INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, FEBRUARY 07 5 Secod Degree Refiemet Jacobi Iteratio Method for Solvig System of Liear Equatio Tesfaye Kebede Abstract Several iterative techiques for the solutio of liear system of equatios have bee proposed i differet literature i the pasti this paper, we preset a Secod degree of refiemet Jacobi Iteratio method for solvig system of liear equatio, Ax = b ad we cosider few umerical examples ad spectral radius to show that the effective of the Secod degree of refiemet Jacobi Iteratio Method (SDRJ) i compariso with other methods of First degree Jacobi (FDJ), First degree Refiemet Jacobi (FDRJ) ad Secod degree Jacobi (SDJ) method Idex Terms Jacobi iteratio, secod degree refiemet, system of liear equatios I INTRODUCTION IN may scietific ad egieerig applicatios, oe ofte comes across with a problem of fidig the solutio of a system of liear equatios writte as the followig equatio i matrix form: Ax = b () where A is a osigular matrix of size, x ad b are -dimesioal vectors Splittig the matrix A [] as : A = D L U () where D is a diagoal matrix ad L ad U are strictly lower ad upper triagular part of A respectively A geeral first degree liear statioary iterative method for the solutio of the system of equatio () may be defied i the form: x (+) = Hx () +C (3) where x (+) ad x () are the approximatio for x at the ( + ) th ad th iteratios respectively, H is called the iterative matrix depedig o matrix A ad C is a colum vector The iteratio system x (+) = Hx () +C is coverge if ad oly if the spectral radius of H are less tha uity, ie σ(h) < The first degree iterative method of Jacobi (FDJ) method for the solutio of () is defied as: x (+) = D (L +U)x () + D b (4) ad the first degree refiemet Jacobi (FDRJ) method ca be obtaied i the form of : where x (+) = (D (L +U)) x () + (I + D (L +U))D b (5) X (+) = H RJ X () +C RJ, (6) H RJ = [D (L +U)],C RJ = [I + D (L +U)]D b (7) Mauscript received December, 06; accepted February, 07 The author is with the Departmet of Mathematics, College of Sciece, Bahir Dar Uiversity, Bahir Dar, Ethiopia Email: tk_ke@yahoocom The liear statioary ay secod degree method is give by [] x (+) = x () + a(x () x ( ) ) + (x (+) x () ) (8) Here x (+) appearig i the right had side as give i equatio (3) is completely cosistet for ay costat a ad such that 0 X (+) = X () + a(x () X ( ) ) + (HX () +C X () ) X (+) = X () + ax () ax ( ) + HX () + C X () X (+) = X () + ax () X () + HX () + C ax ( ) X (+) = ( + a b)x () + HX () ax ( ) + C X (+) = [( + a )I + H]X () ax ( ) + C where X (+) = GX () + H X ( ) + K (9) G = ( + a )I + H (0) H = ai () K = C () The liear statioary ay secod degree method is give by [] ca be writte i umber (6) or umber (7) with (8), (9) ad (0) coditios O the other way equatio () ca be solved usig the secod degree Jacobi(SDJ) statioary iterative method usig x (+) = D (L +U)x () ax ( ) + k x (+) = [D (L +U)x () + k ] ax ( ) (3) For optimal values of a ad Where k = D b If A is a row strictly diagoal domiat (SDD) matrix, the the Jacobi method coverges for ay arbitrary choice of the iitial approximatio [3] I this paper, we costruct a ew