The Inversion-Free Iterative Methods for Solving the Nonlinear Matrix Equation 𝑋+𝐴^𝐻𝑋⁻¹𝐴+𝐵^𝐻𝑋⁻¹𝐵=𝐼

Abstract

We present two inversion-free iterative methods for computing the maximal positive definite solution of the equation 𝑋+𝐴^𝐻𝑋⁻¹𝐴+𝐵^𝐻𝑋⁻¹𝐵=𝐼. We prove that the sequences generated by the two iterative schemes are monotonically increasing and bounded above. We also present some numerical results to compare our proposed methods with some previously developed inversion-free techniques for solving the same matrix equation.

1. Introduction

In this paper, we consider the nonlinear matrix equation: 𝑋+𝐴^𝐻𝑋⁻¹𝐴+𝐵^𝐻𝑋⁻¹𝐵=𝐼, (1) where 𝐴,𝐵∈ℂ^𝑛×𝑛, 𝐼 is the identity matrix, and a Hermitian positive definite solution 𝑋 is required.

Specifically, if 𝐵=0, the nonlinear matrix equation (1) reduces to 𝑋+𝐴^𝐻𝑋⁻¹𝐴=𝐼. (2) The nonlinear matrix equation (2) has many applications in nano research, control theory, dynamic programming, statistics, ladder networks, stochastic filtering, and so forth (see [1–7]). The special case (2) has been widely studied by some authors (see [8–27]). Different iterative methods for computing the positive definite solutions of (2) have been proposed, for example, the fixed-point iteration (see [15]), structure-preserving doubling algorithm (see [7, 16]), and some inversion-free iterations (see [17, 20, 23, 27]). Among them, structure-preserving doubling algorithm has been seen as one of the most efficient algorithms as it has quadratic convergence rate.

However, very little research has been done on the solutions to (1) in the case 𝐵≠0. In [28], Long et al. stated the application background of (1) and presented some conditions for the existence of the positive definite solution of this equation. Moreover, they proposed some iterative algorithms to find the positive definite solution. Popchev et al. made a complete perturbation analysis of (1) (see [29]). In [30], Liu and Chen considered the nonlinear matrix equation 𝑋^𝑠+𝐴^𝐻𝑋^−𝑡1𝐴+𝐵^𝐻𝑋^−𝑡2𝐵=𝑄. They studied the existence of the positive definite solution of this equation.

Motivated and inspired by the works mentioned above, in this paper, we propose two new inversion-free iterative methods for obtaining the maximal positive definite solution of (1). We prove that the sequences generated by the two iterative schemes are monotonically increasing and bounded above. In addition, we also provide some numerical results to illustrate the effectiveness of the proposed algorithm.

Throughout this paper, we use the following notations: for 𝐴∈ℂ^𝑛×𝑛, we write 𝐴^𝐻, 𝐴⁻¹ and ‖𝐴‖ to denote the conjugate transpose, the inverse, and the Frobenius norm of the matrix 𝐴, respectively. For any 𝐴=(𝑎_𝑖𝑗), 𝐵=(𝑏_𝑖𝑗), we write 𝐴≥𝐵 (or 𝐵≤𝐴) if 𝐴−𝐵 is a Hermitian positive semidefinite matrix. And we write 𝐴>𝐵 (or 𝐵<𝐴) if 𝐴−𝐵 is a Hermitian positive definite matrix. We use 0 to denote the zero matrix of size implied by context and 𝐼 to denote the identity matrix of size implied by context.

This paper is organized as follows. In Section 2, we present two new iterative methods to solve the nonlinear matrix equation (1). In Section 3, the convergence analysis of the proposed methods is given. Some numerical experiments are reported in Section 4. Finally, we conclude this paper in Section 5.

