key: cord-0056070-3ss7mvo5
authors: Shekhar, Vaibhav; Giri, Chinmay Kumar; Mishra, Debasisha
title: On convergence of two-stage iterative scheme
date: 2021-02-06
journal: J Anal
DOI: 10.1007/s41478-021-00306-9
sha: f9c2d9e57cc64dfb1621894b523116f60c12d8f9
doc_id: 56070
cord_uid: 3ss7mvo5

Climent and Perea [Journal of Computational and Applied Mathematics 58:43–48, 2003; MR2013603] proposed first the convergence theory of two-stage iterative scheme for solving real rectangular linear systems. In this article, we revisit the same theory. The first main result provides some sufficient conditions which guarantee that the induced splitting from a two-stage iterative scheme is a proper weak regular splitting. We then establish a few comparison results. Out of these, many are even new in nonsingular matrix setting. Further, we study the monotone convergence theory of the two-stage iterative method. Besides these, we also prove the uniqueness of a proper splitting of a rectangular matrix under certain assumptions.

Consider the problem of finding a numerical solution of a linear system of the form

where A 2 R mÂn is large and sparse, and b 2 R m . Iterative methods are more useful for solving such equations as it avoids the use of the normal system A T Ax ¼ A T b, where A T A is frequently ill-conditioned and influenced greatly by roundoff errors (see [12] ). Here, A T denotes the transpose of the matrix A. The linear system (1.1) appears in many scientific and engineering problems. In particular, such a system appears in deconvolution problems with a smooth kernel. Singular linear systems also appear in finite difference representation of Neumann problems (see [22] ). Berman and Plemmons [4] introduced the following iterative scheme

x kþ1 ¼ U y Vx k þ U y b; k ¼ 0; 1; 2; . . .; ð1:2Þ

to find an approximate solution of the system (1.1) using a proper splitting A ¼ U À V, where U y denotes the Moore-Penrose inverse of U. A splitting A ¼ U À V of A 2 R mÂn is called a proper splitting [4] if RðUÞ ¼ RðAÞ and NðUÞ ¼ NðAÞ, where R(B) and N(B) denote the range space and the null space of a matrix B, respectively. The authors of [4] showed that the iterative scheme (1.2) converges to A y b, the least-squares solution of minimum norm of (1.1) for every initial vector x o if and only if qðU y VÞ\1 (see Corollary 1, [4] ). Here, qðBÞ denotes the spectral radius of a matrix B. Later on, many authors introduced different sub-classes of proper splittings and studied their convergence theory (see [9, 19] ). However, the classical iterative methods are computationally expensive, which attracts the researcher to develop fast iterative solvers. In this context, Nichols [21] proposed the notion of the twostage iterative method to find an iterative solution of a nonsingular linear system. Lanzkron et al. [15] and, Frommer and Szyld [11] then studied further its convergence theory. The convergence of two-stage iterative scheme in case of positive definite linear systems is studied extensively in [6, 8, 16, 17] , while in [7] and [26] , the authors considered the singular linear systems. Climent and Perea [10] generalized the notion of the two-stage iterative technique to rectangular matrix case that is recalled next. Let us consider an iterative scheme of the form

where A ¼ U À V is a proper splitting of A for solving a consistent linear system of the form (1.1). The scheme (1.3) is called outer iteration. At each step of (1.3), we must solve the inner equations Uy ¼b;b ¼ Vx k þ b: ð1:4Þ

Solving (1.4) by another iterative scheme (called inner iteration) which is formed by using a proper splitting U ¼ F À G that performs s(k) inner iterations, we get the two-stage iterative scheme [10] , i.e.,

x kþ1 ¼ ðF y GÞ sðkÞ x k þ X sðkÞÀ1 j¼0 ðF y GÞ j F y ðVx k þ bÞ; k ¼ 1; 2; . . .; ð1:5Þ

where s(k) is the number of inner iteration which depends on k. We say that a twostage iterative scheme (1.5) is stationary when sðkÞ ¼ s for all k, while it is called non-stationary when s(k) changes with k. If A ¼ U À V is a proper regular splitting and U ¼ F À G is a proper weak regular splitting of type I (see Definition 2.4), the authors of [10] established that the iterative scheme (1.5) converges to A y b, the minimum norm least-squares solution of system (1.1) for any initial vector x o . In this article, we further establish the proper regularity of the induced splitting induced from the two-stage iterative scheme (1.5) and obtain various comparison results for the faster convergence of the two-stage iterative scheme. Motivation of the current work comes from the results of Bai and Wang [1] , and Wang and Zhao [25] . Although most of our results in this paper are for the rectangular linear system, many results are even new in the nonsingular matrix setting and are stated as corollaries.

