
A principle for study of quasigradient methods of approximate solving operator equations in Hilbert spaces
O. N. Evkhuta^{a}, P. P. Zabreiko^{b} ^{a} SouthRussia State Technical University
^{b} Belarusian State University
Abstract:
The article deals with nonlinear operator equations $f(x)=0$ with operators $f$ defined on a ball $B(x_0,R)$ in a Hilbert space $X$ and taking values from $X$. It is considered iterative methods of type
$x_{n+1}=x_n\Lambda(x_n)T(x_n)$, $n=0,1,2,ldots$, where $T(\xi)$ is an operator from $B(x_0,R)$ into $X$ and $\Lambda(\xi)$ a real functional on on $B(x_0,R)$. It is described conditions under that there is a phenomenon of relaxation of residuals: $\f(x_{n+1}\<\f(x_n)\$. The study of the convergence of iterations and its rate us reduce to the analysis of a scalar function; the graph of this function determines as the conditions of the convergence of iterations well as the rate of this convergence; moreover, it allows to write simple a priori and a posteriori estimates of errors. The general scheme covers classical methods of minimal residuals, of steepest descent, of minimal errors and some others.
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UDC:
513.88 Received: 01.09.2010
Citation:
O. N. Evkhuta, P. P. Zabreiko, “A principle for study of quasigradient methods of approximate solving operator equations in Hilbert spaces”, Tr. Inst. Mat., 19:1 (2011), 32–44
Citation in format AMSBIB
\Bibitem{EvkZab11}
\by O.~N.~Evkhuta, P.~P.~Zabreiko
\paper A principle for study of quasigradient methods of approximate solving operator equations in Hilbert spaces
\jour Tr. Inst. Mat.
\yr 2011
\vol 19
\issue 1
\pages 3244
\mathnet{http://mi.mathnet.ru/timb137}
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