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<a name="Functions-and-Variables-for-QUADPACK"></a>
<div class="header">
<p>
Previous: <a href="maxima_76.html#Introduction-to-QUADPACK" accesskey="p" rel="previous">Introduction to QUADPACK</a>, Up: <a href="maxima_73.html#Integration" accesskey="u" rel="up">Integration</a> &nbsp; [<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="Functions-and-Variables-for-QUADPACK-1"></a>
<h3 class="section">18.4 Functions and Variables for QUADPACK</h3>


<a name="quad_005fqag"></a><a name="Item_003a-Integration_002fdeffn_002fquad_005fqag"></a><dl>
<dt><a name="index-quad_005fqag"></a>Function: <strong>quad_qag</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qag</tt> (<var>f(x)</var>, <var>x</var>, <var>a</var>, <var>b</var>, <var>key</var>, [<var>epsrel</var>, <var>epsabs</var>, <var>limit</var>]) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qag</tt> (<var>f</var>, <var>x</var>, <var>a</var>, <var>b</var>, <var>key</var>, [<var>epsrel</var>, <var>epsabs</var>, <var>limit</var>])</em></dt>
<dd>
<p>Integration of a general function over a finite interval.  <code>quad_qag</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>
<p><code>quad_qag</code> computes the integral
</p>
$$
\int_a^b f(x)\, dx
$$

<p>The function to be integrated is <em>f(x)</em>, with dependent
variable <em>x</em>, and the function is to be integrated between the
limits <em>a</em> and <em>b</em>.  <var>key</var> is the integrator to be used
and should be an integer between 1 and 6, inclusive.  The value of
<var>key</var> selects the order of the Gauss-Kronrod integration rule.
High-order rules are suitable for strongly oscillating integrands.
</p>
<p>The integrand may be specified as the name of a Maxima or Lisp function or
operator, a Maxima lambda expression, or a general Maxima expression.
</p>
<p>The numerical integration is done adaptively by subdividing the
integration region into sub-intervals until the desired accuracy is
achieved.
</p>
<p>The keyword arguments are optional and may be specified in any order.
They all take the form <code>key=val</code>.  The keyword arguments are:
</p>
<dl compact="compact">
<dt><code>epsrel</code></dt>
<dd><p>Desired relative error of approximation.  Default is 1d-8.
</p></dd>
<dt><code>epsabs</code></dt>
<dd><p>Desired absolute error of approximation.  Default is 0.
</p></dd>
<dt><code>limit</code></dt>
<dd><p>Size of internal work array.  <var>limit</var> is the
maximum number of subintervals to use.  Default is 200.
</p></dd>
</dl>

<p><code>quad_qag</code> returns a list of four elements:
</p>
<ul>
<li> an approximation to the integral,
</li><li> the estimated absolute error of the approximation,
</li><li> the number integrand evaluations,
</li><li> an error code.
</li></ul>

<p>The error code (fourth element of the return value) can have the values:
</p>
<dl compact="compact">
<dt><code>0</code></dt>
<dd><p>if no problems were encountered;
</p></dd>
<dt><code>1</code></dt>
<dd><p>if too many sub-intervals were done;
</p></dd>
<dt><code>2</code></dt>
<dd><p>if excessive roundoff error is detected;
</p></dd>
<dt><code>3</code></dt>
<dd><p>if extremely bad integrand behavior occurs;
</p></dd>
<dt><code>6</code></dt>
<dd><p>if the input is invalid.
</p>
</dd>
</dl>


<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) quad_qag (x^(1/2)*log(1/x), x, 0, 1, 3, 'epsrel=5d-8);
(%o1)    [.4444444444492108, 3.1700968502883E-9, 961, 0]
</pre><pre class="example">(%i2) integrate (x^(1/2)*log(1/x), x, 0, 1);
                                4
(%o2)                           -
                                9
</pre></div>





