I. V. Kurbatova

V. G. Kurbatov, I. V. Kurbatova

It is well known that the equation $x'(t)=Ax(t)+f(t)$, where $A$ is a square matrix, has a unique bounded solution $x$ for any bounded continuous free term $f$, provided the coefficient $A$ has no eigenvalues on the imaginary axis. This solution can be represented in the form \begin{equation*} x(t)=\int_{-\infty}^{\infty}\mathcal G(t-s)x(s)\,ds. \end{equation*} The kernel $\mathcal G$ is called Green's function. In the paper, a representation of Green's function in the form of the Newton interpolating polynomial is used for approximate calculation of $\mathcal G$. An estimate of the sensitivity of the problem is given.

V. G. Kurbatov, I. V. Kurbatova

Let $A$ be a square complex matrix, $z_1$, ..., $z_{n}\in\mathbb C$ be (possibly repetitive) points of interpolation, $f$ be analytic in a neighborhood of the convex hull of the union of the spectrum of $A$ and the points $z_1$, ..., $z_{n}$, and $p$ be the interpolation polynomial of $f$, constructed by the points $z_1$, ..., $z_{n}$. It is proved that under these assumptions $$\Vert f(A)-p(A)\Vert\le\frac1{n!} \max_{t\in[0,1];\,μ\in\text{co}\{z_1,z_{2},\dots,z_{n}\}}\bigl\VertΩ(A)f^{(n)} \bigl((1-t)μ\mathbf1+tA\bigr)\bigr\Vert,$$ where $Ω(z)=\prod_{k=1}^n(z-z_k)$.