# An Iterative Algorithm for the Least Squares Generalized Reflexive

Authors: Feng Yin; Guang-Xin Huang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The generalized coupled Sylvester systems play a fundamental role in wide applications in several areas, such as stability theory, control theory, perturbation analysis, and some other fields of pure and applied mathematics. The iterative method is an important way to solve the generalized coupled Sylvester systems. In this paper, an iterative algorithm is constructed to solve the minimum Frobenius norm residual problem:∥(AXBCXD)-(EF)∥= min over generalized reflexive matrix X. For any initial generalized reflexive matrix X1, by the iterative algorithm, the generalized reflexive solution X* can be obtained within finite iterative steps in the absence of round-off errors, and the unique least-norm generalized reflexive solution X* can also be derived when an appropriate initial iterative matrix is chosen. Furthermore, the unique optimal approximate solution X^ to a given matrix X0 in Frobenius norm can be derived by finding the least-norm generalized reflexive solution X~* of a new corresponding minimum Frobenius norm residual problem: min∥(AX~BCX~D)-(E~F~)∥ with E~=E-AX0B, F~=F-CX0D. Finally, several numerical examples are given to illustrate that our iterative algorithm is effective.


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## Body

## 1. Introduction

A matrixP∈ℛn×n is said to be a generalized reflection matrix if P satisfies that PT=P,P2=I. Let P∈ℛm×m and Q∈ℛn×n be two generalized reflection matrices. A matrix A∈ℛm×n is called generalized reflexive (or generalized anti-reflexive) with respect to the matrix pair (P,Q) if PAQ=A(orPAQ=-A). The set of all m-by-n generalized reflexive matrices with respect to matrix pair (P,Q) is denoted by ℛrm×n(P,Q). The generalized reflexive and anti-reflexive matrices have many special properties and usefulness in engineering and scientific computations [1–3].In this paper, we will consider the minimum Frobenius norm residual problem and its optimal approximation problem as follows.Problem 1.
For given matricesA∈ℛp×m,B∈ℛn×q,C∈ℛs×m, D∈ℛn×t,E∈ℛp×q,F∈ℛs×t, find matrix X∈ℛrm×n(P,Q) such that
(1.1)‖(AXBCXD)-(EF)‖=min⁡.Problem 2.
LetSE denote the set of the generalized reflexive solutions of Problem 1. For a given matrix X0∈ℛrm×n(P,Q), find X̂∈SE such that
(1.2)‖X̂-X0‖=min⁡X∈SE‖X-X0‖.Problem1 plays a fundamental role in wide applications in several areas, such as Pole assignment, measurement feedback, and matrix programming problem.Liao and Lei [4]presented some examples to show a motivation for studying Problem 1. Problem 2 arises frequently in experimental design. Here the matrix X0 may be a matrix obtained from experiments, but it may not satisfy the structural requirement (generalized reflexive) and/or spectral requirement (the solution of Problem 1). The best estimate X̂ is the matrix that satisfies both requirements and is the best approximation of Ã in the Frobenius norm.Least-squares-based iterative algorithms are very important in system identification, parameter estimation, and signal processing, including the recursive least squares (RLS) and iterative least squares (ILS) methods for solving the solutions of some matrix equations, for example, the Lyapunov matrix equation, Sylvester matrix equations, and coupled matrix equations as well. Some related contributions in solving matrix equations and parameter identification/estimation should be mentioned in this paper. For example, novel gradient-based iterative (GI) method [5–9] and least-squares-based iterative methods [5, 9, 10] with highly computational efficiencies for solving (coupled) matrix equations are presented and have good stability performances, based on the hierarchical identification principle [11–13] which regards the unknown matrix as the system parameter matrix to be identified.The explicit and numerical solutions of matrix equation pairAXB=E,CXD=F have been addressed in a large body of the literature. Peng et al. [14] presented