method of solvig a liear system of the form Ax = b that arise i ay egieerig ad applied sciece The outlie of this paper is as follows: we itroduce secod degree refiemet Jacobi (SDRJ) iterative method i accordace this we will see the relatioship betwee spectral radius of first degree Jacobi (FDJ), first degree refiemet Jacobi (FDRJ), Secod degree Jacobi (SDJ) methods ad Secod degree refiemet Jacobi iteratio (SDRJ) methods are give Based o the methods ad results, few umerical examples are cosidered to show that the efficiecy of the ew method i compariso with the existig FDJ, FDRJ ad SDJ methods Fially discussio ad coclusio made at Sectio V
6 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, FEBRUARY 07 II SECOND DEGREE REFINEMENT JACOBI (SDRJ) ITERATIVE METHOD Theorem : If matrix A is o sigular PD ad SDD matrix with A = D L U, the the Secod degree of Refiemet Jacobi iterative method is: [x (+) = [D (L +U)] x () aix ( ) + (I + D (L +U))D b] for ay iitial guess ad the optimal values for a ad Give: A is o sigular PD ad SDD matrix ad A = D-L- U Required: the secod degree of refiemet Jacobi iterative method is: x (+) = [D (L +U)+k ]x () a x ( ) Proof: ow cosider equatio (5)ad (6), so oe ca get : x (+) = x () +a(x () x ( ) )+ (H RJ x () +C RJ x () ) (4) This also ca be writte as follows after some computatio: X (+) = G RJ x () + F RJ x ( ) + K RJ (5) where G RJ = ( + a )I + H RJ,F RJ = ai ad K RJ = C RJ By usig Golub ad Varga [] ( ) ( )( ) ( ) x () 0 I x ( ) 0 x (+) = F RJ G RJ x () + (6) K RJ ( x () x (+) ) ( x ( ) = Ĝ x () ( 0 I where Ĝ = F RJ G RJ ) ( 0 + K RJ ) ), (7) The above equatio coverges to the exact solutio if σ(ĝ) <, ie the spectra radius of Ĝ is less tha oe I order to solve the spectra radius of Ĝ, first we have to solve the eigevalues λ of Ĝ σ(ĝ) < iff all roots λ RJ of det(λ RJ I λ RJ G RJ F RJ ) = 0 (8) ie det(λ RJ I λ RJ G RJ F RJ ) = 0 det(λ RJ I λ RJ [( + a )I + H RJ ] + ai) = 0 det(λ RJ I λ RJ ( + a )I λ RJ H RJ + ai) = 0 det( λ RJ [ λ RJ I + ( + a ) I + H RJ a λ RJ I] = 0 ( λ RJ ) det(h RJ + ( + a ) I λ RJ + a I) = 0 λ RJ det(h RJ + ( + a ) I λ RJ + a I) = 0,sice( λ RJ ) 0 λ RJ (9) Thus, the eigevalues λ RJ of Ĝ are related to the eigevalues of H RJ with H J is + ( + a ) = (a + λ RJ ) (0) λ RJ As the image of the circle, Let the eigevalue λ RJ = ve iθ = v(cosθ +isiθ) is the ellipse, the substitutig this i equatio (8), we obtai + ( + a ) + + a + + a = (veiθ ) + a ve iθ = veiθ + a ve iθ = = ( + a ) v(cosθ + isiθ) + vcosθ + + i vsiθ a(cosθ isiθ) v + acosθ v i asiθ v = (v+ a v )cosθ + a +i (v a )siθ () v ie Re = (v + a v )cosθ (+a ) Im = (v a v )siθ cosθ = Re + (+a ) (v + a v ) () siθ = We kow that cos θ + si θ = [ Re + (+a ) (v + a v ) ] + [ IM (v + a v ) (3) Im (v a v )] = (4) cetre = c(h,k) = ( ( + a ),0) (5) cetre = c(h,k) = ( ( + a ),0) (6) Legth of semi-major axis = a / = (v + a v ) (7) Legth of semi-mior axis = b / = (v a v ) (8) Foci = F = (h c,0) = ( + a a,0) = (α,0) (9) Foci = F = (h + c,0) = ( + + a,0) = (β,0) v = (h a,0) = ( + a (v + a v ),0) v = (h + a,0) = ( + a + (v + a v ),0) v 3 = (h,k + b ) = ( + a, (v a v )) v 4 = (h,k b ) = ( + a, (v a v )) (30)