2. New Inversion-Free Iterative Methods

In this section, we present two new inversion free iterative algorithms for solving problem (1). Let 𝑌=𝑋⁻¹; then the nonlinear matrix equation (1) is equivalent to 𝑌⁻¹+𝐴^𝐻𝑌𝐴+𝐵^𝐻𝑌𝐵=𝐼. (3) Obviously, if 𝑌_∗ is a Hermitian positive definite solution of (3), we have 0<𝑌_∗⁻¹≤𝐼. That is, 𝑌_∗≥𝐼.

Premultiplying and postmultiplying by 𝑌 on (3) simultaneously, we get 𝑌−𝑌𝐼−𝐴^𝐻𝑌𝐴−𝐵^𝐻𝑌𝐵𝑌=0. (4) Adding 𝑌 to both sides of the above equation, we have 𝑌=2𝑌−𝑌𝐼−𝐴^𝐻𝑌𝐴−𝐵^𝐻𝑌𝐵𝑌. (5) Obviously, 𝑌 solves (5) if 𝑌 is a solution of (3). Conversely, if 𝑌 is a nonsingular solution of (5), 𝑌 solves (3) as well.

Thus, we just need to solve the matrix equation (5) if we want to get a Hermitian positive definite solution of (3). By this point, we present the following iterative scheme.

Algorithm 1 (an inversion-free iterative algorithm for (1)).
Step 1. Input the matrix 𝐴,𝐵∈ℂ^𝑛×𝑛. Take initial matrix 𝑌₀=𝐼 and tolerance error 𝜀≥0. Set 𝑘∶=0.
Step 2. Obtain 𝑌_𝑘+1 by the following iterative scheme: 𝑌_𝑘+1=2𝑌_𝑘−𝑌_𝑘𝑍_𝑘𝑌_𝑘, (6) where 𝑍_𝑘=𝐼−𝐴^𝐻𝑌_𝑘𝐴−𝐵^𝐻𝑌_𝑘𝐵.
Step 3. Stop if ‖𝑌_𝑘+1−𝑌_𝑘‖_𝐹≤𝜀. Otherwise, 𝑘∶=𝑘+1, go to Step 2.

Remark 2. If 𝐵=0, Algorithm 1 reduces to Algorithm 2.1 in [23] with 𝛼=1 and 𝑄=𝐼. Moreover, as 𝑌₀=𝐼 is a Hermitian matrix, from (6) we know that 𝑌_𝑘 is also a Hermitian matrix, for all 𝑘≥0.

On the other hand, premultiplying and postmultiplying by 𝑌 on (3), respectively, we have 𝐼+𝑌𝐴^𝐻𝑌𝐴+𝑌𝐵^𝐻𝑌𝐵=𝑌,𝐼+𝐴^𝐻𝑌𝐴𝑌+𝐵^𝐻𝑌𝐵𝑌=𝑌. (7) Adding the above two equations, we can get 2𝐼+𝑌𝐴^𝐻𝑌𝐴+𝑌𝐵^𝐻𝑌𝐵+𝐴^𝐻𝑌𝐴𝑌+𝐵^𝐻𝑌𝐵𝑌=2𝑌. (8) By some simple calculating, we obtain 1𝑌=𝐼+2×𝑌𝐴^𝐻𝑌𝐴+𝐵^𝐻𝑌𝐵+𝐴^𝐻𝑌𝐴+𝐵^𝐻𝑌𝐵𝑌. (9) By the above analysis, we know that if 𝑌 is a solution of (3), 𝑌 is a solution of (9). Now we prove that a Hermitian positive definite solution of (9) is also a solution of (3).

Theorem 3. Let 𝑌 be a Hermitian positive definite solution of the nonlinear matrix equation (9). Then 𝑌 is also a positive definite solution of the nonlinear matrix equation (3).

