Sage Journals: Discover world-class research

Abstract

The two-parameter generalized Hermitian and skew-Hermitian splitting (TGHSS) iteration method is applied to solve the continuous Sylvester equation $AX + XB = C$ , where $A \in C^{m \times m}$ , $B \in C^{n \times n}$ and $C \in C^{m \times n}$ . The matrix-vector form of the equation is considered to present convergence properties of the TGHSS method. Some numerical examples are given to show the effectiveness and accuracy of the method. Comparison with other methods is also included.

Keywords

Continuous Sylvester equation HSS method GHSS method TGHSS method convergence

Get full access to this article

View all access options for this article.

References

Aghazadeh

Bastani

Salkuyeh

(2015) Generalized Hermitian and skew-Hermitian splitting iterative method for image restoration. Applied Mathematical Modelling 39: 6126–6138.

Aghazadeh

Salkuyeh

Bastani

(2016) Two-parameter generalized Hermitian and skew-Hermitian splitting iteration method. International Journal of Computer Mathematics 93: 1119–1139.

Anderson

BDO

Agathoklis

Jury

et al . (1986) Stability and the matrix Lyapunov equation for discrete 2-dimensional system. IEEE Transactions on Circuits and Systems 33: 261–267.

Bai

Z-Z

(2009) Optimal parameters in the HSS-like methods for saddle-point problems. Numerical Linear Algebra with Applications 16: 447–479.

Bai

Z-Z

(2011) On Hermitian and skew-Hermitian splitting iteration methods for continuous Sylvester equations. Journal of Computational Mathematics 29: 185–198.

Bai

Z-Z

Golub

(2007) Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems. IMA Journal of Numerical Analysis 27: 1–23.

Bai

Z-Z

Hadjidimos

(2014) Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix. Linear Algebra and its Applications 463: 322–339.

Bai

Z-Z

(2012) Erratum. Numerical Linear Algebra with Applications 19: 891.

Bai

Z-Z

Yang

(2009) On HSS-based iteration methods for weakly nonlinear systems. Applied Numerical Mathematics 59: 2923–2936.

10.

Bai

Z-Z

Deng

Y-B

Gao

Y-H

(2006a) Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations. Numerical Linear Algebra with Applications 13: 801–823.

11.

Bai

Z-Z

Golub

C-K

(2006b) Optimal parameter in Hermitian and skew-Hermitian splitting method for certain two-by-two block matrices. SIAM Journal on Scientific Computing 28: 583–603.

12.

Bai

Z-Z

Golub

(2003) Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM Journal on Matrix Analysis and Applications 24: 603–626.

13.

Bai

Z-Z

Golub

(2007) On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iterations. Numerical Linear Algebra with Applications 14: 319–335.

14.

Bai

Z-Z

Golub

(2008) On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. Linear Algebra and its Applications 428: 413–440.

15.

Bai

Z-Z

Golub

Pan

J-Y

(2004) Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numerische Mathematik 98: 1–32.

16.

Bai

Z-Z

Golub

L-Z

et al . (2005) Block triangular and skew-Hermitian splitting methods for positive-definite linear systems. SIAM Journal on Scientific Computing 26: 844–863.

17.

Bartels

Stewart

(1972) Solution of the matrix equation AX+XB = C. Communications of the ACM 15: 820–826.

18.

Beckermann

(2011) An error analysis for rational Galerkin projection applied to the Sylvester equation. SIAM Journal on Numerical Analysis 49: 2430–2450.

19.

Benner

Truhar

(2009) On the ADI method for Sylvester equations. Journal of Computational and Applied Mathematics 233: 1035–1045.

20.

Benzi

(2009) A generalization of the Hermitian and skew-Hermitian splitting iteration. SIAM Journal on Matrix Analysis and Applications 31: 360–374.

21.

Benzi

Golub

(2004) A preconditioner for generalized saddle point problems. SIAM Journal on Matrix Analysis and Applications 26: 20–41.

