An estimate of approximation of a matrix-valued function by an interpolation polynomial
| Autor: | Kurbatov, V. G., Kurbatova, I. V. |
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| Rok vydání: | 2018 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $$\Vert f(A)-p(A)\Vert\le\frac1{n!} \max_{t\in[0,1];\,\mu\in\text{co}\{z_1,z_{2},\dots,z_{n}\}}\bigl\Vert\Omega(A)f^{{(n)}} \bigl((1-t)\mu\mathbf1+tA\bigr)\bigr\Vert,$$ where $\Omega(z)=\prod_{k=1}^n(z-z_k)$. Comment: 8 pages, 1 figure |
| Databáze: | arXiv |
| Externí odkaz: |
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