# Mixed Simultaneous Iterative Algorithms for the Extended Multiple-Set Split Equality Common Fixed-Point Problem with Lipschitz Quasi-Pseudocontractive Operators

Authors: Meixia Li; Xueling Zhou; Haitao Che
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper, we study a kind of extended multiple-set split equality common fixed-point problem with Lipschitz quasi-pseudocontractive operators, which is an extension of multiple-set split equality common fixed-point problem with quasi-nonexpansive operator. We propose two mixed simultaneous iterative algorithms, in which the selecting of the stepsize does not need any priori information about the operator norms. Furthermore, we prove that the sequences generated by the mixed simultaneous iterative algorithms converge weakly to the solution of this problem. Some numerical results are shown to illustrate the feasibility and efficiency of the proposed algorithms.


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## Body

## 1. Introduction

Throughout this paper, we always assume thatH1, H2, and H3 are real Hilbert spaces with inner product 〈·,·〉 and induced norm ∥·∥. Let C and Q be nonempty closed and convex subsets of H1 and H2, respectively. A:H1→H2 is a bounded linear operators. Let I denote the identity operator. The split feasibility problem (SFP) originally introduced by Censor and Elfving [1] which is to find(1)x∈CsuchthatAx∈Q.It can be used in various disciplines such as image restoration, radiation therapy treatment planning [2, 3]. These applications are in finite-dimensional Hilbert spaces [4–7]. It also can be found in an infinite-dimensional real Hilbert space [8–10] and references therein.In [11], Censor et al. introduce the multiple-set split feasibility problem (MSFP) which finds application in intensity-modulated radiation therapy. It is formulated as finding a point x satisfying the following property:(2)x∈⋂i=1pCisuchthatAx∈⋂j=1rQj,where p,r≥1 are integers and {Ci}i=1p and {Qj}j=1r are nonempty closed convex subsets of H1 and H2, respectively. When p=r=1, then MSFP (2) is known as SFP (1). About the corresponding results in finite-dimensional spaces, we can see [12–14] and references therein.Since every closed convex subset of a Hilbert space is the fixed-point set of its associating projection, problems (1) and (2) are all special cases of the so-called multiple-set split common fixed-point problem (MSCFP) which is to find(3)x∈⋂i=1pFixSisuchthatAx∈⋂j=1rFixTj,where p,r≥1 are integers, {Si}i=1p:H1→H1 and {Tj}j=1r:H2→H2 are nonlinear operators. In particular, if p=r=1, then MSCFP (3) reduces to the split common fixed-point problem (SCFP) [15–17] which is to find(4)x∈FixSsuchthatAx∈FixT,where S:H1→H1 and T:H2→H2 are nonlinear operators.Recently, Moudafi [18] introduced the split equality common fixed-point problem (SECFP) which is to find(5)x∈FixS,y∈FixTsuchthatAx=By,where A:H1→H3 and B:H2→H3 are two bounded linear operators. It allows asymmetric and partial relations between the variables x and y. The interest is to cover many situations, for instance, applications in decomposition methods for PDEs, in game theory, and in intensity-modulated radiation therapy (IMRT). At the same time, Moudafi [18] introduced the following iterative method(6)xn+1=Sxn-γnA∗Axn-Byn,yn+1=Tyn+γnB∗Axn+1-Bynfor firmly quasi-nonexpansive mappings S and T. The weak convergence property was proved where γn was dependent in the spectral radiuses of A∗A and B∗B. Furthermore, Che and Li [19] proposed the following iterative algorithm:(7)un=xn-γnA∗Axn-Byn,xn+1=αnxn+1-αnSun,vn=yn+γnB∗Axn-Byn,yn+1=αnyn+1-αnTvnfor quasi-nonexpansive mappings S and T and the weak convergence of scheme (7) can also be established. Chang, Wang, and Qin [20] modified the iterative scheme (7) and provided the following framework:(8)un=xn-γnA∗Axn-Byn,xn+1=αnxn+1-αn1-ξI+ξS1-ηI+ηSun,vn=yn+γnB∗Axn-Byn,yn+1=αnyn+1-αn1-ξI+ξT1-ηI+ηTvnfor quasi-pseudocontractive mappings S and T with weak convergence property.In [21], Zhao et al. study the multiple-set split equality common fixed-point problem (MSECFP) which is to find(9)x∈⋂i=1pFixSi,y∈⋂j=1rFixTjsuchthatAx=By,where p,r≥1 are integers and {Si}i=1p:H1→H1 and {Tj}j=1r:H2→H2 are nonlinear operators. In particular, if p=r=1, then MSECFP (9) reduces to the split equality common fixed-point problem (SECFP) (5). They proposed the following iterative algorithm:(10)un=xn-γnA∗Axn-Byn,xn+1=αn0xn+αn1S1xn+⋯+αnpSpxnvn=yn+γnB∗Axn-Byn,yn+1=βnvn+1-βnTjnvnfor quasi-nonexpansive mappings, where {αni}⊂[0,1], i=0,1,2,…,p; n=1,2,… such that ∑i=0pαni=1 for every n=1,2,… and {βn}⊂[0,1]. j(n)=n(modr)+1. Weak convergence results were obtained under certain assumption conditions.Recently, Li et al. [22] studied the extended split equality common fixed-point problem (ESECFP) which is to find(11)x∈FixS,y∈FixTsuchthatAx=By∈K,where K⊂H3 is a nonempty set. For L-Lipschitz and quasi-pseudocontractive mappings S and T, as well as a kind of special K={x∈H3:h(x)≤0}, they proposed the following iterative algorithm:(12)ωn=PKnAxn+Byn2,Kn=z∈H3:hAxn+Byn2+ςn,z-Axn+Byn2≤0,un=xn-ξnA∗Axn-ωn,xn+1=αnxn+1-αnUun,vn=yn-ξnB∗Byn-ωn,yn+1=αnyn+1-αnVvn,where U≔(1-ξ)I+ξS((1-η)I+ηS), V≔(1-ξ)I+ξT((1-η)I+ηT), 0<ξ<η<1/1+1+L2, and ςn∈∂h(Axn+Byn/2). They also proved the weak convergence results under mild conditions.Inspired by the above results, in this paper, we study the following extended multiple-set split equality common fixed-point problem (EMSECFP) which is to find(13)x∈⋂i=1pFixSi,y∈⋂j=1rFixTjsuchthatAx=By∈K,where p,r≥1 are integers and Si:H1→H1(i=1,2,…,p) and Tj:H2→H2(j=1,2,…r) are L-Lipschitzian mappings with L≥1 and quasi-pseudocontractive mappings with nonempty fixed-point sets which is more general than the class of quasi-nonexpansive mappings. When K=H3, the EMSECFP (13) is MSECFP (9). When p=r=1, the EMSECFP (13) is ESECFP (11). Therefore, EMSECFP (13) is the extension of MSECFP (9) and ESECFP (11). In this paper, we propose two mixed simultaneous iterative algorithms for solving EMSECFP (13), which avoids using the projection and the selecting of stepsize which do not depend on the operator norms. Furthermore, we prove that the sequences generated by the algorithms weakly converge to a solution of the EMSECFP (13). Numerical examples show the feasibility and efficiency of these algorithms.


## 2. Preliminaries

In this paper, we usexn⇀x and xn→x to indicate that the sequence {xn} converges weakly to x and {xn} converges strongly to x, respectively. We write ωw(xk)={x:∃xkj⇀x} to stand for the weak ω-limit set of {xk} and use Ω to stand for the solution set of the EMSECFP (13).Definition 1.
A mappingT:H→H is called
(i) Nonexpansive ifTx-Ty≤x-y for all x,y∈H.
(ii) Quasi-nonexpansive, ifFix(T)≠∅ and Tx-x∗≤x-x∗ for all x∈H and x∗∈Fix(T).
