2. 4.3 Observations The number of iterations required to obtain a cer tain level of convergence O(72) Gauss-Seidel is about twice as fast as jacobi Whu? 3 Convergence Analysis e= re=rrer The iterative method will converge if mF=0→p(R)=max|(r)<1 T→0 P(R)is the spectral radius The spectral radius is the radius of the smallest circle centered at the origin that contains all the eigenvalues. The above condition indi cates that the nec conditions, is that the largest eienvalue, in modulus, lies within the unit circle p() The condition that if and only if P(r)<10.3 0.2 0.1 0 # Iterations log(||e|| L2 ) n=10 n=20 n=40 0 10 20 30 40 50 60 70 80 90 ➞✺➟❃➠➡➟✜➢ ➤♠➥✵➦✶➧❑➨✶➩✒➫✒➭➔➯✄➲q➳✵➦ ➵q➸❑➺❫➻❖➼✞➽✙➾ ➚↕➪➃➶❴➹✈➘✽➴✒➷❡➬q➹❳➮❍➱❅✃✺❐❃❒✏➹➔➮❆❮❅❒✏❐✜➱✙➘❴❰⑧➮❆➹➔Ï❑➴✒❐✜➮✏➹✶Ð♦❒✏➱❡➱❉➬✔❒❆❮✙❐❃➘Ñ❮❱Ò☞➹➔➮Ó❒✗❮❅❐✜➘❽Ô❃➹➔Õ✙➹➔Ô❴➱❅✃✵Ò☞➱✙➘✽Õ❉➹❳➮❆Ö✙➹❳➘❾Ò☞➹ ❐❄❰✈×❁Ø✄Ù✺Ú✶Û☞Ü ➚✯Ý♣❮✙➴❴❰❆❰❿Þ❿ß✽➹❳❐❄Ð✔➹➔Ô✭❐✜❰⑥❮✙➬❾➱❉➴✔❒➃❒❿à➈❐✜Ò❳➹❱❮✙❰❍✃✄❮❉❰❿❒⑥❮✙❰➈á❑❮✙Ò❳➱✙➬✒❐❥Ü â❽ã✽ä✔å æ ç●è➀é❁ê♦ë⑥ì❙í❡ë✈éïî➈ëñðòéïó❱ô☞õ÷ö➊ø☞ö ➵q➸❑➺❫➻❖➼✆ù✔ú û✒ü✈ý✯þ✞û✒ü❳ÿ✁￾➈ý↕þ✞þ✞û❴ü☞ÿ Ú ý✄✂☎✂✆✂❑ý▲þ❭ü➊û✞✝✠✟ ➪➃➶✒➹♣❐❃❒✏➹❳➮✗❮✁❒❆❐❃Õ❉➹♣➷♦➹❳❒✏➶✒➱✔Ð②à➈❐✜Ô❃Ô❖Ò❳➱✙➘✽Õ✙➹➔➮✏Ö❉➹✈❐❃✃ Ô✜❐❃➷ ü☛✡✌☞ þ♣ü✈ý✎✍✑✏✓✒✕✔ ØþÛ ý ➷❁❮✗✖✙✘ ✚❖ØþÛ☎✘✞✛✢✜ ✟ ✣✥✤ ✔ ØþÛ⑧❐✜❰➃❒❆➶✒➹❱❰✧✦❾➹✶Ò✩❒✏➮✗❮❅Ô✭➮✗❮✙Ð✒❐❃➴❴❰➔Ü ★✇ã✪✩✬✫✮✭✪✩☛✯☎✰✲✱✧✳✗✴✵✱✧✳✷✶✗✸✺✹✻✫✼✸✽✫✾✰❛ã✪✩✬✱✧✳✷✶✗✸✺✹✻✫❀✿❂❁✾✰❛ã✪✩✾✫❄❃✠✳❅✴✺✴❆✩❇✫❄✰✓✯☎✸✺✱✧✯☎✴❆✩❈✯❉✩☎❊❋✰●✩☎✱✧✩☛✶❍✳❅✰■✰❫ã❋✩❏✿✗✱❄✸▲❑✗✸✺❊ ✰❛ã✪✳❅✰✌✯☛✿❅❊▼✰◆✳✗✸✺❊✪✫❖✳✗✴✺✴P✰❛ã✪✩✾✩☎✸▲❑◗✩☎❊❋❘❅✳❅✴❙✹❋✩❇✫☎❚✢★✇ã❋✩✼✳✷❯☛✿❅❘❅✩✼✯☛✿❅❊❱✶✗✸✺✰✲✸❲✿✗❊❳✸✺❊❱✶❅✸❲✯☛✳✗✰◆✩❇✫❨✰❛ã✪✳❅✰✙✰❛ã✪✩❖❊❱✩☛✯☎❩ ✩❇✫❉✫☎✳✗✱✩ä✼✳✗❊❱✶❬✫❄✹❇❭❳✯❇✸❲✩☎❊❋✰❪✯❉✿✗❊❱✶✗✸✺✰✲✸❲✿✗❊❨❁❇✿❅✱✓✰❛ã✪✩✓✫☎✯✗ã✪✩☎❃✓✩❫✰◆✿✼✯☛✿✗❊❋❘❅✩❇✱✲❑❴✩❇❵❛❁❇✿❅✱❫✳❅✱☎❯☎✸✺✰✮✱✧✳❅✱✩ä✾✸✺❊❋✰✮✸❲✳❅✴ ✯☛✿❅❊❜✶❅✸✺✰✮✸❲✿❅❊✪✫☛❵❝✸✽✫❞✰❛ã✪✳✗✰P✰❛ã✪✩❡✴❆✳❅✱✮❑◗✩❇✫❄✰❢✩☎✸❲✩❇❊▼❘❣✳✗✴❙✹❋✩❇❵❝✸✺❊❀❃✠✿✆✶✗✹✞✴❙✹✻✫☛❵❤✴❙✸❲✩❇✫✙✐❥✸✺✰❛ã✻✸✺❊❈✰❛ã✪✩❡✹✞❊❋✸✺✰❦✯☎✸✺✱✧✯☎✴❆✩❧❚ ♠❈♥✁♦❄♣❬q r❡♥✻s❝t✁✉✈♦✈✉❂♥✻s✇♥✞s ✔ ØþