A modification of the Nekrassov method for finding a solution
of a linear system of algebraic equations is given and a numerical example
is shown.* This paper is partly supported by project IS–M–4 of Department for Scientific Research,
Paisii Hilendarski University of Plovdiv

Iliev, Anton

Kyurkchiev, Nikolay

Petkov, Milko

Bulgarian Digital Mathematics Library at IMI-BAS

Serdica J. Computing 3 (2009), 371–380ON SOME MODIFICATIONS OF THE NEKRASSOVMETHOD FOR NUMERICAL SOLUTION OF LINEARSYSTEMS OF EQUATIONS∗Anton Iliev, Nikolay Kyurkchiev, Milko PetkovAbstract. A modification of the Nekrassov method for finding a solutionof a linear system of algebraic equations is given and a numerical exampleis shown.1. Introduction. Let us consider the linear system Ax− b = 0 or(1)ai1x1 + ai2x2 + · · ·+ aiixi + · · ·+ ainxn − bi = 0 = fi(x1, x2, . . . , xn),i = 1, 2, . . . , n.Suppose that the matrix A is diagonally dominant and aii > 0, i =1, . . . , n.ACM Computing Classification System (1998): G.1.3.Key words: Solving linear systems of equations, Jacobi method, Richardson method,Nekrassov method, Chebyshev’s acceleration factors, pseudocode.*This paper is partly supported by project IS–M–4 of Department for Scientific Research,Paisii Hilendarski University of Plovdiv.372 Anton Iliev, Nikolay Kyurkchiev, Milko PetkovOne of the more effective iteration methods for solving the system (1) isthe Jacobi procedure (his method is also known as the method of simultaneousdisplacements):(2)xk+1i = −n∑j 6=iaijaiixkj +biaii= xki −1aiifi(xk1 , . . . , xkn)= xki −fi(xk1 , . . . , xkn)∂fi/∂xki,i = 1, 2, . . . , n; k = 0, 1, 2, . . . ,i.e., (2) is the Newton scheme applied for the equation fi = 0.A more powerful class of methods can be described by the recursion(Richardson iteration):(3) xk+1 = xk − αk(Axk − b),where αi, i = 1, . . . , k are damping factors.For instance, the Richardson iteration (3) with the application of Cheby-shev acceleration factors is defined byαi = 2(a+ b− (b− a) cos(2i+ 1)pi2(k + 1))−1,i = 0, 1, . . . , ka ≤ λi ≤ b, i = 1, . . . , n (λi are the eigenvalues of matrix A).In [8] we give the following modification of the Richardson method:(4) xk+1i = xki −1Mki n∑j=1aijxkj − bi ,i = 1, 2, . . . , n; k = 0, 1, 2, . . . ,whereMki =n∏j 6=i|xki − xkj |, i = 1, 2, . . . , n; k = 0, 1, . . . .On Some Modificationof the Nekrassov Method . . . 373For other contributions see Saad and van der Vorst [14], Freund, Goluband Nachtigal [6], Ishihara, Muroya and Yamamoto [7], Maleev [10], Stork [17],Zawilski [18].One geometric interpretation of method (4) is also given in [8].In a similar manner other iterations can be obtained which are modifica-tions of algorithms which have been explored in details in books by Bjo¨rck [2],Fadeev, D. and Fadeev, V. [4] and Barrett, R., M. Berry and others [1].As an example a scheme of the Gauss–Seidel or the Nekrassov method(see Nekrassov [13], Mehmke [11] and Nekrassov and Mehmke [12]) look thus:(5) xk+1i = −i−1∑j=1aijaiixk+1j −n∑j=i+1aijaiixkj +biaii,i = 1, 2, . . . , n; k = 0, 1, 2, . . . .2. Main results. Let us explore the following modification of theNekrassov method (assume that xi 6= xj and x0i 6= x0j for i 6= j):(6) xk+1i = xki −1Nki i−1∑j=1aijxk+1j + aiixki +n∑j=i+1aijxkj − bi ,i = 1, 2, . . . , n; k = 0, 1, 2, . . . ,whereNki =i−1∏j=1|xki − xk+1j |n∏j=i+1|xki − xkj |, i = 1, 2, . . . , n; k = 0, 1, . . . .Letδki =aiiNki, i = 1, 2, . . . , n; k = 0, 1, 2, . . . .The iteration procedure (6) (successive overrelaxation procedure) can berewritten as(7)xk+1i = xki −aiiNki i−1∑j=1aijaiixk+1j + xki +n∑j=i+1aijaiixkj −biaii= xki (1− δki )− δki i−1∑j=1aijaiixk+1j +n∑j=i+1aijaiixkj −biaii .374 Anton Iliev, Nikolay Kyurkchiev, Milko Petkov1. When δki = 1 from (7) we obtain the Nekrassov method.2. One geometric interpretation of method (7) is the following:LetFk,i = (x− xk+11) . . . (x− xk+1i−1 )(x− xki+1) . . . (x− xkn).ThenF ′k,i(xki ) =i−1∏j=1(xki − xk+1j )n∏j=i+1(xki − xkj )and the previous expression can be used for approximation of aii in the Nekrassovprocedure.We give a convergence theorem for the relaxation method (7).Theorem 1. Let(8)βi =i−1∑j=1|aij |aii, γi =n∑j=i+1|aij |aii, δki ∈ (1, 2),βi + γi ∈(0,1− |1− δki |δki)⊂ (0, 1), i = 1, 2, . . . , n; k = 0, 1, 2, . . . .Then the iteration procedure (7) converges to the unique solution xi, i =1, 2, . . . , n of the system (1).P r o o f. For the error xk+1i − xi, we have(9)xk+1i − xi = xki (1− δki )− xi−δki i−1∑j=1aijaiixk+1j +n∑j=i+1aijaiixkj −i−1∑j=1aijaiixj −n∑j=i+1aijaiixj − xi= (xi − xki )(δki − 1) + δkii−1∑j=1aijaii(xj − xk+1j ) + δkin∑j=i+1aijaii(xj − xkj )and(10)|xk+1i − xi| ≤ |δki − 1||xki − xi|+ δkii−1∑j=1|aij |aii|xj − xk+1j |+ δkin∑j=i+1|aij |aii|xj − xkj |≤ |δki − 1|||x− xk||1 + δki βi||x− xk+1||1 + δki γi||x− xk||1=(|δki − 1|+ γiδki)||x− xk||1 + δki βi||x− xk+1||1.On Some Modificationof the Nekrassov Method . . . 375Letmaxi|xk+1i − xi| = |xk+1i0− xi0|.Then from (10) we get||x− xk+1||1 = maxi|xi − xk+1i | = |xk+1i0− xi0 |≤(|δki0 − 1|+ γi0δki0)||x− xk||1 + δki0βi0 ||x− xk+1||1and(11) ||x− xk+1||1 ≤|δki0 − 1|+ γi0δki01− δki0βi0||x− xk||1 = Ki0 ||x− xk||1.Evidently from (8) we haveKi0 =|δki0 − 1|+ γi0δki01− δki0βi0≤|δki0 − 1|+ δki0(1−|δki0−1|δki0− βi0)1− δki0βi0= 1.This proves Theorem 1. LetL =0 0 · · · 0a21 0 · · · 0.........an1 an2 · · · 0, R =0 a12 · · · a1n0 0 · · · a2n.........0 0 · · · 0, Xk =xk1xk2...xkn ,P =a11 0 · · · 00 a22 · · · 0....... . ....0 0 · · · ann, δk =δk11 0 · · · 00 δk22 · · · 0....... . ....0 0 · · · δknn.Theorem 2. The iteration (6) (or (7)) is convergent when all roots(eigenvalues) of the equation(12)∣∣∣Aδk − (P + δkL)+ t(P + δkL)∣∣∣ = 0376 Anton Iliev, Nikolay Kyurkchiev, Milko Petkovare |ti| < 1, i = 1, . . . , n.P r o o f. In matrix terms the successive overrelaxation procedure (7) canbe written as follows:(13) Xk+1 =(P + δkL)−1 ((I − δk)P − δkR)Xk +(P + δkL)−1δkb,i.e.Xk+1 = BXk + c.Evidently, |B − tI| = 0 can be represented as|B − tI| =∣∣∣∣(P + δkL)−1∣∣∣∣∣∣∣Aδk − (P + δkL)+ t(P + δkL)∣∣∣ = 0,and the statement of Theorem 2 follows from the standard iteration theory. 3. In a number of cases the success of the procedures of type (5) dependson the proper ordering of the equations (and xi, i = 1, . . . , n) in system (1).In spite of this fact the following variant of the Nekrassov method isknown [4]:(14) xk+1i = −i−1∑j=1aijaiixkj −n∑j=i+1aijaiixk+1j +biaii.Further, we are interested in the successive overrelaxation procedure (14)based on the method (7):(15) xk+1i = xki (1− δki )− δki i−1∑j=1aijaiixkj +n∑j=i+1aijaiixk+1j −biaii .In matrix terms the successive overrelaxation procedure (15) can be writ-ten as follows:(16) Xk+1 =(P + δkR)−1 ((I − δk)P − δkL)Xk +(P + δkR)−1δkb.The pseudocode for the modification of Nekrassov method (6) is given inFigure 1.On Some Modificationof the Nekrassov Method . . . 