(iii) Quasi-pseudocontractive, ifFix(T)≠∅ and Tx-x∗2≤x-x∗2+Tx-x2 for all x∈H and x∗∈Fix(T).Definition 2.
An operatorT:H→H is called demiclosed at the origin, for any sequence {xn} which weakly converges to x, and if the sequence {Txn} strongly converges to 0, then Tx=0.In the proof of our results, we need the following lemmas.Lemma 3 (see [23]).
LetH be a real Hilbert space, then the following conclusions hold:(14)x-y2=x2+y2-2x,y,forallx,y∈H,(15)αx+1-αy2=αx2+1-αy2-α1-αx-y,forallx,y∈H,α∈0,1.Lemma 4 (see [20]).
LetH be a real Hilbert space and T:H→H be a L-Lipschitzian mapping with L≥1. Denote(16)U≔1-ξI+ξT1-ηI+ηT.If 0<ξ<η<1/1+1+L2, then the following conclusions hold:
(i)Fix(T)=Fix(T((1-η)I+ηT))=Fix(U).
(ii) IfT is demiclosed at 0, then U is also demiclosed at 0.
(iii) In addition, ifT:H→H is quasi-pseudocontractive, then the mapping U is quasi-nonexpansive; that is,(17)Ux-u∗≤x-u∗,∀x∈H,u∗∈FixT=FixU.Lemma 5 (see [21]).
LetH be a real Hilbert space. Then(18)α0x0+α1x1+⋯+αrxr2≤∑i=0rαixi2-αsαtxs-xt2,for any s,t∈{0,1,2,…,r} and for xi∈H,i=0,1,2,…,r with α0+α1+⋯+αr=1 and 0<αi<1.Lemma 6 (see [16]).
LetH be a real Hilbert space and T:H→H be a quasi-nonexpansive mapping. Set Tα=αI+(1-α)T for α∈[0,1). Then the following properties are reached for all (x,q)∈H×F(T):
(i)〈x-Tx,x-q〉≥1/2x-Tx2 and x-Tx,q-Tx≤1/2x-Tx2;
(ii)Tαx-q2≤x-q2-α(1-α)Tx-x2;
(iii)〈x-Tαx,x-q〉≥1-α/2x-Tx2.


## 3. Main Results

In this section, we give the following assumption.Assumption 7.
Assume that
(i)H1, H2, and H3 are real Hilbert spaces and Rn is n-dimensional Euclidean space. A:H1→H3 and B:H2→H3 are two bounded linear operators; A∗ and B∗ are their adjoint operators, respectively.
(ii)K={x∈H3:h(x)≤0}, where h:H3→R is convex function which is subdifferentiable and its subdifferentials are bounded on bounded sets.
(iii)Si:H1→H1 and Tj:H2→H2 are all L-Lipschitz, quasi-pseudocontractive with L≥1,Fix(Si)≠∅ and Fix(Tj)≠∅ and Si-I and Tj-I are demiclosed at origin, where i=1,2,…,p;j=1,2,…,r.
(iv)Ui=(1-ξ)I+ξSi((1-η)I+ηSi), i=1,2,…,p, and Vj=(1-ξ)I+ξTj((1-η)I+ηTj), j=1,2,…,r, where 0<ξ<η<1/(1+1+L2).Theorem 8.