</dd></dl>


<a name="quad_005fqags"></a><a name="Item_003a-Integration_002fdeffn_002fquad_005fqags"></a><dl>
<dt><a name="index-quad_005fqags"></a>Function: <strong>quad_qags</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qags</tt> (<var>f(x)</var>, <var>x</var>, <var>a</var>, <var>b</var>, [<var>epsrel</var>, <var>epsabs</var>, <var>limit</var>]) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qags</tt> (<var>f</var>, <var>x</var>, <var>a</var>, <var>b</var>, [<var>epsrel</var>, <var>epsabs</var>, <var>limit</var>])</em></dt>
<dd>
<p>Integration of a general function over a finite interval.
<code>quad_qags</code> implements globally adaptive interval subdivision with
extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).
</p>
<p><code>quad_qags</code> computes the integral
</p>
$$
\int_a^b f(x)\, dx
$$

<p>The function to be integrated is <em>f(x)</em>, with
dependent variable <em>x</em>, and the function is to be integrated
between the limits <em>a</em> and <em>b</em>.
</p>
<p>The integrand may be specified as the name of a Maxima or Lisp function or
operator, a Maxima lambda expression, or a general Maxima expression.
</p>
<p>The keyword arguments are optional and may be specified in any order.
They all take the form <code>key=val</code>.  The keyword arguments are:
</p>
<dl compact="compact">
<dt><code>epsrel</code></dt>
<dd><p>Desired relative error of approximation.  Default is 1d-8.
</p></dd>
<dt><code>epsabs</code></dt>
<dd><p>Desired absolute error of approximation.  Default is 0.
</p></dd>
<dt><code>limit</code></dt>
<dd><p>Size of internal work array.  <var>limit</var> is the
maximum number of subintervals to use.  Default is 200.
</p></dd>
</dl>

<p><code>quad_qags</code> returns a list of four elements:
</p>
<ul>
<li> an approximation to the integral,
</li><li> the estimated absolute error of the approximation,
</li><li> the number integrand evaluations,
</li><li> an error code.
</li></ul>

<p>The error code (fourth element of the return value) can have the values:
</p>
<dl compact="compact">
<dt><code>0</code></dt>
<dd><p>no problems were encountered;
</p></dd>
<dt><code>1</code></dt>
<dd><p>too many sub-intervals were done;
</p></dd>
<dt><code>2</code></dt>
<dd><p>excessive roundoff error is detected;
</p></dd>
<dt><code>3</code></dt>
<dd><p>extremely bad integrand behavior occurs;
</p></dd>
<dt><code>4</code></dt>
<dd><p>failed to converge
</p></dd>
<dt><code>5</code></dt>
<dd><p>integral is probably divergent or slowly convergent
</p></dd>
<dt><code>6</code></dt>
<dd><p>if the input is invalid.
</p></dd>
</dl>


<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) quad_qags (x^(1/2)*log(1/x), x, 0, 1, 'epsrel=1d-10);
(%o1)   [.4444444444444448, 1.11022302462516E-15, 315, 0]
</pre></div>

<p>Note that <code>quad_qags</code> is more accurate and efficient than <code>quad_qag</code> for this integrand.
</p>




</dd></dl>


<a name="quad_005fqagi"></a><a name="Item_003a-Integration_002fdeffn_002fquad_005fqagi"></a><dl>
<dt><a name="index-quad_005fqagi"></a>Function: <strong>quad_qagi</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qagi</tt> (<var>f(x)</var>, <var>x</var>, <var>a</var>, <var>b</var>, [<var>epsrel</var>, <var>epsabs</var>, <var>limit</var>]) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qagi</tt> (<var>f</var>, <var>x</var>, <var>a</var>, <var>b</var>, [<var>epsrel</var>, <var>epsabs</var>, <var>limit</var>])</em></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>
<p><code>quad_qagi</code> evaluates one of the following integrals
</p>
$$
\int_a^\infty f(x) \, dx
$$

