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<div class="section">
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<div class="titlepage"><div><div><h3 class="title">
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<a name="math_toolkit.double_exponential.de_caveats"></a><a class="link" href="de_caveats.html" title="Caveats">Caveats</a>
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</h3></div></div></div>
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<p>
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A few things to keep in mind while using the tanh-sinh, exp-sinh, and sinh-sinh
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quadratures:
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</p>
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<p>
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These routines are <span class="bold"><strong>very</strong></span> aggressive about
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approaching the endpoint singularities. This allows lots of significant digits
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to be extracted, but also has another problem: Roundoff error can cause the
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function to be evaluated at the endpoints. A few ways to avoid this: Narrow
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up the bounds of integration to say, [a + ε, b - ε], make sure (a+b)/2 and
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(b-a)/2 are representable, and finally, if you think the compromise between
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accuracy an usability has gone too far in the direction of accuracy, file
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a ticket.
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</p>
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<p>
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Both exp-sinh and sinh-sinh quadratures evaluate the functions they are passed
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at <span class="bold"><strong>very</strong></span> large argument. You might understand
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that x<sup>12</sup>exp(-x) is should be zero when x<sup>12</sup> overflows, but IEEE floating point
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arithmetic does not. Hence <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">pow</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span> <span class="number">12</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">x</span><span class="special">)</span></code> is an indeterminate form whenever <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">pow</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span>
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<span class="number">12</span><span class="special">)</span></code>
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overflows. So make sure your functions have the correct limiting behavior;
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for example
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</p>
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<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>
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<span class="keyword">double</span> <span class="identifier">t</span> <span class="special">=</span> <span class="identifier">exp</span><span class="special">(-</span><span class="identifier">x</span><span class="special">);</span>
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<span class="keyword">if</span><span class="special">(</span><span class="identifier">t</span> <span class="special">==</span> <span class="number">0</span><span class="special">)</span>
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<span class="special">{</span>
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<span class="keyword">return</span> <span class="number">0</span><span class="special">;</span>
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<span class="special">}</span>
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<span class="keyword">return</span> <span class="identifier">t</span><span class="special">*</span><span class="identifier">pow</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span> <span class="number">12</span><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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has the correct behavior for large <span class="emphasis"><em>x</em></span>, but <code class="computeroutput"><span class="keyword">auto</span> <span class="identifier">f</span> <span class="special">=</span> <span class="special">[](</span><span class="keyword">double</span>
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<span class="identifier">x</span><span class="special">)</span> <span class="special">{</span> <span class="keyword">return</span> <span class="identifier">exp</span><span class="special">(-</span><span class="identifier">x</span><span class="special">)*</span><span class="identifier">pow</span><span class="special">(</span><span class="identifier">x</span><span class="special">,</span> <span class="number">12</span><span class="special">);</span> <span class="special">};</span></code> does
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not.
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</p>
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<p>
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Oscillatory integrals, such as the sinc integral, are poorly approximated
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by double-exponential quadrature. Fortunately the error estimates and L1
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norm are massive for these integrals, but nonetheless, oscillatory integrals
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require different techniques.
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</p>
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<p>
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A special mention should be made about integrating through zero: while our
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range adaptors preserve precision when one endpoint is zero, things get harder
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when the origin is neither in the center of the range, nor at an endpoint.
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Consider integrating:
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</p>
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<div class="blockquote"><blockquote class="blockquote"><p>
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<span class="serif_italic">1 / (1 +x^2)</span>
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</p></blockquote></div>
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<p>
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Over (a, ∞). As long as <code class="computeroutput"><span class="identifier">a</span> <span class="special">>=</span> <span class="number">0</span></code> both
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the tanh_sinh and the exp_sinh integrators will handle this just fine: in
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fact they provide a rather efficient method for this kind of integral. However,
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if we have <code class="computeroutput"><span class="identifier">a</span> <span class="special"><</span>
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<span class="number">0</span></code> then we are forced to adapt the range
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in a way that produces abscissa values near zero that have an absolute error
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of ε, and since all of the area of the integral is near zero both integrators
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thrash around trying to reach the target accuracy, but never actually get
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there for <code class="computeroutput"><span class="identifier">a</span> <span class="special"><<</span>
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<span class="number">0</span></code>. On the other hand, the simple expedient
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of breaking the integral into two domains: (a, 0) and (0, b) and integrating
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each seperately using the tanh-sinh integrator, works just fine.
