The Feasibility of Homotopy Continuation Method for a Nonlinear Matrix Equation

Wang, Mengran; Li, Jing

doi:https://doi.org/10.1155/2020/1787058

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 1787058 | https://doi.org/10.1155/2020/1787058

The Feasibility of Homotopy Continuation Method for a Nonlinear Matrix Equation

Mengran Wang¹and Jing Li¹

Academic Editor: Nicolae Herisanu

Received15 Nov 2019

Accepted04 Mar 2020

Published25 Apr 2020

Abstract

In this paper, we discuss the feasibility of homotopy continuation method for the nonlinear matrix equations with . This iterative method does not depend on a good initial approximation to the solution of matrix equation.

1. Introduction

In this paper, we consider the Hermitian positive definite (HPD) solutions of the nonlinear matrix equation:where are complex nonsingular matrices, is an identity matrix, and are positive integers. Here, denotes the conjugate transpose of the matrix .

The nonlinear matrix equations of (1) or some special cases are applicable to many fields such as nanoresearch, ladder networks, dynamic programming, control theory, stochastic filtering, and statistics [1–8].

Equation (1) is recognized as playing an important role in solving a system of linear equations. For example, in many physical calculations, one must solve the system of linear equation , where and are column vectors, andarises in a finite difference approximation to an elliptic partial differential equation (for more information, refer to [2]). We can rewrite as , where

can be factored as if and only if is a solution of equation , where

Some special cases of matrix equation (1) and related matrix equations were studied on the solvability, numerical methods, and perturbation analysis by many scholars: [9–12]; [13–17]; [18]; [19, 20]; [21, 22]; [23]; [24]; and [25–29]. In addition, many homotopy approaches oriented toward engineering applications were investigated by many scholars: the homotopy analysis method (HAM) [30], the Optimal Homotopy Asymptotic Method (OHAM) [31], the Optimal Homotopy Perturbation Method (OHPM) [32], the homotopy asymptotic method [33], and so on.

According to our knowledge, the matrix equation (1) has not been treated explicitly in the literatures. The reason is that (1) does not always have a unique Hermitian positive definite solution. It is hard to find sufficient conditions for the existence of a unique Hermitian positive definite solution because the map is not monotonic. There are two difficulties for discussing the solvability and iterative method for the matrix equation (1). One is how to find a suitable set and some reasonable restrictions on the coefficient matrices ensuring this equation has a unique Hermitian positive definite solution in this set. The other one is how to find a reasonable expression of , which was important for discussing the feasibility of the homotopy continuation iterative method (see [34], for more details), which was not dependent on a good initial approximation to the solution of matrix equation.

In this paper, firstly, we derive necessary and sufficient conditions for the existence of Hermitian positive definite solutions to equation (1) in Section 3. Then, discuss the homotopy continuation methods for obtaining the unique Hermitian positive definite solution in Section 4.

The following notations are used throughout this paper. We denote by , , and the set of all complex matrices, Hermitian matrices, and unitary matrices, respectively. For and a matrix , is a Kronecker product, and is a vector defined by . The symbol stands for the spectral norm. We denote by the eigenvalues of , by and the maximal and minimal eigenvalues of , respectively. For , we write , if is a Hermitian positive semidefinite (definite) matrix. For , the sets and are defined by and .

2. Preliminaries

In this section, we present some lemmas that will be needed to develop this paper.

Lemma 1. (see [35]). If and , then .

Lemma 2. (see [24], Lemma 3.3). For every Hermitian positive definite matrix and , it yields that .

Lemma 3. (see [36], Theorem 1.9.1). Let . Then,

Lemma 4. (see [36], Lemma 1.9.1). Let . Then,

Lemma 5. (see [37], Theorem 6.19). Let and with eigenvalues and , respectively. Then, the eigenvalues of are and .

Lemma 6. (see [24], Lemma 3.2). Suppose that , and . Then,

3. Hermitian Positive Definite Solutions

In this section, some sufficient and necessary conditions for the existence and uniqueness of HPD solution of (1) are derived.

Theorem 1. (1) has a HPD solution if and only if there exist , , and diagonal matrices such thatwhere and . In this case, is a HPD solution of (1).

Proof. Assume is an HPD solution of (1). According to the spectral decomposition theorem, we have that there exists and a diagonal matrix such that . Then, (1) can be expressed asMultiplying the left side of (8) by and the right side by , we obtainRecall that are nonsingular matrices. Then,Therefore, we can rewrite (10) asLetIt is easy to verify that andBy (11), we obtain .
Conversely, assume there exist , , , and diagonal matrices , such thatLet such thatwhich means is an HPD solution of (1).

Theorem 2. If , then (1) has a unique HPD solution on .

Proof.

Step 1. We will prove that (1) has a HPD solution on under the assumption .
Let . DefineObviously, is a bounded convex closed set and is continuous on .
Note that . For any , it follows from Lemma 1 that , which impliesRecall that , then . That is, . By Brouwer’s fixed point theorem, the map has a fixed point , which is a HPD solution of (1).