iterative methods to obtain the symmetric solutions of the matrix equation pair. Sheng and Chen [15] presented a finite iterative method when the matrix equation pair is consistent. Liao and Lei [4] presented an analytical expression of the least squares solution and an algorithm for the matrix equation pair with the minimum norm. Peng et al. [16] presented an efficient algorithm for the least squares reflexive solution. Dehghan and Hajarian [17] presented an iterative algorithm for solving a pair of the matrix equations over generalized centrosymmetric matrices. Cai and Chen [18] presented an iterative algorithm for the least squares bisymmetric solutions of the matrix equations. By applying the hierarchical identification principle,Kılıçman and Zhour [19]developed an iterative algorithm for obtaining the weighted least squares solution. Dehghan and Hajarian [20] constructed an iterative algorithm to solve the generalized coupled Sylvester matrix equations (AXB+CYD,EXF+GYH)=(M,N) over generalized bisymmetric matrices. Wu et al. [21, 22] gave the finite iterative solutions to coupled Sylvester-conjugate matrix equations. Wu et al. [23] gave the finite iterative solutions to a class of complex matrix equations with conjugate and transpose of the unknowns. Jonsson and Kågström [24, 25] proposed recursive block algorithms for solving the coupled Sylvester matrix equations and the generalized Sylvester and Lyapunov Matrix equations. Very recently, Huang et al. [26] presented a finite iterative algorithms for the one-sided and generalized coupled Sylvester matrix equations over generalized reflexive solutions. Yin et al. [27] presented a finite iterative algorithms for the two-sided and generalized coupled Sylvester matrix equations over reflexive solutions. For more studies on the matrix equations, we refer to [1–4, 16, 17, 28–40]. However, the problem of finding the least squares generalized reflexive solution of the matrix equation pair has not been solved.The following notations are also used in this paper. Letℛm×n denote the set of all m×n real matrices. We denote by the superscript T the transpose of a matrix. In matrix space ℛm×n, define inner product as 〈A,B〉=trace(BTA) for all A,B∈ℛm×n, and ∥A∥ represents the Frobenius norm of A. R(A) represents the column space of A. vec(·) represents the vector operator, that is, vec(A)=(a1T,a2T,…,anT)T∈ℛmn for the matrix A=(a1,a2,…,an)∈ℛm×n,ai∈ℛm,i=1,2,…,n. A⊗B stands for the Kronecker product of matrices A and B.This paper is organized as follows. In Section2, we will solve Problem 1 by constructing an iterative algorithm, that is, for an arbitrary initial matrix X1∈ℛrm×n(P,Q), we can obtain a solution X*∈ℛrm×n(P,Q) of Problem 1 within finite iterative steps in the absence of round-off errors. The convergence of the algorithm is also proved. Let X1=ATHBT+CTĤDT+PATHBTQ+PCTĤDTQ, where H∈ℛp×q,Ĥ∈ℛs×t are arbitrary matrices, or more especially, let X1=0∈ℛrm×n(P,Q);we can obtain the unique least-norm solution X* of Problem 1. Then in Section 3, we give the optimal approximate solution of Problem 2 by finding the least-norm generalized reflexive solution of a corresponding new minimum Frobenius norm residual problem. In Section 4, several numerical examples are given to illustrate the application of our iterative algorithm.


## 2. Solution of Problem1

In this section, we firstly introduce some definitions, lemmas, and theorems which are required for solving Problem1. Then we present an iterative algorithm to obtain the solution of Problem 1. We also prove that it is convergent. The following definitions and lemmas come from [41], which are needed for our derivation.Definition 2.1.
A set of matricesS∈ℛm×n is said to be convex if for X1,X2∈S and α∈(0,1),αX1+(1-α)X2∈S. Let ℛc denote a convex subset of ℛm×n.Definition 2.2.
A matrix functionf:ℛc→ℛ is said to be convex if
(2.1)f(αX1+(1-α)X2)≤αf(X1)+(1-α)f(X2)
for X1,X2∈ℛc and α∈(0,1).Definition 2.3.
Letf:ℛc→ℛ be a continuous and differentiable function. The gradient of f is defined as ∇f(X)=(∂f(X)/∂xij).Lemma 2.4.
