Convergence and Comparison Theorems for Three-Step Alternating Iteration Method for Rectangular Linear System

Abstract

The three-step alternating iteration method was introduced by Nandi et al. (J. Appl. Math. Comput. 60:485–515, 2019) for solving a rectangular linear system $Ax=b$ by using proper weak regular splittings of type I. In this article, we expand the theory of such alternating iterations by using proper weak regular splittings of type II.

1 Introduction

Let us consider a system of linear equations of the form

$$\begin{aligned} Ax=b,~~A \in {{\mathbb R}}^{m \times n},~x \in {{\mathbb R}}^{n}, ~b \in {{\mathbb R}}^{m}. \end{aligned}$$

(1.1)

When the coefficient matrix A is very large and sparse, iterative methods become more efficient. In this direction, by using the notion of proper splitting Berman and Plemmons [4] proposed the following iterative method:

$$\begin{aligned} x^{k+1}=U^{\dagger }Vx^k+U^{\dagger }b,\quad k=0,1,2,\ldots , \end{aligned}$$

(1.2)

where $U^{\dagger }V$ is the iteration matrix and $U^{\dagger }$ is the Moore–Penrose inverse of U. The same authors also proved that the above iterative method converges to $A^{\dagger }b$ for any initial guess $x^{0}$ if and only if the spectral radius of the iteration matrix $U^{\dagger }V$ is less than 1 (see Corollary 1, [4] for instance). Therefore, the rate of convergence of the iterative method (1.2) depends on $\rho (U^{\dagger }V)$ and so, the spectral radius of the iteration matrix plays an important role in the comparison of the rate of convergence of different iterative methods of the same system. Many authors such as Berman and Plemmons [4], Climent et al. [7], Climent and Perea [8], Jena et al. [13], Mishra [14, 18], Mishra and Mishra [17], Baliarsingh and Mishra [1], Giri and Mishra [9,10,11,12], Shekhar et al. [21], and others have introduced several convergence and comparison results for different subclasses of a proper splitting.

In particular, Mishra [14] proposed the concept of alternating iteration method for rectangular matrices by extending the work of Benzi and Szyld [2]. Mishra [14] considered two proper splittings of $A \in {{\mathbb R}}^{m \times n}$, namely $A=M-N=U-V$, and proposed the following iterative method

$$x^{k+1}=U^{\dagger }VM^{\dagger }Nx^{k}+U^{\dagger }(VM^{\dagger }+I)b,\quad k=0,1,2,\ldots $$

to solve (1.1). Recently, Nandi et al. [19] introduced the three-step alternating iteration method by extending the work of Mishra [14]. Now we recall the same. Let $A=M-N=P-Q=U-V$ be proper splittings of $A \in {{\mathbb R}}^{m \times n}$. The same authors considered

$$\begin{aligned}&x^{k+1/3}=M^{\dagger }Nx^{k}+M^{\dagger }b, \end{aligned}$$

(1.3)

$$\begin{aligned}&x^{k+1/2}=P^{\dagger }Qx^{k+1/3}+P^{\dagger }b, \end{aligned}$$

(1.4)

$$\begin{aligned}&x^{k+1}=U^{\dagger }Vx^{k+1/2}+U^{\dagger }b,\quad k=0, 1, 2, \ldots . \end{aligned}$$

(1.5)

By simplifying (1.3), (1.4), and (1.5), one can formulate the following iterative method known as the three-step alternating iterative method for rectangular matrices

$$\begin{aligned} x^{k+1}=U^{\dagger }VP^{\dagger }QM^{\dagger }Nx^{k}+U^{\dagger }(VP^{\dagger }QM^{\dagger }+VP^{\dagger }+I)b,\quad k=0,1,2,\ldots , \end{aligned}$$

(1.6)

where $H=U^{\dagger }VP^{\dagger }QM^{\dagger }N$ is the iteration matrix of the iterative method (1.6). The same authors [19] then studied the convergence criteria for the above iterative method by assuming the splittings $A=M-N=P-Q=U-V$ are proper weak regular of type I (see Theorem 3.1). The convergence of three-step alternating iterations for a singular linear system using the Group Inverse (see [3] for the definition) for type II matrix splitting is studied in [22]. However, convergence of three-step alternating for a rectangular linear system in the case of type II splittings is not yet considered in the literature.

The main objective of this paper is to study the convergence of three-step alternating iteration method by considering the splittings are proper weak regular of type II. To fulfill this objective, we organize the contents of the paper as follows. In Sect. 2, we introduce notations, definitions, and some preliminary results that are frequently used while proving our main results. In Sect. 3, we prove our main results. Here, we derived convergence and comparison results for the three-step alternating iteration method by considering splittings $A=M-N=P-Q=U-V$ are proper weak regular of type II. The findings are verified through numerical examples in Sect. 4. Finally, we concluded this article in Sect. 5.

2 Preliminaries

Throughout the article, all the matrices are considered as real of order $m \times n$, unless stated otherwise. The symbol $\mathbb {R}^{m\times n}$ denotes the set of all real matrices of order $m\times n$ and by $\mathbb {R}^{n}$ we mean an n-dimensional Euclidean space. The transpose, the range space and the null space of a matrix A are denoted by $A^\textrm{T},~\mathcal {R}(A)$ and $\mathcal {N}(A)$, respectively. Let L and M be complementary subspaces of $\mathbb {R}^{n}$, and $P_{L,M}$ be a projector onto L along M. Then $P_{L,M}A=A$ if and only if $\mathcal {R}(A)\subseteq L$ and $AP_{L,M}=A$ if and only if $\mathcal {N}(A)\supseteq M$. If $L \bot M$, then we denote $P_{L,M}$ by $P_{L}$. Let $\lambda _{1}, \lambda _{2}, \ldots , \lambda _{n}$ be the eigenvalues of $A \in {{\mathbb R}}^{n \times n}$. Then the spectral radius of $A \in {{\mathbb R}}^{n \times n}$ is denoted by $\rho (A)$ and is defined by $\rho (A)=\max \{ \vert \lambda _{1}\vert ,\vert \lambda _{2}\vert , \ldots , \vert \lambda _{n}\vert \},$ whereas $\sigma \{A\}$ denotes the set of all eigenvalues of A.

2.1 Nonnegative Matrices

$A\in \mathbb {R}^{m\times n}$ is called nonnegative (positive) if each entry of A is nonnegative (positive) and is denoted by $A\ge 0~(A>0)$. For $A,~B\in \mathbb {R}^{m\times n}$, $A\ge B$ means $A-B\ge 0$. The same notation and nomenclature are also used for vectors. The next results deal with the nonnegativity of a matrix and its spectral radius.

Theorem 2.1

(Theorem 2.20, [23])

Let $B \in {{\mathbb R}}^{n \times n}$ and $B \ge 0$. Then

• (i) B has a nonnegative real eigenvalue equal to its spectral radius.

• (ii) To $\rho (B)$, there corresponds an eigenvector $x\ge 0.$

Theorem 2.2

(Theorem 2.1.11, [6])

Let $B \in {{\mathbb R}}^{n \times n}$, $B \ge 0$, $x\ge 0$ $(x\ne 0)$, and $\alpha$ is a positive scalar.

• (i) If $\alpha x\le Bx$, then $\alpha \le \rho (B)$.

• (ii) If $Bx \le \alpha x$, $x>0$, then $\rho (B)\le \alpha$.

Theorem 2.3

(Theorem 3.15, [23])

Let $B\in {{\mathbb R}}^{n\times n}$ and $B\ge 0$. Then $\rho (B)<1$ if and only if $(I-B)^{-1}$ exists and $(I-B)^{-1}=\displaystyle \sum \nolimits _{n=0}^{\infty }B^{n}\ge 0$.

