span>
|
|
<span class="special">{</span> <span class="comment">// Always use try'n'catch blocks with Boost.Math to get any error messages.</span>
|
|
|
|
<span class="keyword">int</span> <span class="identifier">bits</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits</span><span class="special">/</span><span class="number">2</span><span class="special">;</span> <span class="comment">// Maximum is digits/2;</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">streamsize</span> <span class="identifier">prec</span> <span class="special">=</span> <span class="keyword">static_cast</span><span class="special"><</span><span class="keyword">int</span><span class="special">>(</span><span class="number">2</span> <span class="special">+</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">bits</span><span class="special">));</span> <span class="comment">// Number of significant decimal digits.</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">streamsize</span> <span class="identifier">precision</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">prec</span><span class="special">);</span> <span class="comment">// Save.</span>
|
|
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"\n\nFor type "</span> <span class="special"><<</span> <span class="keyword">typeid</span><span class="special">(</span><span class="identifier">T</span><span class="special">).</span><span class="identifier">name</span><span class="special">()</span>
|
|
<span class="special"><<</span> <span class="string">",\n epsilon = "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()</span>
|
|
<span class="comment">// << ", precision of " << bits << " bits"</span>
|
|
<span class="special"><<</span> <span class="string">",\n the maximum theoretical precision from Brent minimization is "</span> <span class="special"><<</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">())</span>
|
|
<span class="special"><<</span> <span class="string">"\n Displaying to std::numeric_limits<T>::digits10 "</span> <span class="special"><<</span> <span class="identifier">prec</span> <span class="special"><<</span> <span class="string">" significant decimal digits."</span>
|
|
<span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
|
|
<span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">maxit</span> <span class="special">=</span> <span class="number">20</span><span class="special">;</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">it</span> <span class="special">=</span> <span class="identifier">maxit</span><span class="special">;</span>
|
|
<span class="comment">// Construct using string, not double, avoids loss of precision.</span>
|
|
<span class="comment">//T bracket_min = static_cast<T>("-4");</span>
|
|
<span class="comment">//T bracket_max = static_cast<T>("1.3333333333333333333333333333333333333333333333333");</span>
|
|
|
|
<span class="comment">// Construction from double may cause loss of precision for multiprecision types like cpp_bin_float.</span>
|
|
<span class="comment">// but brackets values are good enough for using Brent minimization.</span>
|
|
<span class="identifier">T</span> <span class="identifier">bracket_min</span> <span class="special">=</span> <span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">T</span><span class="special">>(-</span><span class="number">4</span><span class="special">);</span>
|
|
<span class="identifier">T</span> <span class="identifier">bracket_max</span> <span class="special">=</span> <span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">T</span><span class="special">>(</span><span class="number">1.3333333333333333333333333333333333333333333333333</span><span class="special">);</span>
|
|
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">T</span><span class="special">,</span> <span class="identifier">T</span><span class="special">></span> <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">brent_find_minima</span><span class="special"><</span><span class="identifier">func</span><span class="special">,</span> <span class="identifier">T</span><span class="special">>(</span><span class="identifier">func</span><span class="special">(),</span> <span class="identifier">bracket_min</span><span class="special">,</span> <span class="identifier">bracket_max</span><span class="special">,</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">it</span><span class="special">);</span>
|
|
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">" x at minimum = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">", f("</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span><span class="special">;</span>
|
|
<span class="keyword">if</span> <span class="special">(</span><span class="identifier">it</span> <span class="special"><</span> <span class="identifier">maxit</span><span class="special">)</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">",\n met "</span> <span class="special"><<</span> <span class="identifier">bits</span> <span class="special"><<</span> <span class="string">" bits precision"</span> <span class="special"><<</span> <span class="string">", after "</span> <span class="special"><<</span> <span class="identifier">it</span> <span class="special"><<</span> <span class="string">" iterations."</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
<span class="keyword">else</span>
|
|