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</p>
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<p>
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Finally, some endpoint singularities are too strong to be handled by <code class="computeroutput"><span class="identifier">tanh_sinh</span></code> or equivalent methods, for example
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consider integrating the function:
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</p>
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<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">p</span> <span class="special">=</span> <span class="identifier">some_value</span><span class="special">;</span>
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<span class="identifier">tanh_sinh</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">integrator</span><span class="special">;</span>
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<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="keyword">return</span> <span class="identifier">pow</span><span class="special">(</span><span class="identifier">tan</span><span class="special">(</span><span class="identifier">x</span><span class="special">),</span> <span class="identifier">p</span><span class="special">);</span> <span class="special">};</span>
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<span class="keyword">auto</span> <span class="identifier">Q</span> <span class="special">=</span> <span class="identifier">integrator</span><span class="special">.</span><span class="identifier">integrate</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="number">0</span><span class="special">,</span> <span class="identifier">constants</span><span class="special">::</span><span class="identifier">half_pi</span><span class="special"><</span><span class="keyword">double</span><span class="special">>());</span>
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</pre>
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<p>
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The first problem with this function, is that the singularity is at π/2, so
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if we're integrating over (0, π/2) then we can never approach closer to the
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singularity than ε, and for p less than but close to 1, we need to get <span class="emphasis"><em>very</em></span>
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close to the singularity to find all the area under the function. If we recall
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the identity <code class="literal">tan(π/2 - x) == 1/tan(x)</code> then we can rewrite
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the function like this:
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</p>
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<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="keyword">return</span> <span class="identifier">pow</span><span class="special">(</span><span class="identifier">tan</span><span class="special">(</span><span class="identifier">x</span><span class="special">),</span> <span class="special">-</span><span class="identifier">p</span><span class="special">);</span> <span class="special">};</span>
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</pre>
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<p>
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And now the singularity is at the origin and we can get much closer to it
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when evaluating the integral: all we have done is swap the integral endpoints
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over.
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</p>
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<p>
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This actually works just fine for p < 0.95, but after that the <code class="computeroutput"><span class="identifier">tanh_sinh</span></code> integrator starts thrashing around
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and is unable to converge on the integral. The problem is actually a lack
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of exponent range: if we simply swap type double for something with a greater
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exponent range (an 80-bit long double or a quad precision type), then we
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can get to at least p = 0.99. If we want to go beyond that, or stick with
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type double, then we have to get smart.
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</p>
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<p>
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The easiest method is to notice that for small x, then <code class="literal">tan(x) ≅ x</code>,
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and so we are simply integrating x<sup>-p</sup>. Therefore we can use this approximation
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over (0, small), and integrate numerically from (small, π/2), using ε as a suitable
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crossover point seems sensible:
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</p>
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<pre class="programlisting"><span class="keyword">double</span> <span class="identifier">p</span> <span class="special">=</span> <span class="identifier">some_value</span><span class="special">;</span>
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<span class="keyword">double</span> <span class="identifier">crossover</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>
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<span class="identifier">tanh_sinh</span><span class="special"><</span><span class="keyword">double</span><span class="special">></span> <span class="identifier">integrator</span><span class="special">;</span>
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<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="keyword">return</span> <span class="identifier">pow</span><span class="special">(</span><span class="identifier">tan</span><span class="special">(</span><span class="identifier">x</span><span class="special">),</span> <span class="identifier">p</span><span class="special">);</span> <span class="special">};</span>
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<span class="keyword">auto</span> <span class="identifier">Q</span> <span class="special">=</span> <span class="identifier">integrator</span><span class="special">.</span><span class="identifier">integrate</span><span class="special">(</span><span class="identifier">f</span><span class="special">,</span> <span class="identifier">crossover</span><span class="special">,</span> <span class="identifier">constants</span><span class="special">::</span><span class="identifier">half_pi</span><span class="special"><</span><span class="keyword">double</span><span class="special">>())</span> <span class="special">+</span> <span class="identifier">pow</span><span class="special">(</span><span class="identifier">crossover</span><span class="special">,</span> <span class="number">1</span> <span class="special">-</span> <span class="identifier">p</span><span class="special">)</span> <span class="special">/</span> <span class="special">(</span><span class="number">1</span> <span class="special">-</span> <span class="identifier">p</span><span class="special">);</span>
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</pre>
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<p>
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There is an alternative, more complex method, which is applicable when we
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are dealing with expressions which can be simplified by evaluating by logs.