$$
\int_\infty^a f(x) \, dx
$$

$$
\int_{-\infty}^\infty f(x) \, dx
$$

<p>using the Quadpack QAGI routine.  The function to be integrated is
<em>f(x)</em>, with dependent variable <em>x</em>, and the function is to
be integrated over an infinite range.
</p>
<p>The integrand may be specified as the name of a Maxima or Lisp function or
operator, a Maxima lambda expression, or a general Maxima expression.
</p>
<p>One of the limits of integration must be infinity.  If not, then
<code>quad_qagi</code> will just return the noun form.
</p>
<p>The keyword arguments are optional and may be specified in any order.
They all take the form <code>key=val</code>.  The keyword arguments are:
</p>
<dl compact="compact">
<dt><code>epsrel</code></dt>
<dd><p>Desired relative error of approximation.  Default is 1d-8.
</p></dd>
<dt><code>epsabs</code></dt>
<dd><p>Desired absolute error of approximation.  Default is 0.
</p></dd>
<dt><code>limit</code></dt>
<dd><p>Size of internal work array.  <var>limit</var> is the
maximum number of subintervals to use.  Default is 200.
</p></dd>
</dl>

<p><code>quad_qagi</code> returns a list of four elements:
</p>
<ul>
<li> an approximation to the integral,
</li><li> the estimated absolute error of the approximation,
</li><li> the number integrand evaluations,
</li><li> an error code.
</li></ul>

<p>The error code (fourth element of the return value) can have the values:
</p>
<dl compact="compact">
<dt><code>0</code></dt>
<dd><p>no problems were encountered;
</p></dd>
<dt><code>1</code></dt>
<dd><p>too many sub-intervals were done;
</p></dd>
<dt><code>2</code></dt>
<dd><p>excessive roundoff error is detected;
</p></dd>
<dt><code>3</code></dt>
<dd><p>extremely bad integrand behavior occurs;
</p></dd>
<dt><code>4</code></dt>
<dd><p>failed to converge
</p></dd>
<dt><code>5</code></dt>
<dd><p>integral is probably divergent or slowly convergent
</p></dd>
<dt><code>6</code></dt>
<dd><p>if the input is invalid.
</p>
</dd>
</dl>


<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) quad_qagi (x^2*exp(-4*x), x, 0, inf, 'epsrel=1d-8);
(%o1)        [0.03125, 2.95916102995002E-11, 105, 0]
</pre><pre class="example">(%i2) integrate (x^2*exp(-4*x), x, 0, inf);
                               1
(%o2)                          --
                               32
</pre></div>





</dd></dl>


<a name="quad_005fqawc"></a><a name="Item_003a-Integration_002fdeffn_002fquad_005fqawc"></a><dl>
<dt><a name="index-quad_005fqawc"></a>Function: <strong>quad_qawc</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qawc</tt> (<var>f(x)</var>, <var>x</var>, <var>c</var>, <var>a</var>, <var>b</var>, [<var>epsrel</var>, <var>epsabs</var>, <var>limit</var>]) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qawc</tt> (<var>f</var>, <var>x</var>, <var>c</var>, <var>a</var>, <var>b</var>, [<var>epsrel</var>, <var>epsabs</var>, <var>limit</var>])</em></dt>
<dd>
<p>Computes the Cauchy principal value of <em>f(x)/(x - c)</em> over a finite
interval.  The strategy is globally adaptive, and modified
Clenshaw-Curtis integration is used on the subranges
which contain the point <em>x = c</em>.
</p>
<p><code>quad_qawc</code> computes the Cauchy principal value of
</p>
$$
\int_{a}^{b}{{{f\left(x\right)}\over{x-c}}\>dx}
$$