Proof. By the nonlinear matrix equation (9), we have 𝑌𝑌⁻¹+𝐴^𝐻𝑌𝐴+𝐵^𝐻+𝑌𝑌𝐵−𝐼₋₁+𝐴^𝐻𝑌𝐴+𝐵^𝐻𝑌𝐵−𝐼𝑌=0. (10) This implies that 𝑌⁻¹+𝐴^𝐻𝑌𝐴+𝐵^𝐻𝑌𝐵−𝐼 is a solution of the matrix equation 𝑌𝐻+𝐻𝑌=0, where 𝐻∈ℂ^𝑛×𝑛 is unknown. Since 𝑌 is positive definite, 𝑌 and −𝑌 have no common eigenvalue and the matrix equation 𝑌𝐻+𝐻𝑌=0 has a unique solution (see [31]). As 0 solves the equation 𝑌𝐻+𝐻𝑌=0, we must have 𝑌⁻¹+𝐴^𝐻𝑌𝐴+𝐵^𝐻𝑌𝐵−𝐼=0. Namely, 𝑌 is a positive definite solution of (3).

Thus, we just need to solve the matrix equation (9) if we want to get a Hermitian positive definite solution of (3). By this point, we present the following iterative scheme.

Algorithm 4 (another inversion free iterative algorithm).
Step 1. Input the matrix 𝐴,𝐵∈ℂ^𝑛×𝑛. Take initial matrix 𝑌₀=𝐼 and tolerance error 𝜀≥0. Set 𝑘∶=0.
Step 2. Obtain 𝑌_𝑘+1 by the following iterative scheme: 𝑍_𝑘=12𝑌_𝑘𝐴^𝐻𝑌_𝑘𝐴+𝐵^𝐻𝑌_𝑘𝐵,𝑌_𝑘+1=𝐼+𝑍^𝐻_𝑘+𝑍_𝑘. (11)
Step 3. Stop if ‖𝑌_𝑘+1−𝑌_𝑘‖_𝐹≤𝜀. Otherwise, 𝑘∶=𝑘+1 go to Step 2.

Obviously, for all 𝑘≥0, sequence {𝑌_𝑘} generated by Algorithm 4 with the initial matrix 𝑌₀=𝐼 are all Hermitian matrices.

3. Convergence Analysis

In this section, we will prove that the sequences {𝑌_𝑘} generated by Algorithms 1 and 4 with the initial matrix 𝑌₀=𝐼 converge to the minimal positive definite solution of (3). In the first place, we introduce the following lemmas.

Lemma 5 (see [28]). If (1) has a positive definite solution 𝑋, then 𝐴^𝐻𝐴+𝐵^𝐻𝐵<𝐼.

Lemma 6 (see [23]). If 𝐶 and 𝑃 are Hermitian matrices of the same order with 𝑃>0, then 2𝐶−𝐶𝑃𝐶≤𝑃⁻¹.

Lemma 7. If 𝑀,𝑃∈ℂ^𝑛×𝑛 are both Hermitian positive semidefinite, then 𝑀𝑃+𝑃𝑀 is a Hermitian positive semidefinite matrix as well.

Proof. Obviously, 𝑀𝑃+𝑃𝑀 is a Hermitian matrix. If one of 𝑀 and 𝑃 is positive definite, without loss of generality, suppose that 𝑀>0. Then by the assumption of the lemma, we know that 𝑀^1/2, 𝑃^1/2 are well defined and 𝑀^1/2>0, 𝑃^1/2≥0. Then we get 𝑀𝑃+𝑃𝑀=𝑀^1/2𝑀^1/2𝑃+𝑃𝑀^1/2𝑀^1/2=𝑀^1/2𝑀^1/2𝑃𝑀^−(1/2)+𝑀^−(1/2)𝑃𝑀^1/2𝑀^1/2≥0. (12) Assume that neither of 𝑀 and 𝑃 is positive definite. Then for 𝛽>0, we have 𝑀+𝛽𝐼>0. Thus by the above analysis, we have (𝑀+𝛽𝐼)𝑃+𝑃(𝑀+𝛽𝐼)≥0. Let 𝛽→0⁺; we get 𝑀𝑃+𝑃𝑀≥0. This completes the proof.

