The positive definite solutions for the system of nonlinear matrix equations X + A^∗Y⁻ⁿA = I, Y + B^∗X^−mB = I are considered, where n, m are two positive integers and A, B are nonsingular complex matrices. Some sufficient conditions for the existence of positive definite solutions for the system are derived. Under some conditions, an iterative algorithm for computing the positive definite solutions for the system is proposed. Also, the estimation of the error is obtained. Finally, some numerical examples are given to show the efficiency of the proposed iterative algorithm.

1. Introduction

Linear and nonlinear matrix equations have been widely used for solving many problems in several areas such as control theory, optimal control, optimization control, stability theory, communication system, dynamic programming, signal processing, and stochastic filtering and statistics, [1–3]. Many authors studied the existence of solutions for several classes of the matrix equations (see, e.g., [4–14]), in particular, Lyapunov matrix equation [15], Sylvester matrix equations [11, 14], algebraic Riccati equations [3], some special case of linear and nonlinear matrix equations [16–21], and coupled matrix equations [22–24].

In recent years, many types of algebraic Riccati equations have been the subject of great activity, the aim being to achieve a fast and reliable algorithm that generates numerical positive definite solutions.

In this paper, we will consider the system (Sys.) of nonlinear matrix equations that can be expressed in the form:

()

where n, m are two positive integers, X, Y are r × r unknown matrices, I is the r × r identity matrix, and A, B are nonsingular matrices. All matrices are defined over the complex field. The system of nonlinear matrix equations with the form of (1) is a special case of the system of algebraic discrete-type Riccati equations of the form:

()

where i = 1, 2, … , k, [2, 3]. The efficient numerical solutions for some special case of the system (2) have been extensively studied by several authors [4–10, 22–26]. For example, Mukaidani [22] proposed a new algorithm for solving cross-coupled sign-indefinite algebraic Riccati equations for weakly coupled large-scale systems, while in [4, 5] Al-Dubiban has studied special cases of Sys. (2) by obtained sufficient conditions for the existence of positive definite solutions for the systems and proposed iterative algorithms to calculate the solutions. In [10], Davies proposed upper bounds for the sum of the maximal eigenvalues of the solutions of the continuous and discrete coupled algebraic Riccati equations. In [25], Ivanov has studied a set of discrete-time coupled algebraic Riccati equations which arise in quadratic optimal control and proposed two iterations for computing a symmetric solution of this system.

In this paper, we derive the sufficient conditions of the existence of solutions for the Sys. (1). We introduce an iterative algorithm to obtain the positive definite solutions of Sys. (1). We discuss the convergence of this iterative algorithm. Finally, some numerical examples are given to illustrate the efficiency for suggested algorithm.

The following notations are used throughout the rest of the paper. The notation A ≥ 0 (A > 0) means that A is positive semidefinite (positive definite), A^⋆ denotes the complex conjugate transpose of A, and I is the identity matrix. Moreover, A ≥ B (A > B) is used as a different notation for A − B ≥ 0 (A − B > 0). We denote by ρ(A) the spectral radius of A; λ_r(X), μ_r(Y) represent the eigenvalues of X and Y, respectively. The norm used in this paper is the spectral norm of the matrix A; that is, unless otherwise noted.

2. Main Theorems

In this section, we will introduce an iterative algorithm which is applicable for computing the positive definite solutions of the Sys. (1). We start with some results which will be used throughout this paper.

Lemma 1 (see [27], [28].)If P > Q > 0 (or P ≥ Q > 0), then P^α > Q^α (or P^α ≥ Q^α > 0) for all α ∈ (0, 1], and P^α < Q^α (or 0 < P^α ≤ Q^α) for all α ∈ [−1, 0).

Theorem 2 (see [29].)Let the matrices P, Q, and R be positive definite r × r matrices, such that the integral

()

exists and

()

then the matrix

()

is the solution of the matrix equation:

()

The solution of Sys. (1) can be found by the following iterative algorithm.

Algorithm 3.

()

Theorem 4. If there exist numbers α, β satisfying 0 < α < β ≤ min {n/(n + 1), m/(m + 1)}, and the following conditions hold:

(i)
αⁿ(1 − α)I < AA^* < βⁿ(1 − β)I,
(ii)
α^m(1 − α)I < BB^* < β^m(1 − β)I,

then the sequences {X_s}, {Y_s} defined by Algorithm 3 converge to a positive definite solution (X, Y) of Sys. (1).

