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<div class="section">
<div class="titlepage"><div><div><h2 class="title" style="clear: both">
<a name="math_toolkit.diff"></a><a class="link" href="diff.html" title="Numerical Differentiation">Numerical Differentiation</a>
</h2></div></div></div>
<h4>
<a name="math_toolkit.diff.h0"></a>
<span class="phrase"><a name="math_toolkit.diff.synopsis"></a></span><a class="link" href="diff.html#math_toolkit.diff.synopsis">Synopsis</a>
</h4>
<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">differentiation</span><span class="special">/</span><span class="identifier">finite_difference</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
<span class="keyword">namespace</span> <span class="identifier">boost</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">math</span> <span class="special">{</span> <span class="keyword">namespace</span> <span class="identifier">differentiation</span> <span class="special">{</span>
<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">Real</span><span class="special">></span>
<span class="identifier">Real</span> <span class="identifier">complex_step_derivative</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x</span><span class="special">);</span>
<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">Real</span><span class="special">,</span> <span class="identifier">size_t</span> <span class="identifier">order</span> <span class="special">=</span> <span class="number">6</span><span class="special">></span>
<span class="identifier">Real</span> <span class="identifier">finite_difference_derivative</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">x</span><span class="special">,</span> <span class="identifier">Real</span><span class="special">*</span> <span class="identifier">error</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">);</span>
<span class="special">}}}</span> <span class="comment">// namespaces</span>
</pre>
<h4>
<a name="math_toolkit.diff.h1"></a>
<span class="phrase"><a name="math_toolkit.diff.description"></a></span><a class="link" href="diff.html#math_toolkit.diff.description">Description</a>
</h4>
<p>
The function <code class="computeroutput"><span class="identifier">finite_difference_derivative</span></code>
calculates a finite-difference approximation to the derivative of a function
<span class="emphasis"><em>f</em></span> at point <span class="emphasis"><em>x</em></span>. A basic usage is
</p>
<pre class="programlisting"><span class="keyword">auto</span> <span class="identifier">f</span> <span class="special">=</span> <span class="special">[](</span><span class="keyword">double</span> <span class="identifier">x</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="identifier">std</span><span class="special">::</span><span class="identifier">exp</span><span class="special">(</span><span class="identifier">x</span><span class="special">);</span> <span class="special">};</span>
<span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">1.7</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">dfdx</span> <span class="special">=</span> <span class="identifier">finite_difference_derivative</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
Finite differencing is complicated, as differentiation is <span class="emphasis"><em>infinitely</em></span>
ill-conditioned. In addition, for any function implemented in finite-precision
arithmetic, the "true" derivative is <span class="emphasis"><em>zero</em></span> almost
everywhere, and undefined at representables. However, some tricks allow for
reasonable results to be obtained in many cases.
</p>
<p>
There are two sources of error from finite differences: One, the truncation
error arising from using a finite number of samples to cancel out higher order
terms in the Taylor series. The second is the roundoff error involved in evaluating
the function. The truncation error goes to zero as <span class="emphasis"><em>h</em></span> →
0, but the roundoff error becomes unbounded. By balancing these two sources
of error, we can choose a value of <span class="emphasis"><em>h</em></span> that minimizes the
maximum total error. For this reason boost's <code class="computeroutput"><span class="identifier">finite_difference_derivative</span></code>
does not require the user to input a stepsize. For more details about the theoretical
error analysis involved in finite-difference approximations to the derivative,
see <a href="http://web.archive.org/web/20150420195907/http://www.uio.no/studier/emner/matnat/math/MAT-INF1100/h08/kompendiet/diffint.pdf" target="_top">here</a>.
</p>
<p>
Despite the effort that has went into choosing a reasonable value of <span class="emphasis"><em>h</em></span>,
the problem is still fundamentally ill-conditioned, and hence an error estimate
is essential. It can be queried as follows
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">error_estimate</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">d</span> <span class="special">=</span> <span class="identifier">finite_difference_derivative</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="identifier">x</span><span class="special">,</span> <span class="special">&</span><span class="identifier">error_estimate</span><span class="special">);</span>
</pre>
<p>
N.B.: Producing an error estimate requires additional function evaluations
and as such is slower than simple evaluation of the derivative. It also expands
the domain over which the function must be differentiable and requires the
function to have two more continuous derivatives. The error estimate is computed
under the assumption that <span class="emphasis"><em>f</em></span> is evaluated to 1ULP. This
might seem an extreme assumption, but it is the only sensible one, as the routine
cannot know the functions rounding error. If the function cannot be evaluated
with very great accuracy, Lanczos's smoothing differentiation is recommended
as an alternative.
</p>
<p>
The default order of accuracy is 6, which reflects that fact that people tend
to be interested in functions with many continuous derivatives. If your function
does not have 7 continuous derivatives, is may be of interest to use a lower
order method, which can be achieved via (say)
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">d</span> <span class="special">=</span> <span class="identifier">finite_difference_derivative</span><span class="special"><</span><span class="keyword">decltype</span><span class="special">(</span><span class="identifier">f</span><span class="special">),</span> <span class="identifier">Real</span><span class="special">,</span> <span class="number">2</span><span class="special">>(</span><span class="identifier">f</span><span class="special">,</span> <span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
This requests a second-order accurate derivative be computed.
</p>
<p>
It is emphatically <span class="emphasis"><em>not</em></span> the case that higher order methods
always give higher accuracy for smooth functions. Higher order methods require
more addition of positive and negative terms, which can lead to catastrophic
cancellation. A function which is very good at making a mockery of finite-difference
differentiation is exp(x)/(cos(x)<sup>3</sup> + sin(x)<sup>3</sup>). Differentiating this function
by <code class="computeroutput"><span class="identifier">finite_difference_derivative</span></code>
in double precision at <span class="emphasis"><em>x=5.5</em></span> gives zero correct digits
at order 4, 6, and 8, but recovers 5 correct digits at order 2. These are dangerous
waters; use the error estimates to tread carefully.