Now we are in a position to prove that {𝑌_𝑘} generated by Algorithm 1 with the initial matrix 𝑌₀=𝐼 converges to the minimal positive definite solution of (3).

Theorem 8. The nonlinear matrix equation (3) has a positive definite solution and the sequence {𝑌_𝑘} is generated by Algorithm 1 with the initial matrix 𝑌₀=𝐼. Let 𝑌 be the minimal positive definite solution of (3). Then the sequence {𝑌_𝑘} is well defined, 𝑌₀≤𝑌₁≤𝑌₂≤⋯<𝑌_𝑘≤⋯≤𝑌, and lim_𝑘→∞𝑌_𝑘=𝑌. (13)

Proof. Let 𝑌_∗ be any positive definite solution of (3). Firstly, we will prove 0<𝑌_𝑘≤𝑌_𝑘+1≤𝑌_∗ for all 𝑘≥0 by mathematical induction.
For 𝑘=0, we have 𝑌₀=𝐼>0. By (6) we have 𝑌₁=2𝐼−𝐼+𝐴^𝐻𝐴+𝐵^𝐻𝐵=𝐼+𝐴^𝐻𝐴+𝐵^𝐻𝐵≥𝐼=𝑌₀. (14) On the other hand, by Lemma 5, we have 𝐼−𝐴^𝐻𝐴−𝐵^𝐻𝐵>0. This together with Lemma 6 and (3) yields 𝑌₁=2𝐼−𝐼𝐼−𝐴^𝐻𝐴−𝐵^𝐻𝐵𝐼≤𝐼−𝐴^𝐻𝐴−𝐵^𝐻𝐵⁻¹≤𝐼−𝐴^𝐻𝑌_∗𝐴−𝐵^𝐻𝑌_∗𝐵⁻¹=𝑌_∗, (15) where the second inequality follows from the fact that 𝑌_∗≥𝐼. Hence 0<𝑌_𝑘≤𝑌_𝑘+1≤𝑌_∗ holds for 𝑘=0.
Assume that 0<𝑌_𝑘≤𝑌_𝑘+1≤𝑌_∗ holds for 𝑘=𝑖≥0. Since 𝑍_𝑖+1=𝐼−𝐴^𝐻𝑌_𝑖+1𝐴−𝐵^𝐻𝑌_𝑖+1𝐵≥𝐼−𝐴^𝐻𝑌_∗𝐴−𝐵^𝐻𝑌_∗𝐵=𝑌_∗⁻¹>0, (16) by (6), Lemma 6, and the fact that 𝑌_𝑖+1>0, we get 𝑌_𝑖+2=2𝑌_𝑖+1−𝑌_𝑖+1𝑍_𝑖+1𝑌_𝑖+1≤𝑍⁻¹_𝑖+1≤𝑌_∗. (17) Moreover, it follows from (6) that 𝑌_𝑖+2−𝑌_𝑖+1=𝑌_𝑖+1−𝑌_𝑖+1𝑍_𝑖+1𝑌_𝑖+1=𝑌_𝑖+1𝑌⁻¹_𝑖+1−𝑍_𝑖+1𝑌_𝑖+1. (18) As 𝑍_𝑖=𝐼−𝐴^𝐻𝑌_𝑖𝐴−𝐵^𝐻𝑌_𝑖𝐵≥𝐼−𝐴^𝐻𝑌_𝑖+1𝐴−𝐵^𝐻𝑌_𝑖+1𝐵=𝑍_𝑖+1>0, this together with (6) and Lemma 6 follows that 𝑌_𝑖+1=2𝑌_𝑖−𝑌_𝑖𝑍_𝑖𝑌_𝑖≤𝑍_𝑖⁻¹≤𝑍⁻¹_𝑖+1. (19) This implies that 𝑌⁻¹_𝑖+1≥𝑍_𝑖+1. Thus, by (18) and the fact that 𝑌_𝑖+1>0, we obtain 𝑌_𝑖+2≥𝑌_𝑖+1. Therefore, 0<𝑌_𝑘≤𝑌_𝑘+1≤𝑌_∗ holds for 𝑘=𝑖+1.
By the principle of mathematical induction, 0<𝑌_𝑘≤𝑌_𝑘+1≤𝑌_∗ is true for all 𝑘≥0. The sequence {𝑌_𝑘} is now well defined, monotonically increasing, and bounded above. Let lim_𝑘→∞𝑌_𝑘=𝑌. Then 𝑌 is a positive definite solution of the nonlinear matrix equation (3) by (6). Since 𝑌≤𝑌_∗ for any positive definite solution 𝑌_∗ of (3), 𝑌 is the minimal positive definite solution of (3). This completes the proof.