<span class="special">{</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">",\n did NOT meet "</span> <span class="special"><<</span> <span class="identifier">bits</span> <span class="special"><<</span> <span class="string">" bits precision"</span> <span class="special"><<</span> <span class="string">" after "</span> <span class="special"><<</span> <span class="identifier">it</span> <span class="special"><<</span> <span class="string">" iterations!"</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
<span class="comment">// Check that result is that expected (compared to theoretical uncertainty).</span>
|
|
<span class="identifier">T</span> <span class="identifier">uncertainty</span> <span class="special">=</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">());</span>
|
|
<span class="comment">//std::cout << std::boolalpha << "x == 1 (compared to uncertainty " << uncertainty << ") is " << close(static_cast<T>(1), r.first, uncertainty) << std::endl;</span>
|
|
<span class="comment">//std::cout << std::boolalpha << "f(x) == (0 compared to uncertainty " << uncertainty << ") is " << close(static_cast<T>(0), r.second, uncertainty) << std::endl;</span>
|
|
<span class="comment">// Problems with this using multiprecision with expression template on?</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">precision</span><span class="special">);</span> <span class="comment">// Restore.</span>
|
|
<span class="special">}</span>
|
|
<span class="keyword">catch</span> <span class="special">(</span><span class="keyword">const</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exception</span><span class="special">&</span> <span class="identifier">e</span><span class="special">)</span>
|
|
<span class="special">{</span> <span class="comment">// Always useful to include try & catch blocks because default policies</span>
|
|
<span class="comment">// are to throw exceptions on arguments that cause errors like underflow, overflow.</span>
|
|
<span class="comment">// Lacking try & catch blocks, the program will abort without a message below,</span>
|
|
<span class="comment">// which may give some helpful clues as to the cause of the exception.</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span>
|
|
<span class="string">"\n"</span><span class="string">"Message from thrown exception was:\n "</span> <span class="special"><<</span> <span class="identifier">e</span><span class="special">.</span><span class="identifier">what</span><span class="special">()</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="special">}</span>
|
|
<span class="special">}</span> <span class="comment">// void show_minima()</span>
|
|
</pre>
|
|
<p>
|
|
We can use this with all built-in floating-point types, for example
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">show_minima</span><span class="special"><</span><span class="keyword">float</span><span class="special">>();</span>
|
|
<span class="identifier">show_minima</span><span class="special"><</span><span class="keyword">double</span><span class="special">>();</span>
|
|
<span class="identifier">show_minima</span><span class="special"><</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">>();</span>
|
|
</pre>
|
|
<p>
|
|
and, on platforms that provide it, a <a href="http://en.wikipedia.org/wiki/Quadruple-precision_floating-point_format" target="_top">128-bit
|
|
quad</a> type. (See <a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/float128.html" target="_top">float128</a>).
|
|
</p>
|
|
<p>
|
|
For this optional include, the build should define the macro BOOST_HAVE_QUADMATH:
|
|
</p>
|
|
<pre class="programlisting"><span class="preprocessor">#ifdef</span> <span class="identifier">BOOST_HAVE_QUADMATH</span> <span class="comment">// Define only if GCC or Intel and have quadmath.lib or .dll library available.</span>
|
|
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">float128</span><span class="special">;</span>
|
|
<span class="preprocessor">#endif</span>
|
|
</pre>
|
|
<p>
|
|
or
|
|
</p>
|
|
<pre class="programlisting"><span class="comment">// #ifndef _MSC_VER</span>
|
|
<span class="preprocessor">#ifdef</span> <span class="identifier">BOOST_HAVE_QUADMATH</span> <span class="comment">// Define only if GCC or Intel and have quadmath.lib or .dll library available.</span>
|
|
<span class="identifier">show_minima</span><span class="special"><</span><span class="identifier">float128</span><span class="special">>();</span> <span class="comment">// Needs quadmath_snprintf, sqrtQ, fabsq that are in in quadmath library.</span>
|
|
<span class="preprocessor">#endif</span>
|
|
</pre>
|
|
<h6>
|
|
<a name="math_toolkit.brent_minima.h4"></a>
|
|
<span class="phrase"><a name="math_toolkit.brent_minima.multiprecision"></a></span><a class="link" href="brent_minima.html#math_toolkit.brent_minima.multiprecision">Multiprecision</a>
|
|
</h6>
|
|
<p>
|
|
If a higher precision than <code class="computeroutput"><span class="keyword">double</span></code>
|
|
(or <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>
|
|
if that is more precise) is required, then this is easily achieved using <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
|
|
with some includes from
|
|
</p>
|
|
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">cpp_dec_float</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> <span class="comment">// For decimal boost::multiprecision::cpp_dec_float_50.</span>
|
|
<span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">multiprecision</span><span class="special">/</span><span class="identifier">cpp_bin_float</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> <span class="comment">// For binary boost::multiprecision::cpp_bin_float_50;</span>
|
|
</pre>
|
|
<p>
|
|
and some <code class="computeroutput"><span class="identifier">typdef</span></code>s.