KEBEDE et al: SECOND DEGREE REFINEMENT JACOBI ITERATION METHOD FOR SOLVING SYSTEM OF LINEAR EQUATION 7 Before we prove the theorem let us prove the followig Lemmas Lemma : If is real, the α β <, for ay foci α ad β which are real Proof: We kow that is a real umber We require that α β <, for ay foci α ad β which are real The proof is as follows: = (v + a v )cosθ + a, sice is a real umber I this equatio θ varies (v+ a v ) + a µ m (v+ a v )cosθ + a, sice cosθ α µ J β < (3) Because α ad β from equatio (6) ad (7) ad to be coverget ρ(h RJ ) < so all the eigevalues must be less tha Lemma 3: If the eigevalues of H RJ < are real ad lie i the iterval Proof: α µ J β <, the the optimal choices of a ad must satisfy the followig coditios: b) α + β c) a) v = a (3) = + a (33) = v (34) d) v = (α + β) ( + v ) (35) Give: the eigevalues of H RJ < are real ad lie i the iterval α µ J β < Required: proof of a) util d) Proof : a) we kow µ m is real, the (v a v )siθ = 0, we have siθ = 0 or (v a v ) = 0, so we get v = a or siθ = 0, from the secod equatio we have θ = π, = 0,, Therefore V = a b) From the (6) ad (7) ad from Lemma (a),we get: α = v + a ad β = v + a α + β = + a (mid poit formula) c) We kow from (b) above we have α = v +a β = v, the oe ca get +a d) From Lemma (b), we have α + β α + β (α + β) = v = + a Divide equatio (3) by (33),we get β α (α+β) = ( + a ) ad = + ( + a ) = + a (36) = v +a (α + β) = v + a v = (α + β) ( + v ) Lemma 4: If is the spectral radius of H RJ, the = (α + β) (37) Proof: Give: is the spectral radius of H RJ Required: = β α (α+β) Proof: we kow that = (v + a v )cosθ +a By defiitio of derivative of fuctios i calculus d dθ = d dθ [ (v+ a v )cosθ + a ] = (v+ a v )siθ To calculate the maximum ad miimum value, the above equatio equates to zero (v + a v )siθ = 0 siθ = 0 θ = 0,π,π Whe θ = 0, the = (v + a v ) +a Whe θ = π, the = (v + a v ) +a Whe θ = π, the = (v + a v ) +a From the above the maximum value occurs at θ = 0 ad π max i= () = (v + a v ) + a The miimum value occurs at θ = π = β mi i= = (v + a v ) + a = α = max = max (v + a i= i= v )cosθ + a = (v + a v ) + a,sice cosθ ad
8 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, FEBRUARY 07 = mi i= ( ) = mi (v + a v )cosθ + a = (v + a v ) + a = v + a = v + a + = (α + β) + = β by equatio (3) ad (33) Therefore = β = mi i= ( ) = (v + a v ) + a = v + a + = α Therefore = α From the previous two results, we obtai = ad = (α + β) The divide the previous two equatios, we get = β α (α+β) Now let us determie the values of a ad First, let us fid a from Lemma 3d equatio (3) v = by lemma 3d, we have (α + β) ( + v ) v = ( + v ) v v + = 0 This is the equatio of quadratic whose graph is a parabola ad the miimum value occurs at p=( µ, RJ µ ) sice µ RJ RJ > 0 Oe ca solve by quadratic formula of the above equatio: v = ± 4 4 = ± v = + ad v = The smallest value is v = Let + v = ω a = ω + v = + µ RJ µ RJ a =, ( + ) sice a = v Secodly, let us fid = = 4v β α 4v = ( + v ) by usig equatio (7) 4 = ( + µ RJ )( (α + β)) Lemma 5: If matrix A is positive defiite matrix ad if H RJ is Jacobi iterative matrix, the β = α = = σ(h RJ ) Proof: Give: matrix A is positive defiite matrix ad if H RJ is Jacobi