# Article

 Article title SOLUTION OF SYSTEMS OF LINEAR ALGEBRAIC EQUATIONS BY THE METHOD OF SUMMATION OF DIVERGENT SERIES Authors V.V. Selyankin, V.I. Shmoylov Section SECTION II. MODELING OF COMPLEX SYSTEMS Month, Year 06, 2015 @en Index UDC 512.644:517.524 DOI Abstract Provides a method for solving systems of linear algebraic equations, which is based on the summation of the corresponding continued fractions. The resulting algorithm simple iteration approximation xi(0), xi(1), xi(2),..., xi(k) shall be considered as partial sums that converges if the iterative process converges and diverges - otherwise. For these partial sums constructed series, the first element of which pi(0) will coincide with хi(0), and subsequent members of the series pi(k) are difference xi(k)-xi(k-1). A number of construction, or associated with this next iteration process may converge slowly or even diverge. The summation of these series generated simple iteration algorithm, implemented the correct C-fractions, and constructed by series regular C-fractions are finite. Thus, the proposed algorithm for solving linear algebraic equation is classified as direct algorithms providing an exact solution of a finite number of operations. For exact solutions of linear algebraic equation is necessary to use the results of 2n iterations, where n - the dimension of the linear algebraic equation. At the same time, the proposed algorithm can provide and approximate solutions of system of linear equations when the number of iterations N<<2n, so we consider the algorithm as an iterative, with the high rate of convergence. Examples of solutions SLAE and evaluated the effectiveness of the algorithm. Download PDF Keywords Acceleration of the convergence of iterative algorithms; continued fractions; and the summation of divergent series; exact methods SLAE References 1. Il'in V.P. Metody nepolnoy faktorizatsii dlya resheniya algebraicheskikh system [Methods incomplete factorization for the solution of algebraic systems]. Moscow: Fizmatlit, 1995, 233 p. 2. Skorobogat'ko V.Ya. Teoriya vetvyashchikhsya tsepnykh drobey i ee prilozhenie v vychislitel'noy matematike [Branching continued fractions and its application in computational mathematics]. Moscow: Nauka, 1983, 312 p. 3. Khardi G. Raskhodyashchiesya ryady [Divergent series]. Moscow: Izd-vo inostr. lit-ry, 1951, 504 p. 4. Shmoylov V.I. Nepreryvnye drobi [Continuous fractions]. In 3 vol. Vol. 3. Iz istorii nepreryvnykh drobey [From the history of continued fractions]. Lviv: Merkator, 2004, 520 p. 5. Dzhouns U., Tron V. Nepreryvnye drobi. Analiticheskaya teoriya i prilozheniya [Continuous fractions. Analytic theory and applications]. Moscow: Mir, 1985, 414 p. 6. Khovanskiy A.N. Primenenie tsepnykh drobey i ikh obobshcheniy k voprosam priblizhennogo analiza [The application of continued fractions and their generalizations to the issues of approximate analysis]. Moscow: Gostekhizdat, 1956, 203 p. 7. Shmoylov V.I. Nepreryvnye drobi [Continuous fractions]. In 3 vol. Vol. 2. Raskhodyashchiesya nepreryvnye drobi [Divergent continuous fractions]. Lviv: Merkator, 2004, 558 p. 8. Khloponin S.S. Priblizhenie funktsiy tsepnymi drobyami [Approximation of functions chain fractions]. Stavropol': Izd-vo SGPI, 1977, 102 p. 9. Rutiskhauzer G. Algoritm chastnykh i raznostey [The algorithm and private differences]. Moscow: IIL, 1960, 93 p. 10. Shmoylov V.I. Summirovanie raskhodyashchikhsya tsepnykh drobey [Summation of divergent continued fractions]. Lviv: IPPMMNAN Ukrainy, 1997, 23 p. 11. Shmoylov V.I., Marchuk M.V., Tuchapskiy R.I. Nepreryvnye drobi i nekotorye ikh primeneniya [Continuous fractions and their application.]. Lviv: Merkator, 2003, 784 p. 12. Shmoylov V.I. Nepreryvnye drobi [Continuous fractions]. In 3 vol. Vol. 1. Periodicheskie nepreryvnye drobi [Periodic continuous fractions], Nats. akad. nauk Ukrainy, In-t prikladnykh problem mekhaniki i matematiki [National Academy of Sciences of Ukraine, Institute of applied problems of mechanics and mathematics]. Lviv: Yu 2004, 645 p. 13. Shmoylov V.I. Raskhodyashchiesya sistemy lineynykh algebraicheskikh uravneniy [Divergent systems of linear algebraic equations]. Taganrog: Izd-vo TTI YuFU, 2010, 205 p. 14. Shmoylov V.I. Reshenie algebraicheskikh uravneniy pri pomoshchi r/φ-algoritma [The solution of algebraic equations using r/φ-algorithm]. Taganrog: Izd-vo TTI YuFU, 2011, 330 p. 15. Shmoylov V.I. Nepreryvnye drobi i r/φ-algoritm [Continuous fractions and r/φ-algorithm]. Taganrog: Izd-vo TTI YuFU, 2012, 608 p. 16. Shmoylov V.I, Kovalenko V.B. Nekotorye primeneniya algoritma summirovaniya raskhodyashchiesya nepreryvnykh drobey [Some applications of the algorithm of summation of divergent continued fractions], Vestnik Yuzhnogo nauchnogo tsentra RAN [Bulletin of the South scientific center of RAS], 2012, Vol. 8, No. 4, pp. 3-13. 17. Shmoylov V.I.,Savchenko D.I. Algoritm summirovaniya raskhodyashchikhsya nepreryvnykh drobey [The algorithm of summation of divergent continued fractions], Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya: Fizika. Matematika [Proceedings of Voronezh State University. Series: Physics. Mathematics], 2013, No. 2, pp. 258-276. 18. Guzik V.F., Shmoylov V.I., Kirichenko G.A. Nepreryvnye drobi i ikh primenenie v vy-chislitel'noy matematike [Continuous fractions and their application in computational mathematics], Izvestiya YuFU. Tekhnicheskie nauki [Izvestiya SFedU. Engineering Sciences], 2014, No. 1 (150), pp. 158-174. 19. Shmoylov V.I., Redin A.A., Nikulin N.A. Nepreryvnye drobi v vychislitel'noy matematike [Continuous fractions in computational mathematics]. Rostov-on-Don: Izd-vo YuFU, 2015, 228 p. 20. Kirichenko G.A., Shmoylov V.I. Algoritm summirovaniya raskhodyashchikhsya nepreryvnykh drobey i nekotorye ego primeneniya [The algorithm of summation of divergent continued fractions and its application], ZhVM i MF [Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki], 2015, Vol. 55, No. 4, pp. 558-573.