<p>using the Quadpack QAWC routine.  The function to be integrated is
<em>f(x)/(x-c)</em>, with dependent variable <em>x</em>, and the
function is to be integrated over the interval <em>a</em> to <em>b</em>.
</p>
<p>The integrand may be specified as the name of a Maxima or Lisp function or
operator, a Maxima lambda expression, or a general Maxima expression.
</p>
<p>The keyword arguments are optional and may be specified in any order.
They all take the form <code>key=val</code>.  The keyword arguments are:
</p>
<dl compact="compact">
<dt><code>epsrel</code></dt>
<dd><p>Desired relative error of approximation.  Default is 1d-8.
</p></dd>
<dt><code>epsabs</code></dt>
<dd><p>Desired absolute error of approximation.  Default is 0.
</p></dd>
<dt><code>limit</code></dt>
<dd><p>Size of internal work array.  <var>limit</var> is the
maximum number of subintervals to use.  Default is 200.
</p></dd>
</dl>

<p><code>quad_qawc</code> returns a list of four elements:
</p>
<ul>
<li> an approximation to the integral,
</li><li> the estimated absolute error of the approximation,
</li><li> the number integrand evaluations,
</li><li> an error code.
</li></ul>

<p>The error code (fourth element of the return value) can have the values:
</p>
<dl compact="compact">
<dt><code>0</code></dt>
<dd><p>no problems were encountered;
</p></dd>
<dt><code>1</code></dt>
<dd><p>too many sub-intervals were done;
</p></dd>
<dt><code>2</code></dt>
<dd><p>excessive roundoff error is detected;
</p></dd>
<dt><code>3</code></dt>
<dd><p>extremely bad integrand behavior occurs;
</p></dd>
<dt><code>6</code></dt>
<dd><p>if the input is invalid.
</p>
</dd>
</dl>

<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) quad_qawc (2^(-5)*((x-1)^2+4^(-5))^(-1), x, 2, 0, 5,
                 'epsrel=1d-7);
(%o1)    [- 3.130120337415925, 1.306830140249558E-8, 495, 0]
</pre><pre class="example">(%i2) integrate (2^(-alpha)*(((x-1)^2 + 4^(-alpha))*(x-2))^(-1),
      x, 0, 5);
Principal Value
                       alpha
        alpha       9 4                 9
       4      log(------------- + -------------)
                      alpha           alpha
                  64 4      + 4   64 4      + 4
(%o2) (-----------------------------------------
                        alpha
                     2 4      + 2

       3 alpha                       3 alpha
       -------                       -------
          2            alpha/2          2          alpha/2
    2 4        atan(4 4       )   2 4        atan(4       )   alpha
  - --------------------------- - -------------------------)/2
              alpha                        alpha
           2 4      + 2                 2 4      + 2
</pre><pre class="example">(%i3) ev (%, alpha=5, numer);
(%o3)                    - 3.130120337415917
</pre></div>





</dd></dl>


<a name="quad_005fqawf"></a><a name="Item_003a-Integration_002fdeffn_002fquad_005fqawf"></a><dl>
<dt><a name="index-quad_005fqawf"></a>Function: <strong>quad_qawf</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qawf</tt> (<var>f(x)</var>, <var>x</var>, <var>a</var>, <var>omega</var>, <var>trig</var>, [<var>epsabs</var>, <var>limit</var>, <var>maxp1</var>, <var>limlst</var>]) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qawf</tt> (<var>f</var>, <var>x</var>, <var>a</var>, <var>omega</var>, <var>trig</var>, [<var>epsabs</var>, <var>limit</var>, <var>maxp1</var>, <var>limlst</var>])</em></dt>
<dd>
<p>Calculates a Fourier cosine or Fourier sine transform on a semi-infinite
interval using the Quadpack QAWF function.  The same approach as in
<code>quad_qawo</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>
<p><code>quad_qawf</code> computes the integral
</p>
$$
\int_a^\infty f(x) \, w(x) \, dx
$$

<p>The weight function <em>w</em> is selected by <var>trig</var>:
</p>
<dl compact="compact">
<dt><code>cos</code></dt>
<dd>
\(w(x) = \cos\omega x\)
</dd>
<dt><code>sin</code></dt>
<dd>
\(w(x) = \sin\omega x\)
</dd>
</dl>

