On rational approximations of functions and eigenvalue selection in Werner algorithm

O. E. Galkin ; S. Yu. Galkina

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 25 (2015) no. 3, pp. 297-305

Résumé

The paper deals with the best uniform rational approximations (BURA) of continuous functions on compact (and even finite) subsets of real axis $\mathbb R$. The authors show that BURA does not always exist. They study the algorithm of Helmut Werner in more detail. This algorithm serves to search for BURA of the type $P_m/Q_n=\sum_{i=0}^ma_ix^i\big/\sum_{j=0}^nb_jx^j$ for functions on a set of $N=m+n+2$ points $x_1\dots$. It can be used within the Remez algorithm of searching for BURA on a segment. The Werner algorithm calculates $(n+1)$ real eigenvalues $h_1,\dots,h_{n+1}$ for the matrix pencil $A-hB$, where $A$ and $B$ are some symmetric matrices. Each eigenvalue generates a rational fraction of the type $P_m/Q_n$ which is a candidate for the best approximation. It is known that at most one of these fractions is free from poles on the segment $[x_1,x_N]$, so the following problem arises: how to determine the eigenvalue which generates the rational fraction without poles? It is shown that if $m=0$ and all values $f(x_1),-f(x_2),\dots,(-1)^{n+2}f(x_{n+2})$ are different and the approximating function is positive (negative) at all points $x_1,\dots,x_{n+2}$, then this eigenvalue ranks $[(n+2)/2]$-th ($[(n+3)/2]$-th) in value. Three numerical examples illustrate this statement.

Détail

Keywords: best uniform rational approximations, rational approximations on finite sets, Werner algorithm, selection of eigenvalues in Werner algorithm.
Mots-clés : Remez algorithm