orthogonal polynomial for Gaussian quadratures More...
#include <ql/math/integrals/gaussianorthogonalpolynomial.hpp>
Public Member Functions | |
virtual Real | mu_0 () const =0 |
virtual Real | alpha (Size i) const =0 |
virtual Real | beta (Size i) const =0 |
virtual Real | w (Real x) const =0 |
Real | value (Size i, Real x) const |
Real | weightedValue (Size i, Real x) const |
orthogonal polynomial for Gaussian quadratures
References: Gauss quadratures and orthogonal polynomials
G.H. Gloub and J.H. Welsch: Calculation of Gauss quadrature rule. Math. Comput. 23 (1986), 221-230
"Numerical Recipes in C", 2nd edition, Press, Teukolsky, Vetterling, Flannery,
The polynomials are defined by the three-term recurrence relation
\[ P_{k+1}(x)=(x-\alpha_k) P_k(x) - \beta_k P_{k-1}(x) \]
and
\[ \mu_0 = \int{w(x)dx} \]