Letf:ℛc→ℛ be a continuous and differentiable function. Then  f is convex on  ℛc if and only if
(2.2)f(Y)≥f(X)+〈∇f(X),Y-X〉
for all X,Y∈ℛc.Lemma 2.5.
Letf:ℛc→ℛ be a continuous and differentiable function, and there exists  X* in the interior of  ℛc such that  f(X*)=minX∈ℛcf(X), then  ∇f(X*)=0.Note that the setℛrm×n(P,Q) is unbounded, open, and convex. Denote
(2.3)F(X)=‖(AXBCXD)-(EF)‖2,
then F(X) is a continuous, differentiable, and convex function on ℛrm×n(P,Q). Hence, by applying Lemmas 2.4 and 2.5, we obtain the following lemma.Lemma 2.6.
LetF(X) be defined by (2.3), then there exists  X*∈ℛrm×n(P,Q) if and only if  F(X*)=minX∈ℛrm×n(P,Q)F(X), then  ∇F(X*)=0.From the Taylor series expansion, we have(2.4)F(X+ϵY)=F(X)+ϵ〈∇F(X),Y〉+o(ϵ),∀X,Y∈Rrm×n(P,Q),ϵ∈R.
On the other hand, by the basic properties of Frobenius norm and the matrix inner product, we get the expression(2.5)F(X+ϵY)=‖(A(X+ϵY)BC(X+ϵY)D)-(EF)‖2=〈AXB-E+ϵAYB,AXB-E+ϵAYB〉+〈CXD-F+ϵCYD,CXD-F+ϵCYD〉=〈AXB-E,AXB-E〉+〈CXD-F,CXD-F〉+2ϵ〈AXB-E,AYB〉+2ϵ〈CXD-F,CYD〉+ϵ2(〈AYB,AYB〉+〈CYD,CYD〉)=F(X)+2ϵ〈AXB-E,AYB〉+2ϵ〈CXD-F,CYD〉+ϵ2(〈AYB,AYB〉+〈CYD,CYD〉).
Note that
(2.6)2ϵ〈AXB-E,AYB〉=2ϵ〈ATAXBBT-ATEBT,Y〉=2ϵ〈ATAXBBT-ATEBT+P(ATAXBBT-ATEBT)Q2,Y〉+2ϵ〈ATAXBBT-ATEBT-P(ATAXBBT-ATEBT)Q2,Y〉=ϵ〈ATAXBBT-ATEBT+PATAXBBTQ-PATEBTQ,Y〉+ϵ〈ATAXBBT-ATEBT-P(ATAXBBT-ATEBT)Q,Y〉=ϵ〈ATAXBBT-ATEBT+PATAXBBTQ-PATEBTQ,Y〉,2ε〈CXD-F,CYD〉=2ϵ〈CTCXDDT-CTFDT,Y〉=2ϵ〈CTCXDDT-CTFDT+P(CTCXDDT-CTFDT)Q2,Y〉+2ϵ〈CTCXDDT-CTFDT-P(CTCXDDT-CTFDT)Q2,Y〉=ϵ〈CTCXDDT-CTFDT+PCTCXDDTQ-PCTFDTQ,Y〉+ϵ〈CTCXDDT-CTFDT-P(CTCXDDT-CTFDT)Q,Y〉=ϵ〈CTCXDDT-CTFDT+PCTCXDDTQ-PCTFDTQ,Y〉.
Thus, we have
(2.7)F(X+ϵY)=F(X)+ϵ〈ATAXBBT-ATEBT+PATAXBBTQ-PATEBTQ+CTCXDDT-CTFDT+PCTCXDDTQ-PCTFDTQ,Y〉+ϵ2(〈AYB,AYB〉+〈CYD,CYD〉).
By comparing (2.4) with (2.7), we have
(2.8)∇F(X)=ATAXBBT-ATEBT+PATAXBBTQ-PATEBTQ+CTCXDDT-CTFDT+PCTCXDDTQ-PCTFDTQ.According to Lemma2.6 and (2.8), we obtain the following theorem.Theorem 2.7.