Remark 9. Let 𝑋 be the maximal positive definite solution of the nonlinear matrix equation (1). By the relationship between (1) and (3), we get that 𝑋⁻¹ is the minimal positive definite solution of (3). So from Theorem 8, we know that the sequence {𝑌_𝑘} generated by Algorithm 1 with the initial matrix 𝑌₀=𝐼 converges to the inverse of the maximal positive definite solution of (1).

Now we consider the convergence theorem of Algorithm 4.

Theorem 10. The nonlinear matrix equation (3) has a positive definite solution and the sequence {𝑌_𝑘} is generated by Algorithm 4 with the initial matrix 𝑌₀=𝐼. Let 𝑌 be the minimal positive definite solution of (3). Then the sequence {𝑌_𝑘} is well defined, 𝑌₀≤𝑌₁≤𝑌₂<⋯≤𝑌_𝑘≤⋯≤𝑌, and lim_𝑘→∞𝑌_𝑘=𝑌.

Proof. Let 𝑌_∗ be any positive definite solution of (3). Firstly, we will prove 0<𝑌_𝑘≤𝑌_𝑘+1≤𝑌_∗ for all 𝑘≥0 by induction.
For 𝑘=0, we have 𝑌₀=𝐼>0; then 𝑌₁=𝐼+𝐴^𝐻𝐴+𝐵^𝐻𝐵≥𝐼. On the other hand, by the fact that 𝑌_∗≥𝐼, (3), and Lemma 7, we get 𝑌₁=𝐼+𝐴^𝐻𝐴+𝐵^𝐻𝐵≤𝐼+𝐴^𝐻𝑌_∗𝐴+𝐵^𝐻𝑌_∗𝐵1=𝐼+2𝐴^𝐻𝑌_∗𝐴𝐼+𝐼𝐴^𝐻𝑌_∗𝐴+12𝐵^𝐻𝑌_∗𝐵𝐼+𝐼𝐵^𝐻𝑌_∗𝐵1≤𝐼+2𝐴^𝐻𝑌_∗𝐴𝑌_∗+𝑌_∗𝐴^𝐻𝑌_∗𝐴+12𝐵^𝐻𝑌_∗𝐵𝑌_∗+𝑌_∗𝐵^𝐻𝑌_∗𝐵1=𝐼+2𝐴^𝐻𝑌_∗𝐴+𝐵^𝐻𝑌_∗𝐵𝑌_∗+12𝑌_∗𝐴^𝐻𝑌_∗𝐴+𝐵^𝐻𝑌_∗𝐵1=𝐼+2𝐼−𝑌_∗⁻¹𝑌_∗+12𝑌_∗𝐼−𝑌_∗⁻¹=𝑌_∗. (20) Hence 0<𝑌_𝑘≤𝑌_𝑘+1≤𝑌_∗ is true for 𝑘=0.
