Geodesic
Convergence of extrapolation coefficients
Applications of Mathematics, Tome 29 (1984) no. 2, pp. 114-133
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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 $p\geq 1$.
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 $p\geq 1$.
DOI : 10.21136/AM.1984.104075
Classification : 47A50, 65J10
Keywords: iterative methods; convergence acceleration; Hilbert space 
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     author = {Z{\'\i}tko, Jan},
     title = {Convergence of extrapolation coefficients},
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     year = {1984},
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     zbl = {0577.65044},
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     url = {http://geodesic.mathdoc.fr/articles/10.21136/AM.1984.104075/}
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Zítko, Jan. Convergence of extrapolation coefficients. Applications of Mathematics, Tome 29 (1984) no. 2, pp. 114-133. doi: 10.21136/AM.1984.104075

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