2.2 The Moore–Penrose Inverse of a Matrix

For $A\in \mathbb {R}^{m\times n}$, the unique matrix $X\in \mathbb {R}^{n\times m}$ satisfying the following four equations known as Penrose equations:

$$AXA = A, \;\; XAX = X, \;\; (AX)^\textrm{T} = AX \; \text {and} \; (XA)^\textrm{T} = XA$$

is called the Moore–Penrose inverse of A. It always exists and is denoted by $A^{\dagger }$ (see [20]). Next, we collect some well-known properties of the Moore–Penrose inverse of $A \in {{\mathbb R}}^{m \times n}$ which will be used frequently in this article, namely: $\mathcal {R}(A^{\dagger })=\mathcal {R}(A^\textrm{T});~\mathcal {N}(A^{\dagger })=\mathcal {N}(A^\textrm{T});~AA^{\dagger }=P_{\mathcal {R}(A^{\dagger })};~A^{\dagger }A=P_{\mathcal {R}(A^\textrm{T})}.$ In particular, if $x\in \mathcal {R}(A)$, then $x = A^{\dagger }Ax$ (for more details, see [3]). $A \in {{\mathbb R}}^{m \times n}$ is called semimonotone, if $A^{\dagger } \ge 0.$

2.3 Proper Splittings

A splitting $A=U-V$ of $A \in {{\mathbb R}}^{m \times n}$ is called a proper splitting if $\mathcal {R}(U)=\mathcal {R}(A)$ and $\mathcal {N}(U)=\mathcal {N}(A).$ A few properties of a proper splitting are summarized below.

Theorem 2.4

(Theorem 1, [4])

Let $A=U-V$ be a proper splitting of $A\in \mathbb {R}^{m\times n}$. Then

• (a) $A=U(I-U^{\dagger }V),$

• (b) $(I-U^{\dagger }V)$ is nonsingular,

• (c) $A^{\dagger }=(I-U^{\dagger }V)^{-1}U^{\dagger }.$

Theorem 2.5

(Theorem 1, [7])

Let $A=U-V$ be a proper splitting of $A\in \mathbb {R}^{m\times n}$. Then

• (a) $A=(I-VU^{\dagger })U,$

• (b) $(I-VU^{\dagger })$ is nonsingular,

• (c) $A^{\dagger }=U^{\dagger }(I-VU^{\dagger })^{-1}.$

Theorem 2.6

(Theorem 1, [15])

Let $A=U-V$ be a proper splitting of $A\in \mathbb {R}^{m\times n}$. Then

• (a) $AA^{\dagger }=UU^{\dagger }$ and $A^{\dagger }A=U^{\dagger }U,$

• (b) $U^{\dagger }VA^{\dagger }=A^{\dagger }VU^{\dagger },$

• (c) $U^{\dagger }VA^{\dagger }V=A^{\dagger }VU^{\dagger }V,$

• (d) $VU^{\dagger }VA^{\dagger }=VA^{\dagger }VU^{\dagger }.$

We refer [5, 16, 17, 21] for methods of construction of a proper splitting for a given $A \in {{\mathbb R}}^{m \times n}$. Different subclasses of a proper splitting are recalled next.

Definition 2.7

A proper splitting $A=U-V$ of $A\in {{\mathbb R}}^{m\times n}$ is called a

• (i) proper regular splitting [13], if $U^{\dag }\ge 0$ and $V\ge 0$.

• (ii) proper weak regular splitting of type I [13], if $U^{\dag }\ge 0$ and $U^{\dag }V\ge 0$.

• (iii) proper weak regular splitting of type II [12], if $U^{\dag }\ge 0$ and $VU^{\dag }\ge 0$.

For the above class of proper splitting, we have the following convergence result.

Theorem 2.8

(Theorem 2.4, [17])

Let $A=U-V$ be any of the above class of splittings of $A\in \mathbb {R}^{m\times n}$. Then $A^{\dagger }\ge 0$ if and only if $\rho (VU^{\dagger })=\rho (U^{\dagger }V)<1$.

It is well known that matrix splittings having a smaller radius of iteration matrix gives a faster rate of convergence for (1.2). Therefore, we have the following comparison result for proper weak regular splittings of different types.

Theorem 2.9

(Theorem 3.3, [9])

Let $A=U_1-V_1=U_2-V_2$ be two proper weak regular splitting of different types a semimonotone matrix $A \in {{\mathbb R}}^{m \times n}$. If $U_1^{\dagger } \ge U_2^{\dagger },$ then $\rho (U_1^{\dagger }V_1) \le \rho (U_2^{\dagger }V_2)<1.$

3 Classical Three-Step Alternating Iteration Method

This section deals with the convergence of the three-step alternating iteration method, when the splittings $A=M-N=P-Q=U-V$ are proper weak regular of type II. Before proving our main results, we recall the following results.

Theorem 3.1

(Theorem 15, [19])

Let $A=M-N=P-Q=U-V$ be three proper weak regular splittings of type I of a semimonotone matrix $A \in {{\mathbb R}}^{m \times n}$. Then $\rho (H)=\rho (U^{\dagger }VP^{\dagger }QM^{\dagger }N)<1$.

Theorem 3.2

(Theorem 16, [19])

Let $A=M-N=P-Q=U-V$ be three proper weak regular splittings of type I of a semimonotone matrix $A \in {{\mathbb R}}^{m \times n}$. Then the unique splitting $A=B-C$ induced by H with $B=M(M+U-A+VP^{\dagger }N)^{\dagger }U$ is a proper weak regular splitting of type I if $\mathcal {R}(M+U-A+VP^{\dagger }N)=\mathcal {R}(A)$ and $\mathcal {N}(M+U-A+VP^{\dagger }N)=\mathcal {N}(A)$.

A few properties of the matrix B mentioned in the above theorem are obtained next.

Lemma 3.3

Let $A=M-N=P-Q=U-V$ be three proper splittings of $A \in {{\mathbb R}}^{m \times n}$. If $B=M(M+U-A+VP^{\dagger }N)^{\dagger }U$, $\mathcal {R}(M+U-A+VP^{\dagger }N)=\mathcal {R}(A)$ and $\mathcal {N}(M+U-A+VP^{\dagger }N)=\mathcal {N}(A)$, then $B^{\dagger }=U^{\dagger }(M+U-A+VP^{\dagger }N)M^{\dagger }$, $\mathcal {R}(B)=\mathcal {R}(A)$ and $\mathcal {N}(B)=\mathcal {N}(A).$

Proof

Let $X=U^{\dagger }(M+U-A+VP^{\dagger }N)M^{\dagger }.$ Since $\mathcal {R}(M+U-A+VP^{\dagger }N)=\mathcal {R}(A)$, $\mathcal {N}(M+U-A+VP^{\dagger }N)=\mathcal {N}(A)$ and $A=M-N=P-Q=U-V$ are proper splittings, we have $(M+U-A+VP^{\dagger }N)(M+U-A+VP^{\dagger }N)^{\dagger }=AA^{\dagger }=MM^{\dagger }=PP^{\dagger }=UU^{\dagger }$. Hence,

$$\begin{aligned} XB&=U^{\dagger }(M+U-A+VP^{\dagger }N)M^{\dagger }M(M+U-A+VP^{\dagger }N)^{\dagger }U\\&=U^{\dagger }(M+U-A+VP^{\dagger }N)(M+U-A+VP^{\dagger }N)^{\dagger }U\\&=U^{\dagger }U\\&=(U^{\dagger }U)^\textrm{T}\\&=(XB)^\textrm{T}, \end{aligned}$$

and

$$\begin{aligned} BX&=M(M+U-A+VP^{\dagger }N)^{\dagger }UU^{\dagger }(M+U-A+VP^{\dagger }N)M^{\dagger }\\&=M(M+U-A+VP^{\dagger }N)^{\dagger }(M+U-A+VP^{\dagger }N)M^{\dagger }\\&=MM^{\dagger }\\&=(MM^{\dagger })^\textrm{T}\\&=(BX)^\textrm{T}. \end{aligned}$$

Therefore, $XBX=U^{\dagger }UU^{\dagger }(M+U-A+VP^{\dagger }N)M^{\dagger } =U^{\dagger }(M+U-A+VP^{\dagger }N)M^{\dagger }=X$ and $BXB=MM^{\dagger }M(M+U-A+VP^{\dagger }N)^{\dagger }U=M(M+U-A+VP^{\dagger }N)^{\dagger }U=B.$ Hence, $X=B^{\dagger }.$