The structure of this paper is as follows. In Sect. 2, we introduce some notations and definitions which are helpful in proving the main results. The uniqueness of a proper splitting for a certain class of real rectangular matrices is also proved in Sect. 2. After revisiting the two-stage iterative scheme and its convergence criteria established in [10] , we analyze the proper regularity of the induced splitting by the two-stage iterative scheme and derive a few comparison theorems of different splittings for two-stage iterative schemes in Sect. 3. In Sect. 4, we set up the monotone convergence theorem for a stationary two-stage iterative scheme.

Throughout this paper, all matrices considered are real, and R mÂn denotes the set of all real m  n matrices. A matrix A 2 R mÂn is called nonnegative(positive), if each entry of A is nonnegative(positive), and is denoted by

The spectral radius of a matrix A 2 R nÂn is equal to the maximum of the moduli of the eigenvalues of A. Next, we recall a few results that deal with nonnegativity of a matrix and the spectral radius. If B 2 R nÂn with qðBÞ\1; then I À B is nonsingular, and ðI À BÞ À1 ¼ I þ B þ B 2 þ Á Á Á, the series on the right converging. Conversely, if the series on the right converges, then qðBÞ\1:

A nonempty subset K of R n is called a convex cone if k ! 0 implies kK K and K þ K K, pointed if K \ ÀK 6 ¼ /, and solid if interior of K is nonempty. A pointed, solid, closed convex cone is called a full cone. The next result is about a full cone in R n .

Let K be a full cone in R n and fs i g 1 i¼0 be a K-monotone non-decreasing sequence. Let t 2 R n be such that t À s i 2 K for every positive integer i. Then the sequence fs i g 1 i¼0 converges. 

and is denoted by A y . It always exists. When A is nonsingular, then the Moore-Penrose inverse of A coincides with the usual matrix inverse, i.e., A y ¼ A À1 . A matrix A 2 R mÂn is called semimonotone if A y ! 0 (see [5] ). A few properties of A y which will be frequently used in this paper are: RðA y Þ ¼ RðA T Þ; NðA y Þ ¼ NðA T Þ; AA y ¼ P RðAÞ and A y A ¼ P RðA T Þ (see [2] for more details). Further, if A and B are such that the matrix product AB is defined, then ðABÞ y ¼ B y A y if and only if A y ABB T A T ¼ BB T A T and BB y A T AB ¼ A T AB (see [13] ). The following theorem that is a part of Theorem 1, [4] collects some properties of a proper splitting.

Let A ¼ U À V be a proper splitting of A 2 R mÂn . Then

In 2018, Mishra and Mishra [20] proved the uniqueness of proper splitting for a given real square matrix (see Theorem 2.2, [20] ) under the assumption of a few conditions. We next extend the same result to real rectangular matrices. Theorem 2.5 Let A 2 R mÂn : If there is a real n  n matrix H such that RðHÞ RðA T Þ; NðAÞ NðHÞ and qðHÞ\1, then A ¼ U À V is a unique proper splitting and H ¼ U y V:

