Second degree generalized gauss-seidel iteration method for solving linear system of equations. ABSTRACT

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1 Ethiop. J. Sci. & Technol. 7( 5-, 0 5 Second degree generlized guss-seidel itertion ethod for solving liner syste of equtions Tesfye Keede Bhir Dr University, College of Science, Deprtent of Mthetics tk_ke@yhoo.co ABSTRACT In this pper, second degree generlized Guss Seidel itertion (SDGGS ethod for solving liner syste of equtions whose itertive trix hs rel nd coplex eigenvlues re less thn unity in gnitude is presented. Few nuericl exples re considered to show the efficiency of the new ethod copred to first degree Guss-Seidel (GS, first degree Generlized Guss-Seidel (GGS nd Second degree Guss-Seidel (SDGS ethods. It is oserved tht the spectrl rdius of the new Second degree Generlized Guss-Seidel (SDGGS ethod is less thn the spectrl rdius of the ethods GS, GGS nd SDGS. By use of second degree itertion (SD ethod, it is possile to ccelerte the convergence of ny itertive ethod. Key words: Guss-Seidel ethod (GS; Generlized Guss-Seidel ethod (GGS; Strictly Digonlly Doinnt Mtrix. DOI: INTRODUCTION Consider clss of liner sttionry second degree ethods for solving liner syste A x ( where A is given rel non singulr n x n trix nd is given vector or nx (colun trix (Sd,995. Itertive ethods, sed on splitting A into A D-L-U, copute successive pproxitions to otin ore ccurte solutions to liner syste t ech itertion step n. This process cn e written in the for of the generl itertion trix eqution s x Gx + C ( n+ ( n In nuericl liner lger the Guss Seidel ethod, lso known s the Lienn ethod or the ethod of successive displceent, is n itertive ethod used to solve liner syste of equtions (Khn, 958. It is ned fter the Gern theticins Crl Friedrich Guss nd Philipp Ludwig Von Seidel, nd is siilr to the Jcoi ethod. Though it cn e pplied to ny trix with non-zero eleents on the digonls, convergence is only gurnteed if the trix is either digonlly doinnt, or syetric nd positive definite. The coputed Guss-Seidel iterte successively for ech coponent. It hs een proved tht, if A is strictly digonlly doinnt (SDD or irreducily digonlly doinnt, then the ssocited Jcoi nd Guss-Seidel itertions converge for ny initil guess X (0 (Li,005. If A is syetric positive definite (SPD trix, then the Guss-Seidel ethod lso converges for ny initil guess X (0 (Dvid, 007. The Guss-Seidel itertion (GS ethod for first degree is ( n+ ( n x ( D L Ux ( D L + ( This is n Open Access rticle distriuted under the ters of the Cretive Coons Attriution License ( BY.0

2 6 Tesfye Keede The Generlized Guss Seidel (GGS itertive ethod for first degree stted y (Dvid, 007 is x ( D L U x + ( D L (3 ( n+ ( n where, G ( D L U is the itertion trix of the GGS ethod. C ( D L is colun vector. In the next section, review of Second degree generlized Guss-Seidel itertive ethod (SDGGS is presented. Following this, the reltionship etween spectrl rdius of first degree Guss-Seidel (GS, first degree generlized Guss-Seidel (GGS nd Second degree Guss-Seidel (SDGS ethods nd Second degree generlized Guss-Seidel itertion ethods is given. Finlly, sed on the results on the nuericl exples considered, discussion nd conclusion de. SECOND DEGREE GENERALIZED GAUSS-SEIDEL ITERATIVE METHOD The liner sttionry second degree ethod is given y (Dvid, 970 is x x + ( x x + ( x x ( ( n+ ( n ( n ( n ( n+ ( n Here, X (n+ ppering in the right hnd side s given in ( is copletely consistent for ny constnt nd such tht 0. x x + ( x x + ( Gx + C x ( n+ ( n ( n ( n ( n ( n x x + x x + Gx + C x ( n+ ( n ( n ( n ( n ( n x [( + I + G] x x + C ( n+ ( n ( n ( n+ ( n ( n Therefore, x Gx + Hx + K (5 Where G ( + I + G (6 H I (7 K C (8 Theore.:- If trix A is strictly digonlly doinnt, then the ssocited generlized Guss-Seidel itertion ethod converges for ny initil pproxition, x (0. Proof: Since trix A is strictly digonlly doinnt, we hve the itertion trix Tking the nor t infinity of oth sides we hve U G D L U < D L U D L. U <. ( ( ( D L G ( D L U. Tht is G <. Thus, the generlized Guss-Seidel itertive ethod converges for ny initil pproxition The second degree generlized Guss - Seidel (GGS ethod is defined s (0 x.