</p>
<p>
For a finite-difference method of order <span class="emphasis"><em>k</em></span>, the error is
<span class="emphasis"><em>C</em></span> ε<sup>k/k+1</sup>. In the limit <span class="emphasis"><em>k</em></span> →
∞, we see that the error tends to ε, recovering the full precision
for the type. However, this ignores the fact that higher-order methods require
subtracting more nearly-equal (perhaps noisy) terms, so the constant <span class="emphasis"><em>C</em></span>
grows with <span class="emphasis"><em>k</em></span>. Since <span class="emphasis"><em>C</em></span> grows quickly
and ε<sup>k/k+1</sup> approaches ε slowly, we can see there is a compromise
between high-order accuracy and conditioning of the difference quotient. In
practice we have found that <span class="emphasis"><em>k=6</em></span> seems to be a good compromise
between the two (and have made this the default), but users are encouraged
to examine the error estimates to choose an optimal order of accuracy for the
given problem.
</p>
<div class="table">
<a name="math_toolkit.diff.id"></a><p class="title"><b>Table 14.1. Cost of Finite-Difference Numerical Differentiation</b></p>
<div class="table-contents"><table class="table" summary="Cost of Finite-Difference Numerical Differentiation">
<colgroup>
<col>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
<p>
Order of Accuracy
</p>
</th>
<th>
<p>
Function Evaluations
</p>
</th>
<th>
<p>
Error
</p>
</th>
<th>
<p>
Continuous Derivatives Required for Error Estimate to Hold
</p>
</th>
<th>
<p>
Additional Function Evaluations to Produce Error Estimates
</p>
</th>
</tr></thead>
<tbody>
<tr>
<td>
<p>
1
</p>
</td>
<td>
<p>
2
</p>
</td>
<td>
<p>
ε<sup>1/2</sup>
</p>
</td>
<td>
<p>
2
</p>
</td>
<td>
<p>
1
</p>
</td>
</tr>
<tr>
<td>
<p>
2
</p>
</td>
<td>
<p>
2
</p>
</td>
<td>
<p>
ε<sup>2/3</sup>
</p>
</td>
<td>
<p>
3
</p>
</td>
<td>
<p>
2
</p>
</td>
</tr>
<tr>
<td>
<p>
4
</p>
</td>
<td>
<p>
4
</p>
</td>
<td>
<p>
ε<sup>4/5</sup>
</p>
</td>
<td>
<p>
5
</p>
</td>
<td>
<p>
2
</p>
</td>
</tr>
<tr>
<td>
<p>
6
</p>
</td>
<td>
<p>
6
</p>
</td>
<td>
<p>
ε<sup>6/7</sup>
</p>
</td>
<td>
<p>
7
</p>
</td>
<td>
<p>
2
</p>
</td>
</tr>
<tr>
<td>
<p>
8
</p>
</td>
<td>
<p>
8
</p>
</td>
<td>
<p>
ε<sup>8/9</sup>
</p>
</td>
<td>
<p>
9
</p>
</td>
<td>
<p>
2
</p>
</td>
</tr>
</tbody>
</table></div>
</div>
<br class="table-break"><p>
Given all the caveats which must be kept in mind for successful use of finite-difference
differentiation, it is reasonable to try to avoid it if possible. Boost provides
two possibilities: The Chebyshev transform (see <a class="link" href="sf_poly/chebyshev.html" title="Chebyshev Polynomials">here</a>)
and the complex step derivative. If your function is the restriction to the
real line of a holomorphic function which takes real values at real argument,
then the <span class="bold"><strong>complex step derivative</strong></span> can be used.
The idea is very simple: Since <span class="emphasis"><em>f</em></span> is complex-differentiable,
<span class="emphasis"><em>f(x+ⅈ h) = f(x) + ⅈ hf'(x) - h<sup>2</sup>f''(x) + 𝑶(h<sup>3</sup>)</em></span>.
As long as <span class="emphasis"><em>f(x)</em></span> ∈ ℝ, then <span class="emphasis"><em>f'(x)
= ℑ f(x+ⅈ h)/h + 𝑶(h<sup>2</sup>)</em></span>. This method requires a single
complex function evaluation and is not subject to the catastrophic subtractive
cancellation that plagues finite-difference calculations.
</p>
<p>
An example usage:
</p>
<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">x</span> <span class="special">=</span> <span class="number">7.2</span><span class="special">;</span>
<span class="keyword">double</span> <span class="identifier">e_prime</span> <span class="special">=</span> <span class="identifier">complex_step_derivative</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">exp</span><span class="special"><</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">complex</span><span class="special"><</span><span class="keyword">double</span><span class="special">>>,</span> <span class="identifier">x</span><span class="special">);</span>
</pre>
<p>
References:
</p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
Squire, William, and George Trapp. <span class="emphasis"><em>Using complex variables to
estimate derivatives of real functions.</em></span> Siam Review 40.1 (1998):
110-112.
</li>
<li class="listitem">
Fornberg, Bengt. <span class="emphasis"><em>Generation of finite difference formulas on
arbitrarily spaced grids.</em></span> Mathematics of computation 51.184
(1988): 699-706.
</li>
<li class="listitem">
Corless, Robert M., and Nicolas Fillion. <span class="emphasis"><em>A graduate introduction
to numerical methods.</em></span> AMC 10 (2013): 12.
</li>
</ul></div>
</div>
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Walker and Xiaogang Zhang<p>
Distributed under the Boost Software License, Version 1.0. (See accompanying
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