Again, pre-multiplying AðI À HÞ to the above expression, we get Ax ¼ 0; which implies NðUÞ NðAÞ: Let us now consider x 2 NðAÞ: Then 0 ¼ Ax ¼ UðI À HÞx ¼ Ux À UHx ¼ Ux: Hence, NðUÞ ¼ NðAÞ: Next, we prove U y ¼ ðI À HÞA y : To do this, we need to verify that A y AðI À HÞ À1 ððI À HÞ À1 Þ T A T ¼ ðI À HÞ À1 ððI À HÞ À1 Þ T A T and ðI À HÞ À1 ðI À HÞA T AðI À HÞ À1 ¼ A T AðI À HÞ À1 . Clearly, ðI À HÞ À1 ðI À HÞA T AðI À HÞ À1 ¼ A T AðI À HÞ À1 . Using the fact that A y AðI À HÞ À1 ¼ ðI À HÞ À1 A y A and ðA y AÞ T ¼ A y A, we have A y AðI À HÞ À1 ððI À HÞ À1 Þ T A T ¼ ðI À HÞ À1 A y AððI À HÞ À1 Þ T A T ¼ ðI À HÞ À1 ðAðI À HÞ À1 A y AÞ T ¼ ðI À HÞ À1 ðAA y AðI À HÞ À1 Þ T ¼ ðI À HÞ À1 ðAðI À HÞ À1 Þ T ¼ ðI À HÞ À1 ððI À HÞ À1 Þ T A T :

We refer the reader to [3, 19] for different methods of construction of a proper splitting. We next recall the definitions of two sub-classes of proper splittings.

The next result combines the convergence criteria for both the above classes of proper splittings (see Theorem 1.3, [14] and Corollary 4, [4] ).

Theorem 2.7 Let A ¼ U À V be any of the above class of splittings of A 2 R mÂn : Then A is semimonotone if and only if qðU y VÞ\1:

A comparison of the spectral radii of two iterative schemes arising from two proper splittings is useful for improving the speed of convergence of the iteration scheme (1.2) . In this direction, several comparison results have been introduced in the literature both for rectangular and nonsingular matrices. Recently, Mishra and Mishra [20] proved the following comparison result.

: We now recall another comparison result proved by Mishra [18] .

be two proper weak regular splittings of a semimonotone matrix A 2 R mÂn . If either of the following cases holds,

In this section, we present some sufficient conditions for the convergence of two stage-iterative schemes and then obtain comparison of the spectral radii of two different iteration matrices arising out of two matrix splittings which is useful for improving the speed of the two-stage iterative scheme. For convenience, let us rewrite the two-stage iterative scheme introduced by Climent and Perea [10] as

ðF y GÞ j F y ðVx k þ bÞ; k ¼ 1; 2; . . .: ð3:1Þ

For a given initial vector x o , the two-stage iterative scheme (3.1) produces the sequence of vectors

ðF y GÞ j F y V and P y sðkÞ ¼ X sðkÞÀ1 j¼0 ðF y GÞ j F y : ð3:3Þ

For stationary two-stage iteration scheme, Climent and Perea [10] proved the following convergence result. For non-stationary two-stage scheme, we have the following result that generalizes Theorem 4.4, [11] . The proof is similar to Theorem 3, [10] , therefore we omit it.

Theorem 3.2 Let A ¼ U À V be a proper regular splitting of a semimonotone matrix A 2 R mÂn . Let U ¼ F À G be a proper weak regular splitting. Then the nonstationary two-stage iterative method (3.1) is convergent for any sequence sðkÞ ! 1; k ¼ 0; 1; 2; . . ..

The next result shows that the iteration matrix T sðkÞ induces a unique proper weak regular splitting A ¼ P sðkÞ À P sðkÞ T sðkÞ under a few sufficient conditions. Theorem 3.3 Let A ¼ U À V be a proper regular splitting of a semimonotone matrix A 2 R mÂn . Let U ¼ F À G be a proper weak regular splitting. Then the unique splitting A ¼ P sðkÞ À P sðkÞ T sðkÞ induced by the iteration matrix T sðkÞ is a proper weak regular splitting.

ðF y GÞ j F y V ¼ I À ðI À ðF y GÞ sðkÞ ÞðI À U y VÞ and P y sðkÞ ¼ X sðkÞÀ1 j¼0 ðF y GÞ j F y ¼ ðI À ðF y GÞ sðkÞ ÞðI À F y GÞ À1 F y ¼ ðI À ðF y GÞ sðkÞ ÞU y :

