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- <div class="titlepage"><div><div><h2 class="title" style="clear: both">
- <a name="math_toolkit.gauss_kronrod"></a><a class="link" href="gauss_kronrod.html" title="Gauss-Kronrod Quadrature">Gauss-Kronrod Quadrature</a>
- </h2></div></div></div>
- <h4>
- <a name="math_toolkit.gauss_kronrod.h0"></a>
- <span class="phrase"><a name="math_toolkit.gauss_kronrod.overview"></a></span><a class="link" href="gauss_kronrod.html#math_toolkit.gauss_kronrod.overview">Overview</a>
- </h4>
- <p>
- Gauss-Kronrod quadrature is an extension of Gaussian quadrature which provides
- an a posteriori error estimate for the integral.
- </p>
- <p>
- The idea behind Gaussian quadrature is to choose <span class="emphasis"><em>n</em></span> nodes
- and weights in such a way that polynomials of order <span class="emphasis"><em>2n-1</em></span>
- are integrated exactly. However, integration of polynomials is trivial, so
- it is rarely done via numerical methods. Instead, transcendental and numerically
- defined functions are integrated via Gaussian quadrature, and the defining
- problem becomes how to estimate the remainder. Gaussian quadrature alone (without
- some form of interval splitting) cannot answer this question.
- </p>
- <p>
- It is possible to compute a Gaussian quadrature of order <span class="emphasis"><em>n</em></span>
- and another of order (say) <span class="emphasis"><em>2n+1</em></span>, and use the difference
- as an error estimate. However, this is not optimal, as the zeros of the Legendre
- polynomials (nodes of the Gaussian quadrature) are never the same for different
- orders, so <span class="emphasis"><em>3n+1</em></span> function evaluations must be performed.
- Kronrod considered the problem of how to interleave nodes into a Gaussian quadrature
- in such a way that all previous function evaluations can be reused, while increasing
- the order of polynomials that can be integrated exactly. This allows an a posteriori
- error estimate to be provided while still preserving exponential convergence.
- Kronrod discovered that by adding <span class="emphasis"><em>n+1</em></span> nodes (computed
- from the zeros of the Legendre-Stieltjes polynomials) to a Gaussian quadrature
- of order <span class="emphasis"><em>n</em></span>, he could integrate polynomials of order <span class="emphasis"><em>3n+1</em></span>.
- </p>
- <p>
- The integration routines provided here will perform either adaptive or non-adaptive
- quadrature, they should be chosen for the integration of smooth functions with
- no end point singularities. For difficult functions, or those with end point
- singularities, please refer to the <a class="link" href="double_exponential.html" title="Double-exponential quadrature">double-exponential
- integration schemes</a>.
- </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">math</span><span class="special">/</span><span class="identifier">quadrature</span><span class="special">/</span><span class="identifier">gauss_kronrod</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">></span>
- <span class="keyword">template</span> <span class="special"><</span><span class="keyword">class</span> <span class="identifier">Real</span><span class="special">,</span> <span class="keyword">unsigned</span> <span class="identifier">N</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../policy.html" title="Chapter 20. Policies: Controlling Precision, Error Handling etc">Policy</a> <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">policies</span><span class="special">::</span><span class="identifier">policy</span><span class="special"><></span> <span class="special">></span>
- <span class="keyword">class</span> <span class="identifier">gauss_kronrod</span>
- <span class="special">{</span>
- <span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">RandomAccessContainer</span><span class="special">&</span> <span class="identifier">abscissa</span><span class="special">();</span>
- <span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">RandomAccessContainer</span><span class="special">&</span> <span class="identifier">weights</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">static</span> <span class="keyword">auto</span> <span class="identifier">integrate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
- <span class="identifier">Real</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">b</span><span class="special">,</span>
- <span class="keyword">unsigned</span> <span class="identifier">max_depth</span> <span class="special">=</span> <span class="number">15</span><span class="special">,</span>
- <span class="identifier">Real</span> <span class="identifier">tol</span> <span class="special">=</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">root_epsilon</span><span class="special"><</span><span class="identifier">Real</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="identifier">Real</span><span class="special">*</span> <span class="identifier">pL1</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">)-></span><span class="keyword">decltype</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">F</span><span class="special">>()(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>()));</span>
- <span class="special">};</span>
- </pre>
- <h4>
- <a name="math_toolkit.gauss_kronrod.h1"></a>
- <span class="phrase"><a name="math_toolkit.gauss_kronrod.description"></a></span><a class="link" href="gauss_kronrod.html#math_toolkit.gauss_kronrod.description">Description</a>
- </h4>
- <pre class="programlisting"><span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">RandomAccessContainer</span><span class="special">&</span> <span class="identifier">abscissa</span><span class="special">();</span>
- <span class="keyword">static</span> <span class="keyword">const</span> <span class="identifier">RandomAccessContainer</span><span class="special">&</span> <span class="identifier">weights</span><span class="special">();</span>
- </pre>
- <p>
- These functions provide direct access to the abscissa and weights used to perform
- the quadrature: the return type depends on the <span class="emphasis"><em>Points</em></span>
- template parameter, but is always a RandomAccessContainer type. Note that only
- positive (or zero) abscissa and weights are stored, and that they contain both
- the Gauss and Kronrod points.