Assume that 0<𝑌_𝑘≤𝑌_𝑘+1≤𝑌_∗ holds for 𝑘=𝑖≥0. Then by Lemma 7 we have 𝑌_𝑖+1𝐴^𝐻𝑌_𝑖+1𝐴+𝐴^𝐻𝑌_𝑖+1𝐴𝑌_𝑖+1≥𝑌_𝑖𝐴^𝐻𝑌_𝑖+1𝐴+𝐴^𝐻𝑌_𝑖+1𝐴𝑌_𝑖≥𝑌_𝑖𝐴^𝐻𝑌_𝑖𝐴+𝐴^𝐻𝑌_𝑖𝐴𝑌_𝑖. (21) Similarly, we get 𝑌_𝑖+1𝐵^𝐻𝑌_𝑖+1𝐵+𝐵^𝐻𝑌_𝑖+1𝐵𝑌_𝑖+1≥𝑌_𝑖𝐵^𝐻𝑌_𝑖𝐵+𝐵^𝐻𝑌_𝑖𝐵𝑌_𝑖. This together with (11) yields 𝑌_𝑖+21=𝐼+2𝑌_𝑖+1𝐴^𝐻𝑌_𝑖+1𝐴+𝑌_𝑖+1𝐵^𝐻𝑌_𝑖+1𝐵+𝐴^𝐻𝑌_𝑖+1𝐴𝑌_𝑖+1+𝐵^𝐻𝑌_𝑖+1𝐵𝑌_𝑖+11=𝐼+2𝑌_𝑖+1𝐴^𝐻𝑌_𝑖+1𝐴+𝐴^𝐻𝑌_𝑖+1𝐴𝑌_𝑖+1+12𝑌_𝑖+1𝐵^𝐻𝑌_𝑖+1𝐵+𝐵^𝐻𝑌_𝑖+1𝐵𝑌_𝑖+11≥𝐼+2𝑌_𝑖𝐴^𝐻𝑌_𝑖𝐴+𝐴^𝐻𝑌_𝑖𝐴𝑌_𝑖+12𝑌_𝑖𝐵^𝐻𝑌_𝑖𝐵+𝐵^𝐻𝑌_𝑖𝐵𝑌_𝑖1=𝐼+2𝑌_𝑖𝐴^𝐻𝑌_𝑖𝐴+𝑌_𝑖𝐵^𝐻𝑌_𝑖𝐵+𝐴^𝐻𝑌_𝑖𝐴𝑌_𝑖+𝐵^𝐻𝑌_𝑖𝐵𝑌_𝑖1=𝐼+2𝑍_𝑖+𝑍^𝐻_𝑖=𝑌_𝑖+1. (22) On the other hand, by Lemma 7 we obtain 𝑌_𝑖+1𝐴^𝐻𝑌_𝑖+1𝐴+𝐴^𝐻𝑌_𝑖+1𝐴𝑌_𝑖+1≤𝑌_∗𝐴^𝐻𝑌_𝑖+1𝐴+𝐴^𝐻𝑌_𝑖+1𝐴𝑌_∗≤𝑌_∗𝐴^𝐻𝑌_∗𝐴+𝐴^𝐻𝑌_∗𝐴𝑌_∗. (23) Similarly, we get 𝑌_𝑖+1𝐵^𝐻𝑌_𝑖+1𝐵+𝐵^𝐻𝑌_𝑖+1𝐵𝑌_𝑖+1≤𝑌_∗𝐵^𝐻𝑌_∗𝐵+𝐵^𝐻𝑌_∗𝐵𝑌_∗. (24) Using the above inequalities, we can deduce that 𝑌_𝑖+21=𝐼+2𝑌_𝑖+1𝐴^𝐻𝑌_𝑖+1𝐴+𝐴^𝐻𝑌_𝑖+1𝐴𝑌_𝑖+1+12𝑌_𝑖+1𝐵^𝐻𝑌_𝑖+1𝐵+𝐵^𝐻𝑌_𝑖+1𝐵𝑌_𝑖+11≤𝐼+2𝑌_∗𝐴^𝐻𝑌_∗𝐴+𝐴^𝐻𝑌_∗𝐴𝑌_∗+12𝑌_∗𝐵^𝐻𝑌_∗𝐵+𝐵^𝐻𝑌_∗𝐵𝑌_∗1=𝐼+2𝑌_∗𝐴^𝐻𝑌_∗𝐴+𝐵^𝐻𝑌_∗𝐵+12𝐴^𝐻𝑌_∗𝐴+𝐵^𝐻𝑌_∗𝐵𝑌_∗1=𝐼+2𝑌_∗𝐼−𝑌_∗⁻¹+12𝐼−𝑌_∗⁻¹𝑌_∗=𝑌_∗, (25) where the first equality follows from (11) and the third equality follows from (3). Hence, 0<𝑌_𝑘≤𝑌_𝑘+1≤𝑌_∗ holds for 𝑘=𝑖+2.
Thus, by the principle of induction, 0<𝑌_𝑘≤𝑌_𝑘+1≤𝑌_∗ is true for all 𝑘≥0. The sequence {𝑌_𝑘} is now well defined, monotonically increasing, and bounded above. Let lim_𝑘→∞𝑌_𝑘=𝑌. Then 𝑌 is a positive definite solution of the nonlinear matrix equation (3) by (11) and Theorem 3. Since 𝑌≤𝑌^𝐻 for any positive definite solution 𝑌^𝐻 of (3), 𝑌 is the minimal positive definite solution of (3). This completes the proof.

