The Method of Organizing the Iterative Process of the System of the Linear Algebraic Equations Solution Excluding the Multidigit Multiplication Operation

Pavel Pavlovich Kravchenko and Liubov Vladimirovna Pirskaya

Southern Federal University, Russia, 344006, Rostov-on-Don, Bolshaya Sadovaya str., 105/42,

ABSTRACT: An iterative method of the systems of the linear algebraic equations solution, excluding the multidigit multiplication operation while designing the special-purpose computing facilities is considered. Algorithmization of the iteration process is based on the use of the first order delta-modulations with the variable quantum that can significantly reduce the number of iterations in comparison with the using of the constant quantum. The main theoretical principles justifying the approximate solution of the problem of minimizing the number of iterations when using the variable quantum are highlighted for the first time. The optimal estimations characterizing the duration of idealized iterative loops with constant quantum of a certain value are formed in theoretical justification quasi-optimal conditions determining a way of the representation a sequence of variable quanta values in the loops and based on using two or four idealized iterations within each loop are developed for the purpose of the real processes. Besides the quanta values shall be represented in the form of 2-S, S E N, that allows at each iteration submitting the multiplication of the matrix coefficient by the quantum represent in the form of a shift operation by S binary digits. The introduction of this way of representing the quantum allows realizing the implementation of the iterative process without using the multiplication of the multi-digit codes. There are also effective ways to complete the iteration in each loop, allowing, in particular, to reduce the number of iterations in the loop per unit. The findings of investigation of the iterative solution of various systems of the linear algebraic equations different with convergence rate are given in the work. The possibility of reduction the number of iterations in comparison with the using the delta-modulation with the constant quantum to hundreds & thousands of times is shown when ensuring the identical accuracy. When performing the experiments the attention to the manifestation of the effective usage of the first proposed method of four idealized iterations in the loop is paid. The number of iterations with variable quantum is to a considerable extent approximate the number of iterations to the simple iteration method.

KEYWORDS: Iteration methods; systems of the linear algebraic equations; the first order delta-modulation; special-purpose control unit

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