|
|
</p>
|
|
<pre class="programlisting"><span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_bin_float_50</span><span class="special">;</span> <span class="comment">// binary.</span>
|
|
|
|
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">number</span><span class="special"><</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_bin_float</span><span class="special"><</span><span class="number">50</span><span class="special">>,</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">et_on</span><span class="special">></span>
|
|
<span class="identifier">cpp_bin_float_50_et_on</span><span class="special">;</span> <span class="comment">// et_on is default so is same as cpp_bin_float_50.</span>
|
|
|
|
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">number</span><span class="special"><</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_bin_float</span><span class="special"><</span><span class="number">50</span><span class="special">>,</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">et_off</span><span class="special">></span>
|
|
<span class="identifier">cpp_bin_float_50_et_off</span><span class="special">;</span>
|
|
|
|
<span class="keyword">using</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float_50</span><span class="special">;</span> <span class="comment">// decimal.</span>
|
|
|
|
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">number</span><span class="special"><</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float</span><span class="special"><</span><span class="number">50</span><span class="special">>,</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">et_on</span><span class="special">></span> <span class="comment">// et_on is default so is same as cpp_dec_float_50.</span>
|
|
<span class="identifier">cpp_dec_float_50_et_on</span><span class="special">;</span>
|
|
|
|
<span class="keyword">typedef</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">number</span><span class="special"><</span><span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">cpp_dec_float</span><span class="special"><</span><span class="number">50</span><span class="special">>,</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">et_off</span><span class="special">></span>
|
|
<span class="identifier">cpp_dec_float_50_et_off</span><span class="special">;</span>
|
|
</pre>
|
|
<p>
|
|
Using thus
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">cpp_bin_float_50</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">);</span>
|
|
|
|
<span class="identifier">cpp_bin_float_50</span> <span class="identifier">fpv</span><span class="special">(</span><span class="string">"-1.2345"</span><span class="special">);</span>
|
|
<span class="identifier">cpp_bin_float_50</span> <span class="identifier">absv</span><span class="special">;</span>
|
|
|
|
<span class="identifier">absv</span> <span class="special">=</span> <span class="identifier">fpv</span> <span class="special"><</span> <span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">cpp_bin_float_50</span><span class="special">>(</span><span class="number">0</span><span class="special">)</span> <span class="special">?</span> <span class="special">-</span><span class="identifier">fpv</span> <span class="special">:</span> <span class="identifier">fpv</span><span class="special">;</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">fpv</span> <span class="special"><<</span> <span class="char">' '</span> <span class="special"><<</span> <span class="identifier">absv</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
|
|
|
|
<span class="keyword">int</span> <span class="identifier">bits</span> <span class="special">=</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">cpp_bin_float_50</span><span class="special">>::</span><span class="identifier">digits</span> <span class="special">/</span> <span class="number">2</span> <span class="special">-</span> <span class="number">2</span><span class="special">;</span>
|
|
|
|
<span class="identifier">cpp_bin_float_50</span> <span class="identifier">bracket_min</span> <span class="special">=</span> <span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">cpp_bin_float_50</span><span class="special">>(</span><span class="string">"-4"</span><span class="special">);</span>
|
|
<span class="identifier">cpp_bin_float_50</span> <span class="identifier">bracket_max</span> <span class="special">=</span> <span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">cpp_bin_float_50</span><span class="special">>(</span><span class="string">"1.3333333333333333333333333333333333333333333333333"</span><span class="special">);</span>
|
|
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="identifier">bracket_min</span> <span class="special"><<</span> <span class="string">" "</span> <span class="special"><<</span> <span class="identifier">bracket_max</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
|
<span class="keyword">const</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">maxit</span> <span class="special">=</span> <span class="number">20</span><span class="special">;</span>
|
|
<span class="identifier">boost</span><span class="special">::</span><span class="identifier">uintmax_t</span> <span class="identifier">it</span> <span class="special">=</span> <span class="identifier">maxit</span><span class="special">;</span>
|
|
<span class="identifier">std</span><span class="special">::</span><span class="identifier">pair</span><span class="special"><</span><span class="identifier">cpp_bin_float_50</span><span class="special">,</span> <span class="identifier">cpp_bin_float_50</span><span class="special">></span> <span class="identifier">r</span> <span class="special">=</span> <span class="identifier">brent_find_minima</span><span class="special">(</span><span class="identifier">func</span><span class="special">(),</span> <span class="identifier">bracket_min</span><span class="special">,</span> <span class="identifier">bracket_max</span><span class="special">,</span> <span class="identifier">bits</span><span class="special">,</span> <span class="identifier">it</span><span class="special">);</span>
