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 errorsç➉è❩é②êìë ísî❈ïtð❳ñ❷ò➫ê✂ó③ç➉è✈ôõí✟î❈ï❊ð➉ñ❷ò✗ê➓ö çè❬é②ê ëõ÷❖ø✫ù❰çè ôõí✟î❥ïtð❳ñ ò✗ê ö ÷ø✗ù ë✛ísîúï❊ð➉ñ✝ò➫ê✂óüû➉ý➲þ✡ÿ✁￾✂￾☎✄☎✆✞✝✠✟☛✡✁✝✠☞✍✌☎✎✏✝✒✑✥þ✓✎✏✟☛✔✖✕✘✗✛þ✙✎✏✟✛✚ ✜✣✢☛✤ ✥✧✦✙✦✙★✩✦✪✥✬✫✮✭✰✯✲✱✓✳✴★✍✵ ✶ ✷✒✸✺✹✼✻✪✽✿✾ ❀✞❁❃❂ ç✘❄❁❅❂❇❆✙❁❉❈✴❊●❋✓❂■❍❏❊▲❑▼❊❖◆◗P ç✶ëõö❙❘ ❚✙❊●❯❲❱❖❑✧❱❖❳✙❳❨❯■❊❬❩✓❍❏❭❪❱❖❂■❁❅❈✴❊●❫❴❋✙❂■❍❏❊▲❑ ç➉è✿❵✙❛❁❝❜✓❁☎❞✖❑✙❁ ❡❣❢✐❤☎❥❇❦✿❢♠❧♦♥✿♣✧q✣❥❣❥■♥✿❥ û r✢è❂ëãç●ï ç✈è st❤❣✉❣❧♦✈❖✇❨❦✿① û ②✑è❂ë❄ö ï P ç➉è ③✇❨④❃❢♠❥■❦▲⑤❃❢☛❧✺♣✠⑥❉P çè❲⑦❥■♥✿⑧⑨④⑩♥✿❢✛❶▼✉❣❧♦✈●❤❣✉❷♥⑦ P ç③ë❄ö❹❸ P ç ïP ç✍❺➴ë❜ö ï P ç✍❺ ❻✍❼❲❼❲❽❾❼❿❻✮➀❾➁t➂➄➃✣➅■❽❾➆➈➇ P r✢è❂ë➉②✡è ➆t➊ ➋✬❤❪♣➌♥❖❢➍❤❪❢✛❶✙❦❖❢❸❙➎❶❨❤☎❥■❤⑩❦✿✉➏❢✛❶✙❤❪❧✺❢✐❤❃❥■❦✿❢♠❧♦♥✿♣➐❤❃❥❣❥■♥✿❥➏❧➑✉❪♣➌♥✿❢✰❦❖♣➐❦▲⑤⑩❤☎✉❇✉❣❧♦④❃①❴❤▼➒☎✇❨❦❖♣❨❢♠❧✺❢☛➓❸ ✉❣❧✺♣➌⑤⑩❤❪❧✺❢ ❥■❤⑩➒❃✇✓❧✺❥■❤❣✉ ç❲❸ ❢✛❶✙❤❷❥■❤☎✉❣❧♦✈✿✇❨❦✿①✁❧➑✉❷❤❇❦✿✉❣❧✺①➔➓→⑤⑩♥✿⑧✰➣➌✇✓❢✐❦▲④❃①❴❤➏❦✿♣➌✈▼♥✿♣✖①➔➓↔❥■❤⑩➒❃✇✓❧✺❥■❤❣✉ ç➉è✒↕ ➙➜➛➞➝❣➟✬➠ ➡✬➟❖➢☛➤➞➝■➥✐➛✓➦✁➧✂➨✲➥➑➩➭➫◗➟✙➝❇➯❝➟✒➟❖➦❿➟❖➲❬➲●➛✠➲➳➤✙➦✩➵➐➲●➟❖➧✏➥➸➵➄➺➌➤✓➢■➻ ➼➜❁❉❆❨❱❬➽▲❁❉❈■❁❃❁✏❑✧❂■❆❨❱❖❂❲❂■❆❨❁❉❯■❁✂❈✴❍➑❜✓❋❨❱▲❫✞❍➑❈❲❁✏❱●❈✴❍❏❫❏➾▼➚❃❊▲❭❪❳✙❋✓❂⑩❱❄❫❏❁❉❱❖❑❨❜✧❍❏❑➪❈■❊▲❭❪❁❷❈■❁❃❑❨❈■❁❝❍➶❂❾❭❪❁✏❱✿➹ ❈■❋✙❯■❁✂❈↔❂❇❆✙❁➘❱▲❭➏❊●❋✙❑✒❂ ❄➾ ❛❆❨❍❏➚⑩❆➉❊●❋✙❯→❱▲❳✙❳✙❯❇❊❬❩✓❍❴❭↔❱✿❂❇❁➪❈✴❊●❫❴❋✙❂■❍❏❊▲❑ ç➉è ◆♦❱❖❍❏❫❏❈➳❂■❊➐❈❇❱✿❂❇❍❏❈✴◆✛➾➴❂■❆✙❁ ❊▲❯❇❍❏➷▲❍❏❑❨❱❖❫➬❳✙❯❇❊❄❫❏❁❃❭❘ ➅➍❂❙❍❏❈❙➚❃❫❴❁✂❱❖❯✣❂❇❆❨❱✿❂❙❍❴◆ ②➙ë➱➮✞❵✓❛❁❅❆❨❱❬➽▲❁ r➏ë✃➮✞❘✩❐❊❛❁❃➽▲❁✏❯ ❵ ❍❴❂❙❍❏❈✣❑✙❊▲❂✰❱❖❫❛❱❬➾✓❈✍❂■❆✙❁❝➚✏❱▲❈■❁t❂❇❆❨❱✿❂ ❛❆✙❁✏❑ ② ❍➑❈❙❈✴❭↔❱▲❫❴❫➞❍❏❑✧❑✙❊▲❯❇❭❵✙r ❍➑❈❲❱▲❫❏❈■❊➏❈■❭↔❱❖❫❏❫➞❍❴❑✧❑✙❊●❯■❭❘ ➼➜❁❝❆❨❱❬➽▲❁t❂■❆❨❱❖❂✰P ç③ë❄ö ❱▲❑❨❜→Pò➫ê☎②➙ë➉r➞❘ ➃✲❱❖❒✠❍❏❑✙➷❪❑✙❊▲❯❇❭↔❈ ❵✓❛❁❅❆❨❱❬➽▲❁ ❮❃❰➄❮❃Ï❾ÐÑ❮ P ❮☎Ï❲❮☎Ò◗❮❃Ï✿Ó ❮☎Ô✓❮☎Ï❾Ð❿❮ Pò✗ê ❮☎Ï❲❮☎Õ✙❮☎Ï❪Ö ×❊▲❭❄❍❏❑✙❍❴❑❨➷Ø❂❇❆✙❁✏❈■❁❾❂❛❊❪❁❃❩✠❳❨❯■❁✂❈■❈■❍❴❊●❑❨❈ ❵✒❛❁❅❆❨❱❬➽▲❁ ❮☎Ô✓❮☎Ï ❮❣Ò➄❮☎Ï ÐÑ❮ P ❮ Ï ❮ Pò➫ê ❮ Ï Ù Ú☎Û Ü Ý✐Þ■ß✏à➄á❴â➞ã ❮❣Õ✙❮☎Ï ❮❃❰➄❮☎Ï Ö ❚✙❯❇❊▲❭ä❆✙❁✏❯■❁ ❛❁→❈■❁❃❁å❂■❆❨❱❖❂Ø❍❴◆✰❊▲❋✙❯Ø❭↔❱✿❂■❯❇❍❴❩➜❍➑❈❷❑❨❊❖❂ ❛❁❃❫❏❫❹➚❃❊▲❑❨❜✙❍➶❂❇❍❴❊●❑✙❁✏❜ ❵ ❍ ❘ ❁ ❘❏❵ ➚☎❊▲❑✖❜ íPñ ❍❏❈ ❫➑❱❖❯❇➷▲❁ ❵ ❂❇❆✙❁❃❑✬❈■❭❪❱▲❫❴❫➞❯❇❁✏❈■❍➑❜✓❋❨❱❖❫➑❈❙➚❃❱▲❑▼➚❃❊▲❯❇❯■❁✂❈✴❳➌❊▲❑✖❜↔❂■❊↔❈■❍❴➷●❑✙❍❴❞✖➚❃❱▲❑●❂❙❁✏❯■❯❇❊▲❯⑩❈ ❘ æ