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<a name="Introduction-to-QUADPACK"></a>
<div class="header">
<p>
Next: <a href="maxima_77.html#Functions-and-Variables-for-QUADPACK" accesskey="n" rel="next">Functions and Variables for QUADPACK</a>, Previous: <a href="maxima_75.html#Functions-and-Variables-for-Integration" accesskey="p" rel="previous">Functions and Variables for Integration</a>, Up: <a href="maxima_73.html#Integration" accesskey="u" rel="up">Integration</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Introduction-to-QUADPACK-1"></a>
<h3 class="section">18.3 Introduction to QUADPACK</h3>
<p>QUADPACK is a collection of functions for the numerical
computation of one-dimensional definite integrals.
It originated from a joint project of
R. Piessens <a name="DOCF1" href="#FOOT1"><sup>1</sup></a>,
E. de Doncker <a name="DOCF2" href="#FOOT2"><sup>2</sup></a>,
C. Ueberhuber <a name="DOCF3" href="#FOOT3"><sup>3</sup></a>,
and D. Kahaner <a name="DOCF4" href="#FOOT4"><sup>4</sup></a>.
</p>
<p>The QUADPACK library included in Maxima is an automatic translation (via the
program <code>f2cl</code>) of the Fortran source code of QUADPACK as it appears in
the SLATEC Common Mathematical Library, Version 4.1 <a name="DOCF5" href="#FOOT5"><sup>5</sup></a>.
The SLATEC library is dated July 1993, but the QUADPACK functions
were written some years before.
There is another version of QUADPACK at Netlib <a name="DOCF6" href="#FOOT6"><sup>6</sup></a>;
it is not clear how that version differs from the SLATEC version.
</p>
<p>The QUADPACK functions included in Maxima are all automatic, in the sense that
these functions attempt to compute a result to a specified accuracy, requiring
an unspecified number of function evaluations. Maxima’s Lisp translation of
QUADPACK also includes some non-automatic functions, but they are not exposed
at the Maxima level.
</p>
<p>Further information about QUADPACK can be found in the QUADPACK book
<a name="DOCF7" href="#FOOT7"><sup>7</sup></a>.
</p>
<a name="Overview"></a>
<h4 class="subsection">18.3.1 Overview</h4>
<dl compact="compact">
<dt><code><code><a href="maxima_77.html#quad_005fqag">quad_qag</a></code></code></dt>
<dd><p>Integration of a general function over a finite interval.
<code><a href="maxima_77.html#quad_005fqag">quad_qag</a></code> implements a simple globally adaptive integrator using the
strategy of Aind (Piessens, 1973).
The caller may choose among 6 pairs of Gauss-Kronrod quadrature
formulae for the rule evaluation component.
The high-degree rules are suitable for strongly oscillating integrands.
</p>
</dd>
<dt><code><code><a href="maxima_77.html#quad_005fqags">quad_qags</a></code></code></dt>
<dd><p>Integration of a general function over a finite interval.
<code><a href="maxima_77.html#quad_005fqags">quad_qags</a></code> implements globally adaptive interval subdivision with
extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).
</p>
</dd>
<dt><code><code><a href="maxima_77.html#quad_005fqagi">quad_qagi</a></code></code></dt>
<dd><p>Integration of a general function over an infinite or semi-infinite interval.
The interval is mapped onto a finite interval and
then the same strategy as in <code>quad_qags</code> is applied.
</p>
</dd>
<dt><code><code><a href="maxima_77.html#quad_005fqawo">quad_qawo</a></code></code></dt>
<dd>
<p>Integration of
\(\cos(\omega x) f(x)\) or
\(\sin(\omega x) f(x)\) over a
finite interval, where
\(\omega\) is a constant.
The rule evaluation component is based on the modified Clenshaw-Curtis
technique. <code><a href="maxima_77.html#quad_005fqawo">quad_qawo</a></code> applies adaptive subdivision with extrapolation,
similar to <code><a href="maxima_77.html#quad_005fqags">quad_qags</a></code>.