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<span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> <span class="special"><<</span> <span class="string">"x at minimum = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">", f("</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span> <span class="special"><<</span> <span class="string">") = "</span> <span class="special"><<</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">second</span>
|
|
<span class="comment">// x at minimum = 1, f(1) = 5.04853e-018</span>
|
|
<span class="special"><<</span> <span class="string">", after "</span> <span class="special"><<</span> <span class="identifier">it</span> <span class="special"><<</span> <span class="string">" iterations. "</span> <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">endl</span><span class="special">;</span>
|
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<span class="identifier">close</span><span class="special">(</span><span class="keyword">static_cast</span><span class="special"><</span><span class="identifier">cpp_bin_float_50</span><span class="special">>(</span><span class="number">1</span><span class="special">),</span> <span class="identifier">r</span><span class="special">.</span><span class="identifier">first</span><span class="special">,</span> <span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">cpp_bin_float_50</span><span class="special">>::</span><span class="identifier">epsilon</span><span class="special">()));</span>
|
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</pre>
|
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<p>
|
|
and with our show function
|
|
</p>
|
|
<pre class="programlisting"><span class="identifier">show_minima</span><span class="special"><</span><span class="identifier">cpp_bin_float_50_et_on</span><span class="special">>();</span> <span class="comment">//</span>
|
|
</pre>
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<pre class="programlisting"><span class="identifier">For</span> <span class="identifier">type</span> <span class="keyword">class</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">number</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">backends</span><span class="special">::</span><span class="identifier">cpp_bin_float</span><span class="special"><</span><span class="number">50</span><span class="special">,</span> <span class="number">10</span><span class="special">,</span> <span class="keyword">void</span><span class="special">,</span> <span class="keyword">int</span><span class="special">,</span> <span class="number">0</span><span class="special">,</span> <span class="number">0</span><span class="special">>,</span> <span class="number">1</span><span class="special">>,</span>
|
|
<span class="identifier">epsilon</span> <span class="special">=</span> <span class="number">5.3455294202e-51</span><span class="special">,</span>
|
|
<span class="identifier">the</span> <span class="identifier">maximum</span> <span class="identifier">theoretical</span> <span class="identifier">precision</span> <span class="identifier">from</span> <span class="identifier">Brent</span> <span class="identifier">minimization</span> <span class="identifier">is</span> <span class="number">7.311312755e-26</span>
|
|
<span class="identifier">Displaying</span> <span class="identifier">to</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits10</span> <span class="number">11</span> <span class="identifier">significant</span> <span class="identifier">decimal</span> <span class="identifier">digits</span><span class="special">.</span>
|
|
<span class="identifier">x</span> <span class="identifier">at</span> <span class="identifier">minimum</span> <span class="special">=</span> <span class="number">1</span><span class="special">,</span> <span class="identifier">f</span><span class="special">(</span><span class="number">1</span><span class="special">)</span> <span class="special">=</span> <span class="number">5.6273022713e-58</span><span class="special">,</span>
|
|
<span class="identifier">met</span> <span class="number">84</span> <span class="identifier">bits</span> <span class="identifier">precision</span><span class="special">,</span> <span class="identifier">after</span> <span class="number">14</span> <span class="identifier">iterations</span><span class="special">.</span>
|
|
</pre>
|
|
<pre class="programlisting"><span class="identifier">For</span> <span class="identifier">type</span> <span class="keyword">class</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">number</span><span class="special"><</span><span class="keyword">class</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">multiprecision</span><span class="special">::</span><span class="identifier">backends</span><span class="special">::</span><span class="identifier">cpp_bin_float</span><span class="special"><</span><span class="number">50</span><span class="special">,</span> <span class="number">10</span><span class="special">,</span> <span class="keyword">void</span><span class="special">,</span> <span class="keyword">int</span><span class="special">,</span> <span class="number">0</span><span class="special">,</span> <span class="number">0</span><span class="special">>,</span> <span class="number">1</span><span class="special">>,</span>
|
|
</pre>
|
|
<div class="tip"><table border="0" summary="Tip">
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<tr>
|
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<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../doc/src/images/tip.png"></td>
|
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<th align="left">Tip</th>
|
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</tr>
|
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<tr><td align="left" valign="top"><p>
|
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One can usually rely on template argument deduction to avoid specifying the
|
|
verbose multiprecision types, but great care in needed with the <span class="emphasis"><em>type
|
|
of the values</em></span> provided to avoid confusing the compiler.
