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Тезисы докладов

Вычислительная алгебра

In many problems of processing of the experimental data, parametric identification there is a necessity of the solution of linear algebraic equations systems (LAES) of a kind $K varphi = f$. As a rule, matrix $K$ is ill - conditioned, and instead of an exact vector $f$ the vector $tilde{f}=f+eta$ is given, where $eta$- random vector with zero mean and correlation matrix $V_{eta}=diag{ sigma_{1}^{2} ,; sigma_{2}^2 ,; ... ,; sigma_{N}^{2} }$. For construction of the steady solution will use different regularization algorithms, including recurrent algorithms of a kind: begin{equation} varphi^{(n)} = varphi^{(n-1)} + frac{P^{(n)}}{sigma_{n}^{2}} k_{n}^{T} biggl( tilde{f}_{n} - k varphi^{(n-1)} biggr) ; ; P^{(n)} = P^{(n-1)} - frac{P^{(n-1)}k_{n}^{T}k_{n}P^{(n-1)}}{sigma_{n}^{2} + k_{n}P^{(n-1)}k_{n}^{T}} , end{equation} $n = 1, ; 2,; ...$, with a point of "start" $varphi^{(0)} , ; P^{(0)}$. The main problems arising at usage of these algorithms in practice, are: a task of "starting point " of algorithm and break timing (or break iteration number) of algorithm. In the given report for the solution of these problems are entered following the accuracy characteristics [1]: $U_b (n) , ; U_{xi} (n)$ . Let's $varphi$ -- vector of the exact solution, and $varphi^{(n)}$ -- estimation for $varphi$, constructed by recurrent algorithm (1) on $n-$ ohm a step. Let's define mean-square error (MSE) of solution $varphi^{(n)}$ by a functional $Delta (n) = M Bigl[ Bigl| varphi^{(n)} - varphi Bigr|^2 Bigr]$ , where $M[cdot ]$ -- operator of expectation. Then, using the accuracy characteristics $U_b (n) , ; U_{xi} (n)$ , it is possible to write $$ Delta (n) = U_b (n) cdot Bigl| varphi^{(0)} - varphi Bigr|^2 + U_{xi} (n) cdot sigma_{n}^{2} ; , $$ where $Bigl| varphi^{(0)} - varphi Bigr|^2$ -- MSE of the "starting solution" $varphi^{(0)}$, $sigma_{n}^{2}$ -- the measuring error dispersion of a projection $tilde{f}_{n}$ . The recurrent algorithms of calculation of the accuracy characteristics are offered. In report four variational problems are formulated, the solution which one allows to define number of a step, on which one the recurrent algorithm (1) has demanded the accuracy characteristic. For example, solution of a variational problem $$ minBigl{ U_{b} (n) cdot b_{max} + U_{xi} (n) cdot sigma_{max}^{2} Bigr} $$ gives number of a step, on which one the minimum MSE on the class of the solutions which are satisfying condition $Bigl| varphi^{(0)} - varphi Bigr|^2 le b_{max}$ and on the class of errors $sigma_{n}^{2} le sigma_{max}^{2}$ is reached.[5mm] 1. Voskoboinikov Yu. E. Accuracy Characteristics and Synthesis of Recurrent Signal Restoration Algorithms // Optoelectronics, Instrumentation and Data Processing. 2001.- N 2

Примечание. Тезисы докладов публикуются в авторской редакции

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