377Choose an initial guess x0 for the solution x.for k = 1, 2, . . . ,for i = 1, 2, . . . , nxi = aiixk−1iNk−1i = 1for j = 1, 2, . . . , i− 1Nk−1i = Nk−1i |xk−1i − xkj |xi = xi + aijxkjendfor j = i+ 1, . . . , nNk−1i = Nk−1i |xk−1i − xk−1j |xi = xi + aijxk−1jendxi = (xi − bi)/Nk−1iendxk = xk−1 − xcheck convergence; continue if necessaryendFig. 1. The modification of the Nekrassov method (6)3. Numerical example. As an example we will consider the system:∣∣∣∣∣∣x1 + 3x2 − 2x3 = 53x1 + 5x2 + 6x3 = 72x1 + 4x2 + 3x3 = 8The exact solution of the system is x(−15, 8, 2).For an initial approximation we choose x0(−15.02, 8.02, 2.02).We give the results of numerical experiments (8 iterations) for each ofmethods (5) and (6).In Table 1 the following notations are used:– in the first column the serial number of the iteration is given;– using the modified scheme (6) in the second column the obtained resultsare given (array x[ ]);– using the classical Nekrassov scheme (5) in the third column the ob-tained results are given (array y[ ]).378 Anton Iliev, Nikolay Kyurkchiev, Milko PetkovTable 11 X [1] = −15.02000000000000 Y [1] = −15.02000000000000X [2] = 8.01884259259259 Y [2] = 7.98800000000000X [3] = 2.01906701123844 Y [3] = 2.029333333333332 X [1] = −15.01999590828629 Y [1] = −14.90533333333333X [2] = 8.01776735852021 Y [2] = 7.90800000000000X [3] = 2.01820333133504 Y [3] = 2.059555555555563 X [1] = −15.01998800937772 Y [1] = −14.60488888888888X [2] = 8.01676825375272 Y [2] = 7.69146666666666X [3] = 2.01740388676229 Y [3] = 2.147970370370374 X [1] = −15.01997656688334 Y [1] = −13.77845925925925X [2] = 8.01583967575863 Y [2] = 7.08951111111110X [3] = 2.01666397500912 Y [3] = 2.399624691358035 X [1] = −15.01996182501415 Y [1] = −11.46928395061725X [2] = 8.01497643146312 Y [2] = 5.40202074074072X [3] = 2.01597923762914 Y [3] = 3.110161646090546 X [1] = −15.01994400998709 Y [1] = −4.98573893004107X [2] = 8.01417370750044 Y [2] = 0.65924938271599X [3] = 2.01534563522253 Y [3] = 5.111493443072737 X [1] = −15.01992333132901 Y [1] = 13.24523873799748X [2] = 8.01342704260065 Y [2] = −12.68093537448576X [3] = 2.01475942421623 Y [3] = 10.744421340649368 X [1] = −15.01989998308720 Y [1] = 64.53164880475601X [2] = 8.01273230196133 Y [2] = −50.21229489163284X [3] = 2.01421713531614 Y [3] = 26.595293985673114. A wide area of problems and practical tasks in tomography and imageprocessing are reduced to the problem of solving a system of algebraic equationswith constraint conditions for the initial approximations x0i , i = 1, . . . , n (seeBjo¨rck [2], A. van der Sluis and H. van der Vorst [16], A. Louis and F. Natterer[9] and R. Santos and A. de Pierro [15]).Acknowledgement. The authors are pleased to acknowledge a referee’shelpful remarks to the first version of this work.On Some Modificationof the Nekrassov Method . . . 379REFERENCES[1] Barrett R., M. Berry, T. Chan, J. Demmel, J. Donato, J. Don-garra, V. Eijkhout, R. Pozo, C. Romine, H. Van der Vorst. Tem-plates for the Solution of Linear Systems: Building Blocks for Iterative Meth-ods. SIAM, Philadelphia, 1994.[2] Bjo¨rck A. Numerical methods for least squares problems. SIAM, Philadel-phia, PA, 1996.[3] De Boor C., J. Rice. Extremal polynomials with application to Richard-son iteration for indefinite linear system. Technical Rept., Wisconsin Univ.Madison Mathematics Research Center, 1980.