Positive Definite Solutions for the System of Nonlinear Matrix Equations *X* + *A*^{*}Y^{-n}A = *I*, *Y* + *B*^{*}X^{-m}B = *I*

Affiliation(s)

Department of Scientific Computing, Faculty of Computers and Informatics, Benha University, Benha, Egypt.

Faculty of Science and Arts, Qassim University, Qassim, KSA.

Department of Scientific Computing, Faculty of Computers and Informatics, Benha University, Benha, Egypt.

Faculty of Science and Arts, Qassim University, Qassim, KSA.

ABSTRACT

In this paper, some properties of the positive
definite solutions for the nonlinear system of matrix equations *X *+ *A ^{*}Y*

KEYWORDS

System of Nonlinear Matrix Equations, Iterative Methods, Monotonic Sequence, Positive Definite Matrices

System of Nonlinear Matrix Equations, Iterative Methods, Monotonic Sequence, Positive Definite Matrices

Cite this paper

El-Sayed, S. and Al-Dubiban, A. (2014) Positive Definite Solutions for the System of Nonlinear Matrix Equations*X* + *A*^{*}Y^{-n}A = *I*, *Y* + *B*^{*}X^{-m}B = *I*. *Applied Mathematics*, **5**, 1977-1987. doi: 10.4236/am.2014.513193.

El-Sayed, S. and Al-Dubiban, A. (2014) Positive Definite Solutions for the System of Nonlinear Matrix Equations

References

[1] Anderson Jr., W.N., Morley, T.D. and Trapp, G.E. (1990) Positive Solutions to*X* = *A* _ *BX*^{-1}B^{*}. Linear Algebra and Its Applications, 134, 53-62.

http://dx.doi.org/10.1016/0024-3795(90)90005-W

[2] Lancaster, P. and Rodman, L. (1995) Algebraic Riccati Equations. Oxford Science, Oxford.

[3] Meini, B. (2000) Matrix Equations and Structures: Efficient Solution of Special Discrete Algebraic Riccati Equations. Proceedings of the WLSSCOO, Bulgaria, 2000.

[4] Engwerda, J.C. (1993) On the Existence of a Positive Definite Solution of the Matrix Equation*X* + *A*^{*}X^{-1}A = *I*. Linear Algebra and Its Applications, 194, 91-108.

http://dx.doi.org/10.1016/0024-3795(93)90115-5

[5] Hasanov, V.I. and Ivanov, I.G. (2004) Solutions and Perturbation Estimates for the Matrix Equations*X* ±*A*^{*}X^{-n}A = *Q*. Applied Mathematics and Computation, 156, 513-525.

http://dx.doi.org/10.1016/j.amc.2003.08.007

[6] Hasanov, V.I. and El-Sayed, S.M. (2006) On the Positive Definite Solutions of Nonlinear Matrix Equation*X* + *A*^{*}X^{-δ}A = *Q*. Linear Algebra and Its Applications, 412, 154-160.

http://dx.doi.org/10.1016/j.laa.2005.06.026

[7] Ivanov, I.G. and El-Sayed, S.M. (1998) Properties of Positive Definite Solutions of the Equation*X* + *A*^{*}X^{-2}A = *I*. Linear Algebra and Its Applications, 279, 303-316.

http://dx.doi.org/10.1016/S0024-3795(98)00023-8

[8] Ivanov, I.G. (2006) On Positive Definite Solutions of the Family of Matrix Equations*X* + *A*^{*}X^{-n}A = *Q*. Journal of Computational and Applied Mathematics, 193, 277-301.

http://dx.doi.org/10.1016/j.cam.2005.06.007

[9] Costa, O.L.V. and Marques, R.P. (1999) Maximal and Stabilizing Hermitian Solutions for Discrete-Time Coupled Algebraic Riccati Equations. Mathematics of Control, Signals and Systems, 12, 167-195.

http://dx.doi.org/10.1007/PL00009849

[10] Czornik, A. and Swierniak, A. (2001) Lower Bounds on the Solution of Coupled Algebraic Riccati Equation. Automatica, 37, 619-624.

http://dx.doi.org/10.1016/S0005-1098(00)00196-5

[11] Czornik, A. and Swierniak, A. (2001) Upper Bounds on the Solution of Coupled Algebraic Riccati Equation. Journal of Inequalities and Applications, 6, 373-385.