4. Numerical Experiments

In this section, we will give some numerical examples to support our Algorithms 1 and 4. All experiments were run on a PC with Pentium(R) Dual-Core CPU E5800 @2.40 GHz. All the programming is implemented in MATLAB R2011b (7.13). We report the number of required iterations (Iter.), the norm of the residual (Res.) when the process is stopped, the required CPU time (CPU), and the number of matrix-matrix (MM) products required. In our implementation, we stop all considered algorithms when ‖𝑌_𝑘+1−𝑌_𝑘‖_𝐹≤10⁻¹⁰ with 𝑘≥1. We compare our Algorithm 1 (A1) and Algorithm 4 (A2) with the following inverse-free methods for solving (1).(i)In [28], for finding positive definite solution of (1), Long et al. proposed the following iteration: B𝑋1∶⁰=𝐼,𝑌₀𝑋=𝐼,_𝑘+1=𝐼−𝐴^𝐻𝑌_𝑘𝐴−𝐵^𝐻𝑌_𝑘𝑌𝐵,_𝑘+1=𝑌_𝑘2𝐼−𝑋_𝑘𝑌_𝑘. (26)(ii)In [30], for solving positive definite solution of (1), Liu and Chen proposed the following iteration: B𝑋2∶⁰=𝐼,𝑌₀𝑌=𝐼,_𝑘+1=𝑌_𝑘2𝐼−𝑋_𝑘𝑌_𝑘,𝑋_𝑘+1=𝐼−𝐴^𝐻𝑌_𝑘+1𝐴−𝐵^𝐻𝑌_𝑘+1𝐵. (27)

Example 11. For the first experiment, we consider (1) when 𝐴 and 𝐵 are given as in Example 4.1 from [28]: ,.𝐴=0.010−0.150−0.2590.0150.212−0.0640.025−0.0690.138𝐵=0.160−0.0250.020−0.025−0.288−0.0600.004−0.016−0.120 (28) We could obtain the maximal solution 𝑋₊ (the first 4 digits) by the iterative schemes A1-A2 and B1–B3. The maximal solution is 𝑋₊≈.0.9718−0.0049−0.0046−0.00490.8144−0.0388−0.0046−0.03880.8836 (29) Our numerical results are reported in Table 1.

Table 1 

Numerical results of Example 11.