- </p>
- <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">static</span> <span class="keyword">auto</span> <span class="identifier">integrate</span><span class="special">(</span><span class="identifier">F</span> <span class="identifier">f</span><span class="special">,</span>
- <span class="identifier">Real</span> <span class="identifier">a</span><span class="special">,</span> <span class="identifier">Real</span> <span class="identifier">b</span><span class="special">,</span>
- <span class="keyword">unsigned</span> <span class="identifier">max_depth</span> <span class="special">=</span> <span class="number">15</span><span class="special">,</span>
- <span class="identifier">Real</span> <span class="identifier">tol</span> <span class="special">=</span> <span class="identifier">tools</span><span class="special">::</span><span class="identifier">root_epsilon</span><span class="special"><</span><span class="identifier">Real</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="identifier">Real</span><span class="special">*</span> <span class="identifier">pL1</span> <span class="special">=</span> <span class="keyword">nullptr</span><span class="special">)-></span><span class="keyword">decltype</span><span class="special">(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">F</span><span class="special">>()(</span><span class="identifier">std</span><span class="special">::</span><span class="identifier">declval</span><span class="special"><</span><span class="identifier">Real</span><span class="special">>()));</span>
- </pre>
- <p>
- Performs adaptive Gauss-Kronrod quadrature on function <span class="emphasis"><em>f</em></span>
- over the range (a,b).
- </p>
- <p>
- <span class="emphasis"><em>max_depth</em></span> sets the maximum number of interval splittings
- permitted before stopping. Set this to zero for non-adaptive quadrature. Note
- that the algorithm descends the tree depth first, so only "difficult"
- areas of the integral result in interval splitting.
- </p>
- <p>
- <span class="emphasis"><em>tol</em></span> Sets the maximum relative error in the result, this
- should not be set too close to machine epsilon or the function will simply
- thrash and probably not return accurate results. On the other hand the default
- may be overly-pressimistic.
- </p>
- <p>
- <span class="emphasis"><em>error</em></span> When non-null, <code class="computeroutput"><span class="special">*</span><span class="identifier">error</span></code> is set to the difference between the
- (N-1)/2 point Gauss approximation and the N-point Gauss-Kronrod approximation.
- </p>
- <p>
- <span class="emphasis"><em>pL1</em></span> When non-null, <code class="computeroutput"><span class="special">*</span><span class="identifier">pL1</span></code> is set to the L1 norm of the result,
- if there is a significant difference between this and the returned value, then
- the result is likely to be ill-conditioned.
- </p>
- <h4>
- <a name="math_toolkit.gauss_kronrod.h2"></a>
- <span class="phrase"><a name="math_toolkit.gauss_kronrod.choosing_the_number_of_points"></a></span><a class="link" href="gauss_kronrod.html#math_toolkit.gauss_kronrod.choosing_the_number_of_points">Choosing
- the number of points</a>
- </h4>
- <p>
- The number of points specified in the <span class="emphasis"><em>Points</em></span> template
- parameter must be an odd number: giving a (N-1)/2 Gauss quadrature as the comparison
- for error estimation.