Let Assumption7 be true and δ>0 be small enough. Assume that {αni}⊂[0,1], i=0,1,2,…,p; n=1,2,…, such that ∑i=0pαni=1 for every n=1,2,… and {βn}⊂[0,1].j(n)=n(modr)+1. Suppose that x0∈H1, y0∈H2 are arbitrary and {xn}, {yn}, {ωn}, {un}, and {vn} are generated by the following iteration scheme:(19)ωn=PKnAxn+Byn2,Kn=z∈H3:hAxn+Byn2+ςn,z-Axn+Byn2≤0,un=xn-γnA∗Axn-ωn,xn+1=αn0un+αn1U1un+⋯+αnpUpun,vn=yn-γnB∗Byn-ωn,yn+1=βnvn+1-βnVjnvn,where(20)ςn∈∂hAxn+Byn2and(21)γn∈δ,1-δAxn-ωn2+Byn-ωn2A∗Axn-ωn2+B∗Byn-ωn2.Assume Ω≠∅, liminfn→∞αn0αni>0, i=1,2,…,p, and βn⊂(σ,1-σ) for small enough σ>0. Then (xn,yn)⇀(x~,y~), (un,vn)⇀(x~,y~), and ωn⇀ω~≔Ax~=By~, where (x~,y~)∈Ω.Proof.
It is obvious thatK⊂Kn for any n∈N. Let (x∗,y∗)∈Ω, that is, x∗∈⋂i=1pFix(Si),y∗∈⋂j=1rFix(Tj) and ω∗=Ax∗=By∗∈K. By Lemma 3, we have(22)ωn-ω∗=PKnAxn+Byn2-PKnω∗2≤12Axn-ω∗+12Byn-ω∗2=12Axn-ω∗2+12Byn-ω∗2-14Axn-Byn2.(23)un-x∗2=xn-x∗-γnA∗Axn-ωn2=xn-x∗2+γn2A∗Axn-ωn2-2xn-x∗,γnA∗Axn-ωn=xn-x∗2+γn2A∗Axn-ωn2-γnAxn-ω∗2-γnAxn-ωn2+γnωn-ω∗2.Similarly,(24)vn-y∗2=yn-y∗2+γn2B∗Byn-ωn2-γnByn-ω∗2-γnByn-ωn2+γnωn-ω∗2.By (23) and (24), we have(25)un-x∗2+vn-y∗2=xn-x∗2+yn-y∗2+γn2A∗Axn-ωn2-γnAxn-ω∗2-γnAxn-ωn2+γn2B∗Byn-ωn2-γnByn-ω∗2-γnByn-ωn2+2γnωn-ω∗2.Combining (22) and (25), we have(26)un-x∗2+vn-y∗2≤xn-x∗2+yn-y∗2+γn2A∗Axn-ωn2-γnAxn-ωn2+γn2B∗Byn-ωn2-γnByn-ωn2-12γnAxn-Byn2.Since Si, i=1,2,…,p, are quasi-pseudocontractive, from Lemma 4, Ui, i=1,2,…,p, are quasi-nonexpansive. Furthermore, from Lemma 5, we have(27)xn+1-x∗2≤αn0un-x∗2+αn1U1un-x∗2+⋯+αnpUpun-x∗2-αn0αn1U1un-un2≤αn0un-x∗2+αn1un-x∗2+⋯+αnpun-x∗2-αn0αn1U1un-un2=un-x∗2-αn0αn1U1un-un2.Similarly, since Tj,j=1,2,…,r are quasi-pseudocontractive, from Lemma 4, Vj,j=1,2,…,r, are quasi-nonexpansive. By Lemma 6 (ii), we have(28)yn+1-y∗2≤vn-y∗2-βn1-βnVjnvn-vn2.Combining (27) and (28) with (26), we have(29)xn+1-x∗2+yn+1-y∗2≤un-x∗2+vn-y∗2-αn0αn1U1un-un2-βn1-βnvn-Vjnvn2≤xn-x∗2+yn-y∗2+γn2A∗Axn-ωn2+γn2B∗Byn-ωn2-γnAxn-ωn2-γnByn-ωn2-12γnAxn-Byn2-αn0αn1U1un-un2-βn1-βnVjnvn-vn2.Since γn∈(δ,(1-δ)Axn-ωn2+Byn-ωn2/A∗(Axn-ωn)2+B∗(Byn-ωn)2), one has(30)γn2A∗Axn-ωn2+B∗Byn-ωn2-γnAxn-ωn2+Byn-ωn2=γnγnA∗Axn-ωn2+B∗Byn-ωn2-Axn-ωn2+Byn-ωn2≤-δγnAxn-ωn2+Byn-ωn2≤0.As a result, we have from (29) and (30) that(31)xn+1-x∗2+yn+1-y∗2≤xn-x∗2+yn-y∗2.