3 Ethiop. J. Sci. & Technol. 7( 5-, 0 7 ( n+ ( n ( n x Gx Hx K + + (9 Where G ( + I + G (0 H I ( K C ( G ( D L U e the itertion trix of the GGS ethods. C ( D L is colun vector. Using the ide of (Golu nd Vrg, 96, (9 cn e written in the for. x 0 I x 0 ( n+ + ( n x H G x K ( n ( n The necessry nd sufficient condition for convergence of the ethod is tht spectrl rdius of Ĝ ust e less thn unity in gnitude for ny (0 ( x nd x. Using (0 nd (, for 0 I G H G, σ ( G <,if nd only if ll roots λ of det( λ I λg H 0 re less thn unity in odulus. (3 Sustituting G nd H of (0 nd ( in (3, we hve det( λ I λ[( + I + G ] + I 0 After collecting nd rerrnging, we hve det( λ I λ[( + I + G ] + I 0 ( + ( λ + det λ[ G + I I 0 λ ( + ( λ + det λ[ G + I I 0, Since det( λ 0 ( λ Thus, the eigenvlues λ of G re relted to the eigenvlues of G y ( + ( λ + + (5 λ Let i λ ve θ (6 Sustituting (6 in (5, we hve ( + ( ve + +, then iθ iθ ve ( + ( v cosθ + ivsin θ + + v(cosθ + isin θ

4 8 Tesfye Keede After collecting nd siplifying, we get ( v v + cosθ i + + sinθ v v (7 Fro (7 rel prt of is ( Re v + + cos v θ Now dd oth sides the ter ( +, we get ( + ( ( + v + Re cosθ. v ( So the result is Re + v + + cos v θ. Fro this we get, cosθ ( + Re + v + v Squring oth sides we hve cos ( + Re + θ. (8 v v + Fro (7 iginry prt of is I v v sinθ, then we get sin I θ v v (9

5 Ethiop. J. Sci. & Technol. 7( 5-, 0 9 ( + Re + I Add (8 nd (9, we get + v + v v v (0 Rel cse Coplex cse Figure. Generl regions for eigenvlues of G Fro (0, we hve the following ( + ( v+, ( v,( v v If the eigenvlues of G re rel nd lie in the intervl <, then the choice of nd ust stisfy the following conditions I 0 (since is rel nuer, we hve ( v sin θ 0, v we get v ( nd fro figure ( ove the distnce fro β to α is c, where c Squring oth sides we get (. ( Now sustituting ( v + nd ( v in ( for nd respectively, we hve v v ( [ ( v ] [ ( v ] v v +, then we get oth sides, v (, tking squre root of

6 0 Tesfye Keede we get v (3 Thus we hve fro (5, (6 nd (7, we get ( + + ( v If we let σ, then we get ( + σ( v + v v nd let + v + σ ω ( (5 Fro (6, we hve σ ω + σ [+ σ ] (6 σω ω nd ( ( + [ + σ ][ ( + ] (7 Therefore the spectrl rdius ofg is ( i. e. σ( G ω (8 Thus with this choice of nd, the second degree ethod for ny itertion is given y (Dvid, 007 ( n+ G ( ( + n n ω ω ( ( + ω + x I x x C ( + ( + ( + (9 Where ω + σ, G is the generlized itertion trix nd C is colun vector. If A is positive definite trix nd if is generlized Guss-Seidel itertive trix nd hence ( +, 0, σ, where is spectrl rdius of generlized Jcoi trix ω + σ ( + + ( + (30

7 Ethiop. J. Sci. & Technol. 7( 5-, 0 ω ( + + (3 nd σω ω ( ( ( ( + ( + ( + (3 ω σ( G ( + (33 Therefore, the second degree generlized Guss-Seidel ethod is given y x Gx + C (3 ( (0 ( n+ ω ( n ω ( n ( n ( n x ( G ( ( ( x + C + x + ω x x Where ω, G ( D L U nd C ( D L + Reltionship etween Spectrl Rdiuses Bsed on the results on the spectrl rdius, the following reltions re oserved First degree Jcoi ethod (FDJ is. Second degree Jcoi ethod (SDJ is +. Second degree generlized Jcoi ethod (SDGJ is + Second degree Guss-Seidel ethod (SDGS is ( + Second degree generlized Guss-Seidel ethod (SDGGS is ( + We know (+ + + Since 0 + > nd lso,, 0,,,3... n

