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<p>orthogonal polynomial for Gaussian quadratures
<a href="class_quant_lib_1_1_gaussian_orthogonal_polynomial.html#details">More...</a></p>
<p><code>#include <ql/math/integrals/gaussianorthogonalpolynomial.hpp></code></p>
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Inheritance diagram for GaussianOrthogonalPolynomial:</div>
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<div class="center"><img src="class_quant_lib_1_1_gaussian_orthogonal_polynomial__inherit__graph.png" border="0" usemap="#_gaussian_orthogonal_polynomial_inherit__map" alt="Inheritance graph"/></div>
<map name="_gaussian_orthogonal_polynomial_inherit__map" id="_gaussian_orthogonal_polynomial_inherit__map">
<area shape="rect" id="node3" href="class_quant_lib_1_1_gauss_hermite_polynomial.html" title="Gauss-Hermite polynomial." alt="" coords="265,5,431,35"/><area shape="rect" id="node5" href="class_quant_lib_1_1_gauss_hyperbolic_polynomial.html" title="Gauss hyperbolic polynomial." alt="" coords="257,58,439,89"/><area shape="rect" id="node7" href="class_quant_lib_1_1_gauss_jacobi_polynomial.html" title="Gauss-Jacobi polynomial." alt="" coords="268,111,428,142"/><area shape="rect" id="node9" href="class_quant_lib_1_1_gauss_laguerre_polynomial.html" title="Gauss-Laguerre polynomial." alt="" coords="263,165,433,195"/></map>
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<p><a href="class_quant_lib_1_1_gaussian_orthogonal_polynomial-members.html">List of all members.</a></p>
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Public Member Functions</h2></td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a285550e8d0f3964935a6301d1d2f4037"></a><!-- doxytag: member="QuantLib::GaussianOrthogonalPolynomial::mu_0" ref="a285550e8d0f3964935a6301d1d2f4037" args="() const =0" -->
virtual <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>mu_0</b> () const =0</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a4b62a25af75d692dbf58ce06f96184ca"></a><!-- doxytag: member="QuantLib::GaussianOrthogonalPolynomial::alpha" ref="a4b62a25af75d692dbf58ce06f96184ca" args="(Size i) const =0" -->
virtual <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>alpha</b> (<a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> i) const =0</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="ad9af9b51bda36a25555694b57ed3014c"></a><!-- doxytag: member="QuantLib::GaussianOrthogonalPolynomial::beta" ref="ad9af9b51bda36a25555694b57ed3014c" args="(Size i) const =0" -->
virtual <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>beta</b> (<a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> i) const =0</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="af8bd61e6c7a4d4d245f25e375ff42fa8"></a><!-- doxytag: member="QuantLib::GaussianOrthogonalPolynomial::w" ref="af8bd61e6c7a4d4d245f25e375ff42fa8" args="(Real x) const =0" -->
virtual <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>w</b> (<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> x) const =0</td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="a452af47b58099bbec68537241bd743f7"></a><!-- doxytag: member="QuantLib::GaussianOrthogonalPolynomial::value" ref="a452af47b58099bbec68537241bd743f7" args="(Size i, Real x) const " -->
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>value</b> (<a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> i, <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> x) const </td></tr>
<tr><td class="memItemLeft" align="right" valign="top"><a class="anchor" id="aef33067e7357c2c91caa62c6f38ecaa1"></a><!-- doxytag: member="QuantLib::GaussianOrthogonalPolynomial::weightedValue" ref="aef33067e7357c2c91caa62c6f38ecaa1" args="(Size i, Real x) const " -->
<a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> </td><td class="memItemRight" valign="bottom"><b>weightedValue</b> (<a class="el" href="group__types.html#gaf38bdb4c54463b1f456655efa95b5c77">Size</a> i, <a class="el" href="group__types.html#ga4bdf4bfe76b9ffa6fa64c47d8bfa0c78">Real</a> x) const </td></tr>
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<hr/><a name="details" id="details"></a><h2>Detailed Description</h2>
<div class="textblock"><p>orthogonal polynomial for Gaussian quadratures </p>
<p>References: Gauss quadratures and orthogonal polynomials</p>
<p>G.H. Gloub and J.H. Welsch: Calculation of Gauss quadrature rule. Math. Comput. 23 (1986), 221-230</p>
<p>"Numerical Recipes in C", 2nd edition, Press, Teukolsky, Vetterling, Flannery,</p>
<p>The polynomials are defined by the three-term recurrence relation </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ P_{k+1}(x)=(x-\alpha_k) P_k(x) - \beta_k P_{k-1}(x) \]" src="form_188.png"/>
</p>
<p> and </p>
<p class="formulaDsp">
<img class="formulaDsp" alt="\[ \mu_0 = \int{w(x)dx} \]" src="form_189.png"/>
</p>
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