# Article

Full entry | PDF   (2.2 MB)
Keywords:
iterative methods; convergence acceleration; Hilbert space
Summary:
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 $p\geq 1$.
References:
[1] J. Zítko: Improving the convergence of iterative methods. Apl. Mat. 28 (1983), 215-229. MR 0701740
[2] J. Zítko: Kellogg's iterations for general complex matrix. Apl. Mat. 19 (1974), 342-365. MR 0368406 | Zbl 0315.65025
[3] G. Maess: Iterative Lösung linear Gleichungssysteme. Deutsche Akademie der Naturforscher Leopoldina Halle (Saale), 1979. MR 0558164
[4] G. Maess: Extrapolation bei Iterationsverfahren. ZAMM 56, 121-122 (1976). DOI 10.1002/zamm.19760560210 | MR 0426417
[5] I. Marek J. Zítko: Ljusternik Acceleration and the Extrapolated S.O.R. Method. Apl. Mat. 22 (1977), 116-133. MR 0431667
[6] I. Marek: On a method of accelerating the convergence of iterative processes. Journal Соmр. Math. and Math. Phys. 2 (1962), N2, 963-971 (Russian). MR 0152112
[7] I. Marek: On Ljusternik's method of improving the convergence of nonlinear iterative sequences. Comment. Math. Univ. Carol, 6 (1965), N3, 371-380. MR 0196901
[8] A. E. Taylor: Introduction to Functional Analysis. J. Wiley Publ. New York 1958. MR 0098966 | Zbl 0081.10202

Partner of