We see that the sequence{xn-x∗2+yn-y∗2} is decreasing and lower bounded by 0. Consequently it converges to some finite limit. Hence the sequences {xn} and {yn} are bounded.
According to the assumption of{γn} and (29), we have(32)limn→∞⁡Axn-Byn=0.
Furthermore, by the conditions on{αni}(0≤i≤p) and {βn}, we obtain(33)limn→∞⁡U1un-un=limn→∞⁡Vjnvn-vn=0.
From Lemma5, (19) and the conference of (27) for 2≤i≤p, we have(34)limn→∞⁡Uiun-un=0and(35)limn→∞⁡Axn-ωn=0,limn→∞⁡Byn-ωn=0,which combining (19) implies that(36)limn→∞⁡un-xn=limn→∞⁡γnA∗Axn-ωn=0and(37)limn→∞⁡vn-yn=limn→∞⁡γnB∗Byn-ωn=0.From (33), (34), and (36), we can conclude that(38)limn→∞⁡Uiun-xn=0,for1≤i≤p.
According to (36), (38), and(39)xn+1-xn≤αn0un-xn+αn1U1un-xn+⋯+αnpUpun-xn,we have(40)limn→∞⁡xn+1-xn=0which infers that {xn} is asymptotically regular. From (33) and (37), we have(41)limn→∞⁡Vjnvn-yn=0.
As a result,(42)limn→∞⁡yn+1-yn=limn→∞⁡βnvn-yn+1-βnVjnvn-yn=0,which implies that {yn} is asymptotically regular.
It follows from(43)vn+1-vn≤vn+1-yn+1+yn+1-yn+yn-vnthat(44)limn→∞⁡vn+1-vn=0.
Furthermore,(45)limn→∞⁡vn+j-vn=0,1≤j≤r.Taking (x~,y~)∈wω(xn,yn), from limn→∞un-xn=0 and limn→∞vn-yn=0, we have (x~,y~)∈wω(un,vn). Let an index j∈{1,2,…,r} be fixed. Noting that the pool of indexes is finite, from (45) and the boundedness of {vn}, there exists a subsequence {vnm}⊂{vn} such that vnm⇀y~ as n→∞ and j(nm)=j for all m. It is easy to see from (33) that(46)limm→∞Vjvnm-vnm=limm→∞Vjnmvnm-vnm=0.According to the demiclosednesses of Ui-I and Vj-I at 0, it yields from (33), (34) and (46) that Ui(x~)=x~ and Vj(y~)=y~ for 1≤i≤p and 1≤j≤r. Hence, x~∈⋂i=1pF(Ui) and y~∈⋂j=1rF(Vj). From Lemma 4, we have x~∈⋂i=1pF(Si) and y~∈⋂j=1rF(Tj).
Furthermore,Ax~-By~∈wω(Axn-Byn) and the weakly lower semicontinuity of the norm imply that(47)Ax~-By~≤liminfn→∞⁡Axn-Byn=0,that is, Ax~=By~.
Fromlimn→∞Axn-ωn=0 and limn→∞Byn-ωn=0, we have(48)ωn⇀ω~=Ax~=By~.As follows, we prove ω~∈K.
According to the boundedness of{ςn}, there exists a constant M¯>0, such that ςn≤M¯.
Noting thatωn=PKn(Axn+Byn/2)∈Kn, we obtain(49)hAxn+Byn2+ςn,ωn-Axn+Byn2≤0.As a result,(50)hAxn+Byn2≤-ςn,ωn-Axn+Byn2≤M¯ωn-Axn+Byn2.From the lower semicontinuity of h, (49), and (50), we have(51)hω~≤liminfn→∞⁡hAxn+Byn2≤M¯liminfn→∞⁡ωn-Axn+Byn2=0.Thus ω~∈K.
