An Iterative Method for the Least-Squares Problems of a General Matrix Equation Subjects to Submatrix Constraints

Abstract

An iterative algorithm is proposed for solving the least-squares problem of a general matrix equation ∑^𝑡_𝑖=1𝑀_𝑖𝑍_𝑖𝑁_𝑖=𝐹, where 𝑍_𝑖 (𝑖=1,2,…,𝑡) are to be determined centro-symmetric matrices with given central principal submatrices. For any initial iterative matrices, we show that the least-squares solution can be derived by this method within finite iteration steps in the absence of roundoff errors. Meanwhile, the unique optimal approximation solution pair for given matrices 𝑍_𝑖 can also be obtained by the least-norm least-squares solution of matrix equation ∑^𝑡_𝑖=1𝑀_𝑖𝑍_𝑖𝑁_𝑖=𝐹, in which 𝑍_𝑖=𝑍_𝑖−𝑍_𝑖,𝐹=𝐹−∑^𝑡_𝑖=1𝑀_𝑖𝑍_𝑖𝑁_𝑖. The given numerical examples illustrate the efficiency of this algorithm.

1. Introduction

Throughout this paper, we denote the set of all 𝑚×𝑛 real matrices by 𝑅^𝑚×𝑛. The symbol 𝐴^𝑇 represents the transpose of matrix 𝐴. 𝐽_𝑛 and 𝐼_𝑛 stand for the reverse unit matrix, and identity matrix, respectively. For 𝐴,𝐵∈𝑅^𝑚×𝑛, the inner product of matrices 𝐴 and 𝐵 is defined by ⟨𝐴,𝐵⟩=trace(𝐵^𝑇𝐴), which leads to the Frobenius norm, that is, ‖𝐴‖=⟨𝐴,𝐴⟩.

A matrix 𝐴=(𝑎_𝑖𝑗)∈𝑅^𝑛×𝑛, is called centro-symmetric (centro-skew symmetric) if and only if 𝑎_𝑖𝑗=𝑎_{𝑛−𝑖+1,𝑛−𝑗+1}𝑎_𝑖𝑗=−𝑎_{𝑛−𝑖+1,𝑛−𝑗+1},𝑖,𝑗=1,2,…,𝑛, (1) which can also be characterized equivalently by 𝐽_𝑛𝐴𝐽_𝑛=𝐴(𝐽_𝑛𝐴𝐽_𝑛=−𝐴). The set of all centro-symmetric (centro-skew symmetric) matrices is denoted by CS𝑅^𝑛×𝑛(CAS𝑅^𝑛×𝑛). This kind of matrices plays an important role in many applications (see, e.g., [1–4]), and has been frequently and widely investigated (see, e.g., [5–7]) by using generalized inverse, generalized singular value decomposition (GSVD) [8], and so forth. For more, we refer the readers to [9–16] and therein.

We firstly introduce the concept of the central principal submatrix which is originally put forward by Yin [17].

Definition 1. Let 𝐴∈𝑅^𝑛×𝑛, if 𝑛−𝑝 is even, a 𝑝×𝑝 central principal submatrix of 𝐴, denoted by 𝐴[𝑝], is obtained by deleting the first and last (𝑛−𝑝)/2 rows and columns of 𝐴, namely, 𝐴[𝑝]=(𝑎_𝑖𝑗)_{(𝑛−𝑝)/2≤𝑖,𝑗≤𝑛−(𝑛−𝑝)/2}.

Evidently, a matrix with odd (even) order only has central principal submatrices of odd (even) order.

Now, the first problem to be studied here can be stated as follows.

Problem 2. Given 𝑀_𝑖∈𝑅^{𝑚×𝑝𝑖}, 𝑁_𝑖∈𝑅^{𝑝𝑖×𝑛} and 𝒳_𝑖∈CS𝑅^{𝑞𝑖×𝑞𝑖} (𝑖=1,2,…,𝑡), 𝐹∈𝑅^𝑚×𝑛. Find the least-squares solution 𝑍_𝑖∈Π_𝑖 of matrix equation 𝑀₁𝑍₁𝑁₁+𝑀₂𝑍₂𝑁₂+⋯+𝑀_𝑡𝑍_𝑡𝑁_𝑡=𝐹, (2) in which Π_𝑖={𝑍_𝑖|𝑍_𝑖∈CS𝑅^{𝑝𝑖×𝑝𝑖} with 𝑍_𝑖[𝑞_𝑖]=𝒳_𝑖}, 𝑝_𝑖>𝑞_𝑖, 𝑖∈Γ≜{1,2,…,𝑡}, and 𝑍_𝑖[𝑞_𝑖] represents the 𝑞_𝑖×𝑞_𝑖 central principal submatrix of 𝑍_𝑖.

Problem 2 is the submatrix constrained problem of matrix equation (2), which originally arises from a practical subsystem expansion problem, and has been deeply investigated (see, e.g., [7, 18–22]). In these literatures, the generalized inverses or some complicated matrix decompositions such as canonical correlation decomposition (CCD) [23] and GSVD are employed. However, it is almost impossible to solve (2) by the above methods. The iterative method is an efficient approach. Recently, kinds of iteration methods have been constructed: Zhou and Duan [24] studied the generalized Sylvester matrix equation ^𝜙_𝑖=0𝐴_𝑖𝑋𝐹^𝑖+^𝜓_𝑘=0𝐵_𝑘𝑌𝐹^𝑘=𝑅 (3) by so-called generalized Sylvester mapping that has pretty properties. Wu et al. [25] presented an finite iterative method for a class of complex matrix equations including conjugate and transpose of unknown solution. Motivated by the well-known Jacobi and Gauss-Seidel iterations methods, Ding and Chen, in [26], proposed a general family of iterative methods to solve linear matrix equations; meanwhile, these methods were also extended to solve the following coupled Sylvester matrix equations ^𝑝_𝑗=1𝐴_𝑖𝑗𝑋_𝑗𝐵_𝑖𝑗=𝐶_𝑖𝑗,𝑖=1,2,…. (4)

Although these iterative algorithms are efficient, there still exist some handicaps when meeting the constrained matrix equation problem (i.e., to find the solution of matrix equation in some matrices sets with specifical structure, for instance, symmetric matrices, centro-symmetric matrices, and bi-symmetric matrices sets) and the submatrix constrained problem, since these methods cannot keep the special properties of the unknown matrix in the iterative process. Based on the classical conjugate gradient (CG) method, Peng et al. [27] gave an iterative method to find the bisymmetric solution of matrix equation (2). Similar method was constructed to solve matrix equations (4) with generalized bisymmetric 𝑋_𝑗 in [28]. In particular, Li et al. [29] proposed an elegant algorithm for solving the generalized Sylvester (Lyapunov) matrix equation 𝐴𝑋𝐵+𝐶𝑌𝐷=𝐸 with bisymmetric 𝑋 and symmetric 𝑌, the two unknown matrices include the given central principal submatrix and leading principal submatrix, respectively. This method shunned the difficulties in numerical instability and computational complexity, and solved the problem, completely. By borrowing the thinking of this iterative algorithm, we will solve Problem 2 by iteration method.