<p>The integrand may be specified as the name of a Maxima or Lisp function or
operator, a Maxima lambda expression, or a general Maxima expression.
</p>
<p>The keyword arguments are optional and may be specified in any order.
They all take the form <code>key=val</code>.  The keyword arguments are:
</p>
<dl compact="compact">
<dt><code>epsabs</code></dt>
<dd><p>Desired absolute error of approximation.  Default is 1d-10.
</p></dd>
<dt><code>limit</code></dt>
<dd><p>Size of internal work array.  (<var>limit</var> - <var>limlst</var>)/2 is the
maximum number of subintervals to use.  Default is 200.
</p></dd>
<dt><code>maxp1</code></dt>
<dd><p>Maximum number of Chebyshev moments.  Must be greater than 0.  Default
is 100.
</p></dd>
<dt><code>limlst</code></dt>
<dd><p>Upper bound on the number of cycles.  Must be greater than or equal to
3.  Default is 10.
</p></dd>
</dl>

<p><code>quad_qawf</code> returns a list of four elements:
</p>
<ul>
<li> an approximation to the integral,
</li><li> the estimated absolute error of the approximation,
</li><li> the number integrand evaluations,
</li><li> an error code.
</li></ul>

<p>The error code (fourth element of the return value) can have the values:
</p>
<dl compact="compact">
<dt><code>0</code></dt>
<dd><p>no problems were encountered;
</p></dd>
<dt><code>1</code></dt>
<dd><p>too many sub-intervals were done;
</p></dd>
<dt><code>2</code></dt>
<dd><p>excessive roundoff error is detected;
</p></dd>
<dt><code>3</code></dt>
<dd><p>extremely bad integrand behavior occurs;
</p></dd>
<dt><code>6</code></dt>
<dd><p>if the input is invalid.
</p>
</dd>
</dl>

<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) quad_qawf (exp(-x^2), x, 0, 1, 'cos, 'epsabs=1d-9);
(%o1)   [.6901942235215714, 2.84846300257552E-11, 215, 0]
</pre><pre class="example">(%i2) integrate (exp(-x^2)*cos(x), x, 0, inf);
                          - 1/4
                        %e      sqrt(%pi)
(%o2)                   -----------------
                                2
</pre><pre class="example">(%i3) ev (%, numer);
(%o3)                   .6901942235215714
</pre></div>





</dd></dl>


<a name="quad_005fqawo"></a><a name="Item_003a-Integration_002fdeffn_002fquad_005fqawo"></a><dl>
<dt><a name="index-quad_005fqawo"></a>Function: <strong>quad_qawo</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qawo</tt> (<var>f(x)</var>, <var>x</var>, <var>a</var>, <var>b</var>, <var>omega</var>, <var>trig</var>, [<var>epsrel</var>, <var>epsabs</var>, <var>limit</var>, <var>maxp1</var>, <var>limlst</var>]) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qawo</tt> (<var>f</var>, <var>x</var>, <var>a</var>, <var>b</var>, <var>omega</var>, <var>trig</var>, [<var>epsrel</var>, <var>epsabs</var>, <var>limit</var>, <var>maxp1</var>, <var>limlst</var>])</em></dt>
<dd>
<p>Integration of 
\(\cos(\omega x) f(x)\) or 
\(\sin(\omega x)\) over a finite interval,
where 
\(\omega\) is a constant.
The rule evaluation component is based on the modified
Clenshaw-Curtis technique.  <code>quad_qawo</code> applies adaptive subdivision with
extrapolation, similar to <code>quad_qags</code>.
</p>
<p><code>quad_qawo</code> computes the integral using the Quadpack QAWO
routine:
</p>
$$
\int_a^b f(x) \, w(x) \, dx
$$

<p>The weight function <em>w</em> is selected by <var>trig</var>:
</p>
<dl compact="compact">
<dt><code>cos</code></dt>
<dd>
\(w(x) = \cos\omega x\)
</dd>
<dt><code>sin</code></dt>
<dd>
\(w(x) = \sin\omega x\)
</dd>
</dl>