Proof. From Algorithm 3, we get

()

Also, we have

()

That is, X₀ > X₁ > αI, similarly we get

()

Also, we have

()

That is, Y₀ > Y₁ > αI.

Suppose that

()

Now, we will prove that X_s > X_s+1 > αI and Y_s > Y_s+1 > αI.

By using the inequalities (12), we have

()

Also, we have

()

Similarly, we get

()

Also, we have

()

Therefore, the inequalities (12) are true for all s = 1, 2, 3, ….

Hence, the sequences {X_s}, {Y_s} are monotonically decreasing and bounded from below by the matrix αI. Consequently, the sequences converge to a positive definite limit (X, Y) which is a solution of Sys. (1).

Theorem 5. If there exist numbers α, β satisfying 0 < α < β ≤ min {n/(n + 1), m/(m + 1)}, and the following conditions hold:

(i)
αⁿ(1 − α)I < AA^* < βⁿ(1 − β)I,
(ii)
α^m(1 − α)I < BB^* < β^m(1 − β)I,
(iii)
,

where m = 2^k, n = 2^t, then Sys. (1) has a positive definite solution (X, Y) which satisfies

()

Proof. From Theorem 4, the two sequences {X_s}, {Y_s} defined by Algorithm 3 are convergent to a positive definite solution (X, Y) of Sys. (1). We compute the spectral norm of the matrices X_s − X, Y_s − Y. For that, we have

()

We denote

()

We use the following equality:

()

Since Y_s > Y > 0 for each s = 0, 1, 2, …, then by using Lemma 1 we have the matrix

being a positive definite solution of the matrix equation:

()

According to Theorem 2, we have

()

Since

are positive definite matrices, then the integral (22) exists, and

()

By using (18) and (22), we have

()

However, Y_s > Y ≥ αI; hence,

()

Then, we have

()

After k times as above, we get

()

Let m = 2^k be special case, then we have

()

Since Y < Y_s−1 ≤ βI, then (I − Y) ⁻¹ < (I − Y_s−1) ⁻¹ ≤ (1/(1 − β))I; that is,

()

Therefore, we get

()

Also, we have

()

We denote

()

We use the following equality:

()

Since X_s > X > 0 for each s = 0, 1, 2, … , then by using Lemma 1 we have the matrix being a positive definite solution of the matrix equation:

()

According to Theorem 2, we have

()

Since

are positive definite matrices, then the integral (35) exists, and

()

By using (31) and (35), we have

()

However, X_s > X ≥ αI; hence,

()

Then, we have

()

After t times as above, we get

()

Let n = 2^t be special case, then we have

()

Since X < X_s−1 ≤ βI, then (I − X) ⁻¹ < (I − X_s−1) ⁻¹ ≤ (1/(1 − β))I; that is,

()

Therefore, we get

()

By using (43) in (30) and (30) in (43), we have

()

which completes the proof.

3. Numerical Examples

We will give some numerical examples for computing the positive definite solution of the Sys. (1). The solution is computed for some different matrices A, B with different orders. Denote by X, Y the solutions which are obtained by Algorithm 3 and ϵ₁(X) = ∥X − X_s∥, ϵ₁(Y) = ∥Y − Y_s∥.

For computing Z^1/n for all 1/n ∈ (0, 1], we use the iterative algorithm.

Algorithm 6.

()

See [30].

Example 7. Consider Sys. (1) with n = 3, m = 2, β = 0.5, and normal matrices

()

By using Algorithms 3 and 6, we have

()

The results are given in Table 1.

Table 1. Error analysis for Example 7.

s	ϵ₁(X)	ϵ₁(Y)
0	3.25515E − 01	1.71540E − 01
2	1.90010E − 01	1.28914E − 01
4	4.65789E − 02	3.11069E − 02
6	8.05431E − 03	5.57630E − 03
8	1.25249E − 03	8.71099E − 04
10	1.90796E − 04	1.32785E − 04
12	2.89695E − 05	2.01634E − 05
14	4.39639E − 06	3.06002E − 06
16	6.67142E − 07	4.64353E − 07
18	1.01236E − 07	7.04637E − 08
20	1.53621E − 08	1.06926E − 08

Example 8. Consider Sys. (1) with n = 5, m = 4, β = 0.4 and normal matrices

()

By using Algorithms 3 and 6, we have;

When r = 4,

()

When r = 7,

()

The results are given in Table 2.