Next we will show that $\mathcal {R}(B)=\mathcal {R}(A)$ and $\mathcal {N}(B)=\mathcal {N}(A)$. To do this, we will first prove that $\mathcal {N}(U)=\mathcal {N}(B)$. Clearly, $\mathcal {N}(U)\subseteq \mathcal {N}(B)$. Let $x \in \mathcal {N}(B)$, i.e., $Bx=0.$ By pre-multiplying $M^{\dagger }$ to $Bx=0$ and using the fact that $M^{\dagger }M=P_{\mathcal {R}((M+U-A+VP^{\dagger }N)^\textrm{T})}=P_{\mathcal {R}(A^\textrm{T})}$, we get $M^{\dagger }M(M+U-A+VP^{\dagger }N)^{\dagger }Ux=(M+U-A+VP^{\dagger }N)^{\dagger }Ux=0$. Again, pre-multiplying $(M+U-A+VP^{\dagger }N)$ and using the fact $(M+U-A+VP^{\dagger }N)(M+U-A+VP^{\dagger }N)^{\dagger }=P_{\mathcal {R}(M+U-A+VP^{\dagger }N)}=P_{\mathcal {R}(A)}=P_{\mathcal {R}(U)}$, we get $(M+U-A+VP^{\dagger }N)(M+U-A+VP^{\dagger }N)^{\dagger }Ux=Ux=0$, i.e., $x \in \mathcal {N}(U).$ Hence, $\mathcal {N}(B) \subseteq \mathcal {N}(U).$ Next, we will show that $\mathcal {R}(B)=\mathcal {R}(A)$ which is equivalent to prove $\mathcal {N}(B^\textrm{T})=\mathcal {N}(A^\textrm{T})$. But $B=M(M+U-A+VP^{\dagger }N)^{\dagger }U$ implies $\mathcal {N}(M^\textrm{T})\subseteq \mathcal {N}(B^\textrm{T}).$ So, we need to show the other way, i.e., $\mathcal {N}(B^\textrm{T}) \subseteq \mathcal {N}(M^\textrm{T})$. Let $x \in \mathcal {N}(B^\textrm{T})$, then $(M(M+U-A+VP^{\dagger }N)^{\dagger }U)^\textrm{T}x=0.$ By pre-multiplying $(U^{\dagger })^\textrm{T}$, we get $(UU^{\dagger })^\textrm{T}((M+U-A+VP^{\dagger }N)^{\dagger })^\textrm{T}M^\textrm{T}x=((M+U-A+VP^{\dagger }N)^{\dagger }UU^{\dagger })^\textrm{T}M^\textrm{T}x=0$ and using the fact $UU^{\dagger }=P_{\mathcal {R}(A)}=P_{\mathcal {R}(M+U-A+VP^{\dagger }N)}=(M+U-A+VP^{\dagger }N)(M+U-A+VP^{\dagger }N)^{\dagger }$, we get $((M+U-A+VP^{\dagger }N)^{\dagger })^\textrm{T}M^\textrm{T}x=0.$ Again, pre-multiplying $(M+U-A+VP^{\dagger }N)^\textrm{T}$ and using the fact $(M{+}U-A+VP^{\dagger }N)^{\dagger }(M+U-A+VP^{\dagger }N)=P_{\mathcal {R}((M+U-A+VP^{\dagger }N)^\textrm{T})}=P_{\mathcal {R}(A^\textrm{T})}=M^{\dagger }M$, we obtain $(M+U-A+VP^{\dagger }N)^\textrm{T}((M+U-A+VP^{\dagger }N)^{\dagger })^\textrm{T}M^\textrm{T}x=(M^{\dagger }M)^\textrm{T}M^\textrm{T}x=M^\textrm{T}x=0.$ Hence, $\mathcal {N}(B^\textrm{T}) \subseteq \mathcal {N}(M^\textrm{T})=\mathcal {N}(A^\textrm{T}).$    $\square$

We have the following two expressions for $B^{\dagger }$,

$$\begin{aligned} B^{\dagger }&=U^{\dagger }(M+U-A+VP^{\dagger }N)M^{\dagger }\\&=U^{\dagger }MM^{\dagger }+U^{\dagger }UM^{\dagger }-U^{\dagger }AM^{\dagger }+U^{\dagger }VP^{\dagger }NM^{\dagger }\\&=U^{\dagger }+U^{\dagger }VM^{\dagger }+U^{\dagger }VP^{\dagger }NM^{\dagger } \end{aligned}$$

and

$$\begin{aligned} B^{\dagger }&=U^{\dagger }(M+U-A+VP^{\dagger }N)M^{\dagger }\\&=U^{\dagger }MM^{\dagger }+U^{\dagger }UM^{\dagger }-U^{\dagger }AM^{\dagger }+U^{\dagger }VP^{\dagger }NM^{\dagger }\\&=M^{\dagger }+U^{\dagger }NM^{\dagger }+U^{\dagger }VP^{\dagger }NM^{\dagger }. \end{aligned}$$

The example given below shows that the alternating scheme is convergent even though $A=M-N=P-Q=U-V$ are not proper weak regular splittings of type I.

Example 3.4

Let $A=\begin{bmatrix} 5 &{} -3 &{} 5\\ -3 &{} 5 &{} -3 \end{bmatrix}$. Then, $A^{\dagger }=\begin{bmatrix} 0.1563 &{} 0.0938\\ 0.1875 &{} 0.3125\\ 0.1563 &{} 0.0938 \end{bmatrix} \ge 0.$ Consider $A{=}\begin{bmatrix} 15 &{} -9 &{} 15\\ -3 &{} 10 &{} -3 \end{bmatrix}{-}\begin{bmatrix} 10 &{} -6 &{} 10\\ 0 &{} 5 &{} 0 \end{bmatrix}=M-N.$ Then $\mathcal {R}(M)=\mathcal {R}(A),~\mathcal {N}(M)=\mathcal {N}(A)$, $M^{\dagger }=\begin{bmatrix} 0.0407 &{} 0.0366\\ 0.0244 &{} 0.1220\\ 0.0407 &{} 0.0366 \end{bmatrix}\ge 0,~M^{\dagger }N=$ $\begin{bmatrix} 0.4065 &{} -0.0610 &{} 0.4065\\ 0.2439 &{} 0.4634 &{} 0.2439\\ 0.4065 &{} -0.0610 &{} 0.4065 \end{bmatrix} \ngeq 0$

and $NM^{\dagger }=\begin{bmatrix} 0.6667 &{} 0\\ 0.1220 &{} 0.6098 \end{bmatrix}\ge 0.$ Hence, $A=M-N$ is a proper weak regular splitting of type II but not I.

Further consider $A=\begin{bmatrix} 20 &{} -12 &{} 20\\ -3 &{} 10 &{} -3 \end{bmatrix}-\begin{bmatrix} 15 &{} -9 &{} 15\\ 0 &{} 5 &{} 0 \end{bmatrix}{=}P-Q.$ Then $\mathcal {R}(P){=}\mathcal {R}(A), \mathcal {N}(P)=\mathcal {N}(A)$, $P^{\dagger }=\begin{bmatrix} 0.0305 &{} 0.0366\\ 0.0183 &{} 0.1220\\ 0.0305 &{} 0.0366 \end{bmatrix} \ge 0$, $P^{\dagger }Q=\begin{bmatrix} 0.4573 &{} -0.0915 &{} 0.4573\\ 0.2744 &{} 0.4451 &{} 0.2744\\ 0.4573 &{} -0.0915 &{} 0.4573 \end{bmatrix} \ngeq 0$ and $QP^{\dagger }= \begin{bmatrix} 0.7500 &{} 0\\ 0.0915 &{} 0.6098 \end{bmatrix} \ge 0.$ Hence, $A=P-Q$ is a proper weak regular splitting of type II but not I.