A matrixX*∈ℛrm×n(P,Q) is a solution of Problem  1 if and only if  ∇F(X*)=0.For the convenience of discussion, we adopt the following notations:(2.9)M(X)=ATAXBBT+CTCXDDT+PATAXBBTQ+PCTCXDDTQ,N=ATEBT+CTFDT+PATEBTQ+PCTFDTQ,G(X)=-∇F(X)=N-M(X),Pk=G(Xk).
The following algorithm is constructed to solve Problems 1 and 2.Algorithm 2.8.
Step 1.
Input matricesA∈ℛp×m,B∈ℛn×q,C∈ℛs×m, D∈ℛn×t,E∈ℛp×q,F∈ℛs×t, and two generalized reflection matrix P∈ℛm×m,Q∈ℛn×n;Step 2.
Choose an arbitrary matrixX1∈ℛrm×n(P,Q). Compute
(2.10)P1=N-M(X1),Q1=M(P1),k∶=1.Step 3.
IfP1=0, then stop. Else go to Step 4.Step 4.
Compute(2.11)Xk+1=Xk+‖Pk‖2〈Qk,M(Pk)〉Qk,Pk+1=Pk-‖Pk‖2〈Qk,M(Pk)〉M(Qk),Qk+1=Pk+1-〈Pk+1,M(Qk)〉〈Qk,M(Qk)〉Qk.Step 5.
IfPk+1=0, then stop. Else, let k:=k+1, and go to Step 4.Remark 2.9.
Obviously, it can be seen thatPi∈ℛrm×n(P,Q),Qi∈ℛrm×n(P,Q), and Xi∈ℛrm×n(P,Q), where i=1,2,….Lemma 2.10.
Suppose thatX∈Rrm×n(P,Q),Y∈Rrm×n(P,Q), then
(2.12)〈M(X),Y〉=〈X,M(Y)〉.Proof.
One has(2.13)〈M(X),Y〉=〈ATAXBBT+CTCXDDT+PATAXBBTQ+PCTCXDDTQ,Y〉=〈ATAXBBT,Y〉+〈CTCXDDT,Y〉+〈PATAXBBTQ,Y〉+〈PCTCXDDTQ,Y〉=〈X,ATAYBBT〉+〈X,CTCYDDT〉+〈X,ATAPYQBBT〉+〈X,CTCPYQDDT〉=〈X,ATAYBBT〉+〈X,CTCYDDT〉+〈PXQ,ATAPYQBBT〉+〈PXQ,CTCPYQDDT〉=〈X,ATAYBBT〉+〈X,CTCYDDT〉+〈X,PATAYBBTQ〉+〈X,PCTCYDDTQ〉=〈X,ATAYBBT+CTCYDDT+PATAYBBTQ+PCTCYDDTQ〉=〈X,M(Y)〉.
This completes the proof.Lemma 2.11.
For the sequences{Pi} and {Qi} generated by Algorithm  2.8, if there exists a positive number  k such that  Pi≠0 for all  i=1,2,…,k, then
(2.14)〈Pi,Pj〉=0,〈Qi,M(Qj)〉=0(i,j=1,2,…,k,i≠j).Proof.
Since〈A,B〉=〈B,A〉 holds for all matrices A and B in ℛm×n, we only need prove that 〈Pi,Pj〉=0,〈Qi,M(Qj)〉=0 for all 1≤i<j≤k. We prove the conclusion by induction and two steps are required.
Step 1.
We will show that(2.15)〈Pi,Pi+1〉=0,〈Qi,M(Qi+1)〉=0,i=1,2,…,k-1.
To prove this conclusion, we also use induction.