Table 2. Error analysis for Example 8.

r	s	ϵ₁(X)	ϵ₁(Y)
4	0	3.07524E − 01	1.95686E − 01
	2	2.24755E − 01	1.44449E − 01
	4	5.31418E − 02	5.07784E − 02
	6	1.08011E − 02	1.03261E − 02
	8	1.75364E − 03	1.66592E − 03
	10	2.69238E − 04	2.55471E − 04
	12	4.09499E − 05	3.88490E − 05
	14	6.21930E − 06	5.90006E − 06
	16	9.44353E − 07	8.95875E − 07
	18	1.43388E − 07	1.36027E − 07
	20	2.17715E − 08	2.06538E − 08

7	0	3.23835E − 01	1.93331E − 01
	2	3.64875E − 01	4.33678E − 01
	4	1.13231E − 01	2.06657E − 01
	6	3.42800E − 02	6.90864E − 02
	8	6.95159E − 03	7.88070E − 03
	10	3.42099E − 04	2.93641E − 04
	12	1.18241E − 05	1.00690E − 05
	14	4.03846E − 07	3.43808E − 07
	16	1.37875E − 08	1.17377E − 08

Example 9. Consider Sys. (1) with n = 2, m = 2, β = 0.4, and matrices

()

By using Algorithms 3 and 6, we have

()

The results are given in Table 3.

Table 3. Error analysis for Example 9.

s	ϵ₁(X)	ϵ₁(Y)
0	9.78137E − 02	1.18408E − 01
2	1.11953E − 01	9.96781E − 02
4	2.16876E − 02	1.76104E − 02
6	3.22178E − 03	2.84069E − 03
8	4.83451E − 04	4.38916E − 04
10	7.50050E − 05	6.69635E − 05
12	1.08812E − 05	1.00510E − 05
14	1.70240E − 06	1.53219E − 06
16	2.59464E − 07	2.32406E − 07
18	3.81681E − 08	3.49836E − 08

Example 10. Consider Sys. (1) with n = 2, m = 1, β = 0.65, and matrices

()

By using Algorithms 3 and 6, we have

()

The results are given in Table 4.

Table 4. Error analysis for Example 10.

s	ϵ₁(X)	ϵ₁(Y)
0	6.49247E − 01	2.37069E − 01
2	8.46092E − 02	1.48180E − 01
4	3.51586E − 03	5.04374E − 02
6	8.98887E − 04	1.30368E − 02
8	3.32057E − 04	4.71441E − 03
10	1.12763E − 04	1.57391E − 03
12	3.78219E − 05	5.11273E − 04
14	1.24881E − 05	1.64362E − 04
16	4.06906E − 06	5.26213E − 05
18	1.31421E − 06	1.68193E − 05
20	4.22232E − 07	5.37246E − 06
22	1.35257E − 07	1.71563E − 06
24	4.32590E − 08	5.47811E − 07
26	1.38239E − 08	1.74912E − 07
28	4.41568E − 09	5.58474E − 08

4. Conclusion

In this paper, the positive definite solutions for Sys. (1) have been tackled. We presented sufficient conditions for the existence of positive definite solutions for Sys. (1). Moreover, we discussed an iterative algorithm from which solutions can always be calculated numerically whenever the system is solvable. Finally, we gave numerical examples that illustrated the behavior of the proposed algorithm.

