Steffensen method for solving nonlinear matrix equation $X+A^T X^{(-1)}A=Q$

Subject Areas : Linear and multilinear algebra; matrix theory

A. Nazari ¹ , Kh. Sayehvand ² , M. Rostami ³

1 - Department of Mathematics, Arak University, P.O. Box 38156-8-8349, Arak, Iran
2 - Faculty of Mathematical Sciences, University of Malayer, P. O. Box 16846-13114, Malayer, Iran
3 - Faculty of Mathematical Sciences, University of Malayer, P. O. Box 16846-13114, Malayer, Iran

Received: 2014-10-08 Accepted : 2015-01-17 Published : 2014-12-01

Keywords: fixed point method, Nonlinear matrix equations, Steffensen method,

Abstract :

In this article we study Steffensen method to solve nonlinear matrix equation $X+A^T X^{(-1)}A=Q$,when $A$ is a normal matrix. We establish some conditionsthat generate a sequence of positive definite matrices which converges to solutionof this equation.

References:

[1] R. Bhatia, Matrix analysis , Springer, Berlin, 1997.

 

[2] Burden, Richard. L,Numerical analysis, 6th, ed. 1997.

 

[3] Fuzhen Zhang, Matrix Theory Basic Result and Techniques, Springer, 1999.

 

[4] J. C. Engwerda, On the existence of a positive definite solution of the matrix equation $X + A^TX^{-1}A$ = I$, Linear Algebra App. 194 (1993) 91-108.

 

[5] S. M. El-Sayed, A. M. Al-Dbiban, A new inversion free iteration for solving the equation $X+A^TX^-{1}A = Q$, Linear Algebra App. 181 (2005) 148-156.

 

[6] X. Zhan, J. Xie, On the Matrix Equation X + ATX􀀀1A = I, Linear Algebra Appl. 247, (1996) 337-345.

 

[7] D. V. Ouellette, Schur complements and statistics, Linear Algebru Appl. 36 (1981) 187-295.

 

[8] W. N. Anderson, T. D. Morley, and G. E. Trapp, Ladder networks, futed Points, and the geometric mean, Circuits Systems Signal Process. 3 (1983) 259-268.

 

[9] T. Ando, Limit of cascade iteration of matrices, Numer. Funct. Anal. Optim. 21 (1980) 579-589.

 

[10] R. S. Bucy, A priori bound for the Riccati equation, in Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. III.

 

[11] W. L. Green and E. Kamen, Stabilization of linear Systems over a commutative normed algebra with applications to spatially distributed Parameter dependent Systems, SIAM J. Control Optim. 23 (1985) 1-18.

 

[12] W. Pusz and S. L. Woronowitz, Funcitonal calculus for sequilinear forms and the purication map, Rep. Math. Phys. 8 (1975) 159-170.

 

[13] G. E. Trapp, The Ricatii equation and the geometric mean, Contemp. Math. 47 (1985) 437-445.

 

[14] J. Zabezyk, Remarks on the control of discrete time distributed paramter Systems, SIAM J. Control 12 (1974) 721-735.

 

[15] J. Zemanian, Non-uniform semi-infinite grounded grids, SIAM J. Appl. Math. 13 (1982) 770-788.

 

[16] R. A. Horn and C. A. Johnson, Matrix Analysis, Cambridge, U.P. Londan, 1985.

 

[17] R. Bhatia, C. Davis, More matrix forms of the arithmetic-geometric mean inequality, SIAM J. Matrix Anal. Appl. 14 (1993) 132-136.

 

[18] J. C. Engwerda, A. C. M. Ran, and A. L. Rijkeboer, Necessary and sufficient conditions for the existente of a positive definite solution of the matrix equation X + AX􀀀1A = Q, Linear Algebra Appl. 186 (1993) 255-275.