The second problem to be considered is the optimal approximation problem.

Problem 3. Let 𝑆_𝐸 be the solutions set of Problem 2. For given matrices 𝑍_𝑖∈𝑅^{𝑝𝑖×𝑝𝑖}, find 𝑍_𝑖 such that ^𝑡_𝑖=1𝑍_𝑖−𝑍_𝑖²=min_{(𝑍1,𝑍2,…,𝑍𝑡)∈𝑆𝐸}^𝑡_𝑖=1𝑍_𝑖−𝑍_𝑖². (5)

This problem occurs frequently in experimental design (see for instance [30]). Here, the preliminary estimation 𝑍_𝑖 of the unknown matrix 𝑍_𝑖 can be obtained from experiments, but it may not satisfy the structural requirement and/or spectral requirement. The best estimation of 𝑍_𝑖, is the matrix 𝑍_𝑖 that satisfies both requirements, which is the optimal approximation of 𝑍_𝑖 (see, e.g., [31, 32]). About this problem, we also refer the authors to [9–11, 13, 15, 16, 20–23, 27–29, 33–36] and therein.

The rest of this paper is outlined as follows. In Section 2, an iterative algorithm will be proposed to solve Problem 2, and the properties of which will be investigated. In Section 3, we will consider the optimal approximation Problem 3 by using the iterative algorithm. In Section 4, some numerical examples will be given to verify the efficiency of this algorithm.

2. The Algorithm for Problem 2 and Its Properties

According to the definition of centro-symmetric matrix, when 𝑛−𝑞 is even, a centro-symmetric matrix 𝑍∈CS𝑅^𝑛×𝑛 can be divided into smaller submatrices, namely,𝑍𝑍=¹¹𝑍₁₂𝑍₁₃𝑍₂₁𝑍₂₂𝐽_𝑞𝑍₂₁𝐽_{(𝑛−𝑞)/2}𝐽_{(𝑛−𝑞)/2}𝑍₁₃𝐽_{(𝑛−𝑞)/2}𝐽_{(𝑛−𝑞)/2}𝑍₁₂𝐽_𝑞𝐽_{(𝑛−𝑞)/2}𝑍₁₁𝐽_{(𝑛−𝑞)/2}, (6)where 𝑍₁₁∈𝑅^{(𝑛−𝑞)/2×(𝑛−𝑞)/2}, 𝑍₁₂∈𝑅^{(𝑛−𝑞)/2×𝑞}, 𝑍₁₃∈𝑅^{(𝑛−𝑞)/2×(𝑛−𝑞)/2}, 𝑍₂₁∈𝑅^{𝑞×(𝑛−𝑞)/2}, and 𝑍₂₂∈CS𝑅^𝑞×𝑞.

Now, for some fixed positive integer 𝑖∈Γ, we define two matrix sets. CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖=𝑍_𝑖∣𝑍_𝑖∈CS𝑅^{𝑝𝑖×𝑝𝑖}with𝑍_𝑖𝑞_𝑖=0,CS𝑅^{𝑝𝑖×𝑝𝑖}_⋄,𝑞𝑖=𝑍_𝑖∣𝑍_𝑖=0000𝑋_𝑖0000∈CS𝑅^{𝑝𝑖×𝑝𝑖},𝑋_𝑖=𝑍_𝑖𝑞_𝑖∈CS𝑅^{𝑞𝑖×𝑞𝑖}. (7)

It is clear that both CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖 and CS𝑅^{𝑝𝑖×𝑝𝑖}_⋄,𝑞𝑖 are linear subspaces of 𝑅^{𝑝𝑖×𝑝𝑖}.

In addition, for any matrix 𝑋∈𝑅^{𝑝𝑖×𝑝𝑖}, it has uniquely decomposition in direct sum, that is, 𝑋=𝑋₁⊕𝑋₂, here 𝑋₁=(𝑋+𝐽_𝑝𝑖𝑋𝐽_𝑝𝑖)/2, 𝑋₂=(𝑋−𝐽_𝑝𝑖𝑋𝐽_𝑝𝑖)/2. Furthermore, 𝑋₁=𝑋₁₁+𝑋₁₂ is also the direct sum decomposition of 𝑋₁ if 𝑋₁₁∈CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖, 𝑍₁₂∈CS𝑅^{𝑝𝑖×𝑝𝑖}_⋄,𝑞𝑖, since ⟨𝑋₁₁,𝑋₁₂⟩=0. Hence, we obtain the following.

Lemma 4. Consider 𝑅^{𝑝𝑖×𝑝𝑖}=CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖⊕CS𝑅^{𝑝𝑖×𝑝𝑖}_⋄,𝑞𝑖⊕CAS𝑅^{𝑝𝑖×𝑝𝑖}.

Lemma 4 reveals that any matrix 𝑊∈𝑅^{𝑝𝑖×𝑝𝑖} can be uniquely written as 𝑊=𝑊₁+𝑊₂+𝑊₃, where 𝑊₁∈CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖, 𝑊₂∈CS𝑅^{𝑝𝑖×𝑝𝑖}_⋄,𝑞𝑖, 𝑊₃∈CAS𝑅^{𝑝𝑖×𝑝𝑖}. Then, we can define the following linear projection operators: ℒ_𝑖∶𝑅^{𝑝𝑖×𝑝𝑖}⟶CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖𝑊⟶𝑊₁ (8) for 𝑖∈Γ.

According to the definition of ℒ_𝑖, if 𝑊∈𝑅^{𝑝𝑖×𝑝𝑖} and 𝑌∈CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖, we have ⟨𝑊,𝑌⟩=𝑊₁,𝑌=ℒ_𝑖(𝑊),𝑌. (9) This property will be employed frequently in the residual context.

The following theorem is essential for solving Problem 2, which transforms equivalently Problem 2 into solving the least-square problem of another matrix equation.

Theorem 5. Any solution group of Problem 2 can be obtained by 𝑍₁,𝑍₂,…,𝑍_𝑡=𝑌₁+𝑍^⋄₁,𝑌₂+𝑍^⋄₂,…,𝑌_𝑡+𝑍^⋄_𝑡, (10) where 𝑌_𝑖∈𝐶𝑆𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖 is the least-squares solution of matrix equation 𝑀₁𝑌₁𝑁₁+𝑀₂𝑌₂𝑁₂+⋯+𝑀_𝑡𝑌_𝑡𝑁_𝑡𝑀=𝐺,𝐺=𝐹−₁𝑍^⋄₁𝑁₁+𝑀₂𝑍^⋄₂𝑁₂+⋯+𝑀_𝑡𝑍^⋄_𝑡𝑁_𝑡,𝑍^⋄_𝑖=0000𝒳_𝑖0000∈𝐶𝑆𝑅^{𝑝𝑖×𝑝𝑖}_⋄,𝑞𝑖, (11)𝒳_𝑖∈CS𝑅^{𝑞𝑖×𝑞𝑖} is the given central principal submatrix of 𝑍_𝑖 in Problem 2.

