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| <div class="titlepage"><div><div><h5 class="title"> |
| <a name="boost_multiprecision.tut.floats.fp_eg.nd"></a><a class="link" href="nd.html" title="Calculating a Derivative">Calculating |
| a Derivative</a> |
| </h5></div></div></div> |
| <p> |
| In this example we'll add even more power to generic numeric programming |
| using not only different floating-point types but also function objects |
| as template parameters. Consider some well-known central difference rules |
| for numerically computing the first derivative of a function <span class="emphasis"><em>f′(x)</em></span> |
| with <span class="emphasis"><em>x ∈ ℜ</em></span>: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../floating_point_eg1.png"></span> |
| </p> |
| <p> |
| Where the difference terms <span class="emphasis"><em>m<sub>n</sub></em></span> are given by: |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../floating_point_eg2.png"></span> |
| </p> |
| <p> |
| and <span class="emphasis"><em>dx</em></span> is the step-size of the derivative. |
| </p> |
| <p> |
| The third formula in Equation 1 is a three-point central difference rule. |
| It calculates the first derivative of <span class="emphasis"><em>f′(x)</em></span> to <span class="emphasis"><em>O(dx<sup>6</sup>)</em></span>, |
| where <span class="emphasis"><em>dx</em></span> is the given step-size. For example, if |
| the step-size is 0.01 this derivative calculation has about 6 decimal |
| digits of precision - just about right for the 7 decimal digits of single-precision |
| float. Let's make a generic template subroutine using this three-point |
| central difference rule. In particular: |
| </p> |
| <pre class="programlisting"><span class="keyword">template</span><span class="special"><</span><span class="keyword">typename</span> <span class="identifier">value_type</span><span class="special">,</span> <span class="keyword">typename</span> <span class="identifier">function_type</span><span class="special">></span> |
| <span class="identifier">value_type</span> <span class="identifier">derivative</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">dx</span><span class="special">,</span> <span class="identifier">function_type</span> <span class="identifier">func</span><span class="special">)</span> |
| <span class="special">{</span> |
| <span class="comment">// Compute d/dx[func(*first)] using a three-point</span> |
| <span class="comment">// central difference rule of O(dx^6).</span> |
| |
| <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">dx1</span> <span class="special">=</span> <span class="identifier">dx</span><span class="special">;</span> |
| <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">dx2</span> <span class="special">=</span> <span class="identifier">dx1</span> <span class="special">*</span> <span class="number">2</span><span class="special">;</span> |
| <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">dx3</span> <span class="special">=</span> <span class="identifier">dx1</span> <span class="special">*</span> <span class="number">3</span><span class="special">;</span> |
| |
| <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">m1</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">func</span><span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">dx1</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">func</span><span class="special">(</span><span class="identifier">x</span> <span class="special">-</span> <span class="identifier">dx1</span><span class="special">))</span> <span class="special">/</span> <span class="number">2</span><span class="special">;</span> |
| <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">m2</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">func</span><span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">dx2</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">func</span><span class="special">(</span><span class="identifier">x</span> <span class="special">-</span> <span class="identifier">dx2</span><span class="special">))</span> <span class="special">/</span> <span class="number">4</span><span class="special">;</span> |
| <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">m3</span> <span class="special">=</span> <span class="special">(</span><span class="identifier">func</span><span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">dx3</span><span class="special">)</span> <span class="special">-</span> <span class="identifier">func</span><span class="special">(</span><span class="identifier">x</span> <span class="special">-</span> <span class="identifier">dx3</span><span class="special">))</span> <span class="special">/</span> <span class="number">6</span><span class="special">;</span> |
| |
| <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">fifteen_m1</span> <span class="special">=</span> <span class="number">15</span> <span class="special">*</span> <span class="identifier">m1</span><span class="special">;</span> |
| <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">six_m2</span> <span class="special">=</span> <span class="number">6</span> <span class="special">*</span> <span class="identifier">m2</span><span class="special">;</span> |
