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

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

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

___

  • [1] Autoulas, A.C., Approximation of Large-Scale Dynamical Systems, Advances in Design and Control, Siam, Philadelphia, PA, USA, 2005.
  • [2] Arnoldi, W.E., The principle of minimized iterations in the solution of the matrix eigenvalue problem, Quarterly of applied mathematics, 9(2007), 17–290.
  • [3] Baur, U., Benner, P., Cross-gramian based model reduction for data-sparse systems, Electronic Transactions on Numerical Analysis, 31(2008), 256–270.
  • [4] Ben-Israel, A., Greville, T.N., Generalized Inverses: Theory and Applications, volume 15. Springer Science & Business Media, 2003.
  • [5] Benner, P., Factorized Solution of Sylvester Equations with Applications in Control, Sign (H), 1:2, 2004.
  • [6] Bhatia, R., Rosenthal, P., How and why to solve the operator equation axxb= y, Bulletin of the London Mathematical Society, 29(1997), 1–21.
  • [7] Campbell, S.L., Meyer, C.D., Generalized Inverses of Linear Transformations, Siam, 2009.
  • [8] Chan, T.F., Ng, M.K., Galerkin projection methods for solving multiple linear systems, SIAM Journal on Scientic Computing, 21(1999), 836–850.
  • [9] Dangarra, J., Sullivan, F., Guest Editors Introduction to The Top 10 Algorithms, Comput. Scince. Eng, 2(1):2, 2000.
  • [10] Datta, B.N., Numerical Methods for Linear Control Systems: Design and Analysis, volume 1. Academic Press, 2004.
  • [11] Guennouni, A.E., Jbilou, K., Riquet, A., Block krylov subspace methods for solving large sylvester equations, Numerical Algorithms, 29(2002),
  • [12] 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.
  • [13] Golub, G., Van Loan, C., Matrix Computations, 2nd Missing. This means that the interpolation was to be ed, 1989.
  • [14] Hoskins, W., Meek, D., Walton, D., The numerical solution of the matrix equationxa+ay= f, BIT Numerical Mathematics, 17(1977), 184–190.
  • [15] Jbilou, K., Low rank approximate solutions to large sylvester matrix equations, Applied mathematics and computation, 177(2006), 365–376.
  • [16] Karimi, S., Attarzadeh, F., A new iterative scheme for solving the semi sylvester equation, Applied Mathematics, 4(2013),
  • [17] Lu, L., Wachspress, E.L., Solution of lyapunov equations by alternating direction implicit iteration, Computers & Mathematics with Applications, 21(1991), 43–58.
  • [18] 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.
  • [19] Saad, Y., Iterative Methods for Sparse Linear Systems. SIAM, 2003.
  • [20] 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,
  • [21] 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.
  • [22] 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. [23] Sima, V., Algorithms for Linear-Quadratic Optimization, volume 200. CRC Press, 1996.
  • [24] Sorensen, D.C., Antoulas, A., The sylvester equation and approximate balanced reduction, Linear Algebra and its Applications, 351(2002), 671–700.
  • [25] 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.