[12] Mukaidani, H., Yamamoto, S. and Yamamoto, T. (2008) A Numerical Algorithm for Finding Solution of Cross-Coupled Algebraic Riccati Equations. IEICE Transactions, 91, 682-685.

http://dx.doi.org/10.1093/ietfec/e91-a.2.682

[13] Al-Dubiban, A.M. (2008) Iterative Algorithms for Computing the Positive Definite Solutions for Nonlinear Matrix Equations. Ph.D. Thesis, Riyadh University for Girls, Riyadh.

[14] Al-Dubiban, A.M. (2012) Iterative Algorithm for Solving a System of Nonlinear Matrix Equations. Journal of Applied Mathematics, 2012, Article ID: 461407.

[15] Al-Dubiban, A.M. (2013) On the Iterative Method for the System of Nonlinear Matrix Equations. Abstract and Applied Analysis, 2013, Article ID: 685753.

[1] Anderson Jr., W.N., Morley, T.D. and Trapp, G.E. (1990) Positive Solutions to

http://dx.doi.org/10.1016/0024-3795(90)90005-W

[2] Lancaster, P. and Rodman, L. (1995) Algebraic Riccati Equations. Oxford Science, Oxford.

[3] Meini, B. (2000) Matrix Equations and Structures: Efficient Solution of Special Discrete Algebraic Riccati Equations. Proceedings of the WLSSCOO, Bulgaria, 2000.

[4] Engwerda, J.C. (1993) On the Existence of a Positive Definite Solution of the Matrix Equation

http://dx.doi.org/10.1016/0024-3795(93)90115-5

[5] Hasanov, V.I. and Ivanov, I.G. (2004) Solutions and Perturbation Estimates for the Matrix Equations

http://dx.doi.org/10.1016/j.amc.2003.08.007

[6] Hasanov, V.I. and El-Sayed, S.M. (2006) On the Positive Definite Solutions of Nonlinear Matrix Equation

http://dx.doi.org/10.1016/j.laa.2005.06.026

[7] Ivanov, I.G. and El-Sayed, S.M. (1998) Properties of Positive Definite Solutions of the Equation

http://dx.doi.org/10.1016/S0024-3795(98)00023-8

[8] Ivanov, I.G. (2006) On Positive Definite Solutions of the Family of Matrix Equations

http://dx.doi.org/10.1016/j.cam.2005.06.007

[9] Costa, O.L.V. and Marques, R.P. (1999) Maximal and Stabilizing Hermitian Solutions for Discrete-Time Coupled Algebraic Riccati Equations. Mathematics of Control, Signals and Systems, 12, 167-195.

http://dx.doi.org/10.1007/PL00009849

[10] Czornik, A. and Swierniak, A. (2001) Lower Bounds on the Solution of Coupled Algebraic Riccati Equation. Automatica, 37, 619-624.

http://dx.doi.org/10.1016/S0005-1098(00)00196-5

[11] Czornik, A. and Swierniak, A. (2001) Upper Bounds on the Solution of Coupled Algebraic Riccati Equation. Journal of Inequalities and Applications, 6, 373-385.

[12] Mukaidani, H., Yamamoto, S. and Yamamoto, T. (2008) A Numerical Algorithm for Finding Solution of Cross-Coupled Algebraic Riccati Equations. IEICE Transactions, 91, 682-685.

http://dx.doi.org/10.1093/ietfec/e91-a.2.682

[13] Al-Dubiban, A.M. (2008) Iterative Algorithms for Computing the Positive Definite Solutions for Nonlinear Matrix Equations. Ph.D. Thesis, Riyadh University for Girls, Riyadh.

[14] Al-Dubiban, A.M. (2012) Iterative Algorithm for Solving a System of Nonlinear Matrix Equations. Journal of Applied Mathematics, 2012, Article ID: 461407.

[15] Al-Dubiban, A.M. (2013) On the Iterative Method for the System of Nonlinear Matrix Equations. Abstract and Applied Analysis, 2013, Article ID: 685753.