Proof. Noting that the definition of CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖, we have min_{𝑍𝑖∈Π𝑖}^𝑡_𝑖=1𝑀_𝑖𝑍_𝑖𝑁_𝑖−𝐹⟺min_{𝑌𝑖∈CS𝑅𝑝𝑖𝑖𝑖×𝑝⋆,𝑞}^𝑡_𝑖=1𝑀_𝑖𝑌_𝑖+𝑍_𝑖𝑁_𝑖−𝐹⟺min_{𝑌𝑖∈CS𝑅𝑝𝑖𝑖𝑖×𝑝⋆,𝑞}^𝑡_𝑖=1𝑀_𝑖𝑌_𝑖𝑁_𝑖−𝐺. (12) The proof is completed.

Remark 6. It follows, from Theorem 5, that Problem 2 can be solved completely by finding the least-squares solution of matrix equations (11) in subspaces CS𝑅^{𝑝𝑖×𝑝𝑖}_⋄,𝑞𝑖.

In the next part of this section, we will establish an iterative algorithm for (11) and analysis its properties. For the convenience of expression, we define a matrix function ℱ𝑍₁,𝑍₂,…,𝑍_𝑡=𝑀₁𝑍₁𝑁₁+𝑀₂𝑍₂𝑁₂+⋯+𝑀_𝑡𝑍_𝑡𝑁_𝑡, (13) then matrix equation (11) can be simplified as ℱ𝑍₁,𝑍₂,…,𝑍_𝑡=𝐺. (14) Moreover, we can easily verify that 𝑋,ℱ𝑍₁,𝑍₂,…,𝑍_𝑡=^𝑡_𝑖=1𝑀^𝑇_𝑖𝑋𝑁^𝑇_𝑖,𝑍_𝑖 (15) holds for arbitrary 𝑋∈𝑅^𝑚×𝑛.

The iterative algorithm for the least squares problem of matrix equations (11) can be expressed as follows.

Algorithm 7. Consider the following.
Step 1. Let 𝑀_𝑖∈𝑅^{𝑚×𝑝𝑖}, 𝑁_𝑖∈𝑅^{𝑝𝑖×𝑛}, 𝐹∈𝑅^𝑚×𝑛 and 𝒳_𝑖∈CS𝑅^{𝑞𝑖×𝑞𝑖} for 𝑖∈Γ.
Input arbitrary matrices 𝑌_𝑖⁽⁰⁾∈CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖.
Step 2. Calculate 𝑅₀=𝐺−ℱ𝑌₁⁽⁰⁾,𝑌₂⁽⁰⁾,…,𝑌_𝑡⁽⁰⁾,𝑃_𝑖⁽⁰⁾=ℒ_𝑖𝑀^𝑇_𝑖𝑅₀𝑁^𝑇_𝑖,𝑄_𝑖⁽⁰⁾=𝑃_𝑖⁽⁰⁾,𝑘=0. (16)
Step 3. Calculate 𝑌_𝑖^(𝑘+1)=𝑌_𝑖^(𝑘)+𝛼_𝑘𝑄_𝑖^(𝑘),𝛼_𝑘=∑^𝑡_𝑖=1𝑃_𝑖^(𝑘)2ℱ𝑄₁^(𝑘),𝑄₂^(𝑘),…,𝑄_𝑡^(𝑘)2. (17)
Step 4. Calculate 𝑅_𝑘+1=𝐺−ℱ𝑌₁^(𝑘+1),𝑌₂^(𝑘+1),…,𝑌_𝑡^(𝑘+1)=𝑅_𝑘−𝛼_𝑘ℱ𝑄₁^(𝑘),𝑄₂^(𝑘),…,𝑄_𝑡^(𝑘),𝑃_𝑖^(𝑘+1)=ℒ_𝑖𝑀^𝑇_𝑖𝑅_𝑘+1𝑁^𝑇_𝑖=𝑃_𝑖^(𝑘)−𝛼_𝑘ℒ_𝑖𝑀^𝑇_𝑖ℱ𝑄₁^(𝑘),𝑄₂^(𝑘),…,𝑄_𝑡^(𝑘)𝑁^𝑇_𝑖,𝑄_𝑖^(𝑘+1)=𝑃_𝑖^(𝑘+1)+𝛽_𝑘𝑄_𝑖^(𝑘),𝛽_𝑘=∑^𝑡_𝑖=1𝑃_𝑖^(𝑘+1)2∑^𝑡_𝑖=1𝑃_𝑖^(𝑘)2. (18)
Step 5. If ∑^𝑡_𝑖=1‖𝑃_𝑖^(𝑘)‖²=0, stop. Otherwise, 𝑘∶=𝑘+1, go to Step 3.

From Algorithm 7, we can see that 𝑌_𝑖^(𝑘),𝑃_𝑖^(𝑘),𝑄_𝑖^(𝑘),∈CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖. In particular, (𝑌₁^(𝑘),𝑌₂^(𝑘),…,𝑌_𝑡^(𝑘)) is a least-squares solution group belonging to matrix equation (11) if 𝑃_𝑖^(𝑘)=0 for all 𝑖∈Γ. The following lemma gives voice to the reason.

Lemma 8. If ℒ_𝑖(𝑀^𝑇_𝑖𝑅_𝑘𝑁^𝑇_𝑖)=0(𝑖=1,2,…,𝑡) hold simultaneously for some positive 𝑘, then (𝑌₁^(𝑘),𝑌₂^(𝑘),…,𝑌_𝑡^(𝑘)) generated by Algorithm 7 is a solution group of matrix equation (11).

Proof. Let ℒ={𝐿∣𝐿=ℱ(𝑌_𝑖),𝑌_𝑖∈CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖} and 𝐺=ℱ(𝑌_𝑖^(𝑘)). Obviously, 𝐺∈ℒ. Then, from the Project Theorem, (𝑌₁^(𝑘),𝑌₂^(𝑘),…,𝑌_𝑡^(𝑘)) is a least-square solution group of matrix equation (11) if and only if 𝐺−𝐺⟂ℒ. That is to say, for any matrices 𝑌_𝑖∈CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖, noting that Lemma 4, we have ⟨𝐺−𝐺,ℱ(𝑌₁,𝑌₂,…,𝑌_𝑡)⟩=0, that is, 𝐺−𝐺,ℱ𝑌₁,𝑌₂,…,𝑌_𝑡=𝑅_𝑘+1,ℱ𝑌₁,𝑌₂,…,𝑌_𝑡=^𝑡_𝑖=1𝑀^𝑇_𝑖𝑅_𝑘+1𝑁^𝑇_𝑖,𝑌_𝑖=^𝑡_𝑖=1ℒ_𝑖𝑀^𝑇_𝑖𝑅_𝑘+1𝑁^𝑇_𝑖,𝑌_𝑖=0, (19) which completes the proof.