References

1 AndersonW. N.Jr., Morley T. D., and Trapp G. E., Positive solutions to X = A − BX⁻¹B^∗, Linear Algebra and Its Applications. (1990) 134, 53–62, https://doi.org/10.1016/0024-3795(90)90005-W, MR1060009, ZBL0702.15009.
10.1016/0024-3795(90)90005-W
Web of Science® Google Scholar
2 Lancaster P. and Rodman L., Algebraic Riccati Equations, 1995, Oxford Science, Oxford, UK, MR1367089.
10.1093/oso/9780198537953.001.0001
Google Scholar
3 Meini B., Matrix equations and structures: efficient solution of special discrete algebraic Riccati equations, Proceedings of the WLSSCOO, 2000.
Google Scholar
4 Aldubiban A. M., Iterative algorithms for computing the positive definite solutions for nonlinear matrix equations [Ph.D. thesis], 2008, Riyadh University for Girls, Riyadh, Saudi Arabia.
Google Scholar
5 Aldubiban A. M., Iterative algorithm for solving a system of nonlinear matrix equations, Journal of Applied Mathematics. (2012) 2012, 15, https://doi.org/10.1155/2012/461407, 461407.
10.1155/2012/461407
Google Scholar
6 Costa O. L. V. and Marques R. P., Maximal and stabilizing Hermitian solutions for discrete-time coupled algebraic Riccati equations, Mathematics of Control, Signals, and Systems. (1999) 12, no. 2, 167–195, https://doi.org/10.1007/PL00009849, MR1692164, ZBL0928.93047.
10.1007/PL00009849
Web of Science® Google Scholar
7 Costa O. L. V. and Aya J. C. C., Temporal difference methods for the maximal solution of discrete-time coupled algebraic Riccati equations, Journal of Optimization Theory and Applications. (2001) 109, no. 2, 289–309, https://doi.org/10.1023/A:1017510321237, MR1834177, ZBL0984.93051.
10.1023/A:1017510321237
Web of Science® Google Scholar
8 Czornik A. and Swierniak A., Lower bounds on the solution of coupled algebraic Riccati equation, Automatica. (2001) 37, no. 4, 619–624, https://doi.org/10.1016/S0005-1098(00)00196-5, MR1832523, ZBL0990.93043.
10.1016/S0005-1098(00)00196-5
Web of Science® Google Scholar
9 Czornik A. and Świerniak A., Upper bounds on the solution of coupled algebraic Riccati equation, Journal of Inequalities and Applications. (2001) 6, no. 4, 373–385, https://doi.org/10.1155/S1025583401000224, MR1888431, ZBL1006.93035.
10.1155/S1025583401000224
Web of Science® Google Scholar
10 Davies R., Shi P., and Wiltshire R., Upper solution bounds of the continuous and discrete coupled algebraic Riccati equations, Automatica. (2008) 44, no. 4, 1088–1096, https://doi.org/10.1016/j.automatica.2007.11.001, MR2530953.
10.1016/j.automatica.2007.11.001
Web of Science® Google Scholar
11 Ding F. and Chen T., Iterative least-squares solutions of coupled Sylvester matrix equations, Systems & Control Letters. (2005) 54, no. 2, 95–107, https://doi.org/10.1016/j.sysconle.2004.06.008, MR2109576, ZBL1129.65306.
10.1016/j.sysconle.2004.06.008
Web of Science® Google Scholar
12 Ding F. and Chen T., Gradient based iterative algorithms for solving a class of matrix equations, IEEE Transactions on Automatic Control. (2005) 50, no. 8, 1216–1221, https://doi.org/10.1109/TAC.2005.852558, MR2156053.
10.1109/TAC.2005.852558
Web of Science® Google Scholar
13 Ding F. and Chen T., On iterative solutions of general coupled matrix equations, SIAM Journal on Control and Optimization. (2006) 44, no. 6, 2269–2284, https://doi.org/10.1137/S0363012904441350, MR2248183, ZBL1115.65035.
10.1137/S0363012904441350
Web of Science® Google Scholar
14 Ding F., Liu P. X., and Ding J., Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Applied Mathematics and Computation. (2008) 197, no. 1, 41–50, https://doi.org/10.1016/j.amc.2007.07.040, MR2396289, ZBL1143.65035.
10.1016/j.amc.2007.07.040
Web of Science® Google Scholar
15 Hashemi B. and Dehghan M., The interval Lyapunov matrix equation: analytical results and an efficient numerical technique for outer estimation of the united solution set, Mathematical and Computer Modelling. (2012) 55, no. 3-4, 622–633, https://doi.org/10.1016/j.mcm.2011.08.036, MR2887402, ZBL1255.15002.
10.1016/j.mcm.2011.08.036
Google Scholar
16 Ding J., Liu Y., and Ding F., Iterative solutions to matrix equations of the form A_iXB_i = F_i, Computers & Mathematics with Applications. (2010) 59, no. 11, 3500–3507, https://doi.org/10.1016/j.camwa.2010.03.041, MR2646321, ZBL1197.15009.
