Abstract
Parallelized algorithms for the localization and computation of the extrema of functions applied to find approximate solutions of systems of nonlinear equations are outlined. It is shown that they can be used to find the extrema of difference solutions of systems of ordinary differential equations and to perform an analysis for Lyapunov stability. The algorithms are based on sorting a sequence with biunique correspondence of input and output indices, extrema being localized by comparing indices without error accumulation.
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Ya. E. Romm, “Sorting-based localization and stable computation of zeroes of a polynomial. I,” Cybern. Syst. Analysis, 43, No. 1, 139–154 (2007).
N. Wirth, Algorithms + Data Structures = Programs, Prentice-Hall Series in Automatic Computation, Prentice-Hall (1976).
Ya. E. Romm, “Parallel merge sort using comparisons matrices. I,” Cybern. Syst. Analysis, 30, No. 5, 631–647 (1994).
Ya. E. Romm, “Parallel merge sort using comparisons matrices. II,” Cybern. Syst. Analysis, 31, No. 4, 484–505 (1995).
A. I. Markushevich and L. A. Markushevich, Introduction to the Theory of Analytic Functions [in Russian], Prosveshchenie, Moscow (1997).
I. V. Zaika, Development and Analysis of Sorting-Based Optimization Schemes as Applied to the Identification of the Extrema of Solutions of Differential Equations, [in Russian], Author’s Abstract of PhD Theses, TRTU, Taganrog (2007).
Ya. E. Romm, I. V. Zaika, and A. A. Labintseva, Unconditional Numerical Optimization under Parameter Variations, I, Dep. VINITI March 4, 2008, No. 193-B2008, Taganrog (2008).
Ya. E. Romm, I. V. Zaika, and A. A. Labintseva, Unconditional Numerical Optimization under Parameter Variations, II, Dep. VINITI March 4, 2008, No. 194-B2008, Taganrog (2008).
Ya. E. Romm, I. V. Zaika, and A. A. Labintseva, “Unconditional sorting-based optimization as applied to the computer analysis of control systems stability,” Izv. Vuzov, Sev.-Kavkaz. Region, Tekn. Nauki, Ser. Control, Computer Engineering, Informatics, No. 6, 11–17 (2008).
I. S. Berezin and N. P. Zhidkov, Computing Methods [in Russian], Vol. 2, Fizmatgiz, Moscow (1962).
F. P. Vasil’ev, Numerical Methods to Solve Extremum Problems [in Russian], Nauka, Moscow (1988).
Ya. E. Romm, “Sorting-based localization and stable computation of zeroes of a polynomial. II,” Cybern. Syst. Analysis, 43, No. 2, 291–302 (2007).
Ya. E. Romm, I. V. Zaika, and I. A. Tyushnyakova, “Sorting-based identification of the extrema of functions as applied to computational schemes of algebra and analysis and pattern recognition,” in: Problemy Programmir., Proc. 5th Intern. Sci.-Pract. Conf. on Programming UkrPROG’2006, May 23–25, 2006, Ukraine, Kyiv, No. 2–3 (2006), p. 708–717.
Ya. E. Romm and I. V. Zaika, “Program localization of the extrema of functions and difference approximations of the solutions of differential equations,” Izv. Vuzov, Sev.-Kavkaz. Region, Tekn. Nauki, Special Issue “Mathematical modeling and computer technologies” (2005), pp. 55–61.
Ya. E. Romm, I. V. Zaika, and I. A. Solov’eva, “Software optimization method as applied to mathematical models in economics,” in: Trans. 4th Intern. Sci.-Pract. Conf. “Problems of regional control, economics, law, and innovational processes in education,” September 8–10, 2005, TIUiE, Taganrog, Vol. 2 (2005), pp. 17–26.
B. P. Demidovich and I. A. Maron, Fundamentals of Computational Mathematics [in Russian], Fizmatgiz, Moscow (1963).
Ya. E. Romm, “Multiplicative stability criteria based on difference solutions of ordinary differential equations,” Cybern. Syst. Analysis, 42, No. 1, 111–125 (2006).
Ya. E. Romm, “Modeling Lyapunov stability based on transformations of difference schemes of solutions of ordinary differential equations,” Izv. RAN, Mat. Modelirovanie, 20, No. 12, 105–118 (2008).
L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Ergebnisse Der Mathematik Und Ihrer Grenzgebiete, 16 (3 ed.), Springer-Verlag, Berlin–Heidelberg–New York (1971).
S. A. Katrich, Development and Analysis of Schemes of Program Modeling of the Stability of Solutions of Nonlinear Differential Equations based on Difference Methods [in Russian], Author’s Abstract of PhD Theses, TRTU, Taganrog (2006).
S. G. Bulanov, Development and Analysis of Methods of Program Modeling of the Stability of Systems of Linear Differential Equations based on Matrix Multiplicative Transformations of Difference Schemes: [in Russian], Author’s Abstract of PhD Theses, TRTU, Taganrog (2006).
S. S. Belokonova, Development and Analysis of Deterministic Sorting-Based Search Schemes with Application to the Identification of Various Digitized Objects [in Russian], Author’s Abstract of PhD Theses, TTI YuFU, Taganrog (2007).
O. G. Ryumin, Development and Analysis of Pattern Recognition Algorithms by Sorting-Based Determination of Extremum Indicators of Closed Loops [in Russian], Author’s Abstract of PhD Theses, TTI YuFU, Taganrog (2008).
Ya. E. Romm, A. I. Dordopulo, and V. V. Zayarnyi, Program Localization of the Zeroes of Polynomials with Application to the Identification of Objects from Sonar Data [in Russian], Izd. TGPI, Taganrog (2005).
Ya. E. Romm and V. V. Zayarnyi, “Computer processing of sonar signals and identification of objects of raster images,” Sistemy Upravl. Informats. Tekhnologii, No. 4, Issue 2(30), 290–295 (2007).
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Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 165–180, March–April 2011.
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Romm, Y.E., Zaika, I.V. Numerical sorting-based optimization as applied to general differential and nonlinear equations. Cybern Syst Anal 47, 316–329 (2011). https://doi.org/10.1007/s10559-011-9314-6
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DOI: https://doi.org/10.1007/s10559-011-9314-6