The linear system. The problem: solve

Transcription

1 The ler syste The prole: solve Suppose A s vertle, the there ests uue soluto How to effetly opute the soluto uerlly??? A A A

2 evew of dret ethods Guss elto wth pvotg Meory ost: O^ Coputtol ost: O^ C oly e used for sll whe A s dese, e.g. < deoposto Meory ost: O^ Coputtol ost: O^ C oly e used for sll whe A s dese, e.g. < Good for prole to solve the ler syste wth dfferet rght hd oplete, Cholesy deoposto;

3 evew of dret ethods For tr-dgol tr Thos lgorth sed o Crout ftorzto Meory ost: O & Coputtol ost: O C e eteded to d-lted tr For ler syste fro dsretzto of Posso euto y FM ret Posso solver sed o FFT Meory ost: O & Coputtol ost: O l For ler syste fro dsretzto of ellpt euto y FEM Multgrd ethod MG or Alger Multgrd ethod AMG Meory ost: O & Coputtol ost: O For ler syste fro dsretzto of Posso euto y tegrl forulto Fst Multpole ethod Meory ost: O & Coputtol ost: O

4 tertve ethods A: to solve lrge sprse ler syste Bs tertve ethods A o ethod Guss-Sedel ethod Suessve over-relto ethod SO Krylov suspe oder tertve ethods Steepest deet ethod Cougte grdet CG ethod GMES for osyetr ehtod φ T T α d φ : A

5 Bs tertve ethods ewrte A A

6 o tertve ethod The ler syste Euto for Mtr for, to for : A A &

7 o tertve ethod A eple The ethod tl guess A A

8 o tertve ethod The results Ths ge ot urretly e dsplyed. Ths ge ot urretly e dsplyed

9 Guss-Sedel ethod de: sed the ew vlues whe they re vlle Euto for Mtr for to for :

10 Guss-Sedel ethod A eple The ethod 5 A A tl guess...

11 Guss-Sedel ethod The results Ths ge ot urretly e dsplyed. Ths ge ot urretly e dsplyed

12 SO ethod de: To prove the Guss-Sedel ethod y ler otoof the old vlue d ew vlue Euto for Mtr for to for : ] [ ] [ ] [ SO SO

13 Covergee lyss Geerl for of s tertve ethods Et soluto efe the error t the -th terto e Error eutos e e e e :

14 Covergee lyss Covergee e: For y sure tr, there ests osgulr tr T suh tht ord ol for e e { } { } r r C T T T T,, dg,, dg

15 Covergee lyss efto: Spetrl rdus of ρ Egevlues of e: For y sure tr, ρ < < Theore: The tertve ethod overges to the et soluto of A ρ ff < <

16 Covergee rte Th: For the tertve ethod suppose the The tertve ethod overges er overgee rte wth < Error oud : < &

17 Proof for overgee rte Ft Error oud Aother error oud Error oud

18 Covergee results f A s strtly row dgolly dot, the oth o d Guss-Sedel ethods overge. Guss-Sedel ethod overges f A s syetr postve defte The relto preter e, s the eessry odto for the overgee of SO ethod. ddto, f A s syetr postve defte, the the odto s lso suffet for the overgee of SO ethod

19 Covergee results efto: A s strtly row dgolly dot f Eples Th: f A s strtly row dgolly dot, t s vertle!,, < > B A

20 Covergee results Th: f A s strtly row dgolly dot, the oth o d Guss-Sedel ethods overge. ft, Proof < : ρ det f det s strtly row dgolly dot f det det det det : < < < ρ

21 Covergee results Th: et A e syetr postve defte tr, the the Guss-Sedel ethod overges for y tl guess. Proof: See detls lss er: There re ler syste, for whh the o ethod overges, ut the Guss-Sedel ethod dverges, e.g > < A ρ ρ

22 Covergee results Th: For SO ethod, we hve Thus the relto preter e, s eessry for SO overge Proof: ρ SO SO det det det det ρ SO SO SO

23 Covergee results Th: f A s syetr postve defte, the for,. Tht s, SO overges for ll ρ < SO, Proof: See detls lss er: Over relto: der relto: < < < < Optl relto preter: opt ρ