Fori=1, by Algorithm 2.8 and Lemma 2.10, we have that(2.16)〈P1,P2〉=〈P2,P1〉=〈P1-‖P1‖2〈Q1,M(P1)〉M(Q1),P1〉=‖P1‖2-‖P1‖2〈Q1,M(P1)〉〈M(Q1),P1〉=0,〈Q1,M(Q2)〉=〈Q2,M(Q1)〉=〈P2-〈P2,M(Q1)〉〈Q1,M(Q1)〉Q1,M(Q1)〉=〈P2,M(Q1)〉-〈P2,M(Q1)〉〈Q1,M(Q1)〉〈Q1,M(Q1)〉=0.
Assume (2.15) holds for i=s(1<s<k). For i=s+1, by Lemma 2.10, we have(2.17)〈Ps+1,Ps+2〉=〈Ps+2,Ps+1〉=〈Ps+1-‖Ps+1‖2〈Qs+1,M(Ps+1)〉M(Qs+1),Ps+1〉=‖Ps+1‖2-‖Ps+1‖2〈Qs+1,M(Ps+1)〉〈M(Qs+1),Ps+1〉=‖Ps+1‖2-‖Ps+1‖2〈Qs+1,M(Ps+1)〉〈Qs+1,M(Ps+1)〉=0,〈Qs+1,M(Qs+2)〉=〈Qs+2,M(Qs+1)〉=〈Ps+2-〈Ps+2,M(Qs+1)〉〈Qs+1,M(Qs+1)〉Qs+1,M(Qs+1)〉=〈Ps+2,M(Qs+1)〉-〈Ps+2,M(Qs+1)〉〈Qs+1,M(Qs+1)〉〈Qs+1,M(Qs+1)〉=0.
Hence, (2.15) holds for i=s+1. Therefore, (2.15) holds by the principle of induction.Step 2.
Assume that〈Pi,Ps〉=0,〈Qi,M(Qs)〉=0,i=1,2,…,s-1, then we show that
(2.18)〈Pi,Ps+1〉=0,〈Qi,M(Qs+1)〉=0,i=1,2,…,s.
In fact, by Algorithm 2.8 we have
(2.19)〈Pi,Ps+1〉=〈Pi,Ps-‖Ps‖2〈Qs,M(Ps)〉M(Qs)〉=〈Pi,Ps〉-∥Ps∥2〈Qs,M(Ps)〉〈Pi,M(Qs)〉=-‖Ps‖2〈Qs,M(Ps)〉〈Qi+〈Pi,M(Qi-1)〉〈Qi-1,M(Qi-1)〉Qi-1,M(Qs)〉=-‖Ps‖2〈Qs,M(Ps)〉〈Qi,M(Qi-1〉+‖Ps‖2〈Qs,M(Ps)〉〈Pi,M(Qi-1)〉〈Qi-1,M(Qi-1)〉〈Qi-1,M(Qs)〉=0,〈Qi,M(Qs+1)〉=〈M(Qi),Qs+1〉=〈M(Qi),Ps+1-〈Ps+1,M(Qs)〉〈Qs,M(Qs)〉Qs〉=〈M(Qi),Ps+1〉-〈Ps+1,M(Qs)〉〈Qs,M(Qs)〉〈M(Qi),Qs〉=〈Qi,M(Pi)〉‖Pi‖2〈Pi-Pi+1,Ps+1〉=-〈Qi,M(Pi)〉‖Pi‖2〈Pi+1,Ps+1〉=0.
By the principle of induction, (2.14) is implied in Steps 1 and 2. This completes the proof.Lemma 2.12.
Assume thatX* is an arbitrary solution of Problem 1, then
(2.20)〈X*-Xk,M(Qk)〉=2‖Pk‖2‖P1‖2(‖AP1B‖2+‖CP1D‖2),k=1,2,…,
where the sequences {Xk},{Pk}, and {Qk} are generated by Algorithm 2.8.Proof.
First, by Algorithm2.8, it is easy to verify that
(2.21)-〈Pk+1,M(Qk)〉〈Qk,M(Qk)〉=‖Pk+1‖2‖Pk‖2.
