Computational aspects of S-differentiability of functions of several variables

Abstract

The study of various processes leads to the need to clarify (expand) the boundaries of the applicability of computational structures and modeling tools. The purpose of this article is to develop the Taylor expansion for functions of several variables based on the concept of $S$-differentiability. A function $f$ from $L_1[Q_0]$, where $Q_0$ is an $m$-dimensional cube, is called $S$-differentiable at an interior point $x_0$ of this cube, if there exists an algebraic if there exists an analgebraic polynomial $P(x)$ of degree not greater than first for which it is uniform over all vectors $v$ of the unit sphere ${\mathbb R}^m$ the integral of $t$ within $0$ and $h$ from the expression $f(x_0 + t \cdot v)-P(t \cdot v)$ is $o(h^2)$ for $h \to 0{+}$. It is shown that with this definition, differentiation of a composite function with a linear interior component is valid, and the vector-gradient principle holds. The following result is proved. Let the function $f$ have continuous partial derivatives up to order $n$ inclusive in some neighborhood of the interior point $x_0 \in Q_0$ that are $S$-differentiable at the point $x_0$, then the Taylor expansion the function $f$ with accuracy $o\big(\Vert x - x_0\Vert^{n + 1}\big)$ holds in this neighborhood.