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iterative methods; convergence acceleration; Hilbert space
Let $x_{k+1}=Tx_k+b$ be an iterative process for solving the operator equation $x=Tx+b$ in Hilbert space $X$. Let the sequence $\{x_k\}^\infty _{k=o}$ formed by the above described iterative process be convergent for some initial approximation $x_o$ with a limit $x^*=Tx^*+b$. For given $l>1,m_0,m_1,\dots ,m_l$ let us define a new sequence $\{y_k\}^\infty _{k=m_1}$ by the formula $y_k=\alpha^{(k)}_0x_k+\alpha^{(k)}_1x_{k-m_1}+\ldots +\alpha^{(k)}_lx_{k-m_l}$, where $\alpha^{(k)}_i$ are obtained by solving a minimization problem for a given functional. In this paper convergence properties of $\alpha^{(k)}_i$ are investigated and on the basis of the results thus obtainded it is proved that $\lim_{k\rightarrow \infty} \left\|x^*-y_k\right\|/\left\|x^*-x_k\right\|^p=0$ for some $p\geq 1$.
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