Article

Article title THE DETERMINATION OF COMPLEX SOLUTIONS OF ISLAE WITH A THREE-DIAGONAL MATRIX
Authors V. I. Shmoylov, D. V. Timoshenko, V. V. Grivtsov
Section SECTION I. METHODS, MODELS AND ALGORITHMS OF INFORMATION PROCESSING
Month, Year 05, 2018 @en
Index UDC 517.524
DOI
Abstract The article deals with infinite systems of linear algebraic equations (ISLAE) and gives examples of solutions of such systems. The use of r/φ-algorithm allows to find complex solutions of ISLAE, if they exist, but does not provide known algorithms for solving systems. Discussed is the summation method of divergent continued fractions. This method is different from classic methods of summation, because it makes it possible to get complex number for the real sequence of convergents fractions, which is represented this continued fraction. An indication of the complexity of such divergent continued fractions with real elements are signs changing in its appropriate fractions, and these signs changes occurs any number of times. Differently, complex unit ei take from appropriate fractions of continued fractions “activity”. Parameters of the complex number =r_0 e^(iφ_0 ) : module r_0 and argument φ_0, can be identified by r/φ-algorithm. Differently from a classical defenition of continued fraction convergence, the convergence of continued fractions, identified by r/φ-algorithm, assumes for continued fractions with real elements both real and complex values. The solutions of SLAE with tegialagonal matrices are written according to Cramer"s formulas by the relations of determinants, which are reduced to the ratio of tridiagonal determinants (n + 1) and n-th orders, which are known to be continuous fractions. These continued fractions can be either convergent or divergent, depending on the coefficients of the real SLAE matrix. The summation by the r / φ-algorithm of divergent continued fractions showed that the divergent continuous fractions in the classical sense have complex values. We give a comparative analysis of the effectiveness of two algorithms for solving divergent ISLAE, an algorithm based on the reduction procedure, and an algorithm providing fast calculations of a series of suitable fractions necessary for the realization of the r/φ-algorithm. An algorithm is investigated that provides a quick calculation of a series of suitable fractions.

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Keywords Infinite systems of linear algebraic equations; divergent continuous fractions; r/φ-algorithm.
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