Thus
(2.22)〈X*-Xk,M(Qk)〉=〈M(X*-Xk),Qk〉=〈Pk,Qk〉=〈Pk-1-‖Pk-1‖2〈Qk-1,M(Pk-1)〉M(Qk-1),Qk〉=〈Pk-1,Qk〉=⋯=〈P1,Qk〉=〈P1,Pk-〈Pk,M(Qk-1)〉〈Qk-1,M(Qk-1)〉Qk-1〉=-〈Pk,M(Qk-1)〉〈Qk-1,M(Qk-1)〉〈P1,Qk-1〉=⋯=(-〈Pk,M(Qk-1)〉〈Qk-1,M(Qk-1)〉)⋯(-〈P2,M(Q1)〉〈Q1,M(Q1)〉)〈P1,Q1〉=‖Pk‖2‖Pk-1‖2⋯‖P2‖2‖P1‖2〈P1,M(P1)〉=2‖Pk‖2‖P1‖2(‖AP1B‖2+‖CP1D‖2).
This complete the proof.Remark 2.13.
Lemma2.12 implies that if Pi≠0, then M(Qi)≠0, thus Qi≠0(i=1,2,…).Theorem 2.14.
For an arbitrary initial matrixX1∈ℛrm×n(P,Q), a solution of Problem  1 can be obtained with finite iteration steps in the absence of round-off errors.Proof.
IfPi≠0,i=1,2,…,mn, by Lemma 2.12 we have Qi≠0,i=1,2,…,mn, then we can compute Xmn+1,Pmn+1 by Algorithm 2.8.
By Lemma2.11, we have (2.23)〈Pi,Pmn+1〉=0,i=1,2,…,mn,〈Pi,Pj〉=0,i,j=1,2,…,mn,i≠j.
It can be seen that the set ofP1,P2,…,Pmn is an orthogonal basis of the matrix space ℛrm×n(P,Q), which implies that Pmn+1=0, that is, Xmn+1 is a solution of Problem 1. This completes the proof.To show the least-norm generalized reflexive solution of Problem1, we first introduce the following result.Lemma 2.15 (see [16, Lemma 2.7]).
Suppose that the minimum residual problem∥My-b∥=min has a solution  y*∈R(MT), then  y* is the unique least Frobenius norm solution of the minimum residual problem.By Lemma2.15, the following result can be obtained.Theorem 2.16.
If one chooses the initial iterative matrixX1=ATHBT+CTĤDT+PATHBTQ+PCTĤDTQ, where  H∈ℛp×q,Ĥ∈ℛs×t are arbitrary matrices, especially, let  X1=0∈ℛrm×n, one can obtain the unique least-norm generalized reflexive solution of Problem  1 within finite iteration steps in the absence of round-off errors by using Algorithm 2.8.Proof.
By Algorithm2.8 and Theorem 2.14, if we let X1=ATHBT+CTĤDT+PATHBTQ+PCTĤDTQ, where H∈ℛp×q,Ĥ∈ℛs×t are arbitrary matrices, we can obtain the solution X* of Problem 1 within finite iteration steps in the absence of round-off errors, the solution X* can be represented that
(2.24)X*=ATGBT+CTĜDT+PATGBTQ+PCTĜDTQ.
In the sequel, we will prove thatX* is just the least-norm solution of Problem 1. Consider the following minimum residual problem(2.25)min⁡X∈Rrm×n(P,Q)‖(AXBCXDAPXQBCPXQD)-(EFEF)‖.
Obviously, the solvability of Problem1 is equivalent to that of the minimum residual problem (2.25), and the least-norm solution of Problem 1 must be the least-norm solution of the minimum residual problem (2.25).
In order to prove thatX* is the least-norm solution of Problem 1, it is enough to prove that X* is the least-norm solution of the minimum residual problem (2.25). Denote vec(X)=x,vec(X*)=x*,vec(G)=g1,vec(Ĝ)=g2,vec(E)=e,vec(F)=f, then the minimum residual problem (2.25) is equivalent to the minimum residual problem as follows:(2.26)min⁡x∈Rmn‖(BT⊗ADT⊗C(BTQ)⊗(AP)(DTQ)⊗(CP))x-(efef)‖.
