In this paper we propose a refinement of some successive overrelaxation methods based on the reverse Gauss–Seidel method for solving a system of linear equations Ax = b by the decomposition A = Tm − Em − Fm, where Tm is a banded matrix of bandwidth 2m + 1. We study the convergence of the methods and give software implementation of algorithms in Mathematica package with numerical examples.

ACM Computing Classification System (1998): G.1.3.This paper is partly supported by project NI13 FMI–002 of Department for Scientific Research, Paisii Hilendarski University of Plovdiv

Iliev, Anton

Kyurkchiev, Nikolay

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica J. Computing 7 (2013), No 3, 245–256A REFINEMENT OF SOME OVERRELAXATIONALGORITHMS FOR SOLVING A SYSTEMOF LINEAR EQUATIONS∗Nikolay Kyurkchiev, Anton IlievAbstract. In this paper we propose a refinement of some successive over-relaxation methods based on the reverse Gauss–Seidel method for solving asystem of linear equations Ax = b by the decomposition A = Tm−Em−Fm,where Tm is a banded matrix of bandwidth 2m+ 1.We study the convergence of the methods and give software implemen-tation of algorithms in Mathematica package with numerical examples.ACM Computing Classification System (1998): G.1.3.Key words: reverse Gauss–Seidel method, or Nekrassov–Mehmke 2 method – (NM2), Suc-cessive Overrelaxation method with 1 parameter, based on (NM2) – (SOR1NM2), SuccessiveOverrelaxation method with 2 parameters, based on (NM2) – (SOR2NM2), Refinement of(SOR1NM2) method – (RSOR1NM2), Refinement of (SOR2NM2) method – (RSOR2NM2).*This paper is partly supported by project NI13 FMI–002 of Department for Scientific Re-search, Paisii Hilendarski University of Plovdiv.246 Nikolay Kyurkchiev, Anton Iliev1. Introduction. Let us consider the linear system:(1) Ax− b = 0.Let A = (aij) be an n×n matrix and Tm = (tij) be a banded matrix of bandwidth2m+ 1 defined as:tij ={aij , |i− j| ≤ m,0 otherwise.LetTm =a11 · · · a1,m+1.... . .. . .am+1,1. . . an−m,n. . .. . ....an,n−m · · · an,n,Em =−am+2,1.... . .−an,1 · · · −an,n−m−1andFm =−a1,m+2 · · · −a1,n. . ....−an−m−1,n.In [15] Salkuyeh considers the following overrelaxation method, based on Gauss–Seidel (forward algorithm) [10]–[12]:(2) xk+1 = (Tm−ωEm)−1[ωFm+(1−ω)Tm]xk+(Tm−ωEm)−1ωb, k = 0, 1, 2, . . . ,where A = Tm − Em − Fm.In [22] the following iteration scheme, based on the reverse Gauss–Seidelmethod [1] is proposed:(3)xk+1 = (Tm − ωFm)−1[ωEm + (1− ω)Tm]xk + (Tm − ωFm)−1ωb= BmSOR1NM2xk + cb, k = 0, 1, 2, . . . .A Refinement Of Some Overrelaxation Algorithms . . . 247Henceforth, we shall call the above scheme the Successive Overrelaxation methodwith 1 parameter, based on (NM2) – (SOR1NM2).In [1] D. Faddeev and V. Faddeeva pointed out that such iteration processesin which cycles studied in Gauss–Seidel (forward and reverse) algorithms alter-nate.The following theorem holds true:Theorem A [22]. Let A and Tm be a strictly diagonally dominant (SDD)matrix. Then for every 0 < ω < 2 the (SOR1NM2) method is convergent for anyinitial guess x0.Salkuyeh in [17] proposed the following overrelaxation method, based onGauss–Seidel (forward algorithm):(4) xk+1 = (Tm−γEm)−1[(1−ω)Tm+(ω−γ)Em+ωFm]xk+(Tm−γEm)−1ωb,k = 0, 1, 2, . . . .In [22] Zaharieva and Malinova published the following iteration scheme,based on the reverse Gauss–Seidel method:(5)xk+1 = (Tm − γFm)−1[(1− ω)Tm + (ω − γ)Fm + ωEm]xk + (Tm − γFm)−1ωb,= BmSOR2NM2xk + c1b, k = 0, 1, 2, . . . .We shall call the above scheme the Successive Overrelaxation method with 2 pa-rameters, based on (NM2) – (SOR2NM2).Definition. A is an M - matrix if aij ≤ 0 for i 6= j, A is non-singularand A−1 ≥ 0.The following theorem holds true:Theorem B [22]. If A is an M -matrix and 0 ≤ γ < ω ≤ 1 with ω 6= 0,then the (SOR2NM2) method is convergent, i.e.:ρ (BmSOR2NM2) < 1.For other results, see [3]–[6], [8], [9], [21], and [23].2. Main results. In this paper, following the ideas given in [20] and[7], we propose a refinement of the methods (SOR1NM2) and (SOR2NM2).248 Nikolay Kyurkchiev, Anton IlievI. Let x1 be an initial approximation for the solution of system (1) andb1i =n∑j=1aijx1j , i = 1, 2, . . . , n.After kth step we have: bk+1i =n∑j=1aijxk+1j , i = 1, 2, . . . , n.Now we refine this obtained solution as bk+1i → bi.Assume that x˜k+1 =(x˜k+11, . . . , x˜k+1n)is good approximation for the so-lution of system (1), i.e., x˜k+1 → x, where x is the exact solution of system (1)and bi =n∑j=1aij x˜k+1j , i = 1, 2, . . . , n.Since all x˜k+1t are unknown, we define them as follows, x˜k+1 = xk+1 +bk+1 − b.By the decompositionωA = (Tm − ωFm)− [(1− ω)Tm + ωEm]we have(6)[(Tm − ωFm)− [(1 − ω)Tm + ωEm]]x = ωb(Tm − ωFm)x = [ωEm + (1− ω)Tm]x+ ωb(Tm − ωFm)x = [Tm − ωFm − ωA]x+ ωb(Tm − ωFm)x = (Tm − ωFm)x+ ω(b−Ax)x = x+ ω(Tm − ωFm)−1(b−Ax)i.e.x˜k+1 = xk+1 + ω(Tm − ωFm)−1(b−Axk+1).A Refinement Of Some Overrelaxation Algorithms . . . 249For the method (3) we have(7)xk+1 = (Tm − ωFm)−1[ωEm + (1− ω)Tm]xk + (Tm − ωFm)−1ωb++ (Tm − ωFm)−1[ωb− ωA[(Tm − ωFm)−1[ωEm + (1− ω)Tm]xk++ (Tm − ωFm)−1ωb]]=[(Tm − ωFm)−1[ωEm + (1− ω)Tm]]2xk++[I + (Tm − ωFm)−1[ωEm + (1 − ω)Tm]](Tm − ωFm)−1ωb= BmRSOR1NM2xk + c2b, k = 0, 1, 2, . . . ,We shall call the above scheme the Refinement of (SOR1NM2) method –(RSOR1NM2).The following theorem holds true:Theorem 1. Let A be a strictly diagonally dominant (SDD) matrix.Then for any natural number m < n the (RSOR1NM2) method is conver-gent for any initial guess x0.P r o o f. Assuming x is the real solution of (1), as A is a SDD matrix byTheorem A, a (SOR1NM2) method is convergent.Let xk+1 → x. Then‖x˜k+1 − x‖∞ ≤ ‖xk+1 − x‖∞ + ω‖(Tm − ωFm)−1‖∞‖(b−Axk+1)‖∞.From the fact ‖xk+1 − x‖∞ → 0, we have ‖(b−Axk+1)‖∞ → 0.Therefore, ‖x˜k+1 − x‖∞ → 0 and a (RSOR1NM2) method is conver-gent. II. By the decompositionωA = (Tm − γFm)− [(1− ω)Tm + (ω − γ)Fm + ωEm]250 Nikolay Kyurkchiev, Anton Ilievwe have(8)[(Tm − γFm)− [(1− ω)Tm + (ω − γ)Fm + ωEm]]x = ωb(Tm − γFm)x = [(1 − ω)Tm + (ω − γ)Fm + ωEm]x+ ωb(Tm − γFm)x = [Tm − γFm − ωA]x+ ωb(Tm − γFm)x = (Tm − γFm)x+ ω(b−Ax)x = x+ ω(Tm − γFm)−1(b−Ax)i.e.x˜k+1 = xk+1 + ω(Tm − γFm)−1(b−Axk+1).For the method (5) we have(9)xk+1 = (Tm − γFm)−1[(1 − ω)Tm + (ω − γ)Fm + ωEm]xk++ (Tm − γFm)−1ωb+ (Tm − γFm)−1[ωb− ωA[(Tm − γFm)−1[(1 − ω)Tm++ (ω − γ)Fm + ωEm]xk + (Tm − γFm)−1ωb]]=[(Tm − γFm)−1[(1 − ω)Tm + (ω − γ)Fm + ωEm]]2xk++[I + (Tm − γFm)−1[(1− ω)Tm + (ω − γ)Fm + ωEm]](Tm − γFm)−1ωb= BmRSOR2NM2xk + c3b, k = 0, 1, 2, . . . ,We shall call the above scheme the Refinement of (SOR2NM2) method –(RSOR2NM2).The following theorem holds true:Theorem 2. Let A be an M -matrix. Then for any natural number m < nthe (RSOR2NM2) method is convergent for any initial guess x0.The proof follows the ideas given in [21], and will be omitted.Remark. If the (SOR1NM2) method is convergent, then the (RSOR2NM2)method is also convergent.Evidently, the (RSOR2NM2) method yields considerable improvement inthe rate of convergence for iterative method (SOR2NM2).A Refinement Of Some Overrelaxation Algorithms . . . 251III. We define the new Refinement Symmetric Successive OverrelaxationNekrassov–Mehmke method (RSSOR2NM2) consists the cyclic proceduresxk+1/2 =[(Tm − γEm)−1[(1− ω)Tm + (ω − γ)Em + ωFm]]2xk + αb,xk+1 =[(Tm − γFm)−1[(1 − ω)Tm + (ω − γ)Fm + ωEm]]2xk+1/2 + βb.This gives the recurrencexk+1 = BmRSSOR2NM2xk + δb,whereBmRSSOR2NM2 =[(Tm − γEm)−1[(1− ω)Tm + (ω − γ)Em + ωFm]]2××[(Tm − γFm)−1[(1− ω)Tm + (ω − γ)Fm + ωEm]]2.3. Numerical example. Let A is anM–matrix (example by Salkuyeh[16]): 4 −2 −1 −2−1 5 −5 −1−2 −1 9 −1−1 −1 −1 5 .Let γ = 0.5, ω = 0.9.For algorithms (5) and (9) and m = 1 we have (see Figure 2):ρ(B1RSOR2NM2)= 0.4927 < 0.7019 = ρ(B1SOR2NM2)< 1.For m = 2 we obtain:ρ(B2RSOR2NM2)= 0.245 < 0.495 = ρ(B2SOR2NM2)< 1.These results show that the method (9) is more appropriate in this case.For an implementation of algorithms (5) and (9) in the Mathematica pack-age ([19]), see Figure 1. The results for m = 1 are shown, see Figure 2.For other results, see [2], [13], and [14]. For other iteration schemes withincreased speed of convergence, see [18].Acknowledgements. The authors would like to thank the anonymousreviewers for their helpful and constructive comments that contributed to improv-ing the final version of the paper.252 Nikolay Kyurkchiev, Anton IlievFig. 1A Refinement Of Some Overrelaxation Algorithms . . . 253Fig. 2254 Nikolay Kyurkchiev, Anton IlievREFERENCES[1] Faddeev D., V. Faddeeva. 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Works – Math., 37 (2010), No 3, 121–134.256 Nikolay Kyurkchiev, Anton IlievNikolay KyurkchievFaculty of Mathematics, Informaticsand Information TechnologyPaisii Hilendarski University of Plovdiv24, Tsar Assen Str.4000 Plovdiv, Bulgariae-mail: nkyurk@uni-plovdiv.bgandInstitute of Mathematics and InformaticsBulgarian Academy of SciencesAcad. G. Bonchev Str., Bl. 81113 Sofia, Bulgariae-mail: nkyurk@math.bas.bgAnton IlievFaculty of Mathematics, Informaticsand Information TechnologyPaisii Hilendarski University of Plovdiv24, Tsar Assen Str.4000 Plovdiv, Bulgariae-mail: aii@uni-plovdiv.bgandInstitute of Mathematics and InformaticsBulgarian Academy of SciencesAcad. G. Bonchev Str., Bl. 81113 Sofia, Bulgariae-mail: anton.iliev@gmail.comReceived September 16, 2013Final Accepted November 7, 2013

A Refinement of some Overrelaxation Algorithms for Solving a System of Linear Equations

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A Refinement of some Overrelaxation Algorithms for Solving a System of Linear Equations

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