In addition, the sequences {𝑌_𝑖^(𝑙)}, {𝑃_𝑖^(𝑙)}, {𝑄_𝑖^(𝑙)} generated by Algorithm 7 are self-orthogonal, that is, as follows.

Lemma 9. Suppose that the sequences {𝑌_𝑖^(𝑙)}, {𝑃_𝑖^(𝑙)}, {𝑄_𝑖^(𝑙)} generated by Algorithm 7 not equal null for 𝑙≤𝑘, then (1)^𝑡_𝑖=1𝑃_𝑖^(𝑗),𝑃_𝑖^(𝑙)(2)=0,_𝑡𝑖=1𝑄_𝑖^(𝑗),𝑃_𝑖^(𝑙)(3ℱ𝑄=0,)₁^(𝑗),𝑄₂^(𝑗),…,𝑄_𝑡^(𝑗),ℱ𝑄₁^(𝑙),𝑄₂^(𝑙),…,𝑄_𝑡^(𝑙)=0, (20)(21)(22) where 𝑗,𝑙=1,2,…,𝑘, 𝑗≠𝑙≤𝑘.

Proof. In view of the symmetry of the inner product, we only prove (20)–(22) when 𝑗<𝑙. According to Algorithm 7, when 𝑘=1, we have ^𝑡_𝑖=1𝑃_𝑖⁽⁰⁾,𝑃_𝑖⁽¹⁾=^𝑡_𝑖=1𝑃_𝑖⁽⁰⁾,𝑃_𝑖⁽⁰⁾−𝛼₀ℒ_𝑖𝑀^𝑇_𝑖ℱ𝑄₁⁽⁰⁾,𝑄₂⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾𝑁^𝑇_𝑖=^𝑡_𝑖=1𝑃_𝑖⁽⁰⁾,𝑃_𝑖⁽⁰⁾−𝛼₀𝑃_𝑖⁽⁰⁾,ℒ_𝑖𝑀^𝑇_𝑖ℱ𝑄₁⁽⁰⁾,𝑄₂⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾𝑁^𝑇_𝑖=^𝑡_𝑖=1𝑃_𝑖⁽⁰⁾²−𝛼₀×^𝑡_𝑖=1𝑃_𝑖⁽⁰⁾,𝑀^𝑇_𝑖ℱ𝑄₁⁽⁰⁾,𝑄₂⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾𝑁^𝑇_𝑖=^𝑡_𝑖=1𝑃_𝑖⁽⁰⁾²−𝛼₀^𝑡_𝑖=1𝑀_𝑖𝑃_𝑖⁽⁰⁾𝑁_𝑖,ℱ𝑄₁⁽⁰⁾,𝑄₂⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾=^𝑡_𝑖=1𝑃_𝑖⁽⁰⁾²−𝛼₀ℱ𝑄₁⁽⁰⁾,𝑄₂⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾²=0, (23) which also deduces that ^𝑡_𝑖=1𝑄_𝑖⁽⁰⁾,𝑃_𝑖⁽¹⁾=^𝑡_𝑖=1𝑄_𝑖⁽⁰⁾,𝑃_𝑖⁽⁰⁾−𝛼₀ℒ_𝑖𝑀^𝑇_𝑖ℱ𝑄₁⁽⁰⁾,𝑄₂⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾𝑁^𝑇_𝑖=^𝑡_𝑖=1𝑄_𝑖⁽⁰⁾,𝑃_𝑖⁽⁰⁾−𝛼₀×^𝑡_𝑖=1𝑄_𝑖⁽⁰⁾,𝑀^𝑇_𝑖ℱ𝑄₁⁽⁰⁾,𝑄₂⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾𝑁^𝑇_𝑖=^𝑡_𝑖=1𝑃_𝑖⁽⁰⁾,𝑃_𝑖⁽⁰⁾−𝛼₀×^𝑡_𝑖=1𝑀_𝑖𝑄_𝑖⁽⁰⁾𝑁_𝑖,ℱ𝑄₁⁽⁰⁾,𝑄₂⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾ℱ𝑄=0,₁₍₀₎,𝑄₂⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾,ℱ𝑄₁⁽¹⁾,𝑄₂⁽¹⁾,…,𝑄_𝑡⁽¹⁾=ℱ𝑄₁⁽⁰⁾,𝑄₂⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾,^𝑡_𝑖=1𝑀_𝑖𝑃_𝑖⁽¹⁾+𝛽₀𝑄_𝑖⁽⁰⁾𝑁_𝑖=ℱ𝑄₁⁽⁰⁾,𝑄₂⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾,ℱ𝑃₁⁽¹⁾,𝑃₂⁽¹⁾,…,𝑃_𝑡⁽¹⁾+𝛽₀ℱ𝑄₁⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾²=1𝛼₀𝑅₀−𝑅₁,ℱ𝑃₁⁽¹⁾,𝑃₂⁽¹⁾,…,𝑃_𝑡⁽¹⁾+𝛽₀ℱ𝑄₁⁽⁰⁾,𝑄₂⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾²=1𝛼₀^𝑡_𝑖=1𝑀^𝑇_𝑖𝑅₀𝑁^𝑇_𝑖,𝑃_𝑖⁽¹⁾−^𝑡_𝑖=1𝑀^𝑇_𝑖𝑅₁𝑁^𝑇_𝑖,𝑃_𝑖⁽¹⁾+𝛽₀ℱ𝑄₁⁽⁰⁾,𝑄₂⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾²1=−𝛼₀^𝑡_𝑖=1𝑃_𝑖⁽¹⁾²+𝛽₀ℱ𝑄₁⁽⁰⁾,𝑄₂⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾²=0. (24)
Assume that (20), (21), and (22) hold for positive integer 𝑠 (<𝑘), that is, for 𝑗=1,2,…,𝑠−1, ^𝑡_𝑖=1𝑃_𝑖^(𝑗),𝑃_𝑖^(𝑠)=0,_𝑡𝑖=1𝑄_𝑖^(𝑗),𝑃_𝑖^(𝑠)ℱ𝑄=0,_1(𝑗),𝑄₂^(𝑗),…,𝑄_𝑡^(𝑗),ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠)=0. (25) Then, similar to the above proof, noting that the assumptions, we have ^𝑡_𝑖=1𝑃_𝑖^(𝑗),𝑃_𝑖^(𝑠+1)=^𝑡_𝑖=1𝑃_𝑖^(𝑗),𝑃_𝑖^(𝑠)−𝛼_𝑠ℒ_𝑖𝑀^𝑇_𝑖ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠)𝑁^𝑇_𝑖=^𝑡_𝑖=1𝑃_𝑖^(𝑗),𝑃_𝑖^(𝑠)−𝛼_𝑠×^𝑡_𝑖=1𝑃_𝑖^(𝑗),ℒ_𝑖𝑀^𝑇_𝑖ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠)𝑁^𝑇_𝑖=−𝛼_𝑠^𝑡_𝑖=1𝑀_𝑖𝑃_𝑖^(𝑗)𝑁_𝑖,ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠)=−𝛼_𝑠^𝑡_𝑖=1𝑀_𝑖𝑄_𝑖^(𝑗)−𝛽_𝑗−1𝑄_{𝑗−1,𝑖}𝑁_𝑖,ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠)=−𝛼_𝑠ℱ𝑄₁^(𝑗),𝑄₂^(𝑗),…,𝑄_𝑡^(𝑗),ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄^(𝑠)_t+𝛼_𝑠𝛽_𝑗−1ℱ𝑄₁^(𝑗−1),𝑄₂^(𝑗−1),…,𝑄_𝑡^(𝑗−1),ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠)=0. (26) Furthermore, ^𝑡_𝑖=1𝑄_𝑖^(𝑗),𝑃_𝑖^(𝑠+1)=^𝑡_𝑖=1𝑄_𝑖^(𝑗),𝑃_𝑖^(𝑠)−𝛼_𝑠ℒ_𝑖𝑀^𝑇_𝑖ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠)𝑁^𝑇_𝑖=^𝑡_𝑖=1𝑃_𝑖^(𝑗),𝑃_𝑖^(𝑠)−𝛼_𝑠×^𝑡_𝑖=1𝑀_𝑖𝑄_𝑖^(𝑗)𝑁_𝑖,ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠)ℱ𝑄=0,_1(𝑗),𝑄₂^(𝑗),…,𝑄_𝑡^(𝑗),ℱ𝑄₁^(𝑠+1),𝑄₂^(𝑠+1),…,𝑄_𝑡^(𝑠+1)=ℱ𝑄₁^(𝑗),𝑄₂^(𝑗),…,𝑄_𝑡^(𝑗),ℱ𝑃₁^(𝑠+1),𝑃₂^(𝑠+1),…,𝑃_𝑡^(𝑠+1)+𝛽_𝑠ℱ𝑄₁^(𝑗),𝑄₂^(𝑗),…,𝑄_𝑡^(𝑗),ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠)=1𝛼_𝑗𝑅_𝑗−𝑅_𝑗+1,ℱ𝑃₁^(𝑠+1),𝑃₂^(𝑠+1),…,𝑃_𝑡^(𝑠+1)=1𝛼_𝑗^𝑡_𝑖=1𝑀^𝑇_𝑖𝑅_𝑗𝑁^𝑇_𝑖,𝑃_𝑖^(𝑠+1)−^𝑡_𝑖=1𝑀^𝑇_𝑖𝑅_𝑗+1𝑁^𝑇_𝑖,𝑃_𝑖^(𝑠+1)=1𝛼_𝑗^𝑡_𝑖=1𝑃_𝑖^(𝑗),𝑃_𝑖^(𝑠+1)−^𝑡_𝑖=1𝑃_𝑗+1,𝑖,𝑃_𝑖^(𝑠+1)=0. (27)
The last equal sign “=” holds due to ∑^𝑡_𝑖=1⟨𝑃_𝑖^(𝑠),𝑃_𝑖^(𝑠+1)⟩=0. In fact, from the hypothesis, we deduce ^𝑡_𝑖=1𝑃_𝑖^(𝑠),𝑃_𝑖^(𝑠+1)=^𝑡_𝑖=1𝑃_𝑖^(𝑠),𝑃_𝑖^(𝑠)−𝛼_𝑠ℒ_𝑖𝑀^𝑇_𝑖ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠)𝑁^𝑇_𝑖=^𝑡_𝑖=1𝑃_𝑖^(𝑠)2−𝛼_𝑠^𝑡_𝑖=1𝑀_𝑖𝑄_𝑖^(𝑠)−𝛽_𝑠−1𝑄_𝑖^(𝑗−1)𝑁_𝑖,ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠)=^𝑡_𝑖=1𝑃_𝑖^(𝑠)2−𝛼_𝑠ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠)2+𝛼_𝑠𝛽_𝑠−1ℱ𝑄₁^(𝑠−1),𝑄₂^(𝑠−1),…,𝑄_𝑡^(𝑠−1),ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠)=0. (28) Moreover, ^𝑡_𝑖=1𝑄_𝑖^(𝑠),𝑃_𝑖^(𝑠)=^𝑡_𝑖=1𝑃_𝑖^(𝑠)+𝛽_𝑠−1𝑄_𝑖^(𝑠−1),𝑃_𝑖^(𝑠)=^𝑡_𝑖=1𝑃_𝑖^(𝑠)2+𝛽^𝑡_{𝑠−1𝑖=1}𝑄_𝑖^(𝑠−1),𝑃_𝑖^(𝑠)=^𝑡_𝑖=1𝑃_𝑖^(𝑠)2. (29) It follows from (29) that ^𝑡_𝑖=1𝑄_𝑖^(𝑠),𝑃_𝑖^(𝑠+1)=^𝑡_𝑖=1𝑄_𝑖^(𝑠),𝑃_𝑖^(𝑠)−𝛼_𝑠ℒ_𝑖𝑀^𝑇_𝑖ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠)𝑁^𝑇_𝑖=^𝑡_𝑖=1𝑄_𝑖^(𝑠),𝑃_𝑖^(𝑠)−𝛼_𝑠𝑄_𝑖^(𝑠),𝑀^𝑇_𝑖ℱ𝑄₁^(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠)𝑁^𝑇_𝑖=^𝑡_𝑖=1𝑃_𝑖^(𝑠)2−𝛼_𝑠ℱ𝑄₁^(𝑠+1),𝑄₂^(𝑠+1),…,𝑄_𝑡^(𝑠+1)2ℱ𝑄=0,_1(𝑠),𝑄₂^(𝑠),…,𝑄_𝑡^(𝑠),ℱ𝑄₁^(𝑠+1),𝑄₂^(𝑠+1),…,𝑄_𝑡^(𝑠+1)=ℱ𝑄₁^(𝑠),…,𝑄_𝑡^(𝑠),ℱ𝑃₁^(𝑠+1),𝑃₂^(𝑠+1),…,𝑃_𝑡^(𝑠+1)+𝛽_𝑠ℱ𝑄₁^(𝑠),…,𝑄_𝑡^(𝑠)2=1𝛼_𝑠𝑅_𝑠−𝑅_𝑠+1,ℱ𝑃₁^(𝑠+1),𝑃₂^(𝑠+1),…,𝑃_𝑡^(𝑠+1)+𝛽_𝑠^𝑡_𝑖=1𝑀_𝑖𝑄_𝑖^(𝑠)𝑁_𝑖²1=−𝛼_𝑠^𝑡_𝑖=1𝑀^𝑇_𝑖𝑅_𝑠+1𝑁^𝑇_𝑖,𝑃_𝑖^(𝑠+1)+𝛽_𝑠^𝑡_𝑖=1𝑀_𝑖𝑄_𝑖^(𝑠)𝑁_𝑖²=0. (30) Therefore, for arbitrary integers number 𝑙, the conclusions hold. The proof is completed.