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Let's suppose that as in this case, all the area under the graph is infinitely
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close to zero, now inagine that we could expand that region out over a much
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larger range of abscissa values: that's exactly what happens if we perform
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argument substitution, replacing <code class="computeroutput"><span class="identifier">x</span></code>
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by <code class="computeroutput"><span class="identifier">exp</span><span class="special">(-</span><span class="identifier">x</span><span class="special">)</span></code> (note
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that we must also multiply by the derivative of <code class="computeroutput"><span class="identifier">exp</span><span class="special">(-</span><span class="identifier">x</span><span class="special">)</span></code>).
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Now the singularity at zero is moved to +∞, and the π/2 bound to -log(π/2).
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Initially our argument substituted function looks like:
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</p>
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<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="keyword">return</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="identifier">pow</span><span class="special">(</span><span class="identifier">tan</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="identifier">p</span><span class="special">);</span> <span class="special">};</span>
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</pre>
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<p>
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Which is hardly any better, as we still run out of exponent range just as
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before. However, if we replace <code class="computeroutput"><span class="identifier">tan</span><span class="special">(</span><span class="identifier">exp</span><span class="special">(-</span><span class="identifier">x</span><span class="special">))</span></code> by
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<code class="computeroutput"><span class="identifier">exp</span><span class="special">(-</span><span class="identifier">x</span><span class="special">)</span></code> for
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suitably small <code class="computeroutput"><span class="identifier">exp</span><span class="special">(-</span><span class="identifier">x</span><span class="special">)</span></code>, and
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therefore <code class="literal">x > -log(ε)</code>, we can greatly simplify the expression
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and evaluate by logs:
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</p>
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<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>
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<span class="special">{</span>
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<span class="keyword">static</span> <span class="keyword">const</span> <span class="keyword">double</span> <span class="identifier">crossover</span> <span class="special">=</span> <span class="special">-</span><span class="identifier">log</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>
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<span class="keyword">return</span> <span class="identifier">x</span> <span class="special">></span> <span class="identifier">crossover</span> <span class="special">?</span> <span class="identifier">exp</span><span class="special">((</span><span class="identifier">p</span> <span class="special">-</span> <span class="number">1</span><span class="special">)</span> <span class="special">*</span> <span class="identifier">x</span><span class="special">)</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="identifier">pow</span><span class="special">(</span><span class="identifier">tan</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="identifier">p</span><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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This form integrates just fine over (-log(π/2), +∞) using either the <code class="computeroutput"><span class="identifier">tanh_sinh</span></code> or <code class="computeroutput"><span class="identifier">exp_sinh</span></code>
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classes.
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</p>
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</div>
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<td align="left"></td>
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<td align="right"><div class="copyright-footer">Copyright © 2006-2019 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, Matthew Pulver, Johan
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Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
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Daryle Walker and Xiaogang Zhang<p>
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Distributed under the Boost Software License, Version 1.0. (See accompanying
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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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