Noting that
(2.27)x*=vec⁡(ATGBT+CTĜDT+PATGBTQ+PCTĜDTQ)=(B⊗AT)g1+(D⊗CT)g2+((QB)⊗(PAT))g1+((QD)⊗(PCT))g2=(B⊗ATD⊗CT(QB)⊗(PAT)(QD)⊗(PCT))(g1g2g1g2)=(BT⊗ADT⊗C(BTQ)⊗(AP)(DTQ)⊗(CP))T(g1g2g1g2)∈R((BT⊗ADT⊗C(BTQ)⊗(AP)(DTQ)⊗(CP))T),
by Lemma 2.15 we can see that x* is the least-norm solution of the minimum residual problem (2.26). Since vector operator is isomorphic, X* is the unique least-norm solution of the minimum residual problem (2.25);furthermore X* is the unique least-norm solution of Problem 1.


## 3. Solution of Problem2

Since the solution set of Problem1 is no empty, when X∈SE, then(3.1)min⁡X∈Rrm×n(P,Q)‖(AXBCXD)-(EF)‖⟺min⁡X∈Rrm×n(P,Q)‖(A(X-X0)BC(X-X0)D)-(E-AX0BF-CX0D)‖.LetX̃=X-X0,Ẽ=E-AX0B,F̃=F-CX0D, then Problem 2 is equivalent to finding the least-norm generalized reflexive solution of a new corresponding minimum residual problem(3.2)min⁡X̃∈Rrm×n(P,Q)‖(AX̃BCX̃D)-(ẼF̃)‖.By using Algorithm2.8, let initially iterative matrix X̃1=ATHBT+CTĤDT+PATHBTQ+PCTĤDTQ, or more especially, let X̃1=0∈Rrm×n(P,Q); we can obtain the unique least-norm generalized reflexive solution X̃* of minimum residual problem (3.2), then we can obtain the generalized reflexive solution X̂ of Problem 2, and X̂ can be represented that X̂=X̃*+X0.


## 4. Numerical Examples

In this section, we will show several numerical examples to illustrate our results. All the tests are performed by MATLAB 7.8.Example 4.1.
Consider the generalized reflexive solution of Problem1, where
(4.1)A=(13-57-92046-10-296-836227-13-55-22-1-1184-6-9-19),B=(408-54-150-234-10250392-6-27-8111),C=(632-57-921046-119-1293-8136427-15-515-22-13-1129-6-9-19),D=(718-614-450-233-1208251694-6-58-2917),E=(592-11911216-244-13313054311234-518221814-4071668-11765371434-1794083-1374-808242-3150-13621104-2848423-29091441-182-3326),F=(-288228302992291-48494096701090-783-7933363-1262979-385124626321734553-3709-100-1774-4534-45481256-6896864-2512-1136-1633-5412).
Let
(4.2)P=(000100000100-1001000001000),Q=(0000-10001000-10001000-10000).
We will find the least squares generalized reflexive solution of the matrix equation pairAXB=E,CXD=F by using Algorithm 2.8. Because of the influence of the error of calculation, ∥Pk∥=∥-∇F(Xk)∥ is usually unequal to zero in the process of the iteration, where k=1,2,…. For any chosen positive number ε, however small enough, for example, ε=1.0000e-8, whenever ∥Pk∥<ε, stop the iteration, and Xk is regarded to be the least squares generalized reflexive solution of the matrix equation pair AXB=E,CXD=F.
Let(4.3)X1=(0000000000000000000000000);
by Algorithm 2.8, we have the unique least Frobenius norm generalized reflexive solution of Problem 1(4.4)X139=(5.00003.0000-6.000012.0000-5.0000-11.00008.0000-1.00009.00007.000013.0000-4.0000-8.00004.000013.00005.000012.00006.00003.0000-5.0000-7.00009.00001.00008.000011.0000),‖P139‖=8.1258e-009<ε,min⁡X∈Rrm×n(P,Q)‖(AXBCXD)-(EF)‖=‖(AX139BCX139D)-(EF)‖=1.9595e-011.
