PEMECAHAN SISTEM PERSAMAAN LINIER NON HOMOGEN DENGAN METODE SAPUAN GANDA CHOLESKY. Oleh : Yusup Fakultas Ilmu Komputer, Universitas AKI Semarang

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1 PEMECAHAN SISTEM PERSAMAAN LINIER NON HOMOGEN DENGAN METODE SAPUAN GANDA CHOLESKY Oleh : Yusup Fkults Ilmu Komputer, Uversts AKI Semrg Astrt The frto of No Homoge Lerty Ajustmet System towr Cholesky Doule Sweepg Metho s y susttutg the gre of x row of lerty justmet system. After tht t outs the oefey of P from gre to, s the frst sweepg step. As t rehes pot, the lulto s oute o the otrry, tht s from to, to out ukow x. umer. Keywors : Trgol Mtrx, Ler Equto Systems, Cholesky. Pehulu Persm Ler serg pk lm proses lss, es stess r sstem perekys. Seuh Sstem Persm Ler yg terr r m uh persm ler eg uh lg yg tk kethu pt ytk eg : m x, x,, x lh lglg yg tk kethu, meytk kott-kostt. D pl Sstem Persm Ler terseut ytk eg perkl mtrks AX B, mk pt ytk eg : sehgg peroleh mtrks Jk B (mtrks ol) mk Sstem Persm Ler tu seut eg Sstem Persm Ler Homoge. Segk jk B, mk Sstem Persm Ler terseut seut eg Sstem Persm Ler No Homoge. Persm Ler terseut k mempuy (tg) kemugk peyeles, ytu :. Peyeles yg uk, jk A, --

2 Pemeh SPL No Hmoge egmetoe Spu G Cholesky (Yusup). Tk mempuy peyeles jk A B,. Peyeles leh r stu jk A B. Peyeles (Solus) r Sstem Persm Ler terseut lh jk terpt hmpu lg k, k,, k yg merupk l r vrelvrel yg tk kethu (x), erlku x k, x k,, x k, sehgg semu persm ler terpeuh. Solus terseut pt sjk lm etuk vektor yg seut eg vektor solus. Bg erkut memuhk utuk meptk gmr ser umum tetg Sstem Persm Ler. Sstem Persm Ler AX B Homoge B Tk Homoge B Sellu A Solus Tk A Solus r(a) r(a,b) A Solus r(a) r(a,b) Solus Trvl (Solus Nol) Solus No Trvl Solus Tuggl r Solus Byk r < Gmr. Gmr Umum Sstem Persm Ler Dm : jumlh vrel yg tk kethu. r rk mtrks. Pg mtrks A eroro (mx) eg eleme-elemey lg rl : A m m m Tp-tp rs (kolom) r mtrks A pt pg seg vektor-vektor rs (kolom) r A, yg k -5-

3 Mjlh Ilmh INFORMATKA Vol. No. Jur memetuk rug vektor. Sehgg pt efsk hw rk rs (kolom) r mtrks A tuls r(a) lh mes r rug rs (kolom) mtrks A. Dmes rug vektor rs (kolom) mtrks A efsk seg jumlh mksmum vektor-vektor rs (kolom) yg es ler, segk setp hmpu vektor rs (kolom) yg es ler r rug vektor rs (kolom) ermes seut ss r rug vektor rs (kolom) tu. J rk mtrks meytk jumlh mksmum vektorvektor rs (kolom) yg es ler. Metoe Spu G Cholesky Dseut jug metoe peyeles lgsug, kre pemky muh mtrks trgol yk jump lm erg permslh terutm lm peyeles Sstem Persm Ler. Mtrks, Trgol lh mtrks yg mempuy eleme sm eg, keul p stu jlur yg erpust p gol utm, etuky seg erkut: A Dpg Sstem Persm Ler seg erkut: x x x x x x x x x x x x x () Brs pertm p persm () r sstem memugkk utuk meuls lg tk kethu x seg fugs lg tk kethu x lm etuk: eg P, l l x susttusk ke lm rs keu persm (), mk pt: x x + () tu x P x + -6-