22.

Benzi

(2006) Preconditioned iterative methods for weighted Toeplitz least squares problems. SIAM Journal on Matrix Analysis and Applications 27: 1106–1124.

23.

Bertaccini

Golub

Serra-Capizzano

et al . (2005) Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation. Numerische Mathematik 99: 441–484.

24.

Ding

Chen

(2005) Iterative least-squares solutions of coupled Sylvester matrix equations. Systems & Control Letters 54: 95–107.

25.

Evans

Galligani

(1994) A parallel additive preconditioner for conjugate gradient method for AX+XB = C. Parallel Computing 20: 1055–1064.

26.

Gallivan

Vandendorpe

Dooren

(2004) Sylvester equations and projection-based model reduction. Journal of Computational and Applied Mathematics 162: 213–229.

27.

Gantmacher

(1959) The Theory of Matrices. New York, NY: Chelsea Publishing Company.

28.

Guennouni

Jbilou

Riquet

(2002) Block Krylov subspace methods for solving large Sylvester equations. Numerical Algorithms 29: 75–96.

29.

Hached

(2015) A note on the Davison-Man method for Sylvester matrix equations. Computational and Applied Mathematics. Epub ahead of print 5 June 2015. DOI: 10.1007/s40314-015-0244-1.

30.

Halanay

Răsvan

(1993) Applications of Lyapunov Methods in Stability. Berlin: Kluwer Academic Publishers.

31.

Ilic

M-D

(1989) New approaches to voltage monitoring and control. IEEE Control Systems Magazine 9: 5–11.

32.

Johnson

Krukier

(2009) General resolution of a convergence question of L. Krukier. Numerical Linear Algebra with Applications 16: 949–950.

33.

Lancaster

Tismenetsky

(1985) The Theory of Matrices. 2nd edn. Orlando, FL: Academic Press.

34.

Lewis

F-L

Mertzios

V-G

Vachtsevanos

et al . (1990) Analysis of bilinear systems using Walsh functions. IEEE Transactions on Automatic Control 35: 119–123.

35.

Niu

Wang

L-Z

(2011) A relaxed gradient based algorithm for solving Sylvester equations. Asian Journal of Control 13: 461–464.

36.

Robbé

Sadkane

(2002) A convergence analysis of GMRES and FOM methods for Sylvester equations. Numerical Algorithms 30: 71–89.

37.

Salkuyeh

Toutounian

(2006) New approaches for solving large Sylvester equations. Applied Mathematics and Computation 173: 9–18.

38.

Sanches

Nascimento

Marques

(2008) Medical image noise reduction using the Sylvester-Lyapunov equation. IEEE Transactions on Image Processing 17: 1522–1539.

39.

Schaft

AVD

(2000) L₂-Gain and Passivity Techniques in Nonlinear Control. 2nd edn. London: Springer-Verlag.

40.

Sorensen

Antoulas

(2002) The Sylvester equation and approximate balanced reduction. Linear Algebra and its Applications 351: 671–700.

41.

Wang

Dai

Liao

(2012) A modified gradient based algorithm for solving Sylvester equations. Applied Mathematics and Computation 218: 5620–5628.

42.

Wang

W-W

Mao

L-Z

(2013) On positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equation AX+XB = C. Computers & Mathematics with Applications 66: 2352–2361.

43.

Zheng

Q-Q

C-F

(2014) On normal and skew-Hermitian splitting iteration methods for large sparse continuous Sylvester equations. Journal of Computational and Applied Mathematics 268: 145–154.

44.

Zhou

Wang

Tang

X-B

(2015) A generalization of the Hermitian and skew-Hermitian splitting iteration method for solving Sylvester equations. Applied Mathematics and Computation 271: 609–617.

A new generalization of the Hermitian and skew-Hermitian splitting method for solving the continuous Sylvester equation

Abstract

Keywords

Get full access to this article

References