|
|
</p></td></tr>
|
|
</table></div>
|
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<div class="tip"><table border="0" summary="Tip">
|
|
<tr>
|
|
<td rowspan="2" align="center" valign="top" width="25"><img alt="[Tip]" src="../../../../../doc/src/images/tip.png"></td>
|
|
<th align="left">Tip</th>
|
|
</tr>
|
|
<tr><td align="left" valign="top"><p>
|
|
Using <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">);</span></code>
|
|
or <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span><span class="special">.</span><span class="identifier">precision</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><</span><span class="identifier">T</span><span class="special">>::</span><span class="identifier">max_digits10</span><span class="special">);</span></code>
|
|
during debugging may be wise because it gives some warning if construction
|
|
of multiprecision values involves unintended conversion from <code class="computeroutput"><span class="keyword">double</span></code> by showing trailing zero or random
|
|
digits after <a href="http://en.cppreference.com/w/cpp/types/numeric_limits/max_digits10" target="_top">max_digits10</a>,
|
|
that is 17 for <code class="computeroutput"><span class="keyword">double</span></code>, digit
|
|
18... may be just noise.
|
|
</p></td></tr>
|
|
</table></div>
|
|
<p>
|
|
The complete example code is at <a href="../../../example/brent_minimise_example.cpp" target="_top">brent_minimise_example.cpp</a>.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.brent_minima.h5"></a>
|
|
<span class="phrase"><a name="math_toolkit.brent_minima.implementation"></a></span><a class="link" href="brent_minima.html#math_toolkit.brent_minima.implementation">Implementation</a>
|
|
</h5>
|
|
<p>
|
|
This is a reasonably faithful implementation of Brent's algorithm.
|
|
</p>
|
|
<h5>
|
|
<a name="math_toolkit.brent_minima.h6"></a>
|
|
<span class="phrase"><a name="math_toolkit.brent_minima.references"></a></span><a class="link" href="brent_minima.html#math_toolkit.brent_minima.references">References</a>
|
|
</h5>
|
|
<div class="orderedlist"><ol class="orderedlist" type="1">
|
|
<li class="listitem">
|
|
Brent, R.P. 1973, Algorithms for Minimization without Derivatives, (Englewood
|
|
Cliffs, NJ: Prentice-Hall), Chapter 5.
|
|
</li>
|
|
<li class="listitem">
|
|
Numerical Recipes in C, The Art of Scientific Computing, Second Edition,
|
|
William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P.
|
|
Flannery. Cambridge University Press. 1988, 1992.
|
|
</li>
|
|
<li class="listitem">
|
|
An algorithm with guaranteed convergence for finding a zero of a function,
|
|
R. P. Brent, The Computer Journal, Vol 44, 1971.
|
|
</li>
|
|
<li class="listitem">
|
|
<a href="http://en.wikipedia.org/wiki/Brent%27s_method" target="_top">Brent's method
|
|
in Wikipedia.</a>
|
|
</li>
|
|
<li class="listitem">
|
|
Z. Zhang, An Improvement to the Brent's Method, IJEA, vol. 2, pp. 2 to
|
|
26, May 31, 2011. <a href="http://www.cscjournals.org/manuscript/Journals/IJEA/volume2/Issue1/IJEA-7.pdf" target="_top">http://www.cscjournals.org/manuscript/Journals/IJEA/volume2/Issue1/IJEA-7.pdf</a>
|
|
</li>
|
|
<li class="listitem">
|
|
Steven A. Stage, Comments on An Improvement to the Brent's Method (and
|
|
comparison of various algorithms) <a href="http://www.cscjournals.org/manuscript/Journals/IJEA/volume4/Issue1/IJEA-33.pdf" target="_top">http://www.cscjournals.org/manuscript/Journals/IJEA/volume4/Issue1/IJEA-33.pdf</a>
|
|
Stage concludes that Brent's algorithm is slow to start, but fast to finish
|
|
convergence, and has good accuracy.
|
|
</li>
|
|
</ol></div>
|
|
</div>
|
|
<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
|
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2010, 2012-2014, 2017 Nikhar
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Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
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Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, 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>)
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</p>
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</div></td>
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</tr></table>
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