- </p>
- <p>
- Internally class <code class="computeroutput"><span class="identifier">gauss_kronrod</span></code>
- has pre-computed tables of abscissa and weights for 15, 31, 41, 51 and 61 Gauss-Kronrod
- points at up to 100-decimal digit precision. That means that using for example,
- <code class="computeroutput"><span class="identifier">gauss_kronrod</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">31</span><span class="special">>::</span><span class="identifier">integrate</span></code>
- incurs absolutely zero set-up overhead from computing the abscissa/weight pairs.
- When using multiprecision types with less than 100 digits of precision, then
- there is a small initial one time cost, while the abscissa/weight pairs are
- constructed from strings.
- </p>
- <p>
- However, for types with higher precision, or numbers of points other than those
- given above, the abscissa/weight pairs are computed when first needed and then
- cached for future use, which does incur a noticeable overhead. If this is likely
- to be an issue, then:
- </p>
- <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
- <li class="listitem">
- Defining BOOST_MATH_GAUSS_NO_COMPUTE_ON_DEMAND will result in a compile-time
- error, whenever a combination of number type and number of points is used
- which does not have pre-computed values.
- </li>
- <li class="listitem">
- There is a program <a href="../../../tools/gauss_kronrod_constants.cpp" target="_top">gauss_kronrod_constants.cpp</a>
- which was used to provide the pre-computed values already in gauss_kronrod.hpp.
- The program can be trivially modified to generate code and constants for
- other precisions and numbers of points.
- </li>
- </ul></div>
- <h4>
- <a name="math_toolkit.gauss_kronrod.h3"></a>
- <span class="phrase"><a name="math_toolkit.gauss_kronrod.complex_quadrature"></a></span><a class="link" href="gauss_kronrod.html#math_toolkit.gauss_kronrod.complex_quadrature">Complex
- Quadrature</a>
- </h4>
- <p>
- The Gauss-Kronrod quadrature support integrands defined on the real line and
- returning complex values. In this case, the template argument is the real type,
- and the complex type is deduced via the return type of the function.
- </p>
- <h4>
- <a name="math_toolkit.gauss_kronrod.h4"></a>
- <span class="phrase"><a name="math_toolkit.gauss_kronrod.examples"></a></span><a class="link" href="gauss_kronrod.html#math_toolkit.gauss_kronrod.examples">Examples</a>
- </h4>
- <p>
- We'll begin by integrating exp(-t<sup>2</sup>/2) over (0,+∞) using a 7 term Gauss rule
- and 15 term Kronrod rule, and begin by defining the function to integrate as
- a C++ lambda expression:
- </p>
- <pre class="programlisting"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">quadrature</span><span class="special">;</span>
- <span class="keyword">auto</span> <span class="identifier">f1</span> <span class="special">=</span> <span class="special">[](</span><span class="keyword">double</span> <span class="identifier">t</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">t</span><span class="special">*</span><span class="identifier">t</span> <span class="special">/</span> <span class="number">2</span><span class="special">);</span> <span class="special">};</span>
- </pre>
- <p>
- We'll start off with a one shot (ie non-adaptive) integration, and keep track
- of the estimated error:
- </p>
- <pre class="programlisting"><span class="keyword">double</span> <span class="identifier">error</span><span class="special">;</span>
- <span class="keyword">double</span> <span class="identifier">Q</span> <span class="special">=</span> <span class="identifier">gauss_kronrod</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">15</span><span class="special">>::</span><span class="identifier">integrate</span><span class="special">(</span><span class="identifier">f1</span><span class="special">,</span> <span class="number">0</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">infinity</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="special">&</span><span class="identifier">error</span><span class="special">);</span>
- </pre>
- <p>
- This yields Q = 1.25348207361, which has an absolute error of 1e-4 compared
- to the estimated error of 5e-3: this is fairly typical, with the difference
- between Gauss and Gauss-Kronrod schemes being much higher than the actual error.