</p>
</dd>
<dt><code><code><a href="maxima_77.html#quad_005fqawf">quad_qawf</a></code></code></dt>
<dd><p>Calculates a Fourier cosine or Fourier sine transform on a semi-infinite
interval. The same approach as in <code><a href="maxima_77.html#quad_005fqawo">quad_qawo</a></code> is applied on successive
finite intervals, and convergence acceleration by means of the Epsilon algorithm
(Wynn, 1956) is applied to the series of the integral contributions.
</p>
</dd>
<dt><code><code><a href="maxima_77.html#quad_005fqaws">quad_qaws</a></code></code></dt>
<dd><p>Integration of
\(w(x)f(x)\) over a finite interval <em>[a, b]</em>, where
<em>w</em> is a function of the form
\((x-a)^\alpha (b-x)^\beta v(x)\) and
<em>v(x)</em> is 1 or
\(\log(x-a)\) or
\(\log(b-x)\) or
</p>
\(\log(x-a)\log(b-x)\)<p>, and
\(\alpha > -1\) and
\(\beta > -1\).
</p>
<p>A globally adaptive subdivision strategy is applied, with modified
Clenshaw-Curtis integration on the subintervals which contain <em>a</em>
or <em>b</em>.
</p>
</dd>
<dt><code><code><a href="maxima_77.html#quad_005fqawc">quad_qawc</a></code></code></dt>
<dd><p>Computes the Cauchy principal value of <em>f(x)/(x - c)</em> over a finite
interval <em>(a, b)</em> and specified <em>c</em>.
The strategy is globally adaptive, and modified
Clenshaw-Curtis integration is used on the subranges
which contain the point <em>x = c</em>.
</p>
</dd>
<dt><code><code><a href="maxima_77.html#quad_005fqagp">quad_qagp</a></code></code></dt>
<dd><p>Basically the same as <code><a href="maxima_77.html#quad_005fqags">quad_qags</a></code> but points of singularity or
discontinuity of the integrand must be supplied. This makes it easier
for the integrator to produce a good solution.
</p></dd>
</dl>
<a name="Item_003a-Integration_002fnode_002fFunctions-and-Variables-for-QUADPACK"></a><div class="footnote">
<hr>
<h4 class="footnotes-heading">Footnotes</h4>
<h3><a name="FOOT1" href="#DOCF1">(1)</a></h3>
<p>Applied Mathematics and Programming Division, K.U. Leuven</p>
<h3><a name="FOOT2" href="#DOCF2">(2)</a></h3>
<p>Applied Mathematics and Programming Division, K.U. Leuven</p>
<h3><a name="FOOT3" href="#DOCF3">(3)</a></h3>
<p>Institut für Mathematik, T.U. Wien</p>
<h3><a name="FOOT4" href="#DOCF4">(4)</a></h3>
<p>National Bureau of Standards, Washington, D.C., U.S.A</p>
<h3><a name="FOOT5" href="#DOCF5">(5)</a></h3>
<p><a href="https://www.netlib.org/slatec">https://www.netlib.org/slatec</a></p>
<h3><a name="FOOT6" href="#DOCF6">(6)</a></h3>
<p><a href="https://www.netlib.org/quadpack">https://www.netlib.org/quadpack</a></p>
<h3><a name="FOOT7" href="#DOCF7">(7)</a></h3>
<p>R. Piessens, E. de Doncker-Kapenga, C.W. Uberhuber, and D.K. Kahaner.
<i>QUADPACK: A Subroutine Package for Automatic Integration.</i>
Berlin: Springer-Verlag, 1983, ISBN 0387125531.</p>
</div>
<hr>
<div class="header">
<p>
Next: <a href="maxima_77.html#Functions-and-Variables-for-QUADPACK" accesskey="n" rel="next">Functions and Variables for QUADPACK</a>, Previous: <a href="maxima_75.html#Functions-and-Variables-for-Integration" accesskey="p" rel="previous">Functions and Variables for Integration</a>, Up: <a href="maxima_73.html#Integration" accesskey="u" rel="up">Integration</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_264.html#g_t_0423_043a_0430_0437_0430_0442_0435_043b_044c-_0444_0443_043d_043a_0446_0438_0439-_0438-_043f_0435_0440_0435_043c_0435_043d_043d_044b_0445" title="Index" rel="index">Index</a>]</p>
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