8 Tesfye Keede If 0, we hve 0. Nuericl Exples Exple : Solve the following strictly digonl doinnt (SDD liner syste of equtions: 6x x + x3 x + 7x + x + x 5x3 x 3 5 Using GS GGS c SDGGS Solution:- Let us choose x (0 (0,0,0 t is n initil pproxition vlue nd tolernce nuer is 0-5. Now the spectrl rdius of Guss-Seidel nd generlized Guss-Seidel itertion ethods re r(gs0.396 nd r(ggs i.e r(ggs < r(gs. Exple : Solve the following positive definite (PD liner syste of eqution using GS, GGS, SDGGS ethods. x x x3 5 x + x x 3 x + x3 x 7 x x3 + x 9 Solution: Let us tke the initil pproxition x (0 (0,0,0,0 t with n ccurcy of 0-5. The spectrl rdius of Guss-Seidel (GS nd generlized Guss-Seidel itertion ethods (GGS re r (GS0.333, r(ggs0.. i.e r(ggs < r(gs. RESULTS AND DISCUSSION As presented in Tle, the exct solution for the given liner syste of eqution is (,,. It is oserved fro the tle tht the se solution is otined t the 8 th itertion y Guss-Seidel ethod (GS, t the 6 th itertion y Generlized Guss-Seidel ethod(ggs nd y Second degree Generlized Guss-Seidel (SDGGS ethod y considering only. If one cn see tht t the first itertion exct solution is otined.

9 Ethiop. J. Sci. & Technol. 7( 5-, 0 3 Tle : Solution of GS, GGS nd SDGGS for SDD Mtrix n GS GGS( SDGGS( x x x x x x As presented in Tle, the exct solution for the given liner syste of eqution is (,0,-,. It is oserved fro the tle tht the se solution is otined, using Guss-Seidel ethod (GS t the th itertion, using Generlized Guss-Seidel ethod(ggs t the 9 th itertion nd using Second degree Generlized Guss- Seidel (SDGGS ethod t the 6 th itertion y considering only. If,3, one cn get lost equl to the exct solution with sll nuer of itertion. Hence for Positive definite (PD trix SDGGS ethod is fster thn first degree GS, GGS nd Second degree GS ethod. Tle : Solution GS, GGS nd SDGGS Itertive Methods GS GGS ( SDGGS ( n x x x x x x x x x

10 Tesfye Keede CONCLUSION If trix A is strictly digonl dointe (SDD nd positive definite (PD trix, then y the use of second degree generlized Guss-Seidel itertive ethod(sdggs, it is possile to ccelerte the convergence rte of the solution of liner syste of equtions which hs rel nd coplex eigenvlues tht re less thn unity in gnitude. The nuericl results shows tht the SDGGS ethod is ore effective thn first degree Guss- Seidel (GS, first degree generlized Guss-Seidel (GGS nd second degree Guss-Seidel (GS ethods. Moreover, fro the reltionship of spectrl rdius, the spectrl rdius of SDGGS is less thn SDGS. In generl, the results of nuericl exples nd spectrl rdius coprison considered clerly deonstrte the ccurcy of the ethods developed in this rticle. It is conjectured tht the rte of convergence of soe ethods developed in this pper cn e further enhnced y using extrpolting techniques. Acknowledgeents I would like to express y sincere pprecition to Professor Vtti, Bssv Kur, Deprtent of Engineering thetics, College of Engineering, Andhr University nd Gshye Desslew for their coents nd suggestion on the rticle. REFERENCES Dvid, M. Y. (970. Second-degree itertive ethods for the solution of lrge liner systes. Journl of Approxition Theory 5:37-8. Dvid, K. S. (007. Generlized Jcoi nd Guss-Seidel ethods for solving liner syste of equtions. Nuericl Mthetics, A Journl of Chinese Universities. (English Series 6: Golu, G.H nd Vrg, R.S. (96. Cheyshev sei-itertive ethods, Successive Over-Relxtion itertive ethods nd second-order Richrdson itertive Methods. Nuericl Mthetics. Khn, W. (958. Guss-Seidel ethods for solving lrge systes of liner equtions. PhD Thesis, University of Toronto. Li, W. (005. A note on the preconditioned (GS ethod for Lrge syste of eqution. Journl of Coputtionl Applied Mthetics 8:8-90. Sd, Y. (995. Itertive ethods for sprse liner systes. PWS Press, New York.

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