2 Matrix Form SLIDE 12 Split a L-U L: Lower triangular Au=f becomes (D-L-t)u=f Iterative (D-L)ur+l=Uur+f The above matri form assumes that we are updating through our unknowns in ascending order. If we were to update in reverse order, i.e., the last unknown firs Note 2 Updating order We see that, unlike in the jacobi method. the order in which the unkowns are updated in Gauss-Seidel changes the result of the iterative procedure. One could sweep through the unknowns in ascending, descending, or alternate orders. The dure is called symmetric Gauss-Seidel Another effective strategy, known as red-black Gauss-Seidel iteration, is to up date the even unknowns first(red) (n2+1+u2-1+h2f2) and then the odd components(black) The red-black Gauss-Seidel iteration is popular for parallel computation since he red (black) points only require the black (red) points and these can be updated in any order. The method readily extends to multiple dimensions. In r instance, the red and black points are shown below 排
✕✗✖☞✕✗✖✟✘ ✙✛✚✢✜☎✣✥✤✧✦✩★✫✪✑✣✥✬ ✭✯✮✱✰✳✲✫✴✶✵✸✷ ✹✻✺✢✼✾✽❀✿❂❁ ❁❄❃❆❅❈❇❊❉❊❇●❋ ❍■❏ ❅▲❑◆▼❖✽✾P✸◗✸❘✸❙✑P❚✼ ❉❂❑◆❯✗❘✥❱❳❲☎❨❳✿❩❨❬✽☞P❚❙✢◗✎❭✢✼☞P❚❨ ❋❪❑◆❫❴✺✡✺✑❲✂❨✍✿❬❨❩✽✾P✸❙✢◗✸❭✡✼✾P✸❨ ❁❊❵✶❃❜❛❞❝✯❲✂❡☎❘✸❢❣❲✂❤❥✐✟❅❈❇❊❉❊❇●❋❧❦✯❵✶❃❄❛ ♠✿❬❲✂❨❩P❚✿❬✽✾♥✸❲♦❢❣❲♣✿❬q✡❘✻r ✐s❅❈❇t❉❳❦✻❵✈✉❩✇②①✍❃❄❋③❵❳✉✈④✶❛ ⑤⑦⑥✢⑧❣⑨✸⑩❷❶❚❸❹⑧❻❺✓⑨✥❼❾❽✝❿➁➀❖➂♣❶❚❽✝❺➃⑨✥➄❷➄❷➅➆❺✓⑧♣➄➇❼✧⑥✢⑨❚❼➉➈✈⑧❣⑨❚❽❬⑧❧➅➋➊✡➌✸⑨❚❼s❿✳➍✻➎➏❼✧⑥✻❽❩❶✥➅✸➎❹⑥➐❶❚➅➆❽➑➅➆➍✻➒❚➍✯❶✥➈➓➍✡➄♦❿✳➍ ⑨❹➄☎➔❷⑧☎➍✯➌✥❿✳➍✻➎➐❶❚❽❬➌✸⑧☎❽☎→➏➣✧➂❪➈✈⑧➏➈✈⑧♣❽❬⑧✓❼↔❶↕➅➋➊✡➌✸⑨❚❼↔⑧➏❿✳➍●❽❬⑧☎❸❹⑧☎❽❷➄☎⑧➙❶✥❽❩➌✸⑧☎❽❷➛❂❿✟→✳⑧✂→❀➛❖❼✳⑥✡⑧✓➜❀⑨✥➄❷❼❖➅➆➍✻➒❚➍✯❶❚➈➓➍ ➝❽❷➄✝❼s➛➓❼✳⑥✡⑧❧❿✳❼➞⑧♣❽❩⑨✥❼❾❿✟❶✥➍➐❺✓⑨✥❼❾❽✝❿➁➀➟➈✈❶✥➅➆➜❀➌➟⑩❷⑧❩➔❷❶❚❺❣⑧❖➠❖➡✫➢ ❃➤✐✟❅❈❇➥❋➑❦✝➦ ① ❉ → ➧t➨➫➩✝➭➲➯ ➳❂➵❂➸➫➺✗➩➼➻✟➽②➾✩➨➆➚✸➸✁➭❚➚ ➪❲❻❤❬❲☎❲➑✿❩q✡P✥✿➋➶✑❭✢❙✡✼❀✽✾➹✸❲❧✽✾❙▲✿❩q✢❲❻➘✱P✸❡☎❘✸❝✢✽✗❢❣❲☎✿❬q✢❘➆r➫➶✯✿❬q✡❲➴❘✸❨❷r➆❲✂❨❴✽❀❙▲❱✍q✡✽✾❡❷q▲✿❩q✢❲➴❭✡❙✢➹✸❘✥❱✍❙✡❤❂P❚❨❩❲ ❭✢✺⑦r✢P✥✿❩❲✂r➑✽✾❙❪➷➇P✸❭✡❤❩❤➮➬➮✹✻❲☎✽☞r➆❲✂✼✎❡❷q✡P✸❙✢◗✸❲➋❤➫✿❬q✡❲✈❨❩❲✂❤❬❭✢✼➁✿✁❘❚➱➆✿❩q✢❲➉✽❀✿❬❲✂❨❩P❚✿❬✽✾♥✸❲✃✺✡❨❬❘➆❡♣❲➋r➆❭✢❨❩❲✸❐✁❒❖❙✡❲➉❡♣❘✎❭✢✼☞r ❤❬❱➉❲✂❲☎✺❻✿❬q✢❨❩❘✸❭✢◗✎q❪✿❬q✢❲❴❭✡❙✢➹✻❙✢❘✥❱✍❙✡❤❮✽✾❙➏P✸❤❩❡♣❲☎❙✑r➆✽❀❙✡◗✡➶✸r➆❲➋❤❬❡☎❲☎❙✡r✢✽❀❙✢◗✑➶✸❘✎❨✈P❚✼❀✿❬❲✂❨❬❙✡P❚✿❬❲❰❘✸❨❷r➆❲✂❨❩❤✂❐②Ï❰q✢❲ ✼☞P✥✿➼✿❩❲☎❨✍✺✢❨❩❘➆❡♣❲➋r➆❭✢❨❩❲♦✽☞❤✍❡☎P✸✼❀✼✾❲✂r➲❤❬Ð✱❢❣❢❣❲♣✿❩❨❬✽☞❡❧➷➇P❚❭✑❤❬❤➼➬➞✹➆❲☎✽☞r➆❲☎✼❾❐ Ñ❙✡❘❚✿❬q✡❲☎❨❴❲☎Ò✯❲➋❡✝✿❬✽✾♥✸❲❧❤➼✿❬❨❷P✥✿❩❲☎◗✎Ð✸➶➆➹✻❙✢❘✥❱✍❙↕P✸❤✍❨❩❲✂r✻➬↔❝✢✼☞P✸❡❷➹➙➷➇P✸❭✡❤❬❤➼➬➮✹✻❲☎✽☞r➆❲☎✼➫✽❀✿❬❲✂❨❩P❚✿❬✽✾❘✸❙✗➶✡✽✾❤❰✿❩❘✓❭✢✺➆➬ r✢P❚✿❬❲♦✿❬q✢❲➑❲✂♥✸❲✂❙➟❭✢❙✡➹✱❙✡❘✥❱✍❙✡❤❳Ó✡❨❷❤➮✿➴✐✟❨❬❲➋r✡❦ Ô✉❩✇②① Õ❩Ö ❃➃×Ø❥ÙÔ✉ Õ❩Ö✇②① ④ Ô✉ Õ❩Ö ➦ ① ④✶ÚÕ✃ÛÕ❩Ö↔Ü③Ý P❚❙✑r➏✿❩q✢❲☎❙➲✿❬q✡❲➑❘✻r✡r➲❡☎❘✸❢❣✺✯❘✸❙✢❲✂❙✎✿❷❤➑✐✧❝✢✼☞P✸❡❷➹✢❦ Ô✉❩✇②① Õ❩Ö✇②① ❃ × Ø ÙÔ✉❩✇②① Õ❩Ö✇②① ④ Ô✉❬✇②① Õ❬Ö ④✶ÚÕ✈ÛÕ❩Ö✇②① ÜßÞ Ï❰q✢❲❣❨❩❲✂r✻➬↔❝✢✼☞P✸❡❷➹à➷➇P❚❭✡❤❩❤➼➬➞✹✻❲✂✽✾r✢❲☎✼❮✽➁✿❩❲☎❨❷P✥✿❩✽❀❘✎❙➐✽☞❤➇✺✑❘✎✺✢❭✢✼☞P❚❨♦➱✧❘✸❨➇✺✡P✸❨❩P✸✼❀✼✾❲☎✼➓❡♣❘✎❢❣✺✢❭➆✿❩P❚✿❬✽✾❘✸❙t❤➼✽✾❙✡❡♣❲ ✿❬q✡❲➐❨❩❲✂r➤✐✧❝✡✼✾P✎❡❷➹➆❦✓✺✑❘✎✽❀❙✱✿❩❤➟❘✸❙✡✼❀Ðß❨❬❲➋á✱❭✢✽❀❨❩❲▲✿❩q✢❲➐❝✢✼☞P✸❡❷➹â✐✧❨❩❲✂r✑❦✓✺✯❘✸✽✾❙✎✿❷❤➙P✸❙✡r✩✿❬q✡❲✂❤❬❲➐❡☎P✸❙ã❝✑❲ ❭✢✺⑦r✢P✥✿❩❲✂r▲✽✾❙àP❚❙✻Ð➲❘✸❨❷r➆❲☎❨➋❐❰Ï❰q✢❲➴❢❣❲♣✿❩q✢❘➆r▲❨❩❲✂P✸r✢✽❀✼✾Ð➙❲☎ä✱✿❩❲☎❙✡r✡❤✍✿❬❘➙❢➴❭✢✼❀✿❬✽✾✺✢✼✾❲❻r➆✽✾❢❣❲☎❙✡❤❬✽❀❘✎❙✡❤✂❐ ♠❙ Ø▼❻➶✻➱✧❘✸❨✍✽✾❙✡❤➼✿❩P✸❙✡❡♣❲✎➶➆✿❬q✢❲➑❨❩❲✂r➲P❚❙✑r➙❝✢✼☞P✸❡❷➹➙✺✑❘✎✽❀❙✱✿❩❤❂P❚❨❩❲♦❤❬q✢❘✥❱✍❙➲❝✑❲✂✼❀❘✥❱➑❐ Red Black ✭✯✮✱✰✳✲✫✴✶✵✸å æ
u+I =(D-L)-Uu+(D-l)-if =R RGs=(D-L)-U: Gauss-Seidel Iteration Matix 2.3 Error Equation SLIDE 1 For an approximate solution u, we define Iteration Error Residual: rr=f sabt from both sides of Au=f, ERROR EQUATION→Aer=y We note that, whereas the iteration error is not an acessible quantity, since it requires u, the residual is easiy computable and only requires Note 3 Relationship betwe d residual We have seen that the residual is easily computable and i e sense it sures the amount by which our approximate solution u' Iin some to satisfy the iginal problem It is clear that if r=0, we have e=0. However, it is not always the case that all in We have that A u=f and A-r=e. Taking norms, We have f|l2≤‖4242 el2 rl2 Combining these two expressions, we have lell ond (a) From here we see that if our matrix is not well conditioned, i.e., cond(A)is large, then small residuals can correspond to significant errors
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