Two-frequency-dependent Gauss quadrature rules
The article presents Gauss quadrature rules to evaluate the integral of oscillatory functions of the form \(y(x)=h_1(x)\cos(\omega_1x)+h_2(x)\sin(\omega_1 x)+h_3(x)\cos(\omega_2x)+h_4(x)\sin(\omega_2 x)\), where \(\omega_1\) and \(\omega_2\) are frequencies and \(h_i\) are smooth functions. The problem of solving a nonlinear system of equations to calculate the quadrature coefficients \(w_k\) and the nodes \(x_k\) is posed according to the exponentially fitted approach by \textit{L. Gr. Ixaru} [Comput. Phys. Commun. 105, 1--19 (1997; Zbl 0930.65150)]. In the case of \(\omega_2\to \omega_1\) or \(\omega_1,\omega_2\to 0\) a regularization procedure is provided to avoid the singularity of the Jacobian matrix of the system of equations. In the section devoted to error analysis a theorem is given to describe the first non-vanishing term of the quadrature error. Numerical tests illustrate the power of the two-frequency-dependent quadrature rule in comparison with the classical Gauss rule and one-frequency-dependent rules.
scientific article; zbMATH DE number 2137441