Review Operasi Matriks. Menghitung invers matriks? Determinan? Matriks Singular?

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1 eview Opersi Mtriks Menghitung invers mtriks? Determinn? Mtriks Singulr?

2 Menghitung invers mtriks

3 Determinn Hny untuk squre mtries b d d b d b det Jik determinn mtriks singulr, tidk puny invers det b b b b b b b b b

4 Cri invers ny

5 Sistem Persmn Liner Simultneous Liner Equtions

6 Metode Penyelesin Metode grfik Eliminsi Guss Metode Guss Jourdn Metode Guss Seidel LU deomposition

7 Metode Grfik Det{A} A is nonsingulr so invertible Unique solution -

8 Sistem persmn yng tk terselesikn No solution Det [A], but system is inonsistent Then this system of equtions is not solvble

9 Sistem dengn solusi tk terbts Det{A} A is singulr infinite number of solutions 8 Consistent so solvble

10 Ill-onditioned system of equtions A liner system of equtions is sid to be illonditioned if the oeffiient mtri tends to be singulr

11 Ill-onditioned system of equtions A smll devition in the entries of A mtri, uses lrge devition in the solution

12 Gussin Elimintion Merupkn slh stu teknik pling populer dlm menyelesikn sistem persmn liner dlm bentuk: [ A ][ X] [ C] Terdiri dri du step. Forwrd Elimintion of Unknowns.. Bk Substitution

13 Forwrd Elimintion Tujun Forwrd Elimintion dlh untuk membentuk mtriks koefisien menjdi Upper Tringulr Mtri 8.8..

14 Forwrd Elimintion Persmn liner n persmn dengn n vribel yng tk dikethui... n n b n n... b... n n n nn n b n

15 Contoh 8 mtriks input 8 8

16 Forwrd Elimintion

17 Forwrd Elimintion

18 Bk substitution 9 9

19 Guss - Jourdn 9 8 8

20 Wrning.. Du kemungkinn keslhn -Pembgin dengn nol mungkin terjdi pd lngkh forwrd elimintion. Mislkn: Kemungkinn error kren round-off (keslhn pembultn)

21 Contoh Dri sistem persmn liner Akhir dri Forwrd Elimintion....

22 Keslhn yng mungkin terjdi Bk Substitution

23 Contoh keslhn Bndung-kn solusi et dengn hsil perhitungn [ ] X [ ] X lulted [ ] X et

24 Improvements Menmbh jumlh ngk penting Mengurngi round-off error (keslhn pembultn) Tidk menghindrkn pembgin dengn nol Gussin Elimintion with Prtil Pivoting Menghindrkn pembgin dengn nol Mengurngi round-off error

25 Pivoting Eliminsi Guss dengn prtil pivoting mengubh tt urutn bris untuk bis mengpliksikn Eliminsi Guss ser Norml How? Di wl sebelum lngkh ke-k pd forwrd elimintion, temukn ngk mksimum dri: kk, k, k,......, nk Jik nili mksimumny Pd bris ke p, pk k p n, Mk tukr bris p dn k.

26 Prtil Pivoting Wht does it Men? Gussin Elimintion with Prtil Pivoting ensures tht eh step of Forwrd Elimintion is performed with the pivoting element kk hving the lrgest bsolute vlue. Jdi, Kit mengeek pd setip lngkh pkh ngk pling ts (pivoting element) dlh sellu pling besr

27 Prtil Pivoting: Emple Consider the system of equtions.9.99 In mtri form.99.9 Solve using Gussin Elimintion with Prtil Pivoting using five signifint digits with hopping

28 Prtil Pivoting: Emple Forwrd Elimintion: Step Emining the vlues of the first olumn, -, nd or,, nd The lrgest bsolute vlue is, whih mens, to follow the rules of Prtil Pivoting, we don t need to swith the rows Performing Forwrd Elimintion

29 Prtil Pivoting: Emple Forwrd Elimintion: Step Emining the vlues of the first olumn -. nd. or. nd. The lrgest bsolute vlue is., so row is swithed with row Performing the row swp

30 Prtil Pivoting: Emple Forwrd Elimintion: Step Performing the Forwrd Elimintion results in:....

31 Prtil Pivoting: Emple Bk Substitution Solving the equtions through bk substitution

32 Prtil Pivoting: Emple Compre the lulted nd et solution The ft tht they re equl is oinidene, but it does illustrte the dvntge of Prtil Pivoting [ ] X et [ ] X lulted

33 Summry -Forwrd Elimintion -Bk Substitution -Pitflls -Improvements -Prtil Pivoting

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