Formulated Algorithm for Computing Dominant Eigenvalue. and the Corresponding Eigenvector

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1 Int. J. Contemp. Mth. Scences Vol no HIKARI Ltd Formulted Algorthm for Computng Domnnt Egenlue nd the Correspondng Egenector Igob Dod Knu Deprtment of Mthemtcs/Sttstcs Federl Unersty Wukr Trb Stte Nger mndodossgob@yhoo.com Copyrght 23 Igob Dod Knu. Ths s n open ccess rtcle dstrbuted under the Crete Commons Attrbuton Lcense whch permts unrestrcted use dstrbuton nd reproducton n ny medum proded the orgnl work s properly cted. Abstrct The chllenge of computng the domnnt egenlue nd ts correspondng egenector of rel mtrx s consdered n ths work. An lgorthm wth strght forwrd procedure (whch conerges fster to desred soluton) usng the power method concept for computng the domnnt egenlues of rel mtrx s formulted. Applcton of the lgorthm on 3x3 rel mtrx yeld the sme egenlue nd correspondng egenectors lues s the power method (e ( ) T ). Ths confrmed the sutblty of the lgorthm Keywords: Algorthm power method egenlue egenector

2 9 Igob Dod Knu. INTRODUCTION In system nlyss of mthemtcl equton models the determnton of the egenlues F nd egenectors of the system equtons s mportnt [8]. Reserched work on lgorthm for fndng the egenlues nd the correspondng egenectors of systems equton hs been on the ncresed. Mny mthemtcl scentsts he formulted theores n bd to prode better lgorthms n hndlng ths chllenge (see;[] [3] [4][5] [6]). Aston [2] ews the Jcob s lgorthm s the smplest lgorthm for fndng egenlues of symmetry mtrces. It s relble method tht produces unformly ccurte result for mtrces of up to order 2 x 2. The conceptul frmework of the Jcob s lgorthm s used n the formulton of other more sophstcted lgorthms. Chpr et l [7] stted tht the Jcob s lgorthm s tme consumng snce t requres longer tme for the off dgonl terms to be suffcently smll. Consequently the Gen s lgorthm whch s fnte nd thus more effcent ws preferred to the Jcob s lgorthm n [7]. Other lgorthm for computng egenlues nclude; the Lnczos n [4] the Monte Crlo lgorthm n [3] the QR lgorthm n [] [6]. In the reewed on the rous formulted lgorthms so fr the notceble chllenge s the number of tertons noled n ny method used for the system soluton to conerge to the desred soluton. Thus the m of ths work s to deelop n lgorthm wth strght forwrd procedure (whch conerges fster to desred soluton) usng the power method concept for computng the domnnt egenlues of rel mtrx. Wth slght modfcton the lgorthm cn be used to determne the smllest nd the ntermedte egenlues. It hs the ddtonl dntge tht the correspondng egenector s obtned s by-product of the method.. Power Method The power method s n terte mechnsm for pproxmtng selected egenlues of gen coeffcent mtrx. Oler [6] stted the terte procedure for n n n mtrx A by consderng the egenector bss 2... n wth ts correspondng egenlues.the soluton to the lner system A m+ m (.) s obtned by multplyng the ntl ector by the successe power of the coeffcent mtrx: m m A. Wrtng the ntl ector n terms of the egenector bss s

3 Formulted lgorthm for computng domnnt egenlue 9 c + c c n n (.) then the soluton tkes the form of m m m m m A c + c cnn n. (.2) Suppose A hs sngle domnnt rel egenlue tht s lrger thn others n mgntude such tht > for 2... n then m m > > nd m >. Therefore the soluton of the terton of (.) wll conerge t the domnnt egenector of the coeffcent mtrx. The correspondng egenlue of the th entry of the terte of m s pproxmted by consderng m m c for m > nd m. (.3) The rte of conergence of the method s goerned by the rto nd domnnt egenlues. between the subordnte.2 ALGORITHM FORMULATION In the formulton of ths lgorthm the power method s presented by [6] s used s frme work. It s presented n stges. Stge -noles the settng of ntl lues of the egenectors (SUM SUM 2 SUM 3 SUM n ). Stge 2 -nputs ector mtrx. Stge 3 -nputs rbles for j n. All the elements of the rry s set to. Stge 4 s method coded cll method. It noles mong executon to pre-defned sequence of nstructons - b c d e nd f respectely. Stge 4 4c noles seres of row nd column multplctons where the lues or results re stored by the rbles erler defned n stge. In stge 4 d Sum Sum 2 Sum n re set on 2... n. Stge 4 e s the normlzton stge where decson on the temporry lue of s determned ( s the desred egenlues).

