A polynomial approximating a given function is constructed assuming that the function and a certain set of its derivatives are known at the endpoints of a given interval. Various analytical for mulas are derived for the approximating polynomial. An interpretation of the two point approxima tion of the function is given and its relation to the Taylor series expansion of the function is indicated. A sufficient condition for the convergence of a sequence of two point polynomials to a given function is established. Examples are given in which the sine function is approximated by a sequence of two point Hermite polynomials on given intervals. The errors in the two point and Taylor series approxi mations of the function are compared analytically and numerically.

No abstract

No abstract

The actual problem of ways to represent aerodrome surface route network is considered. Based on the analysis of various options, an approach is proposed for representing route network sections as smooth curves, which are described by parametric vector functions. Each of the vector function components is represented by a two-point Hermite interpolation polynomial, which uses derivatives up to some order inclusive. Within this approach, the optimization problem related to the coefficients selection of these polynomials based on minimizing the distance between the broken line and smooth curve is solved. The problem is reduced to solving a system of linear equations by the derivatives values at the ends of the route network section. The corresponding finite formulas for approximating broken lines by smooth curves are proposed. Based on the formulas obtained, algorithm and program for approximating route network sections using information about taxi lines, which are stored in aerodrome mapping database (AMDB), were developed. The program also allows you to calculate statistical indicators, what allow to get a quantitative approximation estimate. Numerical experiments based on the Sheremetyevo aerodrome dataset have shown the promise of this approach to presenting aerodrome surface route network, which can significantly (2 – 4 times) reduce the amount of data and increase the realism of the aerodrome model.

Рассмотрена задача вычисления определенного интеграла функции, для которой известны значения ее самой и набора производных до заданного порядка в точках отрезка интегрирования. Построены составные квадратурные формулы, которые используют значения функции и ее производных до m-го порядка включительно. Получено представление остаточного члена, выраженное через производную соответствующего порядка и число узловых точек. Приведены примеры интегрирования заданных функций с исследованием погрешности и ее оценки. Дано сравнение с известными численными методами и формулой Эйлера-Маклорена, которое показало повышенную точность и лучшую сходимость метода двухточечного интегрирования The problem of computing a definite integral of a function for which the values of itself and the set of derivatives up to a given order at the points of the interval of integration are known is considered. Composite quadrature formulas are constructed that use the values of the function and its derivatives up to the m-th order inclusive. A representation of the remainder is obtained, expressed in terms of the derivative of the corresponding order and the number of nodal points. Examples of integration of the given functions with the study of the error and its estimation are given. A comparison is made with the known numerical methods and the Euler-Maclaurin formula, which showed increased accuracy and better convergence of the two-point integration method.