| <span class="keyword">const</span> <span class="identifier">value_type</span> <span class="identifier">ten_dx1</span> <span class="special">=</span> <span class="number">10</span> <span class="special">*</span> <span class="identifier">dx1</span><span class="special">;</span> |
| |
| <span class="keyword">return</span> <span class="special">((</span><span class="identifier">fifteen_m1</span> <span class="special">-</span> <span class="identifier">six_m2</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">m3</span><span class="special">)</span> <span class="special">/</span> <span class="identifier">ten_dx1</span><span class="special">;</span> |
| <span class="special">}</span> |
| </pre> |
| <p> |
| The <code class="computeroutput"><span class="identifier">derivative</span><span class="special">()</span></code> |
| template function can be used to compute the first derivative of any |
| function to <span class="emphasis"><em>O(dx<sup>6</sup>)</em></span>. For example, consider the first |
| derivative of <span class="emphasis"><em>sin(x)</em></span> evaluated at <span class="emphasis"><em>x = |
| π/3</em></span>. In other words, |
| </p> |
| <p> |
| <span class="inlinemediaobject"><img src="../../../../../floating_point_eg3.png"></span> |
| </p> |
| <p> |
| The code below computes the derivative in Equation 3 for float, double |
| and boost's multiple-precision type cpp_dec_float_50. |
| </p> |
| <pre class="programlisting"><span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iostream</span><span class="special">></span> |
| <span class="preprocessor">#include</span> <span class="special"><</span><span class="identifier">iomanip</span><span class="special">></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_dec_float</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| <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">constants</span><span class="special">/</span><span class="identifier">constants</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span> |
| |
| |
| <span class="keyword">int</span> <span class="identifier">main</span><span class="special">(</span><span class="keyword">int</span><span class="special">,</span> <span class="keyword">char</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">constants</span><span class="special">::</span><span class="identifier">pi</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">//</span> |
| <span class="comment">// We'll pass a function pointer for the function object passed to derivative,</span> |
| <span class="comment">// the typecast is needed to select the correct overload of std::sin:</span> |
| <span class="comment">//</span> |
| <span class="keyword">const</span> <span class="keyword">float</span> <span class="identifier">d_f</span> <span class="special">=</span> <span class="identifier">derivative</span><span class="special">(</span> |
| <span class="identifier">pi</span><span class="special"><</span><span class="keyword">float</span><span class="special">>()</span> <span class="special">/</span> <span class="number">3</span><span class="special">,</span> |
| <span class="number">0.01F</span><span class="special">,</span> |
| <span class="keyword">static_cast</span><span class="special"><</span><span class="keyword">float</span><span class="special">(*)(</span><span class="keyword">float</span><span class="special">)>(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">sin</span><span class="special">)</span> |
| <span class="special">);</span> |
| |
| <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">d_d</span> <span class="special">=</span> <span class="identifier">derivative</span><span class="special">(</span> |
| <span class="identifier">pi</span><span class="special"><</span><span class="keyword">double</span><span class="special">>()</span> <span class="special">/</span> <span class="number">3</span><span class="special">,</span> |
| <span class="number">0.001</span><span class="special">,</span> |
| <span class="keyword">static_cast</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">std</span><span class="special">::</span><span class="identifier">sin</span><span class="special">)</span> |
| <span class="special">);</span> |
| <span class="comment">//</span> |
| <span class="comment">// In the cpp_dec_float_50 case, the sin function is multiply overloaded</span> |
| <span class="comment">// to handle expression templates etc. As a result it's hard to take its</span> |
| <span class="comment">// address without knowing about its implementation details. We'll use a </span> |
| <span class="comment">// C++11 lambda expression to capture the call.</span> |
| <span class="comment">// We also need a typecast on the first argument so we don't accidentally pass</span> |
| <span class="comment">// an expression template to a template function:</span> |
| <span class="comment">//</span> |
| <span class="keyword">const</span> <span class="identifier">cpp_dec_float_50</span> <span class="identifier">d_mp</span> <span class="special">=</span> <span class="identifier">derivative</span><span class="special">(</span> |
| <span class="identifier">cpp_dec_float_50</span><span class="special">(</span><span class="identifier">pi</span><span class="special"><</span><span class="identifier">cpp_dec_float_50</span><span class="special">>()</span> <span class="special">/</span> <span class="number">3</span><span class="special">),</span> |