Let X ¼ UðI À ðF y GÞ sðkÞ Þ À1 : Then XP y sðkÞ ¼ UU y and P y sðkÞ X ¼ U y U. So, XP y sðkÞ and P y sðkÞ X are symmetric. Now, P y sðkÞ XP y sðkÞ ¼ P y sðkÞ and XP y sðkÞ X ¼ X: Hence X ¼ ðP y sðkÞ Þ y ¼ P sðkÞ ¼ UðI À ðF y GÞ sðkÞ Þ À1 . Clearly, RðP sðkÞ Þ ¼ RðUÞ. It remains to show that NðP sðkÞ Þ ¼ NðUÞ: We can see that P sðkÞ x ¼ 0 ,UðI À ðF y GÞ sðkÞ Þ À1 x ¼ 0 ,ðI À ðF y GÞ sðkÞ ÞU y UðI À ðF y GÞ sðkÞ Þ À1 x ¼ 0 ,U y UðI À ðF y GÞ sðkÞ ÞðI À ðF y GÞ sðkÞ Þ À1 x ¼ 0

Hence NðP sðkÞ Þ ¼ NðUÞ. Now, we have P y sðkÞ P sðkÞ T sðkÞ ¼ P y sðkÞ P sðkÞ ðF y GÞ sðkÞ þ P y sðkÞ P sðkÞ ð X sðkÞÀ1 j¼0 ðF y GÞ j F y VÞ

Then, it holds that P sðkÞ À P sðkÞ T sðkÞ ¼ P sðkÞ ðI À T sðkÞ Þ ¼ UðI À ðF y GÞ sðkÞ Þ À1 ðI À ðF y GÞ sðkÞ ÞðI À U y VÞ ¼ UðI À U y VÞ ¼ A by Theorem 2.4. So P y sðkÞ ¼ P sðkÞÀ1 j¼0 ðF y GÞ j F y ! 0 and P y sðkÞ P sðkÞ T sðkÞ ! 0 implies that the induced splitting A ¼ P sðkÞ À P sðkÞ T sðkÞ is a proper weak regular splitting.

For uniqueness, suppose that T sðkÞ induces another splitting A ¼ R sðkÞ À R sðkÞ T sðkÞ :

Then R sðkÞ ¼ AðI À T sðkÞ Þ À1 ¼ P sðkÞ . Hence, the proper splitting A ¼ P sðkÞ À P sðkÞ T sðkÞ induced by the iteration matrix T sðkÞ is unique. h

We provide below sufficient conditions which guarantee that the induced splitting from the two-stage iterative scheme is a proper regular one.

Theorem 3.4 Let A ¼ U À V be a proper regular splitting of a semimonotone matrix A 2 R mÂn . Let U ¼ F À G be a proper regular splitting. If ðF y GÞ 2 F y G; then the two-stage splitting A ¼ P sðkÞ À P sðkÞ T sðkÞ is a proper regular splitting.

Proof We have P sðkÞ T sðkÞ ¼ P sðkÞ À A ¼ UðI À ðF y GÞ sðkÞ Þ À1 À U þ V ¼ UððI À ðF y GÞ sðkÞ Þ À1 À IÞ þ V ¼ UðF y GÞ sðkÞ ðI À ðF y GÞ sðkÞ Þ À1 þ V:

Now, we only need to show that UðF y GÞ sðkÞ ! 0: It is enough to prove that UðF y GÞ 2 ! 0:

As F y G ! 0 and UðF y GÞ 2 ! 0, thus UðF y GÞ sðkÞ ! 0: Hence P sðkÞ T sðkÞ ! 0: h

The example given below demonstrates the above theorem. The next result is motivated by Corollary 2.7, [25] which says that the conclusion of Theorem 3.4 can also be achieved if we replace the condition ðF y GÞ 2 F y G by GF y G G.

Theorem 3.6 Let A ¼ U À V be a proper regular splitting of a semimonotone matrix A 2 R mÂn . Let U ¼ F À G be a proper regular splitting. If GF y G G; then the two-stage splitting A ¼ P sðkÞ À P sðkÞ T sðkÞ is a proper regular splitting.

Proof By pre-multiplying F y to the given condition GF y G G; we get ðF y GÞ 2 F y G: By Theorem 3.4, we thus have P sðkÞ T sðkÞ ! 0: h If we replace the condition ðF y GÞ 2 F y G by F y UF y ! 0 in Theorem 3.4, then the induced splitting is proper regular. This is shown next which extends Theorem 2.3, [25] to rectangular matrix case.