Next, we prove the uniqueness of the weak cluster point of{(xn,yn)}. Assume that (x¯,y¯) be other weak cluster point of {(xn,yn)}, then (x¯,y¯)∈Γ. Define ρn(x,y)=xn-x2+yn-y2, then(52)ρnx~,y~=xn-x¯2+x¯-x~2+2xn-x¯,x¯-x~+yn-y¯2+y¯-y~2+2yn-y¯,y¯-y~=ρnx¯,y¯+x¯-x~2+2xn-x¯,x¯-x~+y¯-y~2+2yn-y¯,y¯-y~.Without loss of generality, we suppose that xn⇀x¯,yn⇀y¯. By passing to the limit in relation (52), we have(53)ρx~,y~=ρx¯,y¯+x¯-x~2+y¯-y~2.Reserving the role of (x~,y~) and (x¯,y¯), we also have(54)ρx¯,y¯=ρx~,y~+x~-x¯2+y~-y¯2.By adding the two last equalities, we obtain x~=x¯ and y~=y¯, which shows that whole sequence {(xn,yn)} weakly converges to the unique solution of (13). The proof is completed.Similar to the proof of Theorem8, we can easily prove the following theorem.Theorem 9.
Let Assumption7 be true and δ>0 be small enough. Assume that {αn}⊂[0,1]. i(n)=n(mod⁡p)+1 and {βni}⊂[0,1],i=0,1,2,…,r;n=1,2,… such that ∑i=0rβni=1 for every n=1,2,…. Suppose that x0∈H1,y0∈H2 are arbitrary and {xn}, {yn}, {ωn}, {un}, and {vn} are generated by the following iteration scheme:(55)ωn=PKnAxn+Byn2,Kn=z∈H3:hAxn+Byn2+ςn,z-Axn+Byn2≤0,un=xn-γnA∗Axn-ωn,xn+1=αnun+1-αnUinun,vn=yn-γnB∗Byn-ωn,yn+1=βn0vn+βn1V1vn+⋯+βnrVrvn,where(56)ςn∈∂hAxn+Byn2and(57)γn∈δ,1-δAxn-ωn2+Byn-ωn2A∗Axn-ωn2+B∗Byn-ωn2.Assume Ω≠∅, αn⊂(σ,1-σ) for small enough σ>0 and liminfn→∞βn0βni>0,i=1,2,…,r. Then (xn,yn)⇀(x~,y~), (un,vn)⇀(x~,y~), and ωn⇀ω~≔Ax~=By~, where (x~,y~)∈Ω.


## 4. Numerical Examples

In this section, we give an example to show some insight into the behavior of the algorithms presented in this paper. The algorithms presented in Theorems8 and 9 are denoted as Algorithms 1 and 2 separately. The whole codes are written in Matlab 7.0. All the numerical results are carried out on a personal Lenovo Thinkpad computer with Intel(R) Core(TM) i7-6500U CPU 2.50GHz and RAM 8.00GB.In this section, we need the following lemmas.Lemma 10 (see [24, 25]).
For everyk≥0, let xk∈Rn, Ck={x∈Rn|c(xk)+〈ξk,x-xk〉≤0}. Then for any x∈Rn, we have(58)PCkx=x-cxk+ξk,x-xkξk2ξk,ifcxk+ξk,x-xk>0x,otherwise.Lemma 11 (see [26]).
Supposeh:Rn→R is a finite convex function, then it is subdifferentiable everywhere and its subdifferentials are uniformly bounded on any bounded subset of Rn.Example 12.
LetH1=R2, H2=R3, and H3=R4. Assume that S1x=-3x,x∈H1, S2x=-4x,x∈H1, T1x=-6x,x∈H2, T2x=-7x,x∈H2, K={x∈H3|h(x)=x12+2x22+4x32+x42-2≤0}, and(59)A=0.58360.99620.51180.35450.08260.97130.71960.3464,B=0.88650.12570.69380.45470.30890.00980.41340.72610.84320.21770.78290.9223.
