Orthogonal polynomials and generalized Gauss-Rys quadrature formulae

Orthogonal polynomials and the corresponding quadrature formulas of Gaussian type concerni λ ng > the 1 e / v 2 en wei x gh > t f 0 unction ω(t; x) = exp λ (−= xt 1 2) / ( 2 1 − t2)−1/2 on (−1, 1), with parameters − and , are considered. For these quadrature rules reduce to the socalled Gauss-Rys quadrature formulas, which were investigated earlier by several authors, e.g., Dupuis at al 1976 and 1983; Sagar 1992; Schwenke 2014; Shizgal 2015; King 2016; Milovanovic ´ 2018, etc. In this generalized case, the method of modified moments is used, as well as a transformation of quadratures on (−1, 1) with N nodes to ones on (0, 1) with only (N + 1)/2 nodes. Such an approach provides a stable and very efficient numerical construction.

Investigation of the convergence of the integration algorithm with the choice of the variable of integration at each step using the example of a Luneberg lens

The paper describes an integration algorithm with a choice of a variable at each step in the numerical construction of ray trajectories in a medium with a given dependence of the permittivityon coordinates. The convergence of the calculations to the exact solution is estimated using the example of the problem of calculating the trajectories of rays in a Luneberg lens. It is shown that with a decrease in the grid step, convergence to the exact solution is observed. Purpose. Assess the convergence to an exact solution of an integration algorithm with a choice of a variable at each step using the example of the problem of calculating ray trajectories in a Luneberg lens. Results. The trajectories of rays incident parallel to the ordinate axis and the trajectories of rays incident at an angle to the ordinate axis are calculated. It is shown that with a decrease in the grid step, convergence of the results to the exact solution is observed. Practical significance. It is shown that an integration algorithm with a choice of a variable at each step provides the construction of ray trajectories with an error in the coordinate not exceeding the grid step for the problem of ray propagation in a Luneberg lens.

AUUV path planning in threat environment.

The paper considers the proposed statements and solutions of a number of main and auxiliary problems proposed at Trapeznikov Institute of Control Scientes RAS related to the planning of the movement of autonomous uninhabited underwater vehicles (AUUV) when evading systems of homogeneous and/or heterogeneous observers that determine the threat environment. Examples of analytical and numerical construction of traffic routes are given.