Remark 10. We know from Lemma 4 that the matrices sequences 𝑃_0,1⋱𝑃_0,𝑡,𝑃_1,1⋱𝑃_1,𝑡𝑃,…,_𝑘,1⋯𝑃_𝑘,𝑡,… (31) are orthogonal to each other. Hence, it can be regarded as an orthogonal basis of matrix space 𝑅^{∑𝑡𝑖=1𝑝𝑖×∑𝑡𝑖=1𝑝𝑖}. Hence, the iteration will be terminated at most ∑^𝑡_𝑖=1(𝑝_𝑖+𝑞_𝑖)(𝑝_𝑖−𝑞_𝑖)/2 steps in the absence of roundoff errors. Therefore, there exists a positive integer 𝑘≤∑^𝑡_𝑖=1(𝑝_𝑖+𝑞_𝑖)(𝑝_𝑖−𝑞_𝑖)/2 such that ∑^𝑡_𝑖=1‖𝑃_𝑖^(𝑘)‖²=0, in this case, (𝑌₁^(𝑘),𝑌₂^(𝑘),…,𝑌_𝑡^(𝑘)) can be regarded as a least-squares solution group of matrix equation (11).
In addition, we should point out that if 𝛼_𝑘=0 or ∞, the conclusions may not be true, and the iteration will break down before 𝑃_𝑖^(𝑘)=0 for 𝑘<∑^𝑡_𝑖=1(𝑝_𝑖+𝑞_𝑖)(𝑝_𝑖−𝑞_𝑖)/2. Actually, 𝛼_𝑘=0 implies that ∑^𝑡_𝑖=1‖𝑃_𝑖^(𝑘)‖²=0, so 𝑃_𝑖^(𝑘)=0 for 𝑖∈Γ. While 𝛼_𝑘=∞ leads to ℱ(𝑄₁^(𝑘),𝑄₂^(𝑘),…,𝑄_𝑡^(𝑘))=0, making inner product with 𝑅_𝑖 by both sides, it follows from Algorithm 7 that 𝑅_𝑖,ℱ𝑄₁^(𝑘),𝑄₂^(𝑘),…,𝑄_𝑡^(𝑘)=^𝑡_𝑖=1𝑃_𝑖^(𝑘),𝑃_𝑖^(𝑘)+𝛽_𝑘−1𝑄_𝑖^(𝑘−1)=^𝑡_𝑖=1𝑃_𝑖^(𝑘)2+𝛽^𝑡_{𝑘−1𝑖=1}𝑃_𝑖^(𝑘),𝑄_𝑖^(𝑘−1)=^𝑡_𝑖=1𝑃_𝑖^(𝑘)2=0, (32) which also implies the same situation as 𝛼_𝑘=0. Hence, if there exists a positive integer 𝑠 such that the coefficient 𝛼_𝑠=0 or 𝛼_𝑠=∞, then the corresponding matrix group 𝑌₁^(𝑠),𝑌₂^(𝑠),…,𝑌_𝑡^(𝑠) is just the solution of matrix equation (11).

Together with the above analysis and Lemma 9, we can conclude the following theorem.

Theorem 11. For any initial iteration matrices 𝑌_𝑖⁽⁰⁾∈𝐶𝑆𝑅^𝑛×𝑛_⋆,𝑞𝑖, 𝑖=1,2,…,𝑡, the least-squares solution of matrix equation (11) can be obtained within finite iteration steps. Moreover, Suppose that (𝑌̆₁,𝑌̆₂,…,𝑌̆_𝑡) is a least-squares solution group of (11), then the general solution to Problem 2 can be expressed as 𝑍_𝑖=𝑌̆_𝑖+𝑋_𝑖+𝑍^⋄_𝑖(𝑖=1,2,…,𝑡), where 𝑋_𝑖∈𝐶𝑆𝑅^𝑛×𝑛_⋆,𝑞𝑖 satisfy homogeneous equation ℱ𝑋₁,𝑋₂,…,𝑋_𝑡=0, (33)𝑍^⋄_𝑖 as in Theorem 5.

In order to show the validity of Theorem 11, it is adequate to prove the following conclusion.

Proposition 12. The least-squares solution group (𝑌̆₁,𝑌̆₂,…,𝑌̆_𝑡) of matrix equation (11) can be expressed as 𝑌̆_𝑖+𝑋_𝑖, where 𝑋_𝑖 satisfy equality (33).

Proof. According to the assumptions, we obtain min_{𝑌𝑖∈CS𝑅𝑖𝑛×𝑛⋆,𝑞}ℱ𝑌₁,𝑌₂,…,𝑌_𝑡=ℱ𝑌̆−𝐺₁,𝑌̆₂,…,𝑌̆_𝑡−𝐺. (34) On the other hand, noting that ℱ(𝑋₁,𝑋₂,…,𝑋_𝑡)=0, then ℱ𝑌̆₁+𝑋₁,𝑌̆₂+𝑋₂,…,𝑌̆_𝑡+𝑋_𝑡−𝐺²=ℱ𝑌̆₁,𝑌̆₂,…,𝑌̆_𝑡−𝐺+ℱ𝑋₁,𝑋₂,…,𝑋_𝑡²=ℱ𝑌̆₁,𝑌̆₂,…,𝑌̆_𝑡−𝐺². (35) The proof is completed.

Next, we will show that the unique least norm solution of matrix equation (11) can be derived by choosing a special kind of initial iteration matrices.

