Extremal polynomials and methods of optimization of numerical algorithms

V. I. Lebedev

Sbornik. Mathematics, Tome 195 (2004) no. 10, pp. 1413-1459

Résumé

Chebyshëv–Markov–Bernstein–Szegö polynomials $C_n(x)$ extremal on $[-1,1]$ with weight functions $w(x)=(1+x)^\alpha(1- x)^\beta/\sqrt{S_l(x)}$ where $\alpha,\beta=0,\frac12$ and $S_l(x)=\prod_{k=1}^m(1-c_kT_{l_k}(x))>0$ are considered. A universal formula for their representation in trigonometric form is presented. Optimal distributions of the nodes of the weighted interpolation and explicit quadrature formulae of Gauss, Markov, Lobatto, and Rado types are obtained for integrals with weight $p(x)=w^2(x)(1-x^2)^{-1/2}$. The parameters of optimal Chebyshëv iterative methods reducing the error optimally by comparison with the initial error defined in another norm are determined. For each stage of the Fedorenko–Bakhvalov method iteration parameters are determined which take account of the results of the previous calculations. Chebyshëv filters with weight are constructed. Iterative methods of the solution of equations containing compact operators are studied.