can be easily shown for the case where R is diagonalizable. In this case R XAX, where A is the diagonal matrix of eigenvectors. We have that R X-A X and hence we require that which is equivalent to p(r)< 3.1 Theorems SLIDE 21 If the matrix A is strictly diagonally dominant then Jacobi and gauss Seidel iterations converge starting from an arbitrary initial condition. N5 A=aif is strictly diagonally dominant if a|>∑ laiil for all i If the matrix A is symmetric positive definite then Gauss-Seidel iteration converges for any initial solution Note 5 Convergence of Jacobi iteration for diagonally dominant matrices For the Jacobi iteration, RJ=D(L+U), it can be easily seen that R has zero diagonal entries. It can be shown(Gershgorin Theorem-see G Vl that a matrix with zeros in the main diagonal, p(Rj)<Rjlloo. Hence B)=≤B==41 the last inequality follows form the diagonal dominance of A Convergence of Gauss-Seidel for SPD matrices (See Theorem 10.1.2 of GVL④☎⑤✷⑥❏⑦❱⑧✠⑧✆⑤◗⑨✈⑩②❶②❷❏⑨✧❸✪❹❅❺✵⑥❀❻❼❹◗❽✙❾✧❸✪⑧❫④✆⑤✷⑨✧⑧✓❺✵❸✪⑧☎❽❉⑧❫❿➀⑩✽⑨✌➁✞⑩✽⑤✗➂✷❹◗⑥❋⑤✗❶②⑩②➃✆⑤✗⑦❋❶❆⑧◗➄➆➅●⑥❈❾❉❸✪⑩②⑨❡④☎⑤✷⑨✧⑧✠❿➈➇ ➉➋➊➍➌➏➎✬➉❍➐ ❺✵❸✪⑧✆❽✧⑧ ➎ ⑩②⑨➑❾✧❸✪⑧➆➁✞⑩✽⑤✗➂✷❹◗⑥❋⑤✗❶➓➒✓⑤✗❾✧❽❉⑩▲➔✾❹✗❻❥⑧✆⑩❆➂◗⑧☎⑥✻→✷⑧❧④❄❾✧❹◗❽❉⑨✆➄❦➣❏⑧■❸❋⑤❣→✷⑧✙❾❉❸❋⑤❅❾✙❿❪↔✙➇ ➉➋➊➍➌↕➎↔ ➉ ⑤✗⑥❋➁❬❸✪⑧✆⑥❋④❇⑧❞❺❦⑧✙❽✧⑧❧➙◗➛❋⑩❆❽❉⑧✥❾❉❸❋⑤❅❾ ❶②⑩❆➒ ↔☛➜✌➝ ➎↔ ➇➟➞ ❺✵❸✪⑩✽④☛❸❖⑩✽⑨✵⑧❧➙◗➛❋⑩❆→❅⑤✗❶②⑧☎⑥❴❾❢❾✧❹✠➠❜➡❲❿❞➢❦➤➦➥✷➄ ➧❢➨✈➩ ➫❈➭➑➯➍➲❤➳❋➯➓➵✑➸ ➺❱➻❴➼✺➽➓➾❈➚✁➪ ➶ ➅◆❻➹❾✧❸✪⑧✾➒✠⑤❅❾❉❽✧⑩❆➔❍➘➴⑩②⑨❖➷❄➬✲➮❄➱❲✃☎➬✮❐❙❒❨❮❅➱❲❰✆Ï◗Ð✗Ñ❱❰❅❐✺❐▲❒✾❮◗Ð❅Ò➆➱✺Ñ❱❰❅Ñ▼➬❡❾❉❸✪⑧☎⑥❳Ó❴⑤✷④❇❹◗⑦✪⑩❢⑤✗⑥❋➁ÕÔ✙⑤✗➛❋⑨❉⑨❂Ö × ⑧☎⑩✽➁✞⑧☎❶✻⑩❆❾✧⑧☎❽☛⑤❅❾❉⑩❆❹◗⑥❋⑨➓④☎❹✷⑥✻→✷⑧✆❽✧➂◗⑧❥⑨❂❾☛⑤✗❽✧❾✧⑩②⑥✪➂✵❻❼❽❉❹✷➒Ø⑤✗⑥❡⑤✷❽✧⑦❋⑩▲❾❉❽❉⑤✷❽✧❷➑⑩②⑥✪⑩❆❾✧⑩✽⑤✗❶✞④❇❹◗⑥❋➁✞⑩❆❾✧⑩②❹✷⑥➍➄ Ù❪Ú ➘➦➇✢Û❣Ü❴Ý❇Þ✗ß❞⑩②⑨✥⑨❂❾❉❽✧⑩✽④❄❾❉❶❆❷❨➁✪⑩②⑤✷➂✷❹✷⑥▼⑤✗❶②❶❆❷❫➁✞❹◗➒✓⑩❆⑥❋⑤✷⑥❴❾➑⑩▲❻ à ÜÝ☎Þ à✻áãâä Þ✧å ➌ à ÜÝ❇Þ à ❻❼❹✷❽➑⑤✷❶❆❶✁æ☛ç ➶ ➅◆❻↕❾✧❸❋⑧❞➒✓⑤✗❾✧❽❉⑩▲➔❨➘è⑩✽⑨✵⑨✧❷❴➒✓➒✓⑧❇❾❉❽✧⑩✽④❪é▼❹❴⑨✈⑩❆❾✧⑩②→✷⑧❞➁✞⑧☎ê❋⑥✪⑩❆❾✧⑧✙❾❉❸✪⑧☎⑥✼Ô✙⑤✗➛❋⑨❉⑨✈Ö × ⑧✆⑩②➁✪⑧☎❶✁⑩❆❾✧⑧☎❽☛⑤❅❾❉⑩❆❹◗⑥ ④☎❹✷⑥✻→✷⑧✆❽✧➂◗⑧✆⑨❝❻❼❹✷❽➑⑤✷⑥❴❷❨⑩❆⑥✪⑩❆❾✧⑩✽⑤✗❶❛⑨✈❹◗❶❆➛✞❾❉⑩❆❹◗⑥➍➄ Ù➹ë ì❈í✁î❄ï✬ð ñ❡í✻ò❥ó✪ï✗ô✆õ❛ï✗ò❤ö◗ï✬í✪÷❢ø➍ù❱ö✷í✁ú▼û✵û✧î❄ï✗ô❅ù✁î✈û❂í✻ò➟÷✈í✞ô✾ü❜û✮ù✻õ❛í✻ò➏ù✞ý❲ý✺þ✇ü↕í✻ÿ❍û✲ò↕ù✪ò❥î ÿù✁î✧ô❣û●ö✷ï✁ ✂❹◗❽✙❾✧❸✪⑧❖Ó◗⑤◗④❇❹✷⑦❋⑩↕⑩❆❾✧⑧✆❽❉⑤✗❾✧⑩②❹✷⑥ ➐ ❿☎✄❖➇✝✆ ➊✁➌ ➡✟✞✡✠☞☛✌➢ ➐ ⑩❆❾■④✆⑤✗⑥❈⑦▼⑧❫⑧❧⑤✷⑨✧⑩❆❶②❷❀⑨✧⑧☎⑧✆⑥❏❾❉❸❋⑤❅❾❡❿➈❸❋⑤✷⑨ ➃☎⑧✆❽✧❹✓➁✞⑩✽⑤✗➂◗❹✷⑥❋⑤✷❶▼⑧✆⑥◗❾❉❽✧⑩②⑧✆⑨✆➄➏➅◆❾➑④☎⑤✗⑥❬⑦❱⑧✙⑨✧❸✪❹❅❺✵⑥ ➡✲Ô❪⑧✆❽❉⑨✧❸✪➂✷❹◗❽✧⑩②⑥✍✌❢❸✪⑧✆❹✷❽❉⑧☎➒✑Ö❝⑨✧⑧☎⑧✌Ô✏✎✒✑↕➢❝❾✧❸❋⑤✗❾✵⑤ ➒✠⑤❅❾❉❽✧⑩❆➔❨❺✵⑩❆❾✧❸✾➃☎⑧✆❽✧❹❴⑨P⑩②⑥❬❾✧❸✪⑧❞➒✠⑤✷⑩❆⑥✼➁✞⑩✽⑤✗➂◗❹✷⑥❋⑤✷❶ ➐ ➠❜➡❲❿✄ ➢✔✓✖✕❇❿✄ ✕ ➝ ➄✘✗➑⑧☎⑥▼④❇⑧ ➠❱➡✲❿✄ ➢❝➇✙✕❇❿✄ ✕✛✚✜✓✢✕❇❿✄ ✕ ➝ ➇ ➒✓⑤✗➔ ➌✤✣ Þ ✣ â ✥✦ â ✧✩★ ✦✫✪✧✩★ ✬ ✬ ✬ ✬ Ü✻Ý❇Þ Ü✻Ý❧Ý ✬ ✬ ✬ ✬ ➤✇➥✮✭ ❾✧❸❋⑧❞❶②⑤◗⑨❂❾✵⑩②⑥✪⑧✆➙❴➛❋⑤✷❶❆⑩❆❾❂❷✠❻❼❹✷❶②❶②❹❅❺➑⑨❦❻❼❹✷❽❉➒➀❾✧❸✪⑧✌➁✪⑩②⑤✷➂✷❹✷⑥▼⑤✗❶✁➁✞❹✷➒✓⑩②⑥❋⑤✗⑥▼④❇⑧✙❹✗❻❥➘✌➄ ì❈í✁î❄ï✰✯ ñ❡í✞ò❥ó✞ï✗ô❧õ➓ï✗ò❝ö✷ï❏í✞÷✲✱✠ù✴✳✁✵✁✵✶✸✷ï✗û●ü↕ï✗ý❦÷✈í✞ô ✷✺✹✲✻ ÿù✁î✈ô❣û❂ö✷ï✁ ➡ × ⑧☎⑧✼✌❢❸✪⑧✆❹✷❽❉⑧☎➒ ➥✆➞❋➄❆➥◗➄ ✽❡❹✗❻❤Ô✏✎✒✑↕➢ ✾