Again consider $A=\begin{bmatrix} 25 &{} -15 &{} 25\\ -3 &{} 10 &{} -3 \end{bmatrix}-\begin{bmatrix} 20 &{}-12 &{} 20\\ 0 &{} 5 &{} 0 \end{bmatrix}=U-V.$ Then $\mathcal {R}(U)=\mathcal {R}(A),~\mathcal {N}(U)=\mathcal {N}(A)$, $U^{\dagger }=\begin{bmatrix} 0.0244 &{} 0.0366\\ 0.0146 &{} 0.1220\\ 0.0244 &{} 0.0366 \end{bmatrix} \ge 0$,

$U^{\dagger }V=\begin{bmatrix} 0.4878 &{} -0.1098 &{} 0.4878\\ 0.2927 &{} 0.4341 &{} 0.2927\\ 0.4878 &{} -0.1098 &{} 0.4878 \end{bmatrix} \ngeq 0$ and $VU^{\dagger }=\begin{bmatrix} 0.8000 &{} 0\\ 0.0732 &{} 0.6098 \end{bmatrix} \ge 0.$ Hence, $A=U-V$ is a proper weak regular splitting of type II but not I. But $\rho (H)=\rho (U^{\dagger }VP^{\dagger }QM^{\dagger }N)=0.4<1,$ and A is semimonotone.

In the above example, we can see that $H=U^{\dagger }VP^{\dagger }QM^{\dagger }N$

$= \begin{bmatrix} 0.3046 &{} -0.1147 &{} 0.3046\\ 0.3486 &{} 0.0176 &{} 0.3486\\ 0.3046 &{} -0.1147 &{} 0.3046 \end{bmatrix} \ngeq 0$ but $S=VU^{\dagger }QP^{\dagger }NM^{\dagger }=$

$\begin{bmatrix} 0.4 &{} 0\\ 0.1191 &{} 0.2267 \end{bmatrix} \ge 0.$

Motivated by the above example, we will now introduce a similar result as Theorem 3.1 by considering the given splittings are proper weak regular of type II. With this objective, we first derived the following properties of H and S which are useful to prove further results.

Theorem 3.5

Let $A=M-N=P-Q=U-V$ be three proper splittings of $A \in {{\mathbb R}}^{m \times n}$ and $S=VU^{\dagger }QP^{\dagger }NM^{\dagger }.$ Then,

• (i) $AA^{\dagger }S=S=SAA^{\dagger }$ and $A^{\dagger }AH=H=HA^{\dagger }A$, where H is the iteration matrix of the iterative method (1.6).

• (ii) $S=AHA^{\dagger },$ $H=A^{\dagger }SA$ and $\rho (S)=\rho (H).$

• (iii) $I-S$ and $I-H$ are invertible if $\mathcal {R}(M+U-A+VP^{\dagger }N)=\mathcal {R}(A)$ and $\mathcal {N}(M+U-A+VP^{\dagger }N)=\mathcal {N}(A).$

Proof

• (i) $AA^{\dagger }S=AA^{\dagger }VU^{\dagger }QP^{\dagger }NM^{\dagger }=S$ using the fact that $\mathcal {R}(S)\subseteq \mathcal {R}(V)\subseteq \mathcal {R}(A).$ Again $SAA^{\dagger }=VU^{\dagger }QP^{\dagger }NM^{\dagger }AA^{\dagger }=VU^{\dagger }QP^{\dagger }NM^{\dagger }MM^{\dagger }=S.$ Similar argument yields the other equality.

• (ii) By Theorem 2.6, we have $U^{\dagger }VA^{\dagger }=A^{\dagger }VU^{\dagger }$, $P^{\dagger }QA^{\dagger }=A^{\dagger }QP^{\dagger }$ and $M^{\dagger }NA^{\dagger }=A^{\dagger }NM^{\dagger }$. So

  $$\begin{aligned} S&=AA^{\dagger }S\\&=AA^{\dagger }VU^{\dagger }QP^{\dagger }NM^{\dagger }\\&=AU^{\dagger }VA^{\dagger }QP^{\dagger }NM^{\dagger }\\&=AU^{\dagger }VP^{\dagger }QA^{\dagger }NM^{\dagger }\\&=AU^{\dagger }VP^{\dagger }QM^{\dagger }NA^{\dagger }\\&=AHA^{\dagger }. \end{aligned}$$

We then have $A^{\dagger }S=A^{\dagger }AHA^{\dagger }=HA^{\dagger }$. Again, post-multiplying A, we get $H=A^{\dagger }SA$.

Consider $Sx=\lambda x$, where $\lambda$ is an eigenvalue of S, and x is its corresponding eigenvector. Then $x\in \mathcal {R}(S)\subseteq \mathcal {R}(A)$. Now $\lambda x=Sx=AHA^{\dagger }x$. Pre-multiplying $A^{\dagger }$, we get $\lambda y=Hy$, where $y=A^{\dagger }x.$ Therefore, $\lambda$ is an eigenvalue of H if, $y\ne 0$. Suppose that $y=A^{\dagger }x=0$. Then $x \in \mathcal {N}(A^{\dagger })=\mathcal {N}(A^\textrm{T}) .$ So, we have $x\in \mathcal {R}(A)\cap \mathcal {N}(A^\textrm{T})=\{0 \}$ a contradiction. Hence, $y\ne 0$ and so $\sigma \{S\}\subseteq \sigma \{H\}$. For the other way, consider $Hy=\mu y$, where $\mu$ is an eigenvalue of H, and y is its corresponding eigenvector. Then $y\in \mathcal {R}(H)\subseteq \mathcal {R}(A^\textrm{T})$. Also $\mu y=A^{\dagger }SAy$. Pre-multiplying A, we get $\mu z=Sz$, where $z=Ay$. Suppose that $z=0$. Then $y\in \mathcal {N}(A)$. So $y\in \mathcal {R}(A^\textrm{T})\cap \mathcal {N}(A)$ which yields $y=0$, a contradiction. Hence, $\sigma \{H\}\subseteq \sigma \{S\}$. Therefore, $\rho (S)=\rho (H)$.

• (iii) We will prove this by the method of contradiction. Suppose that $I-S$ is not invertible. Then 1 is an eigenvalue of S. Therefore, $x=VU^{\dagger }QP^{\dagger }NM^{\dagger }x \in \mathcal {R}(VU^{\dagger }QP^{\dagger }NM^{\dagger })\subseteq \mathcal {R}(V) \subseteq \mathcal {R}(U)=\mathcal {R}(P)=\mathcal {R}(M)=\mathcal {R}(A)$. So, $x=UU^{\dagger }x=PP^{\dagger }x=MM^{\dagger }x$ and hence

  $$\begin{aligned} x= & {} VU^{\dagger }QP^{\dagger }NM^{\dagger }x\\= & {} (U-A)U^{\dagger }(P-A)P^{\dagger }(M-A)M^{\dagger }x \\= & {} (UU^{\dagger }-AU^{\dagger })(PP^{\dagger }-AP^{\dagger })(MM^{\dagger }-AM^{\dagger })x\\= & {} (AA^{\dagger }-AU^{\dagger })(AA^{\dagger }-AP^{\dagger })(AA^{\dagger }-AM^{\dagger })x\\= & {} (AA^{\dagger }-AU^{\dagger })(AA^{\dagger }-AP^{\dagger })(AA^{\dagger }x-AM^{\dagger }x)\\= & {} (AA^{\dagger }-AU^{\dagger })(AA^{\dagger }-AP^{\dagger })(x-AM^{\dagger }x)\\= & {} (AA^{\dagger }-AU^{\dagger })(AA^{\dagger }x-AA^{\dagger }AM^{\dagger }x-AP^{\dagger }x+AP^{\dagger }AM^{\dagger }x)\\= & {} (AA^{\dagger }-AU^{\dagger })(x-AM^{\dagger }x-AP^{\dagger }x+AP^{\dagger }AM^{\dagger }x)\\= & {} (x-AM^{\dagger }x-AP^{\dagger }x+AP^{\dagger }AM^{\dagger }x-AU^{\dagger }x+AU^{\dagger }AM^{\dagger }x\\{} & {} +AU^{\dagger }AP^{\dagger }x-AU^{\dagger }AP^{\dagger }AM^{\dagger }x)\\= & {} x-A(M^{\dagger }+P^{\dagger }-P^{\dagger }AM^{\dagger }+U^{\dagger }-U^{\dagger }AM^{\dagger }-U^{\dagger }AP^{\dagger }\\{} & {} +U^{\dagger }AP^{\dagger }AM^{\dagger })x\\= & {} x-A(M^{\dagger }MM^{\dagger }+P^{\dagger }PP^{\dagger }+U^{\dagger }-P^{\dagger }AM^{\dagger }-U^{\dagger }AM^{\dagger }-U^{\dagger }AP^{\dagger }\\{} & {} +U^{\dagger }AP^{\dagger }AM^{\dagger })x\\= & {} x-A(U^{\dagger }UM^{\dagger }+U^{\dagger }UP^{\dagger }+U^{\dagger }-P^{\dagger }AM^{\dagger }-U^{\dagger }AM^{\dagger }-U^{\dagger }AP^{\dagger }\\{} & {} +U^{\dagger }AP^{\dagger }AM^{\dagger })x\\= & {} x-A(U^{\dagger }UM^{\dagger }+U^{\dagger }UP^{\dagger }PP^{\dagger }+U^{\dagger }UU^{\dagger }-P^{\dagger }PP^{\dagger }AM^{\dagger }-U^{\dagger }AM^{\dagger }\\{} & {} -U^{\dagger }AP^{\dagger }PP^{\dagger }+U^{\dagger }AP^{\dagger }AM^{\dagger })x\\= & {} x-A(U^{\dagger }UM^{\dagger }+U^{\dagger }UP^{\dagger }MM^{\dagger }+U^{\dagger }MM^{\dagger }-U^{\dagger }UP^{\dagger }AM^{\dagger }-U^{\dagger }AM^{\dagger }\\{} & {} -U^{\dagger }AP^{\dagger }M^{\dagger }+U^{\dagger }AP^{\dagger }AM^{\dagger })x\\= & {} x-A(U^{\dagger }(U+M-A+UP^{\dagger }M-UP^{\dagger }A-AP^{\dagger }M+AP^{\dagger }A)M^{\dagger })x\\= & {} x-A(U^{\dagger }(U+M-A+VP^{\dagger }N)M^{\dagger })x\\= & {} x-AB^{\dagger }x. \end{aligned}$$

