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Chebyshev rational interpolation. (English) Zbl 0890.65009
Denote by $$E_\rho$$, $$0 \leq \rho \leq 1$$, the closed ellipse in $$\mathbb C$$ with the boundary $\partial E_\rho = \{ t+ \rho t^{-1}:\;|t|=1 \},$ and $s_j(z)=w^j+(\rho/w)^j, \quad z=w+\rho/w, \quad j=2, \dots,$ the corresponding Chebyshev polynomials. The paper faces the problem of the efficient computation of all the rational functions $\displaystyle \sum_{i=0}^m a_i s_i(z) \over \displaystyle \sum_{i=0}^n b_i s_i(z) \tag{*}$ which assume given values $$f_0,\dots,f_{n+m}$$ ($$f_i \neq 0$$) at pairwise distinct points $$z_i, 0 \leq i \leq n+m$$. A modification of a general look-ahead version of the Lanczos procedure in order to compute the coefficients of the denominator of the interpolating function (*) is derived; the same algorithm is applied to the interpolation data $$(z_i,1/f_i)$$, which gives the coefficients of the numerator of (*). The proposed approach is particularly suited for the rational interpolation at nodes on the boundary of $$E_\rho$$. Several computational examples are considered, with $$f_i$$ generated at random or representing values of a function analytic in $$E_\rho$$.

##### MSC:
 65D05 Numerical interpolation 41A05 Interpolation in approximation theory 41A20 Approximation by rational functions
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