Theorem 3.7 Let A ¼ U À V be a proper regular splitting of a semimonotone matrix A 2 R mÂn . Let U ¼ F À G be a proper regular splitting. If F y UF y ! 0; then the two-stage splitting A ¼ P sðkÞ À P sðkÞ T sðkÞ is a proper regular splitting.

Proof We have 0 F y UF y ¼ ðI À F y GÞU y UF y ¼ ðI À F y GÞF y ¼ F y À F y GF y ; i.e., F y GF y F y : By post-multiplying G to the expression F y GF y F y ; we obtain ðF y GÞ 2 F y G: We thus have P sðkÞ T sðkÞ ! 0; by Theorem 3.4. h

The problem of comparison of the convergence rate of two different two-stage iterative schemes is not studied much in the literature even for the nonsingular matrix case. One of the main objectives of this paper is to study this problem. The next three results are in this direction. The different two-stage iterative schemes are framed by taking two different proper splittings U ¼

corresponding iteration matrices as T sðkÞ ¼ ðF y GÞ sðkÞ þ X sðkÞÀ1 j¼0 ðF y GÞ j F y V and b

GÞ j F y V and with the same number of inner iterations s(k).

Theorem 3.8 Let A ¼ U À V be a proper regular splitting of a semimonotone matrix A 2 R mÂn :

G be proper weak regular splittings of U. If any one of the following conditions hold:

G and P sðkÞ T sðkÞ ! 0, (iii) P sðkÞ T sðkÞ b P sðkÞ b T sðkÞ , then qðT sðkÞ Þ qð b T sðkÞ Þ\1.

(i) By Theorem 3.2, we have qðT sðkÞ Þ\1 and qð b T sðkÞ Þ\1. By Theorem 3.3, the induced splittings A ¼ P sðkÞ À P sðkÞ T sðkÞ and A ¼ b P sðkÞ À b P sðkÞ b T sðkÞ are proper weak regular splittings. The condition b P sðkÞ b T sðkÞ ! 0 yields that A ¼ b P sðkÞ À b P sðkÞ b T sðkÞ is a proper regular splitting. Since 0 F y G b F y b G, we have ðF y GÞ sðkÞ ð b F y b GÞ sðkÞ which implies ½I À ðF y GÞ sðkÞ U y ! ½I À ð b F y b GÞ sðkÞ U y yielding P y sðkÞ ! b P y sðkÞ . By Theorem 2.8, we therefore obtain qðT sðkÞ Þ qð b T sðkÞ Þ.

(ii) Using F y G b F y b G, we get P y sðkÞ ! b P y sðkÞ . Now, P sðkÞ T sðkÞ ! 0 and P y sðkÞ ! b P y sðkÞ imply qðT sðkÞ Þ qð b T sðkÞ Þ by Theorem 2.9(ii).

(iii) Applying Theorem 2.9(i) to the induced weak regular splittings A ¼ P sðkÞ À P sðkÞ T sðkÞ and A ¼ b P sðkÞ À b P sðkÞ b T sðkÞ , we directly obtain qðT sðkÞ Þ qð b T sðkÞ Þ.

h

The following example illustrates the above theorem. The nonsingular case is obtained next.

G be weak regular splittings of U. If any one of the following conditions hold:

G and P sðkÞ T sðkÞ ! 0, (iii) P sðkÞ T sðkÞ b P sðkÞ b T sðkÞ , then qðT sðkÞ Þ qð b T sðkÞ Þ\1.

The next result shows that the sufficient conditions in Theorem 3.8 can be reduced if we further assume that U is nonnegative.

Proof By Theorem 3.2, we have qðT sðkÞ Þ\1 and qð b T sðkÞ Þ\1. Now, using Theorem 3.3, the induced splittings A ¼ P sðkÞ À P sðkÞ T sðkÞ and A ¼ b P sðkÞ À b P sðkÞ b T sðkÞ become proper weak regular. Since qðF y GÞ\1 and qð b F y b GÞ\1 as U is semimonotone, the condition F y G b F y b G implies that 0 ½I À ðF y GÞ sðkÞ À1 ½I À ð b F y b GÞ sðkÞ À1 which further yields P sðkÞ T sðkÞ ¼ U½I À ðF y GÞ sðkÞ À1 À A U½I À ð b F y b GÞ sðkÞ À1 À A ¼ b P sðkÞ b T sðkÞ as U is nonnegative. Hence, applying Theorem 2.9 to the splittings A ¼ P sðkÞ À P sðkÞ T sðkÞ and A ¼ b P sðkÞ À b P sðkÞ b T sðkÞ , we get qðT sðkÞ Þ qð b T sðkÞ Þ. h

The above result reduces to the following Corollary in the nonsingular matrix setting.

