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## Solving the Singular Semi-Sylvester Equation Using Drazin-Inverse and DGMRES Algorithm

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In this paper, we want to solve the singular semi-Sylvester equation using the Drazin-inverse and the Drazin-inverse generalized minimum residual method (DGMRES(m) algorithm). First, we transform the semi-Sylvester equation into a multiple linear systems. Then, we present the conditions and assumptions needed to apply the DGMRES(m) algorithm. We compare our proposed method with the Galerkin projection method in point of view CPU-time, accuracy and iteration number. Finally, by some numerical experiments, we show the efficiency of the proposed method.

Semi-Sylvester equations, Multiple linear systems, DGM- RES algorithm
• Autoulas, A.C., Approximation of Large-Scale Dynamical Systems, Advances in Design and Control, Siam, Philadelphia, PA, USA, 2005.
• Arnoldi, W.E., The principle of minimized iterations in the solution of the matrix eigenvalue problem, Quarterly of applied mathematics, 9(2007), 17--290.
• Baur, U., Benner, P., Cross-gramian based model reduction for data-sparse systems, Electronic Transactions on Numerical Analysis, 31(2008), 256--270.
• Ben-Israel, A., Greville, T.N., Generalized Inverses: Theory and Applications, volume 15. Springer Science \& Business Media, 2003.
• Benner, P., Factorized Solution of Sylvester Equations with Applications in Control, Sign (H), 1:2, 2004.
• Bhatia, R., Rosenthal, P., How and why to solve the operator equation axxb= y, Bulletin of the London Mathematical Society, 29(1997), 1--21.
• Campbell, S.L., Meyer, C.D., Generalized Inverses of Linear Transformations, Siam, 2009.
• Chan, T.F., Ng, M.K., Galerkin projection methods for solving multiple linear systems, SIAM Journal on Scientic Computing, 21(1999), 836--850.
• Dangarra, J., Sullivan, F., Guest Editors Introduction to The Top 10 Algorithms, Comput. Scince. Eng, 2(1):2, 2000.
• Datta, B.N., Numerical Methods for Linear Control Systems: Design and Analysis, volume 1. Academic Press, 2004.
• Guennouni, A.E., Jbilou, K., Riquet, A., Block krylov subspace methods for solving large sylvester equations, Numerical Algorithms, 29(2002), 1--3.
• Golub, G., Nash, S., Van Loan, C., A hessenberg-schur method for the problem ax+ xb= c, IEEE Transactions on Automatic Control, 24(1979), 909--913.
• Golub, G., Van Loan, C., Matrix Computations, 2nd Missing. This means that the interpolation was to be ed, 1989.
• Hoskins, W., Meek, D., Walton, D., The numerical solution of the matrix equationxa+ay= f, BIT Numerical Mathematics, 17(1977), 184--190.
• Jbilou, K., Low rank approximate solutions to large sylvester matrix equations, Applied mathematics and computation, 177(2006), 365--376.
• Karimi, S., Attarzadeh, F., A new iterative scheme for solving the semi sylvester equation, Applied Mathematics, 4(2013), 1--6.
• Lu, L., Wachspress, E.L., Solution of lyapunov equations by alternating direction implicit iteration, Computers \& Mathematics with Applications, 21(1991), 43--58.
• Robbe, M. anf Sadkane, M., Use of near-breakdowns in the block arnoldi method for solving large sylvester equations, Applied Numerical Mathematics, 58(2008), 486--498.
• Saad, Y., Iterative Methods for Sparse Linear Systems. SIAM, 2003.
• Saad, Y., Schultz, M.H., Gmres: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM Journal on scientic and statistical computing, 7(1986), 856--869,
• Sidi, A., A unied approach to krylov subspace methods for the drazin-inverse solution of singular nonsymmetric linear systems, Linear Algebra and its Applications, 298(1999), 99--113.
• Sidi, A., Dgmres: A gmres-type algorithm for drazin-inverse solution of singular non-symmetric linear systems, Linear Algebra and its Applications, 335(2001), 189--204.
• Sima, V., Algorithms for Linear-Quadratic Optimization, volume 200. CRC Press, 1996.
• Sorensen, D.C., Antoulas, A., The sylvester equation and approximate balanced reduction, Linear Algebra and its Applications, 351(2002), 671--700.
• Wei, Y., Wu, H., Additional results on index splittings for drazin inverse solutions of singular linear systems, Electronic Journal of Linear Algebra, 27(2001), 300--332.
Primary Language en Mathematics Articles Orcid: 0000-0002-3326-9458Author: Majid AdibCountry: Iran Publication Date: June 30, 2019
 Bibtex @research article { tjmcs528355, journal = {Turkish Journal of Mathematics and Computer Science}, issn = {}, eissn = {2148-1830}, address = {Matematikçiler Derneği}, year = {2019}, volume = {11}, pages = {1 - 7}, doi = {}, title = {Solving the Singular Semi-Sylvester Equation Using Drazin-Inverse and DGMRES Algorithm}, key = {cite}, author = {Adib, Majid} } APA Adib, M . (2019). Solving the Singular Semi-Sylvester Equation Using Drazin-Inverse and DGMRES Algorithm. Turkish Journal of Mathematics and Computer Science, 11 (1), 1-7. Retrieved from http://dergipark.org.tr/tjmcs/issue/46614/528355 MLA Adib, M . "Solving the Singular Semi-Sylvester Equation Using Drazin-Inverse and DGMRES Algorithm". Turkish Journal of Mathematics and Computer Science 11 (2019): 1-7 Chicago Adib, M . "Solving the Singular Semi-Sylvester Equation Using Drazin-Inverse and DGMRES Algorithm". Turkish Journal of Mathematics and Computer Science 11 (2019): 1-7 RIS TY - JOUR T1 - Solving the Singular Semi-Sylvester Equation Using Drazin-Inverse and DGMRES Algorithm AU - Majid Adib Y1 - 2019 PY - 2019 N1 - DO - T2 - Turkish Journal of Mathematics and Computer Science JF - Journal JO - JOR SP - 1 EP - 7 VL - 11 IS - 1 SN - -2148-1830 M3 - UR - Y2 - 2019 ER - EndNote %0 Turkish Journal of Mathematics and Computer Science Solving the Singular Semi-Sylvester Equation Using Drazin-Inverse and DGMRES Algorithm %A Majid Adib %T Solving the Singular Semi-Sylvester Equation Using Drazin-Inverse and DGMRES Algorithm %D 2019 %J Turkish Journal of Mathematics and Computer Science %P -2148-1830 %V 11 %N 1 %R %U ISNAD Adib, Majid . "Solving the Singular Semi-Sylvester Equation Using Drazin-Inverse and DGMRES Algorithm". Turkish Journal of Mathematics and Computer Science 11 / 1 (June 2019): 1-7. AMA Adib M . Solving the Singular Semi-Sylvester Equation Using Drazin-Inverse and DGMRES Algorithm. TJMCS. 2019; 11(1): 1-7. Vancouver Adib M . Solving the Singular Semi-Sylvester Equation Using Drazin-Inverse and DGMRES Algorithm. Turkish Journal of Mathematics and Computer Science. 2019; 11(1): 7-1.