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<title>GNU Scientific Library &ndash; Reference Manual: Fixed order Gauss-Legendre integration</title>

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Next: <a href="Numerical-integration-error-codes.html#Numerical-integration-error-codes" accesskey="n" rel="next">Numerical integration error codes</a>, Previous: <a href="CQUAD-doubly_002dadaptive-integration.html#CQUAD-doubly_002dadaptive-integration" accesskey="p" rel="previous">CQUAD doubly-adaptive integration</a>, Up: <a href="Numerical-Integration.html#Numerical-Integration" accesskey="u" rel="up">Numerical Integration</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Gauss_002dLegendre-integration"></a>
<h3 class="section">17.12 Gauss-Legendre integration</h3>

<p>The fixed-order Gauss-Legendre integration routines are provided for fast
integration of smooth functions with known polynomial order.  The
<em>n</em>-point Gauss-Legendre rule is exact for polynomials of order
<em>2*n-1</em> or less.  For example, these rules are useful when integrating
basis functions to form mass matrices for the Galerkin method.  Unlike other
numerical integration routines within the library, these routines do not accept
absolute or relative error bounds.
</p>
<dl>
<dt><a name="index-gsl_005fintegration_005fglfixed_005ftable_005falloc"></a>Function: <em>gsl_integration_glfixed_table *</em> <strong>gsl_integration_glfixed_table_alloc</strong> <em>(size_t <var>n</var>)</em></dt>
<dd><a name="index-gsl_005fintegration_005fglfixed_005ftable"></a>
<p>This function determines the Gauss-Legendre abscissae and weights necessary for
an <em>n</em>-point fixed order integration scheme.  If possible, high precision
precomputed coefficients are used.  If precomputed weights are not available,
lower precision coefficients are computed on the fly.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fintegration_005fglfixed"></a>Function: <em>double</em> <strong>gsl_integration_glfixed</strong> <em>(const gsl_function * <var>f</var>, double <var>a</var>, double <var>b</var>, const gsl_integration_glfixed_table * <var>t</var>)</em></dt>
<dd><p>This function applies the Gauss-Legendre integration rule
contained in table <var>t</var> and returns the result.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fintegration_005fglfixed_005fpoint"></a>Function: <em>int</em> <strong>gsl_integration_glfixed_point</strong> <em>(double <var>a</var>, double <var>b</var>, size_t <var>i</var>, double * <var>xi</var>, double * <var>wi</var>, const gsl_integration_glfixed_table * <var>t</var>)</em></dt>
<dd><p>For <var>i</var> in <em>[0, &hellip;, t-&gt;n - 1]</em>, this function obtains the
<var>i</var>-th Gauss-Legendre point <var>xi</var> and weight <var>wi</var> on the interval
[<var>a</var>,<var>b</var>].  The points and weights are ordered by increasing point
value.  A function <em>f</em> may be integrated on [<var>a</var>,<var>b</var>] by summing
<em>wi * f(xi)</em> over <var>i</var>.
</p></dd></dl>

<dl>
<dt><a name="index-gsl_005fintegration_005fglfixed_005ftable_005ffree"></a>Function: <em>void</em> <strong>gsl_integration_glfixed_table_free</strong> <em>(gsl_integration_glfixed_table * <var>t</var>)</em></dt>
<dd><a name="index-gsl_005fintegration_005fglfixed_005ftable-1"></a>
<p>This function frees the memory associated with the table <var>t</var>.
</p></dd></dl>




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