Theorem 13. Let the initial iteration matrices 𝑌_𝑖⁽⁰⁾=ℒ_𝑖(𝑀^𝑇_𝑖𝐻𝑁^𝑇_𝑖) with arbitrary 𝐻∈𝑅^𝑚×𝑛, 𝑖=1,2,…,𝑡, and then (𝑌^∗₁,𝑌^∗₂,…,𝑌^∗_𝑡) generated by Algorithm 7 is the least-norm least-squares solution group of matrix equation (11). Furthermore, the least-norm solution group to Problem 2 can be expressed by 𝑍^∗₁,𝑍^∗₂,…,𝑍^∗_𝑡=𝑌^∗₁+𝑍^⋄_𝑖,𝑌^∗₂+𝑍^⋄₂,…,𝑌^∗_𝑡+𝑍^⋄_𝑡. (36)

Proof. From Algorithm 7 and Theorem 11, for initial iteration matrices 𝑌_𝑖⁽⁰⁾=ℒ_𝑖(𝑀^𝑇_𝑖𝐻𝑁^𝑇_𝑖), we can obtain a least-squares solution 𝑌^∗_𝑖 of matrix equation (11) and there exists a matrix 𝐻^∗ such that 𝑌^∗_𝑖=ℒ_𝑖(𝑀^𝑇_𝑖𝐻^∗𝑁^𝑇_𝑖). Hence, it is enough to prove that the 𝑌^∗_𝑖 is the least-norm solution. In fact, noting that (33) and Proposition 12, we have ^𝑡_𝑖=1𝑌^∗_𝑖+𝑋_𝑖²=^𝑡_𝑖=1𝑌^∗_𝑖²+𝑋_𝑖²+2𝑌^∗_𝑖,𝑋_𝑖=^𝑡_𝑖=1𝑌^∗_𝑖²+^𝑡_𝑖=1𝑋_𝑖²+2^𝑡_𝑖=1ℒ_𝑖𝑀^𝑇_𝑖𝐻^∗𝑁^𝑇_𝑖,𝑋_𝑖≥^𝑡_𝑖=1𝑌^∗_𝑖², (37) as required.

Theorems 11 and 13 display the efficiency of Algorithm 7. Actually, the iteration sequence {𝑌_𝑖^(𝑘)} converges smoothly to the solution 𝑌_𝑖, that is the minimization property of Algorithm 7.

Theorem 14. For any initial iteration matrices 𝑌_𝑖⁽⁰⁾, the 𝑌_𝑖^(𝑘) generated by Algorithm 7 satisfy the minimization problem ℱ𝑌₁^(𝑘),𝑌₂^(𝑘),…,𝑌_𝑡^(𝑘)−𝐺=min_{𝑌𝑖∈𝕃𝑖}ℱ𝑌₁,𝑌₂,…,𝑌_𝑡−𝐺, (38) where 𝕃_𝑖=𝑌_𝑖⁽⁰⁾+span{𝑄_𝑖⁽⁰⁾,𝑄_𝑖⁽¹⁾,…,𝑄_𝑖^(𝑘−1)}, 𝑖=1,2,…,𝑡.

Proof. From the definition of 𝕃_𝑖, there exist a series of real numbers 𝑎₀,𝑎₁,…,𝑎_𝑘−1 such that 𝑌_𝑖=𝑌_𝑖⁽⁰⁾+∑^𝑘−1_𝑙=0𝑎_𝑙𝑄_𝑖^(𝑙).
Define a function of 𝑘 variables 𝑓(𝑎₀,𝑎₁,…,𝑎_𝑘−1), that is, 𝑓𝑎₀,𝑎₁,…,𝑎_𝑘−1=^𝑡_𝑖=1𝑀_𝑖𝑌_𝑖⁽⁰⁾+^𝑘−1_𝑙=0𝑎_𝑙𝑄_𝑖^(𝑙)𝑁_𝑖−𝐺². (39) In addition, from Algorithm 7, we know that 𝑅₀=𝑅_𝑙+𝛼_𝑙−1ℱ𝑄₁^(𝑙−1),𝑄₂^(𝑙−1),…,𝑄_𝑡^(𝑙−1)+⋯+𝛼₀ℱ𝑄₁⁽⁰⁾,𝑄₂⁽⁰⁾,…,𝑄_𝑡⁽⁰⁾. (40) Noting that (22) and making the inner product with ℱ(𝑄₁^(𝑙),𝑄₂^(𝑙),…,𝑄_𝑡^(𝑙)) on both sides of (40) yield ℱ𝑄₁^(𝑙),𝑄₂^(𝑙),…,𝑄_𝑡^(𝑙),𝑅₀=ℱ𝑄₁^(𝑙),𝑄₂^(𝑙),…,𝑄_𝑡^(𝑙),𝑅_𝑙. (41) Hence, by simple calculation, (40) and (41), the function can be rewritten as 𝑓𝑎₀,𝑎₁,…,𝑎_𝑘−1=𝑅₀²+^𝑘−1_𝑙=0𝑎²_𝑙ℱ𝑄₁^(𝑙),𝑄₂^(𝑙),…,𝑄_𝑡^(𝑙)2−2^𝑘−1_𝑙=0𝑎_𝑙ℱ𝑄₁^(𝑙),𝑄₂^(𝑙),…,𝑄_𝑡^(𝑙),𝑅_𝑙. (42) Then, min_𝑎𝑙𝑓𝑎₀,𝑎₁,…,𝑎_𝑘−1⟺min_{𝑌𝑖∈𝕃𝑖}ℱ𝑌₁,𝑌₂,…,𝑌_𝑡−𝐺. (43)
Since 𝑓(𝑎₀,𝑎₁,…,𝑎_𝑘−1)=min only if 𝜕𝑓/𝜕𝑎_𝑙=0, it follows from (29) that 𝑎_𝑙=ℱ𝑄₁^(𝑙),𝑄₂^(𝑙),…,𝑄_𝑡^(𝑙),𝑅_𝑙ℱ𝑄₁^(𝑙),𝑄₂^(𝑙),…,𝑄_𝑡^(𝑙)2=∑^𝑡_𝑖=1𝑄_𝑖^(𝑙),𝑃_𝑙,𝑖ℱ𝑄₁^(𝑙),𝑄₂^(𝑙),…,𝑄_𝑡^(𝑙)2=∑^𝑡_𝑖=1𝑃_𝑙,𝑖²ℱ𝑄₁^(𝑙),𝑄₂^(𝑙),…,𝑄_𝑡^(𝑙)2=𝛼_𝑙. (44) Combined with (43), we complete the proof.

Theorem 14 reveals that the sequence ℱ𝑌₁^(𝑘),𝑌₂^(𝑘),…,𝑌_𝑡^(𝑘)−𝐺 (45) monotonically decreases with respect to increasing integer 𝑘. The descent property of the residual norm of matrix equation (11) leads to the smoothly convergence of Algorithm 7.

3. The Solution of Problem 3

In this section, we discuss the optimal approximation Problem 3. Since the least squares problem is always consistent, it is easy to verify that the solution set 𝑆_𝐸 of Problem 2 is a nonempty convex cone, so the optimal approximation solution is unique.

Without loss of generality, we can assume that the given matrices 𝑍_𝑖∈CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖. In fact, from Lemma 4, arbitrary 𝑍_𝑖∈𝑅^{𝑝𝑖×𝑝𝑖} can be divided into 𝑍_𝑖=𝑍_𝑖1+𝑍_𝑖2+𝑋_𝑖3,with𝑍_𝑖1∈CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖,𝑍_𝑖2∈CS𝑅^{𝑝𝑖×𝑝𝑖}_⋄,𝑞𝑖,𝑍_𝑖3∈CAS𝑅^{𝑝𝑖×𝑝𝑖}. (46) Furthermore, if 𝑍_𝑖∈CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖, then 𝑍_𝑖−𝑍_𝑖²=𝑍_𝑖−𝑍_𝑖1²+𝑍_𝑖2²+𝑍_𝑖3², (47) which meets the claim.

