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numerical analysis
The convergence of the sequence $x_{n+1}=Ax_n+b$ to $(l-A)^{-1}b$ (where $x_n, b$ are vectors in Banach space, $A$ is a bounded linear operator with bounded $(l-A)^{-1}$) can be accelerated by constructing certain linear combinations of several ordinary successive approximations. A sufficient condition is that the spectrum of $A$ decompose into a finite set and a subset of a sufficiently small neighborhood of zero (e. g., $A$ is compact).
[1] D. K. Faddějev V. N. Faddějevová: Numerické metody lineární algebry. SNTL, Praha 1964 (překlad z ruštiny).
[2] И. С. Березин H. П. Жидков: Методы вычислений, т. II. ФИЗМАТГИЗ, Москва 1962. Zbl 0285.34022
[3] H. Данфорд, Дж. Шварц: Линейные операторы. (общая теория); ИИЛ, Москва 1962 (перевод с английского). Zbl 1005.68507
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