<p>The integrand may be specified as the name of a Maxima or Lisp function or
operator, a Maxima lambda expression, or a general Maxima expression.
</p>
<p>The keyword arguments are optional and may be specified in any order.
They all take the form <code>key=val</code>.  The keyword arguments are:
</p>
<dl compact="compact">
<dt><code>epsrel</code></dt>
<dd><p>Desired relative error of approximation.  Default is 1d-8.
</p></dd>
<dt><code>epsabs</code></dt>
<dd><p>Desired absolute error of approximation.  Default is 0.
</p></dd>
<dt><code>limit</code></dt>
<dd><p>Size of internal work array.  <var>limit</var>/2 is the
maximum number of subintervals to use.  Default is 200.
</p></dd>
<dt><code>maxp1</code></dt>
<dd><p>Maximum number of Chebyshev moments.  Must be greater than 0.  Default
is 100.
</p></dd>
<dt><code>limlst</code></dt>
<dd><p>Upper bound on the number of cycles.  Must be greater than or equal to
3.  Default is 10.
</p></dd>
</dl>

<p><code>quad_qawo</code> returns a list of four elements:
</p>
<ul>
<li> an approximation to the integral,
</li><li> the estimated absolute error of the approximation,
</li><li> the number integrand evaluations,
</li><li> an error code.
</li></ul>

<p>The error code (fourth element of the return value) can have the values:
</p>
<dl compact="compact">
<dt><code>0</code></dt>
<dd><p>no problems were encountered;
</p></dd>
<dt><code>1</code></dt>
<dd><p>too many sub-intervals were done;
</p></dd>
<dt><code>2</code></dt>
<dd><p>excessive roundoff error is detected;
</p></dd>
<dt><code>3</code></dt>
<dd><p>extremely bad integrand behavior occurs;
</p></dd>
<dt><code>6</code></dt>
<dd><p>if the input is invalid.
</p>
</dd>
</dl>

<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) quad_qawo (x^(-1/2)*exp(-2^(-2)*x), x, 1d-8, 20*2^2, 1, cos);
(%o1)     [1.376043389877692, 4.72710759424899E-11, 765, 0]
</pre><pre class="example">(%i2) rectform (integrate (x^(-1/2)*exp(-2^(-alpha)*x) * cos(x),
      x, 0, inf));
                   alpha/2 - 1/2            2 alpha
        sqrt(%pi) 2              sqrt(sqrt(2        + 1) + 1)
(%o2)   -----------------------------------------------------
                               2 alpha
                         sqrt(2        + 1)
</pre><pre class="example">(%i3) ev (%, alpha=2, numer);
(%o3)                     1.376043390090716
</pre></div>





</dd></dl>


<a name="quad_005fqaws"></a><a name="Item_003a-Integration_002fdeffn_002fquad_005fqaws"></a><dl>
<dt><a name="index-quad_005fqaws"></a>Function: <strong>quad_qaws</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qaws</tt> (<var>f(x)</var>, <var>x</var>, <var>a</var>, <var>b</var>, <var>alpha</var>, <var>beta</var>, <var>wfun</var>, [<var>epsrel</var>, <var>epsabs</var>, <var>limit</var>]) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qaws</tt> (<var>f</var>, <var>x</var>, <var>a</var>, <var>b</var>, <var>alpha</var>, <var>beta</var>, <var>wfun</var>, [<var>epsrel</var>, <var>epsabs</var>, <var>limit</var>])</em></dt>
<dd>
<p>Integration of <em>w(x) f(x)</em> over a finite interval, where <em>w(x)</em> is a
certain algebraic or logarithmic function.  A globally adaptive subdivision
strategy is applied, with modified Clenshaw-Curtis integration on the
subintervals which contain the endpoints of the interval of integration.
</p>
<p><code>quad_qaws</code> computes the integral using the Quadpack QAWS routine:
</p>
$$
\int_a^b f(x) \, w(x) \, dx
$$