Findx∈⋂i=12Fix(Si),y∈⋂j=12Fix(Tj) such that Ax=By∈K.It is easy to see thatSi:H1→H1,i=1,2 and Tj:H2→H2,j=1,2 are all Lipschitz quasi-pseudocontractive operators, but not quasi-nonexpansive operators. Therefore, the Example 12 cannot be calculated by the algorithms in [21].In the experiments, we takeδ=0.6, L=7, ξ=1/3L, η=1/2L, αn0=αn1=0.25, αn2=0.5, and βn=0.3. And the stopping criterion is Axn-wn+Byn-wn≤ϵ.Tables1 and 2 show the number of iterations, the cpu time, and the final iteration points for different ϵ when the initial point is x0=(0.1,0.1)T,y0=(0.1,0.1,0.1)T in Algorithms 1 and 2. We denote by Iter., Sec. and Fina. the number of iterations, the cpu time in seconds, and the final iteration points, respectively.Table 1
Number of iterations and the cpu time for differentϵ in Algorithm 1.
ϵ Iter. Sec. Fina. 1 0 - 2 12 0.002 x = 0.0028,0.0014 T, y = 0.0114 , - 0.0064 , - 0.0001 T 1 0 - 4 55 0.004 x = - 0.2230,0.2539 T × 1 0 - 4, y = - 0.0189 , - 0.1302,0.1469 T × 1 0 - 3 1 0 - 6 104 0.006 x = - 0.1723,0.4902 T × 1 0 - 6, y = - 0.0264 , - 0.1526,0.1556 T × 1 0 - 5Table 2
Number of iterations and the cpu time for differentϵ in Algorithm 2.
ϵ Iter. Sec. Fina. 1 0 - 2 12 0.002 x = ( 0.0044,0.0008 ) T, y = ( 0.0106 , - 0.0080,0.0011 ) T 1 0 - 4 50 0.005 x = ( - 0.1526,0.5347 ) T × 1 0 - 4, y = ( - 0.0096 , - 0.1518,0.1633 ) T × 1 0 - 3 1 0 - 6 112 0.014 x = ( - 0.0627,0.4938 ) T × 1 0 - 6, y = ( - 0.0229 , - 0.1262,0.1157 ) T × 1 0 - 5Figures1 and 2 present the behaviors of Axn-Byn for ϵ=10-4 and ϵ=10-6 when the initial point is x0=(0.1,0.1)T, y0=(0.1,0.1,0.1)T in Algorithms 1 and 2.Figure 1
The behaviors ofAxn-Byn for different ϵ in Algorithm 1.Figure 2
The behaviors ofAxn-Byn for different ϵ in Algorithm 2.It is easy to see that the presentation reveals thatAx=By.Furthermore, for testing the stationary property of iterative numbers, we carry out 500 experiments for different initial points which are presented randomly, such as(60)x=0.1∗rand2,1,y=0.1∗rand3,1.Figures 3 and 4 illustrate the behaviors of iterative numbers for ϵ=10-4 and ϵ=10-6, which reveals the stationary property of iterative numbers of the Algorithms 1 and 2.Figure 3
The behaviors of iterative numbers for differentϵ in Algorithm 1.Figure 4
The behaviors of iterative numbers for differentϵ in Algorithm 2.Especially, Figure5 illustrates the behaviors of xn+1-xn and yn+1-yn for ϵ=10-4 in Algorithm 1. Figure 6 shows the behaviors of iterative numbers for different ϵ in Algorithm 1.Figure 5
The behaviors ofxn+1-xn and yn+1-yn in Algorithm 1.Figure 6
The behaviors of iterative numbers for differentϵ in Algorithm 1.



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*Source: 3606294-2019-03-07.xml*