Then $AB^{\dagger }x=0.$ Thus, $B^{\dagger }x\in \mathcal {N}(A)=\mathcal {N}(B)$ and so $BB^{\dagger }x=0.$ But $x\in \mathcal {R}(A)=\mathcal {R}(B).$ (We have used the facts that $\mathcal {N}(B)=\mathcal {N}(A)$ and $\mathcal {R}(B)=\mathcal {R}(A)$ which follows from Lemma 3.3.) Therefore, $x=BB^{\dagger }x=0$, a contradiction. Thus, $I-S$ is invertible.

The other proof is explained next. Let $(I-H)x=(I-U^{\dagger }VP^{\dagger }QM^{\dagger }N)x=0.$ So, $x\in \mathcal {R}(U^{\dagger })=\mathcal {R}(U^\textrm{T})=\mathcal {R}(A^\textrm{T}).$ Substituting $V=U-A$, $Q=P-A$ and $N=M-A$ in $(I-H)x=0$ and then simplifying, we get $B^{\dagger }Ax=0.$ Pre-multiplying B and using the fact $\mathcal {R}(B)=\mathcal {R}(A),$ we get $x \in \mathcal {N}(A).$ Hence, $x=0$ yielding a contradiction. Thus, $I-H$ is invertible.    $\square$

Theorem 3.6

Let $A=M-N=P-Q=U-V$ be three proper weak regular splittings of type II of a semimonotone matrix $A \in {{\mathbb R}}^{m \times n}$ and $S=VU^{\dagger }QP^{\dagger }NM^{\dagger }$. Then $\rho (H)=\rho (U^{\dagger }VP^{\dagger }QM^{\dagger }N)<1$.

Proof

We have

$$\begin{aligned} 0\le S= & {} VU^{\dagger }QP^{\dagger }NM^{\dagger }\\= & {} (U-A)U^{\dagger }(P-A)P^{\dagger }(M-A)M^{\dagger }\\= & {} (UU^{\dagger }-AU^{\dagger })(PP^{\dagger }-AP^{\dagger })(MM^{\dagger }-AM^{\dagger })\\= & {} (AA^{\dagger }-AU^{\dagger })(AA^{\dagger }-AP^{\dagger })(AA^{\dagger }-AM^{\dagger })\\= & {} AA^{\dagger }-AM^{\dagger }-AP^{\dagger }-AU^{\dagger }+AP^{\dagger }AM^{\dagger }+AU^{\dagger }AM^{\dagger }\\{} & {} +AU^{\dagger }AP^{\dagger }-AU^{\dagger }AP^{\dagger }AM^{\dagger } \end{aligned}$$

since $A=M-N=P-Q=U-V$ are proper weak regular splittings of type II. Now

$$\begin{aligned} A^{\dagger }(I-S)= & {} A^{\dagger }-A^{\dagger }S\\= & {} A^{\dagger }AA^{\dagger }-A^{\dagger }S\\= & {} A^{\dagger }(AA^{\dagger }-S)\\= & {} A^{\dagger }(AA^{\dagger }-AA^{\dagger }+AM^{\dagger }+AP^{\dagger }+AU^{\dagger }-AP^{\dagger }AM^{\dagger }-AU^{\dagger }AM^{\dagger }\\{} & {} -AU^{\dagger }AP^{\dagger }+AU^{\dagger }AP^{\dagger }AM^{\dagger })\\= & {} A^{\dagger }AM^{\dagger }+A^{\dagger }AP^{\dagger }+A^{\dagger }AU^{\dagger }-A^{\dagger }AP^{\dagger }AM^{\dagger }-A^{\dagger }AU^{\dagger }AM^{\dagger }\\{} & {} -A^{\dagger }AU^{\dagger }AP^{\dagger }+A^{\dagger }AU^{\dagger }AP^{\dagger }AM^{\dagger }\\= & {} M^{\dagger }+P^{\dagger }+U^{\dagger }-P^{\dagger }AM^{\dagger }-U^{\dagger }AM^{\dagger }-U^{\dagger }AP^{\dagger }+U^{\dagger }AP^{\dagger }AM^{\dagger }\\= & {} M^{\dagger }+P^{\dagger }(M-A)M^{\dagger }+U^{\dagger }(P-A)P^{\dagger }+U^{\dagger }(AP^{\dagger }A-A)M^{\dagger }\\= & {} M^{\dagger }+P^{\dagger }NM^{\dagger }+U^{\dagger }QP^{\dagger }+U^{\dagger }(AP^{\dagger }A-PP^{\dagger }A)M^{\dagger }\\= & {} M^{\dagger }+P^{\dagger }NM^{\dagger }+U^{\dagger }QP^{\dagger }-U^{\dagger }(P-A)P^{\dagger }AM^{\dagger }\\= & {} M^{\dagger }+P^{\dagger }NM^{\dagger }+U^{\dagger }QP^{\dagger }-U^{\dagger }QP^{\dagger }(M-N)M^{\dagger }\\= & {} M^{\dagger }+P^{\dagger }NM^{\dagger }+U^{\dagger }QP^{\dagger }NM^{\dagger }\ge 0. \end{aligned}$$

Hence,

$$\begin{aligned} 0&\le A^{\dagger }(I-S)(I+S+S^{2}+S^{3}+\cdots +S^{m})\\&=A^{\dagger }(I-S^{m+1})\\&\le A^{\dagger } \end{aligned}$$

for each $m\in \mathbb {N}.$ So, the partial sums of the series $\displaystyle \sum \nolimits _{m=0}^{\infty }S^{m}$ is uniformly bounded. Hence, $\rho (S)<1.$ Thus $\rho (H)=\rho (S)<1$ by Theorem 3.5 (ii).    $\square$

The next result showing that the matrix B in the splitting $A=B-C$ induced by H can also be expressed as the product of A and $(I-H)^{-1}.$

Lemma 3.7

Let $A=M-N=P-Q=U-V$ be three proper splittings of $A \in {{\mathbb R}}^{m \times n}$ such that $\rho (H)<1,$ $\mathcal {R}(M+P-A+VP^{\dagger }N)=\mathcal {R}(A)$ and $\mathcal {N}(M+P-A+VP^{\dagger }N)=\mathcal {N}(A).$ Then the unique splitting $A=B-C$ induced by H is a proper splitting such that $H=B^{\dagger }C,$ where $B=A(I-H)^{-1}.$

Proof

Let $B=A(I-H)^{-1}$ and $C=B-A.$ Then

$$\begin{aligned} B^{\dagger }C&=B^{\dagger }(B-A)\\&=B^{\dagger }B-B^{\dagger }A\\&=B^{\dagger }B-(I-H)A^{\dagger }A\\&=A^{\dagger }A-A^{\dagger }A+HA^{\dagger }A\\&=H \end{aligned}$$

using the fact that $A=B-C$ is a proper splitting which is shown next.