Step 2. We will prove that if (1) has a HPD solution on , then the HPD solution is unique.
If is a HPD solution of (1), according to Lemma 3.1, there exist , , and diagonal matrices such thatwhereIn this case, , where with the eigenvalues of . Similarly, if is a HPD solution of (1), then there exist , , and diagonal matrices such thatwhereIn this case, , where with the eigenvalues of. According to in Lemma 2, we haveBywe haveLetThen, (24) can be expressed asBy (26), Lemmas 3 and 4, we have thatAssume thatIt follows from (10), (19), and (21) thatLetThen, (27) can be expressed asBy Lemma 5, we haveIt follows thatNote that is nonsingular. Multiplying the left side of (31) by , we haveA combination of (30) and Lemma 3 givesIt follows (19), (21), and Lemma 5 that . Therefore,By the hypothesis of the theorem, we have , which implies that Note that . Then, . Therefore, it follows (32) and (33) thatwhere is defined in Lemma 6. It is easy to verify thatA combination of Lemma 6, (36)–(38) gives thatwhich implies is nonsingular. It follows (34) that . Recall that . Therefore, , which means the HPD solution on of (1) is unique.

4. The Homotopy Continuation Iterative Method

In this section, by means of the homotopy continuation iterative method (see [34], for more details), we derive a numerical iterative process for solving the matrix equation (1).

Define the nonlinear map byConsider the homotopy :Then, at , the solution of is a known matrix , while at , the solution of also solves . To discuss the numerical method for solving the homotopy equation , we rewrite the homotopy equation as the following fixed point form.

Assume that is a map such thatwhere denotes the solution of . Then, for each , we can consider the iterative process:

Since for a fixed , this process will converge to only for starting values near that point; to overcome the local convergence of iterative process, we consider the following numerical continuation process.

A partition of :and a sequence of integers , , is chosen with the property that the pointsare well-defined and such thatconverges to as .

The main idea is to choose partition (44) so that lies in some domain of attraction , for each , . Then, if , the sequence generated by (43) for must produce an iterate , which in turn can be taken as the starting point for the next iteration involving . Thus, with as initial point, the entire process can be carried out until finally is reached. For , is then in which ensures that (45) converges to as .

To discuss the feasibility of the abovementioned numerical continuation process, we will use the following definition and lemmas which can be found in [34].

Definition 1. (see [34]). If a partition (44) exists so that with some sequence of integers , the entire process (45)(46) is well defined so that (46) converges to , and then the numerical continuation process (45)(46) is called feasible.

Definition 2. (see [34]). Let be a given mapping. Then, any nonempty set is a domain of attraction of the iterative process:with respect to the point , if for any , we have and .
If for some domain of attraction , then is a point of attraction of (47).

Lemma 7. (see [34]). Let be Fréchet differentiable at the fixed point of . If , then is a point of attraction of (47) and, more precisely, there is an open ball with center and radius which is a domain of attraction of (47) with respect to . Here, denotes the spectral radius of .

Lemma 8. (see [34]). Let , where is open and assume that is continuous and satisfies . Let have a Fréchet derivative with respect to at for every . If is continuous on and for all , then the numerical continuation process (45)(46) is feasible.

In what follows, we derive a sufficient condition for the existence of a unique HPD solution of the homotopy equation for all .

Theorem 3. If , then for arbitrary , the homotopy equation has a unique HPD solution on .

Proof. Since , then the homotopy equation can be rewritten asBy the hypothesis of the theorem, we haveIt follows from Theorem 2 that the homotopy equation has a unique HPD solution on .
In the next theorem, the local convergence of the iterative process (43) is obtained.

Theorem 4. If has a unique HPD solution on , then there exists an open ball with center and radius such that, for any starting value , converges to as .

Proof. Recall that . According to Lemma 2, for any , we haveBy the definition of Fréchet derivative, we obtainLet be any eigenvalue of . Then, there exists a nonzero matrix such thatSince is the unique HPD solution of , then by Theorem 1, there exists , , and diagonal matrices such thatwhereIn this case, , where with the eigenvalues of . Therefore, (52) can be rewritten asLet . It follows thatDefine the operator byThen,Using (58) and Lemmas 3 and 4, we can rewrite (57) asLetThen, (59) can be rewritten asAccording to (36), (55), (57), (59), and (61), we have thatAssume thatBy (54), we haveAccording to Lemma 5 and (64), we haveSince , then . By (65)–(67), we have thatwhere is defined in Lemma 6. Combining Lemma 6 (62) with (68) gives thatAccording to Lemma 7, there exists an open ball with center and radius such that, for any starting value , converges to as .
In the next theorem, we will prove the numerical continuation process (45)(46) is feasible.

Theorem 5. If , then the numerical continuation process (45)(46) is feasible.

Proof. Define the map byIn the following, we will prove the numerical continuation processes (45)(46) are feasible.
By Lemma 2, for any , we haveBy the definition of Fréchet derivative, we obtainUsing the same technique described in Theorem 4, we have thatwhere .
According to Lemma 2, the numerical continuation processes (45)(46) are feasible.

5. Conclusions

We have introduced a class of nonlinear matrix equations that is wider than those studied earlier in the literature. We have derived some sufficient and necessary conditions for the existence and uniqueness of HPD solution of (1). In order to apply the homotopy continuation iterative method proposed by Avila [34] to the nonlinear matrix equation (1), we have constructed a related homotopy matrix equation and have derived a sufficient condition for the existence of a unique HPD solution of this equation. We have cited the definition and the judgment theorem of feasible for the numerical continuation process (45)(46), which was proposed in [34]. And then we have derived the condition for feasible of the numerical continuation processes (45)(46). Furthermore, we obtained the feasible conclusion of the numerical method by verifying the conditions in Theorem 2.5 in [34].

Data Availability

No data were used to support our study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Authors’ Contributions

All authors read and approved the final manuscript.

Acknowledgments

This work was supported in part by the National Nature Science Foundation of China (11601277).

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Copyright © 2020 Mengran Wang and Jing Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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