<title>tanh_sinh</title>
<link rel="stylesheet" href="../../math.css" type="text/css">
<meta name="generator" content="DocBook XSL Stylesheets V1.79.1">
-<link rel="home" href="../../index.html" title="Math Toolkit 2.10.0">
+<link rel="home" href="../../index.html" title="Math Toolkit 2.11.0">
<link rel="up" href="../double_exponential.html" title="Double-exponential quadrature">
<link rel="prev" href="de_overview.html" title="Overview">
<link rel="next" href="de_tanh_sinh_2_arg.html" title="Handling functions with large features near an endpoint with tanh-sinh quadrature">
<span class="special">};</span>
</pre>
<p>
- The tanh-sinh quadrature routine provided by boost is a rapidly convergent
- numerical integration scheme for holomorphic integrands. By this we mean
- that the integrand is the restriction to the real line of a complex-differentiable
- function which is bounded on the interior of the unit disk <span class="emphasis"><em>|z|
- < 1</em></span>, so that it lies within the so-called <a href="https://en.wikipedia.org/wiki/Hardy_space" target="_top">Hardy
- space</a>. If your integrand obeys these conditions, it can be shown
- that tanh-sinh integration is optimal, in the sense that it requires the
- fewest function evaluations for a given accuracy of any quadrature algorithm
- for a random element from the Hardy space. A basic example of how to use
- the tanh-sinh quadrature is shown below:
+ The <code class="computeroutput"><span class="identifier">tanh</span><span class="special">-</span><span class="identifier">sinh</span></code> quadrature routine provided by boost
+ is a rapidly convergent numerical integration scheme for holomorphic integrands.
+ By this we mean that the integrand is the restriction to the real line of
+ a complex-differentiable function which is bounded on the interior of the
+ unit disk <span class="emphasis"><em>|z| < 1</em></span>, so that it lies within the so-called
+ <a href="https://en.wikipedia.org/wiki/Hardy_space" target="_top">Hardy space</a>.
+ If your integrand obeys these conditions, it can be shown that <code class="computeroutput"><span class="identifier">tanh</span><span class="special">-</span><span class="identifier">sinh</span></code>
+ integration is optimal, in the sense that it requires the fewest function
+ evaluations for a given accuracy of any quadrature algorithm for a random
+ element from the Hardy space.
+ </p>
+<p>
+ A basic example of how to use the <code class="computeroutput"><span class="identifier">tanh</span><span class="special">-</span><span class="identifier">sinh</span></code> quadrature
+ is shown below:
</p>
<pre class="programlisting"><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>
<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="number">5</span><span class="special">*</span><span class="identifier">x</span> <span class="special">+</span> <span class="number">7</span><span class="special">;</span> <span class="special">};</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">0.0</span><span class="special">,</span> <span class="number">1.1</span><span class="special">);</span>
</pre>
<p>
- The basic idea of tanh-sinh quadrature is that a variable transformation
+ The basic idea of <code class="computeroutput"><span class="identifier">tanh</span><span class="special">-</span><span class="identifier">sinh</span></code> quadrature is that a variable transformation
can cause the endpoint derivatives to decay rapidly. When the derivatives
at the endpoints decay much faster than the Bernoulli numbers grow, the Euler-Maclaurin
summation formula tells us that simple trapezoidal quadrature converges faster
</tbody>
</table></div>
<p>
- Although the tanh-sinh quadrature can compute integral over infinite domains
- by variable transformations, these transformations can create a very poorly
- behaved integrand. For this reason, double-exponential variable transformations
- have been provided that allow stable computation over infinite domains; these
- being the exp-sinh and sinh-sinh quadrature.
+ Although the <code class="computeroutput"><span class="identifier">tanh</span><span class="special">-</span><span class="identifier">sinh</span></code> quadrature can compute integral over
+ infinite domains by variable transformations, these transformations can create
+ a very poorly behaved integrand. For this reason, double-exponential variable
+ transformations have been provided that allow stable computation over infinite
+ domains; these being the exp-sinh and sinh-sinh quadrature.
</p>
<h5>
<a name="math_toolkit.double_exponential.de_tanh_sinh.h0"></a>
integrals</a>
</h5>
<p>
- The tanh_sinh integrator supports integration of functions which return complex
- results, for example the sine-integral <code class="computeroutput"><span class="identifier">Si</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span></code>
- has the integral representation:
- </p>
-<p>
- <span class="inlinemediaobject"><img src="../../../equations/sine_integral.svg"></span>
+ The <code class="computeroutput"><span class="identifier">tanh_sinh</span></code> integrator
+ supports integration of functions which return complex results, for example
+ the sine-integral <code class="computeroutput"><span class="identifier">Si</span><span class="special">(</span><span class="identifier">z</span><span class="special">)</span></code> has
+ the integral representation:
</p>
+<div class="blockquote"><blockquote class="blockquote"><p>
+ <span class="inlinemediaobject"><img src="../../../equations/sine_integral.svg"></span>
+
+ </p></blockquote></div>
<p>
Which we can code up directly as:
</p>
projects / platform / upstream / boost.git / blobdiff