iterative matrix Required: β = α = = σh RJ Proof: I order to prove this Lemma, we have to use Lemma 3 = max i= = (v + a v ) + a = (v + a v ) + a β = α = Now we ca fid the optimal value of a ad µ RJ = α = β ie a = b (+ µ RJ ) =, sice β = α + µ RJ α +β = 0 Now let us fid secod degree of Refiemet Jacobi (SDRJ) method: sice β = α From the secod degree + a + a = α + β = α α ( + a ) = 0 x (+) = G RJ x () + F RJ x ( ) + k RJ x (+) = [( + a)i + H RJ ]x () + ( ai)x ( ) + C RJ where x (+) = H RJ x () ax ( ) + CRJ RJ x (+) = [D (L +U)x () + k ] ax ( ), a =, = ( + µ ) + µ µ III RELATIONSHIP BETWEEN SPECTRAL RADIUS As we have see above the spectral radius of First degree Jacobi method(fdj) is µ First degree Refiemet Jacobi method(fdj) is = µ Secod degree Jacobi method(sdj) is a = µ + µ Secod degree Refiemet Jacobi method(sdgj) is a = µ (+ µ ) µ µ ) That is oe ca see (+ + µ µ sice + µ > 0 ad also µ sice 0 < µ < µ
KEBEDE et al: SECOND DEGREE REFINEMENT JACOBI ITERATION METHOD FOR SOLVING SYSTEM OF LINEAR EQUATION 9 IV NUMERICAL EXAMPLES a Solve the followig SDD matrix usig FDJ,FDGJ,SDJ ad SDRJ iterative methods 4x x x 3 = 3 x + 6x + x 3 = 9 x + x + 7x 3 = 6 Solutio: all results are based o the give data ad we get the spectral radius as follows Method FDJ FDRJ SDJ SDRJ Spectral radius 0495 0845 057 0093 b Solve the followig PD matrix usig FDJ, FDGJ, SDJ ad SDRJ iterative methods 3x x x 3 = x + 3x + x 3 = 3 x + x + 4x 3 = 7 Solutio: all results are based o the give data we get the spectral radius as follows Method FDJ FDRJ SDJ SDRJ Spectral radius 03333 0 076 00557 The detailed experimetal results are writte i the appedix Table I shows that FDRJ method coverges faster tha FDJ method for SDD matrix Table II shows that SDRJ method coverges faster tha the SDJ method for SDD matrix Table III shows that FDRJ method coverges faster tha FDJ method for PD matrix Table IV shows that SDRJ method coverges faster tha the SDJ method for PD matrix REFERENCES [] V K Vatti ad T K Eeyew, A refiemet of gauss-seidel method for solvig of liear system of equatios, It J Cotemp Math Scieces, vol 6, o 3, pp 7, 0 [] D Youg ad D Kicaid, Liear statioary secod-degree methods for the solutio of large liear systems, Uiv of Texas, Ceter for Numerical Aalysis, Austi, TX (Uited States), Tech Rep, 990 [3] F Naeimi Dafchahi, A ew refiemet of jacobi method for solutio of liear system equatios Ax=b, It J Cotemp Math Scieces,, vol 3, o 7, pp 89 87, 008 APPENDIX TABLE I ALL NUMERICAL RESULT OF SDD MATRIX FOR FDJ AND FDRJ OF EXAMPLE First degree Jacobi (FDJ) First degree refiemet Jacobi (FDRJ) () x () x () x () x 075 5-085743 09074 89857-096486 090755 89857-096485 0997 9936-099745 3 09843 96486-0997448 099936 99976-0999683 4 09970 9936-0997448 0999895 00008-099995 5 0999044 9968-00089 0999986 000008-0999993 6 0998998 99988-099968 0999999 00000-000000 7 000035 99963-0006 00000 000000-00000 8 0999873 00003-0999939 000000 000000-000000 9 0003 999948-0000 0 099998 0000-0999989 000006 99999-000003 0999998 000003-0999997 3 00000 999999-000000 4 000000 00000-000000 5 000000 000000-000000 V CONCLUSIONS As we have see i this report for SDD ad PD matrix, we ca