Numerical Method of Simultaneous Solution the Problem of Finding the Distance (Best) Between a Convex Polyhedron and a Finite-Dimensional Subspace of a Linear Normed Space and Dual Task

• Дата: 2021
• Автори: Гудима, Уляна, Гнатюк, Василь
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2021
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/251160
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

One of the most developing areas of mathematics is theory of approximations, including the theory of approxi-mations of functions. It is begins from of task of P. L. Chebyshev on the uniform (Chebyshev) approximation of a continuous on a segment of a real-valued function by a set of algebraic polynomials of degree not exceeding n. Later, a number of other formulations of problems on the best approximation of functions were considered, one of which is the problem of uniform approximation of a function continuous on a compact set by a finite-dimensional subspace generated by other functions continuous on this compact set. An important place in the theory of approximation is occupied by the problem of approximation of an element of linear normed space by the elements of its finite-dimensional subspace, partial cases of which are the problems discussed above. An important questionі of this problem are general theorems of existence of an extremal element, duality theorems and criteria of an extremal element, construction of numerical methods for finding this element and the magnitude of the best approximation, which have been studied by many mathematicians. The paper considers the problem of finding the distance (best) between a convex polyhedron and a finite-dimensional subspace of a linear normalized space, a partial case of which is the problem of the best approximation of an element of a linear normed space by its finitedimensional subspace. For this problem the existence of an extremal element, the ratio of duality, the criterion of an extremal element are established. A convergent numerical method of simultaneous solution of direct and dual problems is constructed, bilateral estimates of convergence are obtained, which allow to find corresponding values with predetermined accuracy.