Информационная система "Конференции"

Abstracts

Approximation of functions and quadrature formulas

This paper proposes a modification of the composite (compound) Gauss-Legendre formula for numerical integration. The question arises which number of subintervals $M$ is to be taken depending on the number of nodes in the quadrature rule (the order $n$ of the integration method). Related to it is the question of the magnitude of the subinterval, i. e. the problem of nodes $x_{i}$ distribution, the so-called problem of splitting (partition) of integration interval into sub-intervals, so that the result of a prescribed accuracy is reached with as little as possible calculations of the integrand. It is shown in which way it is necessary to make an optimal splitting of the integration interval into sub-intervals. Then the optimal splitting of the interval is reached when the nodes are values of inverse function to the primitive function $F^{-1}$ of the integrand $f$ $$x_{i} = F^{-1}left( frac{i}{M}left( f(x_{M})-f(x_{0})right) +f(x_{0})right),$$ where $i = 0, 1, ..., M$, and $x_{0}$=$a$, $x_{M}$=$b$. The general result (adaptive splitting) is applied to some classes of functions on the basis of whose features the method for exact determination of the necessary value for the order of method $n$ was developed, depending on the calculation accuracy (on the error) and some features of subintegral function. The very useful and interesting results are obtained.

Note. Abstracts are published in author's edition

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