The convergence curve for the Frobenius norm of the residual is shown in Figure 1.Figure 1
Convergence curve for the Frobenius norm of the residual for Example4.1 with X1=0.Example 4.2.
Consider the least-norm generalized reflexive solution of the minimum residual problem in Example4.1. Let
(4.5)H=(101020-10101-10012010-301210-10-2-10),Ĥ=(-11-100010-131-10-202010-301210-10-212),X1=ATHBT+CTĤDT+PATHBTQ+PCTĤDTQ.
By using Algorithm 2.8, we have the unique least Frobenius norm generalized reflexive solution of Problem 1(4.6)X118=(5.00003.0000-6.000012.0000-5.0000-11.00008.0000-1.00009.00007.000013.0000-4.0000-8.00004.000013.00005.000012.00006.00003.0000-5.0000-7.00009.00001.00008.000011.0000),‖P118‖=9.8093e-009<ε,min⁡X∈Rrm×n(P,Q)‖(AXBCXD)-(EF)‖=‖(AX118BCX118D)-(EF)‖=1.1235e-010.The convergence curve for the Frobenius norm of the residual is shown in Figure2.Figure 2
Convergence curve for the Frobenius norm of the residual for Example4.2.Example 4.3.
LetSE denote the set of all generalized reflexive solutions of Problem 1 in Example 4.1. For a given matrix
(4.7)X0=(-331110-7161010-90910-11-133-106-1-70)∈Rr5×5(P,Q),
we will find X̂∈SE, such that
(4.8)‖X̂-X0‖=min⁡X∈SE‖X-X0‖,
that is, find the optimal approximate solution to the matrix X0 in SE.LetX̃=X-X0,Ẽ=E-AX0B,F̃=F-CX0D, by the method mentioned in Section 3, we can obtain the least-norm generalized reflexive solution X̃* of the minimum residual problem (3.2) by choosing the initial iteration matrix X̃1=0, and X̃* is such that(4.9)X̃102*=(8.00000.0000-7.000011.0000-6.0000-11.000015.0000-2.00003.0000-3.00003.00005.0000-8.0000-5.00003.00006.000011.00007.00000.0000-8.00003.00003.00002.000015.000011.0000),‖P102‖=8.6456e-009<ε,X̂=X̃102*+X0=(5.00003.0000-6.000012.0000-5.0000-11.00008.0000-1.00009.00007.000013.0000-4.0000-8.00004.000013.00005.000012.00006.00003.0000-5.0000-7.00009.00001.00008.000011.0000).The convergence curve for the Frobenius norm of the residual is shown in Figure3.Figure 3
Convergence curve for the Frobenius norm of the residual for Example4.3.


## 5. Conclusion

This paper mainly solves the minimum Frobenius norm residual problem and its optimal approximate problem over generalized reflexive matrices by constructing an iterative algorithm. We solve the minimum Frobenius norm residual problem by constructing an iterative algorithm, that is, for an arbitrary initial matrixX1∈ℛrm×n(P,Q),we obtain a solution X*∈ℛrm×n(P,Q) of Problem 1 within finite iterative steps in the absence of round-off errors. The convergence of the algorithm is also proved. Let X1=ATHBT+CTĤDT+PATHBTQ+PCTĤDTQ, where H∈ℛp×q,Ĥ∈ℛs×t are arbitrary matrices, or more especially, let X1=0∈ℛrm×n(P,Q); we obtain the unique least-norm solution X* of the minimum Frobenius norm residual problem. Then we give the generalized reflexive solution of the optimal approximate problem by finding the least-norm generalized reflexive solution of a corresponding new minimum Frobenius norm residual problem.Several numerical examples are given to confirm our theoretical results. We can see that our iterative algorithm is effective. We also note that for the minimum Frobenius norm residual problem with large but not sparse matricesA,B,C,D,E, and F, Algorithm 2.8 may be terminated more than mn steps because of computational errors.



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*Source: 857284-2012-03-12.xml*