We next provide an example that demonstrates the above result. Let the square region R ¼ ðx; yÞ : 0 x 1; 0 y 1 f g be covered by a grid with sides parallel to the coordinate axis and with equal grid spacing h ¼ Dx ¼ Dy. If Nh ¼ 1, then the number of internal grid points is ðN À 1Þ 2 . When N ¼ 6, the finite difference method using the Oðh 2 Þ central difference discretization on uniform grids generates the linear system Ax ¼ b, where A is of order 25 and the right hand side vector b ¼ , where I is a 5 Â 5 identity matrix and Setting U ¼ diagðAÞ, we get a regular splitting A ¼ U À V of the monotone matrix A. Further, if F ¼ 1:5U and b F ¼ 1:75U, then U ¼ F À G and U ¼ b

are two weak regular splittings of the nonnegative matrix U. Clearly, F À1 G ¼ diagð0:3333; 0:3333; Á Á Á ; 0:3333Þ b F À1 b G ¼ diagð0:4286; 0:4286; Á Á Á ; 0:4286Þ. Therefore, for sðkÞ ¼ 2, we have qðT sðkÞ Þ ¼ 0:8809 0:8906 ¼ qð b

T sðkÞ Þ\1.

The nonnegative restriction on the matrix U in Theorem 3.11 can also be dropped

and is proved next.

Theorem 3.14 Let A ¼ U À V be a proper regular splitting of a semimonotone matrix A 2 R mÂn :

Proof By Theorem 3.4 and the condition ðF y GÞ 2 F y G, the induced splittings A ¼ P sðkÞ À P sðkÞ T sðkÞ and A ¼ b P sðkÞ À b P sðkÞ b T sðkÞ are proper regular and proper weak regular, respectively. Using the inequality F y G b F y b G, we get P y sðkÞ ! b P y sðkÞ . Thus qðT sðkÞ Þ qð b T sðkÞ Þ by Theorem 2.8. h

As a corollary, we have the following result which is new even in the nonsingular matrix setting.

The next example illustrates the above result.

A ¼ 20 À 0:967 À 1:109 À 0:6948 À 0:2824 À 0:0357 À 1:7314 À 1:654 À 2:4631 À 3:1682 À1:6816 20 À 1:1131 À 0:1384 À 1:1361 À 0:3131 À 1:3545 À 0:1639 À 0:7782 À 0:7888 À0:2902 À 0:3625 20 À 0:7711 À 0:3427 À 0:1397 À 0:6844 À 0:4231 À 0:1587 À 0:1993 À2:421 À 0:4117 À 0:6286 20 À 2:0433 À 0:528 À 0:9069 À 0:449 À 0:9369 À 0:5398 À1:3445 À 1:2228 À 0:0544 À 0:0637 20 À 0:7927 À 0:2768 À 0:1316 À 0:1644 À 0:2638 À1:7024 À 0:0895 À 0:4184 À 0:9671 À 1:3149 20 À 0:6647 À 0:7289 À 0:4031 À 0:72 À1:3714 À 1:6605 À 0:0633 À 0:2552 À 0:0939 À 0:0451 20 À 2:8123 À 2:166 À 3:3831 À1:1986 À 0:4811 À 0:918 À 0:1832 À 0:5206 À 0:1071 À 1:204 20 À 0:6027 À 0:4507 we have two weak regular splittings U ¼

It is easy to observe that U, F and b F are M-matrices and computed as 

Therefore, for sðkÞ ¼ 2, qðT sðkÞ Þ ¼ 0:2135 0:2372 ¼ qð b T sðkÞ Þ\1.

We conclude this section with another comparison result that partially extends Theorem 3.3, [1] to the rectangular case.