[4] Fadeev D., V. Fadeev. Numerical methods of linear algebra. FizmatgizM., 1963.[5] Fisher B., L. Reichel. A stable Richardson iteration method for complexlinear systems. Numer. Math., 54 (1989), No 2, 225–242.[6] Freund R., G. Golub, N. Nachtigal. Iterative solution of linear sys-tems. Acta Numerica, (1992), 1–44.[7] Ishihara K., Y. Muroya, T. Yamamoto. On linear SOR-like methods,II-convergence on the SOR-Newton method for mildly nonlinear equations.Japan J. Indust. Appl. Math., 14 (1997), No 1, 99–110.[8] Kyurkchiev N., M. Petkov, A. Iliev. On a modification of Richardsonmethod for numerical solution of linear system of equations. Compt. rend.Acad. bulg. Sci., 61 (2008), No 10, 1257–1264.[9] Louis A., F. Natterer. Mathematical problems of computerized tomog-raphy. Proceedings of the IEEE, 71 (1983), No 3, 379–389.[10] Maleev A. A lower bound for order of convergence of methods of Jacobi’stype. Comp. Math. and Math. Phys., 46 (2006), 2128–2137 (in Russian).[11] Mehmke R. On the Seidel scheme for iterative solution of linear system ofequations with a very large number of unknowns by successive approxima-tions. Math. Sb., 16 (1892), No 2, 342–345 (in Russian).[12] Mehmke R., P. Nekrassov, Solution of linear system of equations bymeans of successive approximations, Math. Sb., 16 (1892), 437–459 (inRussian).[13] Nekrassov P. Determination of the unknowns by the least squares whenthe number of unknowns is considerable. Math. Sb., 12 (1885), 189–204 (inRussian).380 Anton Iliev, Nikolay Kyurkchiev, Milko Petkov[14] Saad Y., H. van der Vorst. Iterative solution of linear systems in the20th century. JCAM, 123 (2000), No 1–2, 1–33.[15] Santos R., A. de Pierro. The effect of the nonlinearity on GCV appliedto conjugate gradients in computerized tomography. Comp. and Appl. Math.,25 (2006), No 1, 111–128.[16] van der Sluis A., H. van der Vorst. Numerical solution of large, sparselinear algebraic systems arising from tomographic problems. In: Seismic To-mography (Ed. G. Notel ), D. Reidel Publ. Comp., Dordrecht, The Nether-lands (1987), 49–84.[17] Stork Ch. Comparison of Richardson’s iteration with Chebyshev acceler-ation factors to conjugate gradient iteration. Stanford Expl. Proj. SEP-57(1988), 479–503.[18] Zawilski A., Numerical stability of the cyclic Richardson iteration, Numer.Math., 60 (1991), No 1, 251–290.Anton IlievFaculty of Mathematicsand InformaticsPaisii HilendarskiUniversity of Plovdiv24 Tsar Assen Str.4000 Plovdiv, Bulgariae-mail: aii@uni-plovdiv.bgandInstitute of Mathematicsand InformaticsBulgarian Academy of SciencesAcad. G. Bonchev Str., Bl. 81113 Sofia, BulgariaNikolay KyurkchievInstitute of Mathematics and InformaticsBulgarian Academy of SciencesAcad. G. Bonchev Str., Bl. 81113 Sofia, Bulgariae-mail: nkyurk@math.bas.bgMilko PetkovFaculty of Mathematics and InformaticsEpiskop Konstantin Preslavski University of Shumen115 Universitetska Str.9712 Shumen, BulgariaReceived September 7, 2009Final Accepted November 24, 2009

On some Modifications of the Nekrassov Method for Numerical Solution of Linear Systems of Equations

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On some Modifications of the Nekrassov Method for Numerical Solution of Linear Systems of Equations

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