Let $Z=(I-H)A^{\dagger }.$ Then $BZ=AA^{\dagger }$ which is symmetric and $ZBZ=(I-H)A^{\dagger }AA^{\dagger }=(I-H)A^{\dagger }=Z.$ Again

$$\begin{aligned} ZB&=(I-H)A^{\dagger }A(I-H)^{-1}\\&=(A^{\dagger }A-HA^{\dagger }A)(I-H)^{-1}\\&=(A^{\dagger }A-A^{\dagger }AH)(I-H)^{-1}\\&=A^{\dagger }A \end{aligned}$$

which is symmetric and $BZB=AA^{\dagger }A(I-H)^{-1}=A(I-H)^{-1}=B.$ So, we have

$$\begin{aligned} B^{\dagger }= & {} (I-H)A^{\dagger }\\= & {} A^{\dagger }-U^{\dagger }VP^{\dagger }QM^{\dagger }NA^{\dagger }\\= & {} A^{\dagger }-(U^{\dagger }(U-A)P^{\dagger }(P-A)M^{\dagger }(M-A)A^{\dagger })\\= & {} A^{\dagger }-(P^{\dagger }-U^{\dagger }AP^{\dagger })(PM^{\dagger }-AM^{\dagger })(MA^{\dagger }-AA^{\dagger })\\= & {} A^{\dagger }-(M^{\dagger }-P^{\dagger }AM^{\dagger }-U^{\dagger }AM^{\dagger }+U^{\dagger }AP^{\dagger }AM^{\dagger })(MA^{\dagger }-AA^{\dagger })\\= & {} M^{\dagger }+P^{\dagger }-P^{\dagger }AM^{\dagger }+U^{\dagger }-U^{\dagger }AM^{\dagger }-U^{\dagger }AP^{\dagger }+U^{\dagger }AP^{\dagger }AM^{\dagger }\\= & {} M^{\dagger }+U^{\dagger }-U^{\dagger }AM^{\dagger }+P^{\dagger }-P^{\dagger }(P-Q)M^{\dagger }-U^{\dagger }(U-V)P^{\dagger }\\{} & {} +U^{\dagger }(U-V)P^{\dagger }(M-N)M^{\dagger }\\= & {} M^{\dagger }+U^{\dagger }-U^{\dagger }AM^{\dagger }+U^{\dagger }VP^{\dagger }NM^{\dagger }. \end{aligned}$$

Hence, $A=B-C$ is a proper splitting, by Lemma 3.3.

For uniqueness, suppose that there exists another splitting $A=\bar{B}-\bar{C}$ such that $H=\bar{B}^{\dagger }\bar{C}.$ Then $\bar{B}H=\bar{B}\bar{B}^{\dagger }\bar{C}=\bar{C}=\bar{B}-A.$ So $\bar{B}(I-H)=A.$ Hence $\bar{B}=A(I-H)^{-1}=B.$    $\square$

Theorem 3.8

Let $A=M-N=P-Q=U-V$ be three proper weak regular splittings of type II of a semimonotone matrix $A \in {{\mathbb R}}^{m \times n}$ and $S=VU^{\dagger }QP^{\dagger }NM^{\dagger }$. Then, H and S induce the same proper splitting $A=B-C$ if $\mathcal {R}(M+U-A+VP^{\dagger }N)=\mathcal {R}(A)$ and $\mathcal {N}(M+U-A+VP^{\dagger }N)=\mathcal {N}(A)$. Furthermore, the unique proper splitting $A=X-Y$ induced by matrix S is also a proper weak regular splitting of type II.

Proof

By Lemma 3.7, we have $B=A(I-H)^{-1}$. Let us consider $X=(I-S)^{-1}A$ and $Y=X-A.$ We will show that the matrices H and S induce the same proper splitting $A=B-C.$ Since $H=HA^{\dagger }A$ and $S=AHA^{\dagger },$ so $S^{k}=AH^{k}A^{\dagger },$ for any nonnegative integer k. By Theorem 3.6, we have $\rho (S)<1.$ Also, $S\ge 0.$ Therefore, Theorem 2.3 yields

$$\begin{aligned} X&=(I-S)^{-1}A\\&=\sum _{k=o}^{\infty }S^{k}A\\&=\sum _{k=o}^{\infty }AH^{k}A^{\dagger }A\\&=\sum _{k=o}^{\infty }AH^{k}\\&=A(I-H)^{-1}\\&=B. \end{aligned}$$

Then $\mathcal {R}(X)=\mathcal {R}(B)=\mathcal {R}(A)$ and $\mathcal {N}(X)=\mathcal {N}(B)=\mathcal {N}(A)$ which in turn yields $A=X-Y$ is a proper splitting.

To prove $A=X-Y$ is a proper weak regular splitting of type II, consider $Z=A^{\dagger }(I-S).$ Then $ZX=A^{\dagger }(I-S)(I-S)^{-1}A=A^{\dagger }A.$ Hence, ZX is symmetric and $ZXZ=A^{\dagger }AA^{\dagger }(I-S)=A^{\dagger }(I-S)=Z.$ Using the property $AA^{\dagger }S=S=SAA^{\dagger },$ we obtain

$$\begin{aligned} XZ&=(I-S)^{-1}AA^{\dagger }(I-S)\\&=(I-S)^{-1}(AA^{\dagger }-AA^{\dagger }S)\\&=(I-S)^{-1}(AA^{\dagger }-SAA^{\dagger })\\&=AA^{\dagger }. \end{aligned}$$

So, XZ is symmetric and

$$\begin{aligned} XZX&=AA^{\dagger }(I-S)^{-1}A\\&=AA^{\dagger }\sum _{k=0}^{\infty }S^{k}A\\&=\sum _{k=0}^{\infty }S^{k}A\\&=(I-S)^{-1}A\\&=X. \end{aligned}$$

Hence, $X^{\dagger }=A^{\dagger }(I-S)\ge 0$ (see the proof of Theorem 3.6 for $A^{\dagger }(I-S)\ge 0$). Therefore,

$$\begin{aligned} YX^{\dagger }&=(X-A)X^{\dagger }\\&=XX^{\dagger }-AX^{\dagger }\\&=AA^{\dagger }-AA^{\dagger }(I-S)\\&=S\ge 0. \end{aligned}$$

Thus $A=X-Y$ is a proper weak regular splitting of type II induced by S. Let $A=X_{1}-Y_{1}$ be another splitting induced by S such that $S=Y_{1}X_{1}^{\dagger }.$ Then $SX_{1}=Y_{1}X_{1}^{\dagger }X_{1}=Y_{1}=X_{1}-A.$ So $A=X_{1}-SX_{1}=(I-S)X_{1}$ which implies $X_{1}=(I-S)^{-1}A=X.$ Hence, $A=X-Y$ is the unique proper weak regular splitting of type II induced by S.    $\square$

The next results confirm that the proposed alternating iterative scheme converges faster than (1.2) under suitable assumptions.

Theorem 3.9

Let $A = M-N =P-Q =U-V$ be three proper regular splittings of a semimonotone matrix $A \in {{\mathbb R}}^{m \times n}$ with $\mathcal {R}(M+U-A+VP^{\dagger }N)=\mathcal {R}(A)$ and $\mathcal {N}(M+U-A+VP^{\dagger }N)=\mathcal {N}(A)$. Then $\rho (H)\le \min \{\rho (M^{\dagger }N),~ \rho (P^{\dagger }Q ), \rho (U^{\dagger }V )\} < 1$.