4 Pemeh SPL No Hmoge egmetoe Spu G Cholesky (Yusup) ( x + tu ( ) ) + x + x + ) x x + ( pt pul tuls seg: x P x + eg P, persm meujukk hw x merupk fugs r x, lgkh sepert t pt ulg lg utuk semu rs p persm erkuty. Deg emk setp lg tk kethu pt ytk seg lg tk kethu erkuty. Msly telh peroleh persm seg erkut: x P x + Apl l x susttusk ke lm rs ke r sstem persm (), mk: (P x + ) + x + x + ( P + ) x + x + ( ) x x + ( P ) ( P ) Persm terseut ts pt tuls lm etuk: x P x + + () eg: P () ( P ) () ) ( P Utuk, mk persm (), mej: x P x + () eg: P ( P ) ( P ) () () Perg persm () (), meujukk hw: P (5) Persm () (5), memugkk utuk meghtug koefse P sert r l smp, lgkh merupk spu pertm. Setelh smp ttk ke htug lkuk lm rh kelky, ytu r ke, utuk meghtug lg tk kethu x. Utuk tu persm terkhr r sstem persm () tuls lm etuk: -7-

5 Mjlh Ilmh INFORMATKA Vol. No. Jur x + x (6) P sstem persm (), pl, mk: x P x + (7) Susttus r persm (7) ke lm persm (6), k memerk: (P x + ) + x ( P + ) x x ( P ) Sesu eg persm (), mk: x. Nl x pt peroleh, ersrk l x yg pt mk l x pt htug pul eg persm seg erkut: x P x +. Dr l x kemu htug l x, x, seterusy hgg ke l x. x x x x 6x x x x x x 7 () 7 Peyeles: Sstem persm ts pt tuls lm etuk mtrks trgol, yg peyelesy pt lkuk eg megguk persm erkut: x P x + + () eg: P () ( P ) () ) ( P Skem peyeles sstem persm eg metoe spu g seg erkut: Cotoh sol: Selesk sstem persm erkut eg megguk metoe spu g. P, (,,,) P, P, P, P, x x x x x (,,,) -8-

6 Pemeh SPL No Hmoge egmetoe Spu G Cholesky (Yusup) Lgkh pertm htug l P (,,, ) r kr ke k. Setelh smp ke ttk, htug l x. Bersrk l x terseut, kemu htug ljutk r k ke kr utuk meptk l x (,,, ). ) Meghtug koefse P (,,, ) Koefse P htug eg megguk persm () (), ersrk sstem persm (). Utuk, P. P P,5. P 7 7,5. Utuk, P,5,5. P P 6.,5 P,5,5 7. ( ) (,5) (,5) Utuk, P 6 7. P P,9. P 69, (6) ( ) 7 (6( 7)) 6 Utuk, P ( P x, mk: ) ((,9759)) (,9) ( ),588,588,. P Setelh l P (,,, ) pt, llu htug l x (,,, ). ) Meghtug x (,,, ) Vrel x (,,, ) htug eg megguk persm (): x P x + + Utuk, mk x,. -9-

7 Mjlh Ilmh INFORMATKA Vol. No. Jur Utuk, mk x P x + (,9(,)) +,9759 5,. Utuk, mk x P x + (6(5,)) + (7),. Utuk, mk x P x + (,5(,)) +,5,. Deg emk hsl yg peroleh lh: x,; x,; x 5,; x,. Utuk megethu er tu tky hsl yg peroleh, mk l-l terseut msukk ke lm persm yg telh selesk. (,) +, 7 ( 7), +, (5,) ( ) 6 (,) (5,) + (,) 7 ( 7) (5,) (,) ( ) Dftr Pustk Chpr Steve C, Cle Rymo P, 6, Numerl Methos for Egeeers, Ffth Eto, M Grw Hll I, New York. Chrles G. Culle (lh hs oleh Bmg Sumtr, Ir.), 99, Aljr Ler eg Peerpy, PT. Grme Pustk Utm, Jkrt..9.5./uplo/fles/6_B_.o (Se, jur ) --