### Unconventional analogs of single-parametric method of iterational aggregation

#### Abstract

When we solve practical problems that arise, for example, in mathematical economics, in the theory of Markov processes, it is often necessary to use the decomposition of operator equations using methods of iterative aggregation. In the studies of these methods for the linear equation $x = Ax + b$ the most frequent are the conditions of positiveness of the operator $A$, constant $b$ and the aggregation functions, and also the implementation of the inequality $\rho(A) <1$ for the spectral radius $ \rho (A) $ of the operator $ A $.

In this article, for an approximate solution of a system composed of the equation $x=Ax+b$ represented in the form $ x = A_1x + A_2x + b, $ where $ b \in E, $ $ E $ is a Banach space, $ A_1, A_2 $ are linear continuous operators that act from $ E $ to $ E $ and the auxiliary equation $ y = \lambda y - (\varphi, A_2x) - (\varphi, b) $ with a real variable $y$, where $ (\varphi, x) $ is the value of the linear functional $ \varphi \in E ^ * $ on the elements $ x \in E $, $ E^* $ is conjugation with space $ E $, an iterative process is constructed and investigated

$$

\begin{split}

x^{(n+1)}&=Ax^{(n)}+b+\frac{\sum\limits_{i=1}^{m}A^i_1x^{(n)}}{(\varphi, x^{(n)})\sum\limits_{i=0}^{m}\lambda^i}(y^{(n)}-y^{(n+1)}) \quad (m<\infty),\\

y^{(n+1)}&=\lambda y^{(n+1)}-(\varphi,A_2x^{(n)})-(\varphi,b).

\end{split}

$$

The conditions are established under which the sequences $ {x ^ {(n)}}, {y ^ {(n)}}$, constructed with the help of these formulas, converge to $ x ^ *, y ^ * $ as a component of solving the system constructed from equations $ x = A_1 x + A_2 x + b $ and the equation $ y = \lambda y - (\varphi, A_2 x) - (\varphi, b) $ not slower than the rate of convergence of the geometric progression with the denominator less than $1$. In this case, it is required that the operator $ A $ be a compressive and constant by sign, and that the space $ E $ is semi-ordered. The application of the proposed algorithm to systems of linear algebraic equations is also shown.

In this article, for an approximate solution of a system composed of the equation $x=Ax+b$ represented in the form $ x = A_1x + A_2x + b, $ where $ b \in E, $ $ E $ is a Banach space, $ A_1, A_2 $ are linear continuous operators that act from $ E $ to $ E $ and the auxiliary equation $ y = \lambda y - (\varphi, A_2x) - (\varphi, b) $ with a real variable $y$, where $ (\varphi, x) $ is the value of the linear functional $ \varphi \in E ^ * $ on the elements $ x \in E $, $ E^* $ is conjugation with space $ E $, an iterative process is constructed and investigated

$$

\begin{split}

x^{(n+1)}&=Ax^{(n)}+b+\frac{\sum\limits_{i=1}^{m}A^i_1x^{(n)}}{(\varphi, x^{(n)})\sum\limits_{i=0}^{m}\lambda^i}(y^{(n)}-y^{(n+1)}) \quad (m<\infty),\\

y^{(n+1)}&=\lambda y^{(n+1)}-(\varphi,A_2x^{(n)})-(\varphi,b).

\end{split}

$$

The conditions are established under which the sequences $ {x ^ {(n)}}, {y ^ {(n)}}$, constructed with the help of these formulas, converge to $ x ^ *, y ^ * $ as a component of solving the system constructed from equations $ x = A_1 x + A_2 x + b $ and the equation $ y = \lambda y - (\varphi, A_2 x) - (\varphi, b) $ not slower than the rate of convergence of the geometric progression with the denominator less than $1$. In this case, it is required that the operator $ A $ be a compressive and constant by sign, and that the space $ E $ is semi-ordered. The application of the proposed algorithm to systems of linear algebraic equations is also shown.

#### Keywords

aggregating functional, decomposition, iterative aggregation.

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