<p>The weight function <em>w</em> is selected by <var>wfun</var>:
</p>
<dl compact="compact">
<dt><code>1</code></dt>
<dd>
\(w(x) = (x - a)^\alpha (b - x)^\beta\)
</dd>
<dt><code>2</code></dt>
<dd>
\(w(x) = (x - a)^\alpha (b - x)^\beta \log(x - a)\)
</dd>
<dt><code>3</code></dt>
<dd>
\(w(x) = (x - a)^\alpha (b - x)^\beta \log(b - x)\)
</dd>
<dt><code>4</code></dt>
<dd>
\(w(x) = (x - a)^\alpha (b - x)^\beta \log(x - a) \log(b - x)\)
</dd>
</dl>

<p>The integrand may be specified as the name of a Maxima or Lisp function or
operator, a Maxima lambda expression, or a general Maxima expression.
</p>
<p>The keyword arguments are optional and may be specified in any order.
They all take the form <code>key=val</code>.  The keyword arguments are:
</p>
<dl compact="compact">
<dt><code>epsrel</code></dt>
<dd><p>Desired relative error of approximation.  Default is 1d-8.
</p></dd>
<dt><code>epsabs</code></dt>
<dd><p>Desired absolute error of approximation.  Default is 0.
</p></dd>
<dt><code>limit</code></dt>
<dd><p>Size of internal work array.  <var>limit</var>is the
maximum number of subintervals to use.  Default is 200.
</p></dd>
</dl>

<p><code>quad_qaws</code> returns a list of four elements:
</p>
<ul>
<li> an approximation to the integral,
</li><li> the estimated absolute error of the approximation,
</li><li> the number integrand evaluations,
</li><li> an error code.
</li></ul>

<p>The error code (fourth element of the return value) can have the values:
</p>
<dl compact="compact">
<dt><code>0</code></dt>
<dd><p>no problems were encountered;
</p></dd>
<dt><code>1</code></dt>
<dd><p>too many sub-intervals were done;
</p></dd>
<dt><code>2</code></dt>
<dd><p>excessive roundoff error is detected;
</p></dd>
<dt><code>3</code></dt>
<dd><p>extremely bad integrand behavior occurs;
</p></dd>
<dt><code>6</code></dt>
<dd><p>if the input is invalid.
</p>
</dd>
</dl>

<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) quad_qaws (1/(x+1+2^(-4)), x, -1, 1, -0.5, -0.5, 1,
                 'epsabs=1d-9);
(%o1)     [8.750097361672832, 1.24321522715422E-10, 170, 0]
</pre><pre class="example">(%i2) integrate ((1-x*x)^(-1/2)/(x+1+2^(-alpha)), x, -1, 1);
       alpha
Is  4 2      - 1  positive, negative, or zero?

pos;
                          alpha         alpha
                   2 %pi 2      sqrt(2 2      + 1)
(%o2)              -------------------------------
                               alpha
                            4 2      + 2
</pre><pre class="example">(%i3) ev (%, alpha=4, numer);
(%o3)                     8.750097361672829
</pre></div>





</dd></dl>


<a name="quad_005fqagp"></a><a name="Item_003a-Integration_002fdeffn_002fquad_005fqagp"></a><dl>
<dt><a name="index-quad_005fqagp"></a>Function: <strong>quad_qagp</strong> <em><br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qagp</tt> (<var>f(x)</var>, <var>x</var>, <var>a</var>, <var>b</var>, <var>points</var>, [<var>epsrel</var>, <var>epsabs</var>, <var>limit</var>]) <br>&nbsp;&nbsp;&nbsp;&nbsp;<tt>quad_qagp</tt> (<var>f</var>, <var>x</var>, <var>a</var>, <var>b</var>, <var>points</var>, [<var>epsrel</var>, <var>epsabs</var>, <var>limit</var>])</em></dt>
<dd>
<p>Integration of a general function over a finite interval.
<code>quad_qagp</code> implements globally adaptive interval subdivision with
extrapolation (de Doncker, 1978) by the Epsilon algorithm (Wynn, 1956).
</p>
<p><code>quad_qagp</code> computes the integral
</p>
$$
\int_a^b f(x) \, dx
$$

