Computation of Sylvester and Stein matrix equations' solutions by Iterative Decreasing Dimension Method

Year 2019, Volume: 7 Issue: 2, 486 - 491, 15.10.2019

Oğuzer Sinan

Abstract

Sylvester matrix equation  and the  Stein matrix equation which have very important applications in the stability analysis of continuous-time and discrete-time linear systems, respectively are studied. The solutions of these equations  be possible  demonstrated by  a number of methods. Been one of them transformation  to the  system of linear equation  was examined. Algorithms that provide transformation to $Ax=f$ system of linear equation was introduced. Iterative Decreasing Dimension  Method ($IDDM$) algorithm was applicated for the solution of the obtained system. The $IDDM$ algorithm has been evaluated according to the Gaussian elimination type algorithms. The $IDDM$ algorithm is evade from  divisor of zero based on its algorithmic structure. Depending on the study, codes for calculation were prepared in the MAPLE program code development environment.

Keywords

Sylvester matrix equation, Stein matrix equation, IDDM, Kronecker product

References

• [1] Akın, O. and Bulgak, H., Linear Difference Equations and Stability Theory [in Turkish], Selc¸uk University, Research Center of Applied Mathematics, Konya, (1998).
• [2] Alexander, G., Kronecker Products and Matrix Calculus with Applications, John Wiley & Sons, N.Y., (1981).
• [3] K. Aydın, G.C . Kızıltan, A.O. C¸ ıbıkdiken,Generalized iterative decreasing method., European Journal of Pure and Aplied Mathematics, Vol:3, No:5,(2010), 819-830.
• [4] R. Bartels and G. Stewart, Solution Of The Matrix Equation AX+XB=C, Communications of the ACM., Vol.15, No.9, (1972), 820-826.
• [5] A. Bulgak, G. Demidenko, ,I. Matveeva, On location of the matrix spectrum inside an ellipse., Selcuk Journ. Appl. Math., Vol:4, No.1 (2003), 35-41.
• [6] Bulgak, H., Pseudo eigenvalues, spectral portrait of a matrix and their connections with different criteria of stability, in: Error Control and Adaptivity in Scientific Computing, H. Bulgak and C. Zenger, eds., NATO Science Series, Kluwer Academic Publishers, (1999), 95-124.
• [7] H. Bulgak, and D. Eminov, Computer dialogue system MVC, Selcuk Journ. Appl. Math., Vol:2, No.2 (2001), 17-38.
• [8] A. Duman, K. Aydın, Sensitivity of Shur stability of systems of linear difference equations with constant coefficients, Scientific Research and Essays, Vol.6, No.28, (2011), 5846-5854.
• [9] A. Duman, K. Aydın, Sensitivity of Hurwitz stability of linear differential equation systems with constant coefficients, International Journal of Geometric Methods in Modern Physics, Vol.14, No.6, (2017), 1750084
• [10] Godunov, S.K., Antonov, A.G, Kiriljuk O.P and Kostin, V.I.,Guaranteed Accuracy in Numerical Linear Algebra, Kluwer Acaden’c Publishers, Dordrecht, Netherlands, (1993).
• [11] G.H. Golub, S. Nash, Van C.F.Loan, A Hessenberg-Schur Method For The Problem AX +XB = C, IEEE TransAutomat Control, Vol.24, No.6, (1979), 909-913.
• [12] Golub, G.H. and Van Loan, C.F., Matrix Computations, The Johns Hopkins University Press, Baltimore,MD, (2013).
• [13] T. Keskin and K. Aydın, Iterative decreasing dimension algorithm., Computers and Mathematics with Appl., Vol:53, No.1 (2007), 1153-1158.
• [14] A. Sadeghi, A new approach for computing the solution of Sylvester matrix equation., Journal of Interpolation and Approximation in Scientific Computing.,Vol.2016, No.2, (2016), 66-76.
• [15] V. Simoncini, Computational Methods for Linear Matrix Equations., SIAM Review., Vol.58, No.3, (2016), 377–441.