Proof

By Theorem 3.8, $A = B-C$ is a proper weak regular splitting of type II induced by H, and from (1.6), we have

$$B^{\dagger } = U^{\dagger }(VP^{\dagger }QM^{\dagger }+VP^{\dagger }+ I) = U^{\dagger }VP^{\dagger }QM^{\dagger } + U^{\dagger }VP^{\dagger } + U^{\dagger }\ge U^{\dagger }.$$

and

$$\begin{aligned} B^{\dagger }&= U^{\dagger }VP^{\dagger }QM^{\dagger }+U^{\dagger }VP^{\dagger } + U^{\dagger }\\&= U^{\dagger }VP^{\dagger }QM^{\dagger } + U^{\dagger }UP^{\dagger }-U^{\dagger }AP^{\dagger } + U^{\dagger }MM^{\dagger }\\&= U^{\dagger }VP^{\dagger }QM^{\dagger } + P^{\dagger } +U^{\dagger }(M- A)M^{\dagger }\\&= U^{\dagger }VP^{\dagger }QM^{\dagger } + P^{\dagger } +U^{\dagger }NM^{\dagger } \ge P^{\dagger }. \end{aligned}$$

Also,

$$\begin{aligned} B^{\dagger }&= U^{\dagger }(M+U-A+VP^{\dagger }N)M^{\dagger }\\&= U^{\dagger }MM^{\dagger }+U^{\dagger }UM^{\dagger }-U^{\dagger }(M -N)M^{\dagger }+U^{\dagger }VP^{\dagger }NM^{\dagger }\\&= U^{\dagger }+M^{\dagger }-U^{\dagger }MM^{\dagger }+U^{\dagger }NM ^{\dagger }+U^{\dagger }VP^{\dagger }NM^{\dagger }\\&= U^{\dagger }+M^{\dagger }-U^{\dagger }+U^{\dagger }NM^{\dagger }+U^{\dagger }VP^{\dagger }NM^{\dagger }\\&= M^{\dagger }+U^{\dagger }NM^{\dagger }+U^{\dagger }VP^{\dagger }NM^{\dagger }\ge M^{\dagger }. \end{aligned}$$

Hence, by applying Theorem 2.9 to the pair of the splittings $A=B-C$ and $A=P-Q$, $A=B-C$ and $A=U-V$, and $A=B-C$ and $A=B-C$, we have $\rho (H)\le \rho (P^{\dagger }Q)<1$, $\rho (H)\le \rho (M^{\dagger }N)<1$ and $\rho (H)\le \rho (U^{\dagger }V)<1$, respectively. Therefore, $\rho (H)\le \min \{\rho (P^{\dagger }Q),~\rho (M^{\dagger }N), ~\rho (U^{\dagger }V)\}<1.$    $\square$

Theorem 3.10

Let $A=P-Q=M-N=U-V$ be three proper weak regular splittings of type II of a semimonotone matrix $A\in \mathbb {R}^{n\times n}$ with $\mathcal {R}(M+U-A+VP^{\dagger }N)=\mathcal {R}(A)$ and $\mathcal {N}(M+U-A+VP^{\dagger }N)=\mathcal {N}(A)$. Let $A=B-C$ be the proper weak regular splitting of type II induced by H (or S). If $PB^{\dagger }\ge I$, $MB^{\dagger }\ge I$ and $UB^{\dagger }\ge I$, then $\rho (H)\le \min \{\rho (P^{\dagger }Q),~\rho (M^{\dagger }N),~\rho (U^{\dagger }V)\}<1.$

Proof

Consider the pair of splittings $A=B-C$ and $A=P-Q$. By Theorem 2.5, we have

$$\begin{aligned} B^{\dagger }(I-CB^{\dagger })^{-1}=P^{\dagger }(I-QP^{\dagger })^{-1}. \end{aligned}$$

(3.1)

Pre-multiplying (3.1) by P, we obtain

$$\begin{aligned} PB^{\dagger }(I-CB^{\dagger })^{-1}=PP^{\dagger }(I-QP^{\dagger })^{-1}=(I-QP^{\dagger })^{-1}. \end{aligned}$$

(3.2)

As $CB^{\dagger }\ge 0$, there exists an eigenvector $x\ge 0$ such that $CB^{\dagger }x=\rho (CB^{\dagger })x$ by Theorem 2.1 . Post-multiplying (3.2) by x, we get $PB^{\dagger }(I-CB^{\dagger })^{-1}x=(I-QP^{\dagger })^{-1}x$, i.e., $\frac{PB^{\dagger }x}{1-\rho (B^{\dagger }C)}=(I-QP^{\dagger })^{-1}x$. Using the fact $PB^{\dagger }\ge I$, we have

$$\frac{x}{1-\rho (CB^{\dagger })}\le \frac{PB^{\dagger }x}{1-\rho (CB^{\dagger })}= (I-QP^{\dagger })^{-1}x.$$

Thus, $\rho (B^{\dagger }C)\le \rho (P^{\dagger }Q)$ by Theorem 2.2 (i). Similarly, for the pair of splittings $A=B-C$ and $A=M-N$, and $A=B-C$ and $A=U-V$, we have $\rho (H)\le \rho (M^{\dagger }N)<1$ and $\rho (H)\le \rho (U^{\dagger }V)<1$, respectively. Hence, $\rho (H)\le \min \{\rho (P^{\dagger }Q), ~\rho (M^{\dagger }N),~\rho (U^{\dagger }V)\}<1$.    $\square$

Theorem 3.11

Let $A = M-N =P-Q =U-V$ be three proper regular splittings of a semimonotone matrix $A \in {{\mathbb R}}^{m \times n}$ with $\mathcal {R}(M+U-A+VP^{\dagger }N)=\mathcal {R}(A)$ and $\mathcal {N}(M+U-A+VP^{\dagger }N)=\mathcal {N}(A)$. Then $\rho (H)\le \min \{\rho (P^{\dagger }QM^{\dagger }N), ~\rho (U^{\dagger }VP^{\dagger }Q ),~ \rho (U^{\dagger }VM^{\dagger }N)\} < 1$.

Proof

Suppose $A=B-C$ be the splitting induced by the matrix H. Let $A=B_{1}-C_{1}$, $A=B_{2}-C_{2}$, and $A=B_{3}-C_{3}$ be the splitting induced by the matrices $M^{\dagger }NP^{\dagger }Q$, $M^{\dagger }NU^{\dagger }V$, and $P^{\dagger }QU^{\dagger }V$, respectively. Then

$$\begin{aligned} B^{\dagger }&=M^{\dagger }+U^{\dagger }-U^{\dagger }AM^{\dagger }+U^{\dagger } VP^{\dagger }NM^{\dagger }\\&\ge M^{\dagger }+U^{\dagger }-U^{\dagger }AM^{\dagger }&\\&=U^{\dagger }(M+U-A)M^{\dagger }=B^{\dagger }_{2} \end{aligned}$$

and by Theorems 3.8 and 2.9, we have $\rho (B^{\dagger }C)\le \rho (B^{\dagger }_{2}C_{2})$, i.e., $\rho (H)\le \rho (U^{\dagger }VM^{\dagger }N)$. Again,

$$\begin{aligned} B^{\dagger }&=M^{\dagger }+P^{\dagger }+U^{\dagger }-P^{\dagger }AM^{\dagger }-U^{\dagger }AP^{\dagger }+U^{\dagger }AP^{\dagger }AM^{\dagger }\\&=M^{\dagger }+P^{\dagger }-P^{\dagger }AM^{\dagger }+U^{\dagger }-U^{\dagger }AM^{\dagger }-U^{\dagger }AP^{\dagger }+U^{\dagger }AP^{\dagger }AM^{\dagger }\\&=M^{\dagger }+P^{\dagger }-P^{\dagger }AM^{\dagger }+U^{\dagger }-U^{\dagger }(M-N)M^{\dagger }-U^{\dagger }(P-Q)P^{\dagger }\\&\quad +U^{\dagger }(P-Q)P^{\dagger }(M-N)M^{\dagger }\\&=M^{\dagger }+P^{\dagger }-P^{\dagger }AM^{\dagger }+U^{\dagger }NM^{\dagger }-U^{\dagger }+U^{\dagger }QP^{\dagger }\\&\quad +U^{\dagger }(PP^{\dagger }-QP^{\dagger })(MM^{\dagger }-NM^{\dagger })\\&=M^{\dagger }+P^{\dagger }-P^{\dagger }AM^{\dagger }+U^{\dagger }QP^{\dagger }NM^{\dagger }\\&\ge M^{\dagger }+P^{\dagger }-P^{\dagger }AM^{\dagger }\\&=P^{\dagger }(M+P-A)M^{\dagger }=B^{\dagger }_{1} \end{aligned}$$

and by Theorems 3.8 and 2.9, we have $\rho (B^{\dagger }C)\le \rho (B^{\dagger }_{1}C_{1})$, i.e., $\rho (H)\le \rho (P^{\dagger }QM^{\dagger }N)$.

