A New Refinement of Jacobi Method for Solution of Linear System Equations AX=b

Transcription

1 Int J Contemp Math Scences, Vol 3, 28, no 17, A New Refnement of Jacob Method for Soluton of Lnear System Equatons AX=b F Naem Dafchah Department of Mathematcs, Faculty of Scences Unversty of Gulan, PO Box , Rasht, Iran f naemdafchah@yahoocom Abstract The Jacob method s one of the methods wth a few computatons, but ts rate of convergence s low In ths paper we present a refnement for ths method whch ncreases ts rate of convergence up to the rate of convergence of SOR method (e more than Gause-sedel method) Snce fndng optmal parameter ω n SOR method s dffcult, ths method can be used nstead of SORAt the end we apply ths method to solve the Posson s partal dfferental equatons Mathematcs Subject Classfcaton: 65F1, 93C5, 41A25 Keywords: The Jocob method; Convergence; Row strctly dagonally domnant matrx 1 Introducton The basc dea behnd an teratve method s frst to wrte the system AX=b n equvalent form X=BX+d Then startng wth an ntal approxmaton x (1) of the soluton vector x, generate a sequence of approxmatons {x(k)} teratvely defned by = Bx (k) + d,k=1,2, wth a hope that under certan mld condtons, the sequence {x(k)}converges to the soluton as k To solve the lnear system Ax = b teratvely usng the dea, we therefore need to know 1 How to wrte the system Ax=b n the form of x=bx+d, and 2 How X (1) should be chosen so that teraton X (k+1) = BX (k) + d converges to the lmt or under what sort of assumptons the teraton converges to the lmt wth any arbtrary choce of x (1)

2 82 Farhad Naem Dafchah 2 Prelmnary Notes The system AX=b or a 11 x 1 + a 12 x a 1n x n = b 1 a n1 x 1 + a n2 x a nn x n = b n (1) Can be rewrtten (under the assumpton that a =1,, n) as x 1 = 1 a 11 (b 1 a 12 x 2 a 1n x n ) x n = 1 a nn (b n a n1 x 1 a n,n 1 x n 1 ) (2) In matrx notaton [1] x 1 x 2 x n = a 12 a 11 a 1n a 11 a 21 a 22 a 23 a 22 a 2n a 22 a n1 a nn a n,n 1 a nn x 1 x 2 x n + b 1 a 11 b 2 a 22 b n a nn (3) Or X=BX+d If we wrte the matrx A n the form A=L+D+U where L= a 21 a 32 a n1 a n2 a n,n 1 D=dag( a 11 a 22 a nn ) Then t s easy to see that [2],U = a 12 a 1n a n 1,n, B = D 1 (L + U),d= D 1 b (4) We shall call the matrx B = D 1 (L + U)the Jacob teraton matrx and denote t by B J Smlarly; we shall denote the vector D 1 bby b J,whch we call the Jacob vector

3 Jacob method for soluton of lnear system equatons 821 Wth the Jacob teraton matrx and Jacob vector as defned, the Jacob teraton n the matrx and computng form becomes the followng, X (k+1) = D 1 (L + U)X (k) + D 1 b (5) = 1 a (b j =1 j a j x (k) j ),=1n (6) 3 Man Results These are the man results of the paper ALGORITHM 31 ALGORITHM of the Jacob method Step1: X (1) = ( ) x (1) 1 x (1) n Step2: for k from 1 to untl a stoppng crteron s satsfed do = 1 a (b j =1 j a j x (k) j ),=1n end do; In ths formula because there s no exst a new ndex n the rght sde usually the rate of Jacob method s low We solve ths problem by a refnement formula Corollary 32 If A s row strctly dagonally domnant matrx,then the Jacob method converges, for any arbtrary choce of the ntal approxmaton x (1) Proof: See [2] Refnement of Jacob method Assume X (1) be an ntal approxmaton for soluton of the lnear system AX=b and b (1) = j=1 a j x (1) j,=1n After soluton of k-step of algorthm 31 we have,

4 822 Farhad Naem Dafchah X (k+1) = ( ) 1 (k+1) n And follow that we obtan b Now, before gong to the next step of algorthm 31, we refne ths obtaned soluton The man dea n ths paper s to tend b (k+1) to b Now assume X (k+1) = ( 1 n ) be a good approxmaton for soluton of lnear system AX=b Because we want that to be close to the real soluton so we have b = j=1 a j j,=1n (7) But all arraes of are unknown We solve ths problem by usng j, j =1 1, +1 n that obtaned n last step from Jacob method nstead of j, j =1 1, +1 n Now we use from above technc and b (k+1) factor for transformaton (7) formula to a smple refnement of formula b = b (k+1) a + a = + 1 a (b b (k+1) ) Thus,f after every step of Jacob method we use from refnement formula (8) and the obtaned vector was used n next step of teratve Jacob method then Frstly ths method s convergent and Secondly the rate of convergence of ths method s much more than the rate of convergence of Jacob method These clams wll be proved n ths paper ALGORITHM 33 ALGORITHM of refnement of Jacob method Step1: choose X (1) Step 2: for k from1 to untl a stoppng crteron s satsfed do for from 1 to n do = 1 a (b n j =1 j end do; for from 1 to n do b (k+1) = a j x (k) j ),=1n j=1 a j j,=1n (8) end do; end do; = + 1 a (b b (k+1) )