otice that FDJ, FDRJ ad SDJ are reasoable to approximate the exact solutio of system of liear equatios at a certai give coditio But they are relatively slow to coverge to the exact solutio However, the Secod degree of refiemet Jacobi iterative method for solvig system of liear equatios are reasoable ad efficiet way of approximatig the exact solutio of system of liear equatios Numerical results of spectral radius show that, SDRJ methods coverge with a small umber of iteratio steps for solvig systems of liear equatios I geeral, the results of umerical examples cosidered clearly demostrate the accuracy of the methods developed i this paper It is cojectured that the rate of covergece of the method that developed i this paper ca be further ehaced by usig extrapolatig techiques TABLE II ALL NUMERICAL RESULT OF SDD MATRIX FOR SDJ AND SDRJ OF EXAMPLE Secod degree Jacobi (SDJ) Secod degree refiemet Jacobi (SDRJ) () x () x () x () x 075 5-085743 09074 89857-096486 0945649 965467-0076 0996783 00389-0055 3 00094 000589-0084 000043 00083-000045 4 00005 00884-0999995 0999995 000064-099996 5 000739 999994-000097 0999997 000006-0999995 6 099997 0006-0999889 000000 00000-0999999 7 000043 99997-00004 000000 000000-000000 8 099987 00003-0999939 9 000003 999995-000003 0 0999998 00000-0999999 00000 999999-00000 0999999 00000-000000 3 000000 000000-000000 VI ACKNOWLEDGEMENTS I would like to express my sicere appreciatio to Professor Vatti, Bassava Kumar, Departmet of Egieerig mathematics, College of Egieerig, Adhra Uiversity Visakhapatam 530 003, Idia My Studet Gashaye Dessalew ad Yilkal Abiyu ad also my fried Bekalu Tarekeg, Abraham Kassa ad Teshager Assefa for their commets ad suggestio for preparatio of the work
0 INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS, VOL 3, NO, FEBRUARY 07 TABLE III ALL NUMERICAL RESULT OF POSITIVE DEFINITE (PD) MATRIX FOR FDJ AND FDRJ OF EXAMPLE First degree Jacobi (FDJ) First degree refiemet Jacobi (FDRJ) () x () x () x () x 0333333 000000 75 50000 057778 333333 50000 057778 333333 098846 0986883 030093 3 0953704 097 0993056 099839 0998864 000643 4 098846 0986883 030093 0003 0999583 00070 5 005659 0986 009066 0999986 0999967 000039 6 099839 0998864 000643 0999998 0999968 000038 7 0999836 099950 00088 0999999 0999998 0999999 8 0003 0999583 00070 000000 000000 000000 9 099995 0999948 000048 0 0999999 0999968 000038 00000 0999987 000009 0999998 0999997 000005 3 0999999 0999999 000003 4 0999999 0999998 00000 5 000000 0999999 00000 6 000000 000000 000000 TABLE IV ALL NUMERICAL RESULT OF POSITIVE DEFINITE (PD) MATRIX FOR SDJ AND SDRJ OF EXAMPLE Secod degree Jacobi (SDJ) Secod degree refiemet Jacobi (SDFJ) () x () x () x () x 0333333 000000 75 50000 057778 333333 873 0543447 3790 099546 0989998 03334 3 099088 0970548 094744 099680 00843 0998534 4 096337 08536 00098 0004 099965 000073 5 00436 098790 03096 00009 0999970 000030 6 00434 099840 0997707 0999996 000003 000000 7 099899 00638 099936 000000 000000 000000 8 00088 0999700 0005 9 0007 099987 0999949 0 0999930 000070 099995 000004 0999996 00000 000008 099999 000000 3 0999997 000003 0999997 4 000000 000000 00000 5 000000 000000 000000