<p>The function to be integrated is <em>f(x)</em>, with
dependent variable <em>x</em>, and the function is to be integrated
between the limits <em>a</em> and <em>b</em>.
</p>
<p>The integrand may be specified as the name of a Maxima or Lisp function or
operator, a Maxima lambda expression, or a general Maxima expression.
</p>
<p>To help the integrator, the user must supply a list of points where
the integrand is singular or discontinuous.
</p>
<p>The keyword arguments are optional and may be specified in any order.
They all take the form <code>key=val</code>.  The keyword arguments are:
</p>
<dl compact="compact">
<dt><code>epsrel</code></dt>
<dd><p>Desired relative error of approximation.  Default is 1d-8.
</p></dd>
<dt><code>epsabs</code></dt>
<dd><p>Desired absolute error of approximation.  Default is 0.
</p></dd>
<dt><code>limit</code></dt>
<dd><p>Size of internal work array.  <var>limit</var> is the
maximum number of subintervals to use.  Default is 200.
</p></dd>
</dl>

<p><code>quad_qagp</code> returns a list of four elements:
</p>
<ul>
<li> an approximation to the integral,
</li><li> the estimated absolute error of the approximation,
</li><li> the number integrand evaluations,
</li><li> an error code.
</li></ul>

<p>The error code (fourth element of the return value) can have the values:
</p>
<dl compact="compact">
<dt><code>0</code></dt>
<dd><p>no problems were encountered;
</p></dd>
<dt><code>1</code></dt>
<dd><p>too many sub-intervals were done;
</p></dd>
<dt><code>2</code></dt>
<dd><p>excessive roundoff error is detected;
</p></dd>
<dt><code>3</code></dt>
<dd><p>extremely bad integrand behavior occurs;
</p></dd>
<dt><code>4</code></dt>
<dd><p>failed to converge
</p></dd>
<dt><code>5</code></dt>
<dd><p>integral is probably divergent or slowly convergent
</p></dd>
<dt><code>6</code></dt>
<dd><p>if the input is invalid.
</p></dd>
</dl>


<p>Examples:
</p>
<div class="example">
<pre class="example">(%i1) quad_qagp(x^3*log(abs((x^2-1)*(x^2-2))),x,0,3,[1,sqrt(2)]);
(%o1)   [52.74074838347143, 2.6247632689546663e-7, 1029, 0]
</pre><pre class="example">(%i2) quad_qags(x^3*log(abs((x^2-1)*(x^2-2))), x, 0, 3);
(%o2)   [52.74074847951494, 4.088443219529836e-7, 1869, 0]
</pre></div>

<p>The integrand has singularities at <code>1</code> and <code>sqrt(2)</code> so we supply
these points to <code>quad_qagp</code>.  We also note that <code>quad_qagp</code> is
more accurate and more efficient that <code><a href="#quad_005fqags">quad_qags</a></code>.
</p>




</dd></dl>

<a name="quad_005fcontrol"></a><a name="Item_003a-Integration_002fdeffn_002fquad_005fcontrol"></a><dl>
<dt><a name="index-quad_005fcontrol"></a>Function: <strong>quad_control</strong> <em>(<var>parameter</var>, [<var>value</var>])</em></dt>
<dd>
<p>Control error handling for quadpack.  The parameter should be one of
the following symbols:
</p>
<dl compact="compact">
<dt><code>current_error</code></dt>
<dd><p>The current error number
</p></dd>
<dt><code>control</code></dt>
<dd><p>Controls if messages are printed or not.  If it is set to zero or
less, messages are suppressed.
</p></dd>
<dt><code>max_message</code></dt>
<dd><p>The maximum number of times any message is to be printed.
</p></dd>
</dl>

<p>If <var>value</var> is not given, then the current value of the
<var>parameter</var> is returned.  If <var>value</var> is given, the value of
<var>parameter</var> is set to the given value.
</p>




</dd></dl>


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