5 Jacob method for soluton of lnear system equatons 823 Refnement of Jacob method n matrx form AX = b (L + D + U)X = b DX = (L + U)X + b DX =(D A)X + b DX = DX +(b AX) X = X + D 1 (b AX) From above form we obtan the teratve refnement of formula n matrx form X (k+1) = X (k+1) + D 1 (b AX (k+1) ) (9) From (5) obtan the refnement of Jacob method n matrx form X (k+1) =(D 1 (L + U)) 2 X (k) +(I D 1 (L + U))D 1 b (1) We shall call the matrx (D 1 (L + U)) 2 the refnement of Jacob teraton matrx and denote t by B J Smlarly, we shall denote the vector (I D 1 (L + U))D 1 b by b J, whch call the refnement of Jacob vector Relatonshp between B J andb J, b J and b J b J =(I + B J )b J, B J = B 2 J (11) Convergence and Rate Theorems For Refnement Of Jacob Method Theorem 34 If A s row strctly dagonally domnant matrx then the refnement of Jacob method converges for any arbtrary choce of the ntal approxmaton X (1) Proof Assume X s the real soluton of lnear system AX=b Because matrx A n row strctly dagonally domnant, from corollary 32 the Jacob method s convergent then we can close X (k+1) to Xarbtrary thus we have X(k+1) X X (k+1) X + D 1 b AX (k+1) From X (k+1) X we have b AX (k+1) so X (k+1) X So the refnement of Jacob method converges to soluton of lnear system AX=b 1 Theorem 35 If A s row strctly dagonally domnant matrx then B J < Proof See [2]

6 824 Farhad Naem Dafchah Theorem 36 If A s row strctly dagonally domnant matrx, then B J = B J 2 < 1 Proof (D 1 (L + U)) 2 = D 1 (L + U) 2 = B J 2 < 1 Theorem 37 If A s row strctly dagonally domnant matrx then B J < B J Proof From theorem 36 we have B J = B J 2 < B J (12) Theorem 38 When Jacob and refnement of Jacob methods converge,the refnement of Jacob method converges faster than the Jacob method Proof Let X s the real soluton of lnear system AX=baccordng to the theorems 36 and 37 we have Because Xs the real soluton of system AX=b then B J X + b J = X so X (k+1) X = BJ 2(X(k) X)+b J X + B J X X (k+1) X = BJ 2(X(k) X) X (k+1) X B 2 J (X (k) X) X (k+1) X B J 2 (X (k) X) X (k+1) X B J 2k (X (1) X) On the other hand f we use the Jacob method and (5) we have X (k+1) X = B J X (k) + b J X X (k+1) X = B J (X (k) X)+B J X + b J X Because Xs the real soluton of lnear system AX=b, then we have X (k+1) X = B J (X (k) X) X (k+1) X (X B J (k) X) X (k+1) X B J k (X (1) X) Accordng to the coeffcents of above nequaltes and theorems 36 and 37, the refnement of Jacob method converge faster than the Jacob method Applcaton Defnton 39 The ( matrx A ) s 2-cyclc f there s a permutaton matrx P such that PAP T A11 A = 12 where A A 21 A 11 and A 22 are dagonal matrces 22

7 Jacob method for soluton of lnear system equatons 825 Example 31 Consder 2-cyclc matrx T = where T arses n the dscretzaton of the posson s equatons 2 T + 2 T = f x 2 2 y on the unt squarenow consder system TX = b where X = ( ) T ( ) T x 1 x 2 x 3 x 4 x 5 x 6 and b = 1 or x 1 x 2 x 3 x 4 x 5 x 6 = Because matrx T s row strctly dagonally domnant so the refnement of Jacob and Jacob methods are converge By usng from Jacob method and ntal approxmaton X (1) = ( ) T after 16-steps we have X (16) = ( ) T Ths result shows that the rate of convergence of Jacob method s low Now, by new algorthm of ths paper wth the same ntal approxmaton after 6-steps we have X (6) = ( ) T These results show that the rate of convergence of refnement of Jacob method s much more than Jacob method 1 Comparson of numercal tests We test the results of the new algorthm 33 for example 31 and compare them wth results of Jacob and gauss-sedel and SOR methods n tables 1,2,3,4 For ths work we used Maple 95 on a personal computer PENTIUM 4 wth the machne precson ε

8 826 Farhad Naem Dafchah X () Table 1: Jacob method for example X () Table 2: Refnement of Jacob method for example X () Table 3: gauss-sedel method for example31

9 Jacob method for soluton of lnear system equatons X () Table 4: SOR method for example31 Note 1 Tables 1 and 2 and 3 show that the refnement of Jacob method s very faster than Jacob and gauss-sedel methods Note 2 Tables 2 and 4 show that the refnement of Jacob method as fast as SOR method wth optmal parameterω, but n the refnement of Jacob method there s no exst problem of fndngω Conclusons The soluton of lnear systems s one of the most mportant subjects n Appled Scences and EngneerngIn ths paper we found a refnement for Jacob method for usng n soluton of lnear systemsths method n compare wth Jacob and Gauss-Sedel methods s faster and ts error n any level s less than those methodsths refnement of Jacob method s as fast as SOR method but, n compare wth SOR method s easer because we don t requre fndng ω (optmal parameter) References [1] DCLay, Lnear Algebra and ts Applcatons, New York (1994) [2] BNDatta, Numercal Lnear Algebra and Applcatons,Pacfc Grove: Calforna(1999) Receved: December 25, 27