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