3.2 Jacobi fortunately none of the above convergence theorems helps us to show that Jacobi iteration will converge for our model problem. However, since for our model pro lem we can calculate the eigenvalues of A explicitly (see previous lectures), convergence can be shown directly (L+U)=D-1(D-A)=I-D-1A=I 12 入 h2 then RI 2 Av 入(A) h2 Eigenvectors of Rj= Eigenvectors of A SLIDE 23 0 uj=sin(khj)=sin Eigenvalues SPD h2 LIDE 24 (R, < k=1 Jacobi converges for our model problem Rv2=入(B3)✿❁❀❃❂ ❄✔❅❇❆❉❈❋❊✒● ❍❏■✩❑▼▲✺◆P❖◗❖ ❘❚❙✵❯❲❱❨❳✤❩❭❬❪❙❏❫❴❩❛❵✛❜❞❝❡❙❏❱❴❙❏❵P❱❢❯❣❩✐❤◗❵P❫❦❥❧❱❨♠❨❵❡♥✸❱❴❙♦♠❨❵❲❳❭♣q❵✛❙❏♥❧❵✰❩▼❤♦❵✸❱❴❳✫❵✛rts✉❤◗❵✛❜✈❏s✇❬①s②❩③❱Ps✫❤◗❱❴④✝❩▼❤♦❫❨❩ ⑤❪❫❦♥❧❱❦❥✛⑥☎⑥▼❩③❵✛❳✫❫❨❩❭⑥✟❱❨❙⑦④⑧⑥▼❜▼❜⑨♥✸❱❴❙♦♠❨❵❲❳❭♣q❵✜❯❲❱❴❳②❱❨❬❪❳✍r⑩❱✵❶❦❵✛❜✴✈❉❳✫❱❦❥✛❜❷❵✛r⑩❸❡❹❺❱❨④❋❵✛♠❻❵✛❳❧❼✲s✤⑥▼❙❏♥❧❵✏❯❲❱❨❳②❱❴❬❪❳ r✮❱❽❶q❵❲❜❋✈❏❳✫❱q❥❲❜❾❵❲r❿④❋❵❡♥❧❫❨❙✖♥❧❫❨❜❷♥✛❬❪❜❷❫❨❩③❵P❩✐❤◗❵❡❵❲⑥➀♣✩❵✛❙♦♠❨❫❨❜❞❬♦❵❲sP❱❢❯✉➁➂❵✸➃❲✈❏❜❞⑥✟♥✛⑥▼❩❭❜❞❝➅➄❭s✛❵❧❵➆✈❏❳✫❵✛♠❽⑥✟❱❴❬①s ❜❷❵❧♥✛❩❃❬❪❳✫❵❲s❛➇❻❼✘♥❧❱❴❙♦♠❨❵❲❳❃♣✩❵✛❙❏♥❧❵❺♥❧❫❴❙P❥❧❵✲s✫❤♦❱❨④⑧❙P❶❴⑥▼❳✫❵❧♥❲❩❭❜❞❝❦❸ ➈✒➉✏➊➌➋❣➍✴➎❁➏✟➐✰➑➓➒✲➔✘➊☞➋❣➍→➎⑨➏✟➋↔➣↕➁✒➔✘➊☞➙✲➣P➋➛➍→➎❇➁☞➊☞➙✲➣✖➜♦➝➞ ➁ ➟❛➠ ➁❡➡❚➢✜➊➅➤➥➢✔➡✺➢✍➦ ➧✸➨◗➩✛➫ ➈✒➉✘➡✴➭✢➊➲➯◗➙t➣➳➜➥➝➞ ➁➸➵➛➡ ➢ ➊➲➯→➺⑨➣➳➜➥➝➞ ➤➢ ➏✟➁✏➔③➵ ➻ ➼✤➽ ➾ ➚❻➪❻➶❷➹→➘✸➴ ➡ ➢ ➤➢ ➏❃➈☎➉❨➔❋➊✖➺⑨➣➳➜➝ ➞ ➤➢ ➏✟➁✏➔ ➷✘➬❾➮➩✛➫①➱q➩❽✃✤➧✸❐❦❒❧❮✘❐➠❇➈✒➉✏❰Ï➷✘➬❾➮➩❽➫①➱❦➩❽✃❲➧✫❐q❒✸❮Ð❐➠⑧➁ ❍❏■✩❑▼▲✺◆P❖◗Ñ Ò➩✵✃✛Ó❴Ô❾Ô⑧Õ✛Õ❽Õ ➁☞➊ ➺ ➜➝ Ö× × × × × × ×Ø ➞ ➣✲➺ Ù Ú❽Ú✛Ú Ù ➣✲➺ ➞ ➣✲➺ Õ Õ Õ Õ Õ Õ Ù Õ Õ Õ Õ Õ Õ Õ Õ Õ Ù Õ Õ Õ Õ Õ Õ ➣✲➺ ➞ ➣✲➺ Ù Ú✛Ú❽Ú Ù ➣✲➺ ➞ Û✛ÜÜ Ü Ü Ü Ü Ü Ý ÞPß✉Þ à◗á⑧â ➁➸➡ ➢ ➊➅➤➢ ➡ ➢ ã ➊✖➺❦➦✛ä❽ä✛ä❲➦ Þ ➷✘➬❾➮➩❽➫✩➱q➩❽✃❲➧✫❐❦❒❧❮❽å ➜ ➊ ➎ æ❴ç ➎ è ➢ é ➊ ❮ ➬➫ ➏ ã❪ê➜që ➔❋➊ ❮ ➬➫ ➯ ã①êë Þ ➑➌➺ ➵ ➷✘➬❾➮➩❽➫✩➱❨Ó❦Ô❷ì◗➩✵❮✛å ➤➢ ➏❃➁✒➔❋➊ ➞ ➜➝✮í ➺⑨➣ ✃❲❐q❮ ➏ ã①ê➜ ➔❛î ❍❏■✩❑▼▲✺◆P❖①ï ➤➢ ➏❃➈☎➉❴➔ð➊ ➺⑨➣✖➜➥➝ ➞ ➤➢ ➏✟➁✒➔Ð➊✖➺⑨➣ í ➺⑨➣ ✃❲❐✩❮ ã①ê➜ î ➊ ✃✛❐q❮ ã①ê Þ ➑ñ➺↕òôó ➦ ã ➊✖➺q➦✛ä✛ä❽ä❲➦ Þ ⑤❪❫q♥✸❱q❥❲⑥✔♥✸❱❴❙♦♠❨❵❲❳❭♣q❵❲s✘❯❲❱❨❳t❱❨❬❪❳✲r✮❱✵❶❦❵✛❜❪✈❏❳✸❱❦❥✛❜❷❵❲r✮❸ ➈☎➉✵➡ ➢ ➊ñ➤➢ ➏✟➈✒➉❨➔③➡ ➢ ❍❏■✩❑▼▲✺◆P❖◗õ ➺❽Ù