| <span class="identifier">cpp_dec_float_50</span><span class="special">(</span><span class="number">1.0E-9</span><span class="special">),</span> |
| <span class="special">[](</span><span class="keyword">const</span> <span class="identifier">cpp_dec_float_50</span><span class="special">&</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">-></span> <span class="identifier">cpp_dec_float_50</span> |
| <span class="special">{</span> |
| <span class="keyword">return</span> <span class="identifier">sin</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span> |
| <span class="special">}</span> |
| <span class="special">);</span> |
| |
| <span class="comment">// 5.000029e-001</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> |
| <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</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">float</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</span> |
| <span class="special"><<</span> <span class="identifier">d_f</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="comment">// 4.999999999998876e-001</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> |
| <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</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> |
| <span class="special"><<</span> <span class="identifier">d_d</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="comment">// 4.99999999999999999999999999999999999999999999999999e-01</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> |
| <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</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_dec_float_50</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</span> |
| <span class="special"><<</span> <span class="identifier">d_mp</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> |
| </pre> |
| <p> |
| The expected value of the derivative is 0.5. This central difference |
| rule in this example is ill-conditioned, meaning it suffers from slight |
| loss of precision. With that in mind, the results agree with the expected |
| value of 0.5. |
| </p> |
| <p> |
| We can take this a step further and use our derivative function to compute |
| a partial derivative. For example if we take the incomplete gamma function |
| <span class="emphasis"><em>P(a, z)</em></span>, and take the derivative with respect to |
| <span class="emphasis"><em>z</em></span> at <span class="emphasis"><em>(2,2)</em></span> then we can calculate |
| the result as shown below, for good measure we'll compare with the "correct" |
| result obtained from a call to <span class="emphasis"><em>gamma_p_derivative</em></span>, |
| the results agree to approximately 44 digits: |
| </p> |
| <pre class="programlisting"><span class="identifier">cpp_dec_float_50</span> <span class="identifier">gd</span> <span class="special">=</span> <span class="identifier">derivative</span><span class="special">(</span> |
| <span class="identifier">cpp_dec_float_50</span><span class="special">(</span><span class="number">2</span><span class="special">),</span> |
| <span class="identifier">cpp_dec_float_50</span><span class="special">(</span><span class="number">1.0E-9</span><span class="special">),</span> |
| <span class="special">[](</span><span class="keyword">const</span> <span class="identifier">cpp_dec_float_50</span><span class="special">&</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">-></span><span class="identifier">cpp_dec_float_50</span> |
| <span class="special">{</span> |
| <span class="keyword">return</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">gamma_p</span><span class="special">(</span><span class="number">2</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span> |
| <span class="special">}</span> |
| <span class="special">);</span> |
| <span class="comment">// 2.70670566473225383787998989944968806815263091819151e-01</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</span> |
| <span class="special"><<</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">setprecision</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_dec_float_50</span><span class="special">>::</span><span class="identifier">digits10</span><span class="special">)</span> |
| <span class="special"><<</span> <span class="identifier">gd</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="comment">// 2.70670566473225383787998989944968806815253190143120e-01</span> |
| <span class="identifier">std</span><span class="special">::</span><span class="identifier">cout</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">gamma_p_derivative</span><span class="special">(</span><span class="identifier">cpp_dec_float_50</span><span class="special">(</span><span class="number">2</span><span class="special">),</span> <span class="identifier">cpp_dec_float_50</span><span class="special">(</span><span class="number">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> |
| </pre> |
| </div> |
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| <td align="right"><div class="copyright-footer">Copyright © 2002-2013 John Maddock and Christopher Kormanyos<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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