Theorem 3.17 Let A ¼ U À V be a proper regular splitting of a semimonotone matrix A 2 R mÂn :

G be proper weak regular splittings of U. Consider the iterative scheme defined by (3.1) with corresponding iteration matrices T sðkÞ and b T qðkÞ : If the following conditions hold:

(i) b P qðkÞ b T qðkÞ ! 0, k ¼ 0; 1; 2; . . .,

(ii) b F y F y and qðkÞ ! sðkÞ, k ¼ 0; 1; 2; . . .,

Proof Since 0 F y GU y ¼ F y ðF À UÞU y ¼ U y À F y , utilizing condition (ii), we get U y À F y U y À b F y . Also, using condition (iii), we observe that Uð b

we will use induction to show that ðF y GÞ k U y ð b F y b GÞ k U y , k ¼ 0; 1; 2; . . .. For k ¼ 0, the inequality ðF y GÞ k U y ð b F y b GÞ k U y is trivial. Suppose that for k ¼ 0; 1; . . .; p, the inequality holds. Then, for k ¼ p þ 1; we have

Thus ðF y GÞ k U y ð b F y b GÞ k U y holds for all k ¼ 0; 1; 2; . . . which implies that U y À ðF y GÞ k U y ! U y À ð b F y b GÞ k U y for all k ¼ 0; 1; 2; . . ., i.e., P y sðkÞ ! b P y qðkÞ , k ¼ 0; 1; 2; . . .. By Theorem 2.8, we thus have qðT sðkÞ Þ qð b T qðkÞ Þ. h Our next example supports the above theorem. 

In this section, we discuss the monotone convergence theory of the stationary twostage iterative method (3.1) that generalizes Theorem 3.1, [1] .

Theorem 4.1 (Monotone Convergence Theorem) Let A ¼ U À V be a proper regular splitting of a semimonotone matrix A 2 R mÂn . Let U ¼ F À G be a proper weak regular splitting and sðkÞ ! 1, k ¼ 0; 1; 2. . . be the inner iteration sequence. If x 0 and y 0 are initial values that hold

x 0 x 1 ; y 1 y 0 and x 0 A y b y 0 ; ð4:1Þ then the sequences fx k g and fy k g generated by

x kþ1 ¼ T sðkÞ x k þ P y sðkÞ b; y kþ1 ¼ T sðkÞ y k þ P y sðkÞ b;

k ¼ 0; 1; 2; . . . satisfy (i) x k x kþ1 A y b y kþ1 y k ; k ¼ 0; 1; 2; . . ., (ii) lim k!1

(iii) Further, if qðT sðkÞ Þ\1; there exist x 0 and y 0 which satisfy (4.1).

(i) We will show by induction that x k x kþ1 for k ¼ 0; 1; 2; . . .. The case k ¼ 0 is established by the hypothesis. Assume that the result holds for k ¼ 0; 1. . .p [ 0 so that x pþ1 À x p ! 0. Since s(k) is independent of k and T sðkÞ ! 0 for k ¼ 0; 1; . . ., we have T sðpÞ ðx pþ1 À x p Þ ¼ ðT sðpÞ x pþ1 þ X sðpÞÀ1 j¼0 ðF y GÞ j F y bÞ À ðT sðpÞ x p þ X sðpÞÀ1 j¼0 ðF y GÞ j F y bÞ ¼ ðT sðpþ1Þ x pþ1 þ X sðpþ1ÞÀ1 j¼0 ðF y GÞ j F y bÞ À ðT sðpÞ x p þ X sðpÞÀ1 j¼0 ðF y GÞ j F y bÞ ¼ x pþ2 À x pþ1 ! 0:

It follows by induction that x p x pþ1 for each p. Similarly, we can show that y pþ1 y p for each p.

(ii) Applying Lemma 2.3 by considering K ¼ R n þ , the nonnegative orthant of R n , the sequences fx k g and fy k g converge to ðI À T sðkÞ Þ À1 P y sðkÞ b, i.e., A y b. (iii) As T sðkÞ ! 0; there exists x ! 0 such that T sðkÞ x ¼ qðT sðkÞ Þx by Theorem 2.1.