4 92 Igob Dod Knu In stge 4 f the egenlues/egenectors re prnted out. Stge 5 - the lue of s ssgned to rble k. Stge 6 -mples recllng gn on the method cll. Note tht n ths stge new lue of s dered. Stge 7 - stoppng crteron s estblshed. At ths pont the desred domnnt egenlues s cheed nd PRINT sttement to prnts the egenlues/egenectors ssued. Stges 8 -f the condton n stge7 does not hold decson to goto stge 5 s crred out. Stge 9 -termnton stge of the lgorthm. ALGORITHM Stge SUM SUM 2... SUM n Stge 2 For to n For j to n Input A j Stge 3 For to n Stge 4 Method cll Stge 5 * Set k Stge 6 Repet Cll Stge 7 {f (k ) / β prnt the boe s the domnnt egenlues wth ts correspondng ectors } goto * Stge 8 Else Goto * Stge 9 ** End

5 Formulted lgorthm for computng domnnt egenlue 93 For to n { j 4 Sum Sum + ( A j ) For to n { j 4 b Sum 2 Sum 2 + ( A j ) For to n { j 4 c Sum n Sum n + A ) ( j METHOD cll Sum 2 Sum 2 4 d n Sum n < then If 2 If 2 < n then { n Else } 2 2 n n n { 2 4 e n n } Else f < n { n } 2 2 n n n

6 94 Igob Dod Knu Else { 2 2 n n Prnt Egenlues Prnt Prnt Egenector For to n 4 f { Prnt } THE ALGORITHM FLOWCHART

7 Formulted lgorthm for computng domnnt egenlue 95 START Set Sum Sum 2 Sum n For to n j to n Input A j For to n Input A j Method cll Set k Method cll (k ) / β A

8 96 Igob Dod Knu A Set Sum 2 Sum 2 n Sum n If < 2 n <... < 2 <... n 2 n n n 2 2 n n n n n 2 2 Prnt For to n Prnt End Method

9 Formulted lgorthm for computng domnnt egenlue 97.3 ILLUSTRATION Consder 3x3 mtrx:.5 A. (.4) Applyng the power method; A A A Ths contnues untl the rto conerges.

10 98 Igob Dod Knu.4 USING THE FORMULATED ALGORITHM Usng equton (.4) A yelds Sum Sum 2 nd Sum 3 respectely. These rbles re ll sets to zero snce they re not known t the begnnng of computton s n stge of the lgorthm. Stge 2 -elements of A re entered ccordngly for the th row nd j th column. Stge 3 -elements of Stge 4 - the method cll perform the seres of multplcton nd dson opertons whch s subdded nto b c d e nd f. re set to ( ) T. b. c : : : : : : : : : Ψ Sum (.5*) + ( *) + (*) Sum 2.5 Ψ Sum2 Sum2 Ψ Sum ( *) + (*) + ( *) Sum3 2 ( *) + (*) + (*) T d. ( 2.5 ). e & f. - decson mkng stges where the hghest entry of the new ector s computed nd set s. In nomlsng ech entry s dded by nd ts result stored bck s wth the lrgest egenlues new Stge 5 - the lue of s ssgned new rble k. Stge 6 - the method Cll s repeted gn (e stge 4 4 f ) Stge 7 9 k f.

11 Formulted lgorthm for computng domnnt egenlue 99 then prnt ( ( ) T correspondng ectors ) s the egenlues nd the Tble: SUMMARY OF RESULTS Power method conergence condton Formulted lgorthm stoppng k creter.5 CONCLUSION Results of tble shows tht computton of egenlue nd the correspondng egenector usng the formulted lgorthm yeld the sme lues s the power method (e ( ) T ) wth the power method conergng crteron k nd lgorthm stoppng crteron. Ths llustrton confrmed the sutblty of the lgorthm.

12 9 Igob Dod Knu References [] D. S. Wtkns The QR Algorthm rested SIAM reew 5 no. (28) [2] F. S. Aston Numercl Method tht works Second Edton McGrw-Hll New York99. [3] I.Dmo V. Alexndro A. Krno Implementton of Monte Crlo Algorthms for Egenlue problem usng MPI f_monte_cl(998). [4] J. Brkmejer Approxmtng domnnt Egenlues nd Egenectors of the locl forcst error cornce mtrx Tellus 47 (995) [5] N. L.Amy D. M. Crl A surey of Egenector methods for web nformton meyer.mth.ncsu.edu/meyer/ps_fles/ surey.pdf 28. [6] P. J. Oler Numercl Anlyss Lecture note (28). [7] S. C. Chppr R. P.Cnle Numercl method for Engneers Second EdtonMcGrw- Hll New York 985. [8] V. A. Ihegwm J. U. Onwutu Fundmentl of Numercl Anlyss Second Edton Dne Prntng nd Publshng Co. Nger 2. Receed: June 23