- Before moving on to adaptive quadrature, lets try again with more points, in
- fact with the largest Gauss-Kronrod scheme we have cached (30/61):
- </p>
- <pre class="programlisting"><span class="identifier">Q</span> <span class="special">=</span> <span class="identifier">gauss_kronrod</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">61</span><span class="special">>::</span><span class="identifier">integrate</span><span class="special">(</span><span class="identifier">f1</span><span class="special">,</span> <span class="number">0</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">infinity</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="special">&</span><span class="identifier">error</span><span class="special">);</span>
- </pre>
- <p>
- This yields an absolute error of 3e-15 against an estimate of 1e-8, which is
- about as good as we're going to get at double precision
- </p>
- <p>
- However, instead of continuing with ever more points, lets switch to adaptive
- integration, and set the desired relative error to 1e-14 against a maximum
- depth of 5:
- </p>
- <pre class="programlisting"><span class="identifier">Q</span> <span class="special">=</span> <span class="identifier">gauss_kronrod</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">15</span><span class="special">>::</span><span class="identifier">integrate</span><span class="special">(</span><span class="identifier">f1</span><span class="special">,</span> <span class="number">0</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">infinity</span><span class="special">(),</span> <span class="number">5</span><span class="special">,</span> <span class="number">1e-14</span><span class="special">,</span> <span class="special">&</span><span class="identifier">error</span><span class="special">);</span>
- </pre>
- <p>
- This yields an actual error of zero, against an estimate of 4e-15. In fact
- in this case the requested tolerance was almost certainly set too low: as we've
- seen above, for smooth functions, the precision achieved is often double that
- of the estimate, so if we integrate with a tolerance of 1e-9:
- </p>
- <pre class="programlisting"><span class="identifier">Q</span> <span class="special">=</span> <span class="identifier">gauss_kronrod</span><span class="special"><</span><span class="keyword">double</span><span class="special">,</span> <span class="number">15</span><span class="special">>::</span><span class="identifier">integrate</span><span class="special">(</span><span class="identifier">f1</span><span class="special">,</span> <span class="number">0</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">infinity</span><span class="special">(),</span> <span class="number">5</span><span class="special">,</span> <span class="number">1e-9</span><span class="special">,</span> <span class="special">&</span><span class="identifier">error</span><span class="special">);</span>
- </pre>
- <p>
- We still achieve 1e-15 precision, with an error estimate of 1e-10.
- </p>
- <h4>
- <a name="math_toolkit.gauss_kronrod.h5"></a>
- <span class="phrase"><a name="math_toolkit.gauss_kronrod.caveats"></a></span><a class="link" href="gauss_kronrod.html#math_toolkit.gauss_kronrod.caveats">Caveats</a>
- </h4>
- <p>
- For essentially all analytic integrands bounded on the domain, the error estimates
- provided by the routine are woefully pessimistic. In fact, in this case the
- error is very nearly equal to the error of the Gaussian quadrature formula
- of order <code class="computeroutput"><span class="special">(</span><span class="identifier">N</span><span class="special">-</span><span class="number">1</span><span class="special">)/</span><span class="number">2</span></code>, whereas the Kronrod extension converges exponentially
- beyond the Gaussian estimate. Very sophisticated method exist to estimate the
- error, but all require the integrand to lie in a particular function space.
- A more sophisticated a posteriori error estimate for an element of a particular
- function space is left to the user.
- </p>
- <p>
- These routines are deliberately kept relatively simple: when they work, they
- work very well and very rapidly. However, no effort has been made to make these
- routines work well with end-point singularities or other "difficult"
- integrals. In such cases please use one of the <a class="link" href="double_exponential.html" title="Double-exponential quadrature">double-exponential
- integration schemes</a> which are generally much more robust.
- </p>
- <h4>
- <a name="math_toolkit.gauss_kronrod.h6"></a>
- <span class="phrase"><a name="math_toolkit.gauss_kronrod.references"></a></span><a class="link" href="gauss_kronrod.html#math_toolkit.gauss_kronrod.references">References</a>
- </h4>
- <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
- <li class="listitem">
- Kronrod, Aleksandr Semenovish (1965), <span class="emphasis"><em>Nodes and weights of quadrature
- formulas. Sixteen-place tables</em></span>, New York: Consultants Bureau
- </li>
- <li class="listitem">
- Dirk P. Laurie, <span class="emphasis"><em>Calculation of Gauss-Kronrod Quadrature Rules</em></span>,
- Mathematics of Computation, Volume 66, Number 219, 1997
- </li>
- <li class="listitem">
- Gonnet, Pedro, <span class="emphasis"><em>A Review of Error Estimation in Adaptive Quadrature</em></span>,
- https://arxiv.org/pdf/1003.4629.pdf
- </li>
- </ul></div>
- </div>
- <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
- <td align="left"></td>
- <td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar
- Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
- Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, 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
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