Since qðT sðkÞ Þ\1, we have qðT sðkÞ Þx x: Let x 0 ¼ A y b À x: Then

ðF y GÞ j F y b À T sðkÞ x ¼ ½I À ðI À ðF y GÞ sðkÞ ÞðI À U y VÞA y b þ X sðkÞÀ1 j¼0 ðF y GÞ j F y b À qðT sðkÞ Þx ¼ ½I À ðI À ðF y GÞ sðkÞ ÞðI À U y VÞðI À U y VÞ À1 U y b þ X sðkÞÀ1 j¼0 ðF y GÞ j F y b À qðT sðkÞ Þx ¼ ðI À U y VÞ À1 U y b À ðI À ðF y GÞ sðkÞ ÞU y b þ X sðkÞÀ1 j¼0 ðF y GÞ j F y b À qðT sðkÞ Þx ¼ ðI À U y VÞ À1 U y b À X sðkÞÀ1 j¼0 ðF y GÞ j F y b þ X sðkÞÀ1 j¼0 ðF y GÞ j F y b À qðT sðkÞ Þx ¼ A y b À qðT sðkÞ Þx:

Similarly, assuming y 0 ¼ A y b þ x; and proceeding as above, we get y 1 y 0 . Moreover, y 0 À x 0 ¼ 2x ! 0 which implies y 0 ! x 0 .

h Note that condition (iii) in Theorem 4.1 guarantees the existence of x 0 and y 0 which satisfy the inequality (4.1). We conclude the paper with the remark that the convergence theory of two-stage iteration scheme for type II splittings even in the nonsingular matrix setting is not yet established in the literature. Very recently, Shekhar et. al. [23] introduced convergence theory of type II double splittings. Motivated by the same work, we plan to study the problem of the convergence of the two-stage iteration scheme for type II splittings near future.

Funding This study was partially funded by Science and Engineering Research Board

The monotone convergence of the two-stage iterative method for solving large sparse systems of linear equations

Generalized Inverses. Theory and Applications

Proper splittings of rectangular matrices

Cones and iterative methods for best least squares solutions of linear systems

Nonnegative matrices in the mathematical sciences

Convergence of block two-stage iterative methods for symmetric positive definite systems

Convergence of two-stage iterative methods for singular symmetric positive semidefinite systems

On the convergence of nonstationary iterative methods for symmetric positive (semi)definite systems

Convergence results for proper splittings

Iterative methods for least-square problems based on proper splittings

H-splitting and two-stage iterative methods

Numerical methods for solving linear least squares problem

Note on the generalized inverse of a matrix product

Convergence and comparison theorems for single and double decomposition of rectangular matrices

Convergence of nested classical iterative methods for linear systems

The two-stage iterative methods for symmetric positive definite matrices

The monotonicity of two-stage iterative scheme

Further study of alternating iterations for rectangular matrices

On splitting of matrices and nonnegative generalized inverses

Two-stage iterations based on composite splittings for rectangular linear systems

On the convergence of two-stage iterative processes for solving linear equations

Regular splittings and the discrete Neumann problem

A note on double weak splittings of type II. Linear Multilinear Algebra

Matrix Iterative Analysis

Further results on regular splittings and multisplittings

Semiconvergence of two-stage iterative methods for singular linear systems

Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations

Acknowledgements We thank the referee for the constructive and insightful comments, which have helped us to substantially improve our manuscript. We thank the Government of India for introducing the work from home initiative during the COVID-19 pandemic.

U ¼ 20 À 0:967 À 1:109 À 0:6948 À 0:2824 À 0:0357 À 1:7314 À 1:654 À 2:4631 À 3:1682 0 2 0 À 1:1131 À 0:1384 À 1:1361 À 0:3131 À 1:3545 À 0:1639 À 0:7782 À 0:7888 0 0 2 0 : À 0:7711 À 0:3427 À 0:1397 À 0:6844 À 0:4231 À 0:1587 À 0:1993 0 0 0 2 0 À 2:0433 À 0:528 À 0:9069 À 0:449 À 0:9369 À 0:5398 Compliance with ethical standardsConflicts of interest The authors declare that they have no conflict of interest.Human participants This article does not contain any studies with human participants or animals performed by the authors.