Similarly, we can obtain

$$\begin{aligned} B^{\dagger }&=U^{\dagger }+P^{\dagger }-U^{\dagger }AP^{\dagger }+U^{\dagger }VP^{\dagger }QM^{\dagger }\\&\ge U^{\dagger }+P^{\dagger }-U^{\dagger }AP^{\dagger }\\&=B^{\dagger }_{3} \end{aligned}$$

and $\rho (H)\le \rho (U^{\dagger }VP^{\dagger }Q)$. Hence $\rho (H)\le \min \{\rho (P^{\dagger }QM^{\dagger }N), \rho (U^{\dagger }VP^{\dagger }Q),$

$\rho (U^{\dagger }VM^{\dagger }N)\}<1$.    $\square$

The following example demonstrates the above results.

Example 3.12

Consider the matrices in Example 3.4. Then $S=VU^{\dagger }QP^{\dagger }NM^{\dagger }$

$=\begin{bmatrix} 0.4000 &{} 0\\ 0.1191 &{} 0.2267 \end{bmatrix}$ and the splitting induced by S is $A=X-Y$

$= \begin{bmatrix} 8.3333 &{} -5 &{} 8.3333\\ -2.5960 &{} 5.6957 &{} -2.5960 \end{bmatrix}-\begin{bmatrix} 3.3333 &{} -2 &{} 3.3333\\ 0.4040 &{} 0.6957 &{} 0.4040 \end{bmatrix}.$ Here, $\mathcal {R}(A)=\mathcal {R}(X), \mathcal {N}(A)=\mathcal {N}(X),~X^{\dagger }= \begin{bmatrix} 0.0826 &{} 0.0725\\ 0.0753 &{} 0.2417\\ 0.0826 &{} 0.0725 \end{bmatrix} \ge 0$ and $YX^{\dagger }=\begin{bmatrix} 0.4000 &{} 0\\ 0.1191 &{} 0.2267 \end{bmatrix} \ge 0$. Hence, the induced splitting $A=X-Y$ is a proper weak regular splitting of type II. Also, $PX^{\dagger }= \begin{bmatrix} 2.4000 &{} 0\\ 0.2573 &{} 1.9816 \end{bmatrix} \ge I,~MX^{\dagger }=\begin{bmatrix} 1.8000 &{} 0\\ 0.2573 &{} 1.9816 \end{bmatrix} \ge I,~UX^{\dagger }= \begin{bmatrix} 3 &{} 0\\ 0.2573 &{} 1.9816 \end{bmatrix} \ge I,$ respectively. Further $\rho (H)=0.4< \min \{\rho (M^{\dagger }N),~\rho (P^{\dagger }Q),\rho (U^{\dagger }V) \}= \min \{0.6666,~0.75,~0.8 \}<1$ and $\rho (H)=0.4< \min \{\rho (P^{\dagger }QM^{\dagger }N), \rho (U^{\dagger }VP^{\dagger }Q ), \rho (U^{\dagger }VM^{\dagger }N)\}= \min \{0.5, ~0.6, ~0.5333\}< 1$.

4 Numerical Computation

In this section, we demonstrate a few numerical examples to validate the proposed theory. The estimation of error bounds, time in seconds, the number of iterations (IT), and the spectral radius of the corresponding iteration matrix are evaluated using MATLAB. We use MATLAB R2015a for the numerical computations in a Windows operating system with configurations: Intel(R) Xenon(R) E-2224 with 16 GB RAM. To terminate the iterative process, we use the stopping criteria $||x_k-x_{k-1}||<\epsilon$, where $\epsilon =10^{-7}$.

To show that the three-step iterative scheme converges faster than the two-step and single-step iterative schemes, we consider two different inconsistent linear system and we consider three proper splittings of the coefficient matrices.

Example 4.1

Let us consider a linear system of the form (1.1) with $A=\begin{bmatrix} 1 &{}1 &{}0\\ 0 &{}0 &{}8\\ 11 &{}11 &{}0\\ 0 &{}0 &{}2 \end{bmatrix}$ and $b=[1,1,1,1]^\textrm{T}$. We have

$A^{\dagger }=\begin{bmatrix} 0.0041 &{}0 &{}0.0451 &{}0\\ 0.0041 &{}0 &{}0.0451 &{}0\\ 0 &{}0.1176 &{}0 &{}0.0294 \end{bmatrix}\ge 0$. Setting $M=\begin{bmatrix} 1 &{}1 &{}0\\ 0 &{}0 &{}12\\ 11 &{}11 &{}0\\ 0 &{}0 &{}3 \end{bmatrix}$, $P=\begin{bmatrix} 1 &{}1 &{}0\\ 0 &{}0 &{}28\\ 11 &{}11 &{}0\\ 0 &{}0 &{}7 \end{bmatrix}$ and $U=\begin{bmatrix} 1 &{}1 &{}0\\ 0 &{}0 &{}16\\ 11 &{}11 &{}0\\ 0 &{}0 &{}16 \end{bmatrix}$, we get three proper regular splittings $A=M-N=P-Q=U-V$ of A. The comparison of rate of convergence of single-step, two-step, and three-step is given in Table 1.

Table 1 Comparison table for Example 4.1

Example 4.2

Let us consider another linear system of the form (1.1) with $A=\begin{bmatrix} 1.41 &{}0 &{}1.41\\ 0 &{}1.41 &{}0\\ 1.41 &{}0 &{}1.41\\ 2.82 &{}0 &{}2.82 \end{bmatrix}$ and $b=[2,4,1,1]^\textrm{T}$. We have

$A^{\dagger }=\begin{bmatrix} 0.0591 &{}0 &{}0.0591 &{}0.1182\\ 0.0000 &{}0.7092 &{}0 &{}0\\ 0.0591 &{}0 &{}0.0591 &{}0.1182 \end{bmatrix}\ge 0$. Setting $M=\begin{bmatrix} 2.115 &{}0 &{}2.115\\ 0 &{}2.115 &{}0\\ 2.115 &{}0 &{}2.115\\ 4.23 &{}0 &{}4.23\\ \end{bmatrix}$, $P=\begin{bmatrix} 4.512 &{}0 &{}4.512\\ 0 &{}4.5120 &{}0\\ 4.512 &{}0 &{}4.512\\ 9.024 &{}0 &{}9.024 \end{bmatrix}$ and $U=\begin{bmatrix} 3.525 &{}0 &{}3.525\\ 0 &{}3.525 &{}0\\ 3.525 &{}0 &{}3.525\\ 7.05 &{}0 &{}7.05 \end{bmatrix}$, we get three proper regular splittings $A=M-N=P-Q=U-V$ of A. The comparison of rate of convergence of single-step, two-step, and three-step is given in Table 2.

Table 2 Comparison table for Example 4.2

From Tables 1 and 2, we observe that the spectral radius of the iteration matrix of three-step alternating iteration schemes is significantly less which ultimately leads to faster converges of these schemes. One may note that for a small size linear system $4\times 3$, the iteration number (IT) and time in seconds is quite less in case of three-step. So, for large inconsistent linear system, one can expect a significantly less iterations and time consumption which eventually leads to a faster convergence and thus computationally more feasible.

5 Conclusion

In this paper, we have settled the problem of finding the least-squares solution of minimum norm of a given inconsistent linear system using iterative method. In this direction, we have derived some suitable sufficient conditions for the convergence of the three-step alternating iterative scheme in case of proper weak regular splittings of type II (Theorem 3.6) and then we have shown that the three-step alternating iteration scheme converges faster than the usual iteration scheme and the two-step alternating iteration scheme (Theorems 3.9, 3.10 and 3.11). Finally, we have validated our theoretical results by performing numerical computations in Sect. 4. Our results further expand the existing theory and give a faster numerical solution.