Denote 𝑍_𝑖=𝑍_𝑖−𝑍_𝑖, 𝐹=𝐹−ℱ(𝑍₁,𝑍₂,…,𝑍_𝑡), then to solve Problem 3 is equivalent to find the least-norm solution of the new matrix equation ℱ𝑍₁,𝑍₂,…,𝑍_𝑡=𝐹. (48) Furthermore, similar to the construction of (11), Problem 2 is transformed equivalently into finding the least-norm least-squares solution of matrix equation ℱ𝑌₁,𝑌₂,…,𝑌_𝑡=𝐺,with𝑌_𝑖∈CS𝑅^{𝑝𝑖×𝑝𝑖}_⋆,𝑞𝑖, (49) in which 𝐺=𝐹−ℱ(𝑍^⋄₁,𝑍^⋄₂,…,𝑍^⋄_𝑡).

Therefore, we can apply Algorithm 7 to derive the required solution of matrix equation (49). Virtually, it follows from Theorem 13 that if let the initial iteration matrices 𝑌_𝑖⁽⁰⁾=ℒ_𝑖(𝑀^𝑇_𝑖𝐻𝑁^𝑇_𝑖) with arbitrary 𝐻∈𝑅^𝑚×𝑛, or especially 𝑌_𝑖⁽⁰⁾=0∈𝑅^{𝑝𝑖×𝑝𝑖}, then the iteration solutions 𝑌^∗_𝑖 consist of the least-norm least-squares solution of which. In this case, the unique optimal approximation solution to Problem 3 can be obtained by 𝑍₁,𝑍₂,…,𝑍_𝑡=𝑌^∗₁+𝑍₁+𝑍^⋄₁,𝑌^∗₂+𝑍₂+𝑍^⋄₂,…,𝑌^∗_𝑡+𝑍_𝑡+𝑍^⋄_𝑡. (50)

4. Numerical Example

In this section, we illustrate the efficiency and reliability of Algorithm 7 by some numerical experiments.

All the numerical experiments are performed by using Matlab 6.5. In addition, because of the influence of the roundoff errors, 𝑃_𝑖^(𝑘) may not equal zero within finite iteration steps, so the iteration will be terminated if ∑^𝑡_𝑖=1‖𝑃_𝑖^(𝑘)‖²<𝜖, for example, let 𝜖=1.0𝑒−008. At this time, (𝑌₁^(𝑘),𝑌₁^(𝑘),…,𝑌_𝑡^(𝑘)) can be regarded as a solution of matrix equation (11), and 𝑌_𝑖^(𝑘)+𝑍^⋄_𝑖 (𝑖=1,2,…,𝑡) consist of the solution group to Problem 2. In particular, let the initial iteration matrices 𝑌_𝑖⁽⁰⁾=0, then we will obtain the least-norm solution by (36).

Example 1. Input matrices 𝑀₁, 𝑀₂, 𝑀₃, 𝑁₁, 𝑁₂, 𝑁₃, and 𝐹 as follows: 𝑀₁=𝑟hilb2𝑟ones2𝑟hankel1∶2𝑟zeros2,𝑀₂=𝑟toeplitz1∶2𝑟hilb2𝑟ones2𝑟hankel1∶2,𝑀₃=𝑟zeros2𝑟hankel1∶2𝑟hilb2𝑟ones2,𝑁₁=eye(𝑟),𝑁₂(𝑟),𝑁=ones₃]),,=tridiag([7,1,−1,𝑟𝐹=3−2−1−23−2⋱−1−2⋱−2−1⋱−23−2−1−23 (51)where toeplitz(𝑘), hilb(𝑘), hankel(𝑘), zeros(𝑘), and eye(𝑘) denote the Toeplitz matrix, Hilbert matrix, Hankel matrix, null matrix, identity matrix with orders 𝑘, and the elements of matrix ones(⋅) are one, tridiag([7,1,−1],𝑟) represents 𝑟×𝑟 tri-diagonal matrix produced by vector [7,1,−1].

Let the given central principal matrices 𝒳₁𝑟=zeros2,𝒳₂𝑟=ones2∗10,𝒳₃𝑟=toeplitz1∶2. (52) By using the Algorithm 7, we obtain the solution to Problem 2. To save space, we shall not report the explicit datum of the solution, but the bars graphs of the components for the solution matrices will be given. Let 𝑟=20, Figure 1 shows the bars graphs of 𝑍₁, 𝑍₂, 𝑍₃ when we choose the initial iterative matrices 𝑌_𝑖=ones𝑟−𝑞2ones𝑟−𝑞2,𝑞ones𝑟−𝑞2ones𝑞,𝑟−𝑞2𝑟zeros2ones𝑞,𝑟−𝑞2ones𝑟−𝑞2ones𝑟−𝑞2,𝑞ones𝑟−𝑞2,𝑖=1,2,3, (53) and the terminal condition ‖𝑃₁^(𝑘)‖+‖𝑃₂^(𝑘)‖+‖𝑃₃^(𝑘)‖<𝜖=1.0𝑒−012.

(a)

(b)

(c)

Figure 1 

The bars graphs of 𝑍𝑖 when 𝑟=20.

Moreover, when 𝑟=20 and 𝑟=40, the convergence curves for the Frobenius norm of the residual denoted by RES=‖𝑅_𝑘‖ and the termination condition denoted by TC=‖𝑃₁^(𝑘)‖+‖𝑃₂^(𝑘)‖+‖𝑃₃^(𝑘)‖ are plotted in Figures 2 and 3, respectively.

Figure 2 

The convergence curve of the Frobenius norm of the residual.

Figure 3 

The convergence curve of the Frobenius norm of the terminated condition.

From Figure 2, we can see that the residual norm of Algorithm 7 is monotonically decreasing, which is in accordance with the theory established in Theorem 14, namely, this algorithm is numerical stable. While Figure 3 shows that the terminated condition ‖𝑃₁^(𝑘)‖+‖𝑃₂^(𝑘)‖+‖𝑃₃^(𝑘)‖ is oscillating back and forth and approaches to zero as iterative process. Hence, the iterative Algorithm 7 is efficient, but it lacks of smooth convergence. Of course, for a problem with large and sparse matrices, Algorithm 7 may not terminate in a finite number of steps because of roundoff errors. How to establish an efficient and smooth algorithm is an important problem which we should study in a future work.

Conflict of Interests

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

Acknowledgments

The authors would like to express their sincere gratitude to the editor and two anonymous reviewers for their valuable comments and suggestions which have helped immensely improving the quality of the paper. Mao-lin Liang acknowledges the support of the scientific foundation of Tianshui Normal University (no. TSA1104). Young-hong Shen is supported by the “QingLan” Talent Engineering Funds of Tianshui Normal University.