Example 12. In this test, the matrices 𝐴 and 𝐵 are given as in Example 4.2 from [28]: 1𝐴=,168040252335662532274521232728162435451652656621246569𝐵=.40011212325322131604233236034182625421844303233263050 (30) We will obtain the maximal solution 𝑋₊ (the first 4 digits) by the iterative schemes A1-A2 and B1–B3. The maximal solution is 𝑋₊=.0.9437−0.0642−0.0530−0.0691−0.0772−0.06420.9063−0.0739−0.0833−0.0907−0.0530−0.07390.9297−0.0717−0.0763−0.0691−0.0833−0.07170.9080−0.0970−0.0772−0.0907−0.0763−0.09700.8889 (31) Our numerical results are reported in Table 2.

Table 2 

Numerical results of Example 12.

Example 13. In this experiment we solve (1) with the matrices 𝐴 and 𝐵 as follows: 1𝐴=,120200100120010003010100201101030010012𝐵=.50216057347130092478853001250217400149 (32) We could obtain the maximal solution 𝑋₊ (the first 4 digits) by the iterative schemes A1-A2 and B1–B3. The unique positive definite solution is𝑋₊≈.0.9301−0.0443−0.0367−0.0202−0.0411−0.0521−0.04430.9104−0.0395−0.0271−0.0603−0.0781−0.0367−0.03950.9234−0.0110−0.0566−0.0430−0.0202−0.0271−0.01100.9755−0.0224−0.0374−0.0411−0.0603−0.0566−0.02240.9064−0.0858−0.0521−0.0781−0.0430−0.0374−0.08580.8486 (33)The numerical results are listed in Table 3.

Table 3 

Numerical results of Example 13.

Example 14. In this test, the matrices 𝐴 and 𝐵 are provided with the following forms: 1𝐴=𝐼10^𝑛+1𝑅2𝑛^𝑛1,𝐵=𝐼20^𝑛+1𝑛²𝑆_𝑛, (34) where 𝐼_𝑛 is 𝑛×𝑛 identity matrix and 𝑅_𝑛=(𝑟_𝑖𝑗)_𝑛×𝑛 and 𝑆_𝑛=(𝑠_𝑘𝑙)_𝑛×𝑛 are randomly generated with entries 𝑟_𝑖𝑗∈(0,1) and 𝑠_𝑘𝑙∈(−1,1). For different matrix dimension (DIM) 𝑛, the numerical results are reported in Table 4.

Table 4 

Numerical results of Example 14.

From the above experiments, we find that Algorithm 1 has an advantage in the number of iterations and CPU time. Although Algorithm 4 performs worse than B2, it performs better than B1. In Example 14, we can see that the iterations of Algorithms 1 and 4 are invariant with the dimension, which are the same as the performance of B1 and B2. In a word, our algorithms are promising.

5. Conclusion

In this paper, we propose two inversion-free iterative algorithms for obtaining the maximal positive definite solution of the equation 𝑋+𝐴^𝐻𝑋⁻¹𝐴+𝐵^𝐻𝑋⁻¹𝐵=𝐼. We prove that the sequences generated by the proposed iterative schemes are monotonically increasing and bounded above. Some numerical results are also reported in the paper, which confirm the good theoretical properties of our approach. Although we prove that the sequence {𝑌_𝑘} generated by Algorithms 1 and 4 with the initial matrix 𝑌₀=𝐼 converges to the minimal positive definite solution of (3) (or the maximal positive definite solution of (1)), we do not yet give the analysis on the convergence rate of the two methods, which is our work in the future.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Authors’ Contribution

All authors contributed equally and significantly in writing this paper. All authors read and approved the final paper.

Acknowledgments

The project is supported by the National Natural Science Foundation of China (Grant nos. 11071041 and 11201074), Fujian Natural Science Foundation (Grant No. 2013J01006) and The University Special Fund Project of Fujian (Grant no. JK2013060).