DEVELOPING COMPUTER PROGRAM FOR COMPUTING EIGENPAIRS OF 2 2 MATRICES AND 3 3 UPPER TRIANGULAR MATRICES USING THE SIMPLE ALGORITHM

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1 Fr Est Journl o Mthtil Sins (FJMS) Volu 6 Nur Pgs 8- Pulish Onlin: Sptr This ppr is vill onlin t Pushp Pulishing Hous DEVELOPING COMPUTER PROGRAM FOR COMPUTING EIGENPAIRS OF MATRICES AND UPPER TRIANGULAR MATRICES USING THE SIMPLE ALGORITHM IRAWATI n TITO WALUYO PURBOYO Algr Rsrh Group Fulty o Mthtis n Nturl Sins Institut Tknologi Bnung Bnung Inonsi -il: irwti@thiti Astrt In this ppr w invstigt th thory n th pross o oputing ignvlu n ignvtor using sipl lgorith n rliz it in oputr progr W opr th sipl lgorith with th lssil on to s th tivnss o oth lgoriths Th ritri or opring th two lgoriths r th sp n th ury o th oputr progr λ Mthtis Sujt Clssiition: 6F A8 Kywors n phrss: sipl lgorith ignvlus ignvtors Riv My Th Thory For th ollowing trix A lt us s th hrtristi qution ( ) λ ( ) With sustitut λ ( ) into th qution w gt ( ) ( )( ) n w hv Nxt w gt ( )

2 86 IRAWATI n TITO WALUYO PURBOYO So ( ) hs Δ ( ) s isriinnt But th hrtristi qution hs isriinnt Δ ( ) ( ) whih n hng into Δ n w hv Δ So Δ ( ) W n s tht th hrtristi qutions λ ( ) λ ( ) n ( ) hv th s isriinnt So w n us th qution ( ) to in th ignvlu n th ignvtor o A Proposition Lt A with R Thn () is th ignoupl o A i n only i is th init root o ( ) () Th oupl ( ) is th ignoupl o A i n only i is th root o () I t lst on o th oiints o ( ) is not zro thn its ignsp is o insion on I ll o th oiint o ( ) x r zro thn is n ignoupl o A or so x n y not ll y zro Proo Lt v n v For rl nur is th slop o v n th slop o v is ininit () ( ) Lt n ignoupl o A or init nur so w hv

3 DEVELOPING COMPUTER PROGRAM FOR COMPUTING 87 ( ) W gt So W hv ( ) So is init root o ( ) ( ) Lt init root o ( ) W hv ( ) So ( ) Th lst qution n writtn s An w hv ( ) It ns tht is n ignoupl o A () ( ) Lt n ignoupl o A

4 IRAWATI n TITO WALUYO PURBOYO 88 W hv So so th root o is Bus o w gt tht th root o is ininit So is root o ( ) Lt λ th root o Bus th root o is ininit n th root is thn w gt Lt Bus o is th ignoupl o A () Lt hv t lst on non-zro oiint so w hv hs t ost two roots Nxt lt us s λ n λ I v n v r ignvtors thn th ignvlu orrspons to v is n th ignvlu orrspons to v is

5 DEVELOPING COMPUTER PROGRAM FOR COMPUTING 89 Bus tht vry non-zro vtor y x in R n writtn uniquly s onstnt ultipl ithr o v or rl nur or o v tht is R s s y x ; or ; R s s y x Thn vry ignoupl or is on insionl For th son sttnt lt ll o th oiint o r zro n so λ n vry init is th root o In this s A For init w gt [ ] v For n ininit [ ] v So y x is n ignoupl o A or x n y not ll zro In th nxt proposition w xpn th rsult in tringulr tris A with ; R Proposition Lt A with ; R () Th oupl is n ignoupl o A i n only i is root o [ ]

6 IRAWATI n TITO WALUYO PURBOYO 9 () Th oupl is n ignoupl o A i n only i is init root o [ ] () Th oupl is n ignoupl o A i n only i is root o () Th oupl is n ignoupl o A i n only i is root o Proo () ( ) Lt is th ignoupl o A or init suh tht W gt It ns tht : us

