LU FACTORIZATION. ELM1222 Numerical Analysis. ELM1222 Numerical Analysis Dr Muharrem Mercimek

Transcription

1 EM Nmeric Asis Dr Mhrrem Mercimek FACTORIZATION EM Nmeric Asis Some of the cotets re dopted from ree V. Fsett, Appied Nmeric Asis sig MATAB. Pretice H Ic., 999

2 EM Nmeric Asis Dr Mhrrem Mercimek Cotets Fctoriztio from Gssi Eimitio Fctoriztio with Pivotig Direct Fctoriztio Appictios of Fctoriztio

3 Step : 9 Step : Fctoriztio from Gssi Eimitio Empe : EM Nmeric Asis Dr Mhrrem Mercimek r r + r r + r Mtip with (-) Pt to the sme octio t r r r + r Mtip with (-) Pt to the sme octio t

4 A,, Step : 9 Step : Mtip b =. = = A Fctoriztio from Gssi Eimitio Empe : EM Nmeric Asis Dr Mhrrem Mercimek

5 9 9 7 A Step : Row of is chged, d rows - re modified to give / / / 9 / / / Empe : Fctoriztio from Gssi Eimitio EM Nmeric Asis Dr Mhrrem Mercimek

6 Step : Row d of re chged, row d re trsformed sig row 7 Step : The forth row of is modified to compete the forwrd eimitio stge: / / / / / / / / / / / Fctoriztio from Gssi Eimitio EM Nmeric Asis Dr Mhrrem Mercimek 9 / / /

7 For verifictio =. = A Fctoriztio from Gssi Eimitio 7 EM Nmeric Asis Dr Mhrrem Mercimek

8 EM Nmeric Asis Dr Mhrrem Mercimek Fctoriztio from Gssi Eimitio Empe i V R R Fow rod the eft oop ( i - i ) + ( i - i ) = V i R Fow rod pper oop i + ( i - i ) + ( i - i ) = R i R Fow rod ower oop i + ( i - i ) + ( i - i ) = V

9 EM Nmeric Asis Dr Mhrrem Mercimek 9 Fctoriztio from Gssi Eimitio i i i V i i i V i i i V A Step : / / / / / / Step : / / / / /

10 EM Nmeric Asis Dr Mhrrem Mercimek Fctoriztio from Gssi Eimitio from differet poit of view c. = d d d Here, d = c +, d =c +, d d = c +. We write this s CA = D We so eed the iverse of the mtri C c. c = C -. C = I., or

11 Fctoriztio from Gssi Eimitio - Discssio m m. m m = M -. M = I. m. m = M -. M = I. m m m. = m m m M -. M - = EM Nmeric Asis Dr Mhrrem Mercimek

12 Fctoriztio with pivotig P where / / A P / / / / A EM Nmeric Asis Dr Mhrrem Mercimek Empe We cot cotie We hve to pp row pivotig For d d rd rows re iterchged For O ower digo eemets of d d rd rows re iterchged ower digo eemets re the eemets beow the digo?

13 Direct Fctoriztio ) ( ) (. j i j i e e jj ij j j j i j i ij ij ii j i i i j i ij EM Nmeric Asis Dr Mhrrem Mercimek

14 EM Nmeric Asis Dr Mhrrem Mercimek Direct Fctoriztio most commo forms of fctoriztio correspodig to three differet choices of the form of the digo eemets. For Dooitte s Method digo eemets of re s For Crot s Method digo eemets of re s For Chesk s Method digo eemets of d re eq

15 Dooitte empe.... EM Nmeric Asis Dr Mhrrem Mercimek Empe

16 Choesk empe.... EM Nmeric Asis Dr Mhrrem Mercimek Empe

17 EM Nmeric Asis Dr Mhrrem Mercimek 7 Appictios: Sovig Sstems of ier Eqtios A = b = b = b A =, = Sove the sstem = b for Forwrd sbstittio Sove the sstem = for Bckwrd sbstittio - If pivotig hs bee sed A=b PA=Pb, where PA=, Pb=c

18 )() ( /,, o forwrd sbstitti B / / / -/ / - - / / / b A b A b A Appictios: Sovig Sstems of ier Eqtios EM Nmeric Asis Dr Mhrrem Mercimek Empe 7

19 EM Nmeric Asis Dr Mhrrem Mercimek 9 Appictios: Sovig Sstems of ier Eqtios - / / B bckwrd sbstittio / /. ( /)(/) (/)[-(/) i / -(/)]. / i. i

20 EM Nmeric Asis Dr Mhrrem Mercimek Appictios: Determit of mtri If A, the det( A) i ii i ii If pivotig is sed, so tht A P det( A) ( ) i i where k is the mber of row iterchges tht occrred the fctoriztio k ii - ii Empe A det( A) ()( )()

21 EM Nmeric Asis Dr Mhrrem Mercimek Appictios: Iverse of mtri The iverse of -b- mtri A A i =e i (i=,,) e i =[ ], where the ppers i the ith positio X whose coms re the sotio vectors,, is A -

22 correspodig com of o, forwrd sbstitti : Ech com of for Sove Y I Y Y I Y A Appictios: Iverse of mtri EM Nmeric Asis Dr Mhrrem Mercimek Empe 9

23 EM Nmeric Asis Dr Mhrrem Mercimek Appictios: Iverse of mtri Sove X Y for X ; the A - X Ech com of X A / / / X : bck sbstittio, / / / / / / correspodig com of Y

24 EM Nmeric Asis Dr Mhrrem Mercimek MATAB s Method bit-i fctios decompositio of sqre mtri A [, ] = (A), [,, P] = (A) Choesk fctoriztio cho Determit of mtri det Iverse of mtri iv