10.1016/j.camwa.2010.03.041
Web of Science® Google Scholar
17 Liu A. and Chen G., On the Hermitian positive definite solutions of nonlinear matrix equation , Mathematical Problems in Engineering. (2011) 2011, 18, 163585, MR2837449, ZBL1235.15033.
10.1155/2011/163585
Web of Science® Google Scholar
18 Xie L., Ding J., and Ding F., Gradient based iterative solutions for general linear matrix equations, Computers & Mathematics with Applications. (2009) 58, no. 7, 1441–1448, https://doi.org/10.1016/j.camwa.2009.06.047, MR2555281, ZBL1189.65083.
10.1016/j.camwa.2009.06.047
Web of Science® Google Scholar
19 Xie L., Liu Y., and Yang H., Gradient based and least squares based iterative algorithms for matrix equations AXB + CX^TD = F, Applied Mathematics and Computation. (2010) 217, no. 5, 2191–2199, https://doi.org/10.1016/j.amc.2010.07.019, MR2727965, ZBL1210.65097.
10.1016/j.amc.2010.07.019
Web of Science® Google Scholar
20 Yin F. and Huang G.-X., An iterative algorithm for the least squares generalized reflexive solutions of the matrix equations AXB = E, CXD = F, Abstract and Applied Analysis. (2012) 2012, 18, https://doi.org/10.1155/2012/857284, 857284, MR2898044, ZBL1242.65085.
10.1155/2012/857284
Google Scholar
21 Zhao W., Li H., Liu X., and Xu F., Necessary and sufficient conditions for the existence of a Hermitian positive definite solution of a type of nonlinear matrix equations, Mathematical Problems in Engineering. (2009) 2009, 13, https://doi.org/10.1155/2009/672695, 672695.
10.1155/2009/672695
Web of Science® Google Scholar
22 Mukaidani H., Newton′s method for solving cross-coupled sign-indefinite algebraic Riccati equations for weakly coupled large-scale systems, Applied Mathematics and Computation. (2007) 188, no. 1, 103–115, https://doi.org/10.1016/j.amc.2006.09.100, MR2327097, ZBL1118.65048.
10.1016/j.amc.2006.09.100
Web of Science® Google Scholar
23 Mukaidani H., Yamamoto S., and Yamamoto T., A Numerical algorithm for finding solution of cross-coupled algebraic Riccati equations, IEICE Transactions. (2008) 91, 682–685.
10.1093/ietfec/e91-a.2.682
Google Scholar
24 Mukaidani H., Numerical computation of cross-coupled algebraic Riccati equations related to H_∞-constrained LQG control problem, Applied Mathematics and Computation. (2008) 199, no. 2, 663–676, https://doi.org/10.1016/j.amc.2007.10.026, MR2420595.
10.1016/j.amc.2007.10.026
Web of Science® Google Scholar
25 Ivanov I. G., A method to solve the discrete-time coupled algebraic Riccati equations, Applied Mathematics and Computation. (2008) 206, no. 1, 34–41, https://doi.org/10.1016/j.amc.2008.08.034, MR2474953, ZBL1162.65021.
10.1016/j.amc.2008.08.034
Web of Science® Google Scholar
26 Ivanov I. G., Stein iterations for the coupled discrete-time Riccati equations, Nonlinear Analysis. Theory, Methods & Applications. (2009) 71, no. 12, 6244–6253, https://doi.org/10.1016/j.na.2009.06.025, MR2566530, ZBL1190.65066.
10.1016/j.na.2009.06.025
Web of Science® Google Scholar
27 Bhatia R., Matrix Analysis, 1997, 169, Springer, New York, NY, USA, Graduate Texts in Mathematics, https://doi.org/10.1007/978-1-4612-0653-8, MR1477662.
10.1007/978-1-4612-0653-8
Google Scholar
28 Fujii M. and Seo Y., Reverse inequalities of Araki, Cordes and Löwner-Heinz inequalities, Nihonkai Mathematical Journal. (2005) 16, no. 2, 145–154, MR2217555, ZBL1111.47011.
Google Scholar
29 Lancaster P., Theory of Matrices, 1969, Academic Press, New York, NY, USA, MR0245579.
Google Scholar
30 Guo C.-H. and Higham N. J., A Schur-Newton method for the matrix pth root and its inverse, SIAM Journal on Matrix Analysis and Applications. (2006) 28, no. 3, 788–804, https://doi.org/10.1137/050643374, MR2262981, ZBL1128.65030.
10.1137/050643374
Web of Science® Google Scholar

Citing Literature

All articles

On the Iterative Method for the System of Nonlinear Matrix Equations

Abstract

1. Introduction

2. Main Theorems

3. Numerical Examples

4. Conclusion

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

On the Iterative Method for the System of Nonlinear Matrix Equations

Abstract

1. Introduction

2. Main Theorems

3. Numerical Examples

4. Conclusion

References

Citing Literature

References

Related

Information