7 DEVELOPING COMPUTER PROGRAM FOR COMPUTING 9 By writing w hv (us ) [ ] So [ ] An w gt is root o [ ] ( ) Lt is th root o [ ] W hv tht (us ) (us ) So

8 IRAWATI n TITO WALUYO PURBOYO 9 An w gt So is n ignoupl o A () ( ) Lt is n ignoupl o A or init rl nur suh tht So w hv This ns So [ ]

9 DEVELOPING COMPUTER PROGRAM FOR COMPUTING 9 W hv tht is init root o [ ] ( ) Lt is init root o [ ] Thn w hv So W hv An inlly w gt So is n ignoupl o A () ( ) Lt n ignoupl o A Thn So

10 IRAWATI n TITO WALUYO PURBOYO 9 As th root o w gt Bus o w gt tht hs n ininit root ( ) Lt λ th root o Fro w gt Lt So is n ignoupl o A () ( ) Lt n ignoupl o A W hv So

11 DEVELOPING COMPUTER PROGRAM FOR COMPUTING 9 Th roots o n gottn ro Bus o hs n ininit root So is root o ( ) Lt λ root o Bus o is th root o w gt Lt Bus o w hv is n ignsp o A Th Coputr Progr Th progr is writtn in Dlphi lngug Th oputr progr or lssil lgorith Th qution or th lssil lgorith onsists o:

12 96 IRAWATI n TITO WALUYO PURBOYO Th hrtristi qution λ t λ Th qution or gtting ignvlus λ ( ) ( ) For vry ignvlu λ w in th ignvtor ro th ollowing qution x Av λv whr v with x y R y Th oputr progr or th sipl lgorith Th qution or th sipl lgorith onsists o: Th qution Th root o th qution in ( ) ( ) ( ) I is init thn th ignoupl is I is ininit thn th ignoupl is

13 DEVELOPING COMPUTER PROGRAM FOR COMPUTING 97 Th sp oprison o th two progrs Th tris Mtrix Dt Th root in Th root o th hr polynoil Eignoupl Ti or sipl lgorith in illison Ti or lssil lgorith in illison Ininit n Ininit 9 9 n n 9 6 n n 6 9 i n i i n i i i i i 8 i n i i n i i i i i 6 It n sn tht th ti n or th sipl lgorith is shortr thn or th lssil lgorith

14 IRAWATI n TITO WALUYO PURBOYO 98 In th ollowing tl w opr th two lgoriths or tris Mtrix Dt Roots in Th root o th hrtristi polynoil Th ignoupl Ti or sipl progr in illison Ti or lssil progr in illison Ininit

15 DEVELOPING COMPUTER PROGRAM FOR COMPUTING Ininit It lso n sn tht th ti n or th sipl lgorith is shortr thn or th lssil lgorith Rrns [] R B Ash A Prir o Astrt Mthtis Th Mthtil Assoition o Ari Wshington 998 pp 7- [] D S Duit n R M Foot Astrt Algr Prnti Hll Nw Jrsy 99 pp -8 [] J L Golrg Mtrix Thory with Applition MGrw-Hill Nw York 99 pp -9

16 IRAWATI n TITO WALUYO PURBOYO [] M Goluitsky n M Dllnitz Linr Algr n Dirntil Equtions using MATLAB Brooks/Col Pulishing Copny Cliorni 999 pp -67 [] B Jo Linr Algr W H Frn n Copny Nw York 99 pp 6-6 [6] A Jons S A Morris n K R Prson Astrt Algr n Fous Ipossiilitis Springr-Vrlg Nw York 99 pp - [7] S Lng Introution to Linr Algr Springr-Vrlg Nw York 98 pp -8 [8] S Lipshutz n M L Lipson Aljr Linir Pnrit Erlngg Jkrt pp 7-68 [9] T A Nwton A sipl lgorith or ining ignvlus n ignvtors or tris Ar Mth Monthly 97() (99) [] S Ron Avn Linr Algr Springr-Vrlg Nw York Vol 99 pp -6 [] D S Wtkins Funntls o Mtrix Coputtions John Wily n Sons In Cn 99 pp 99-9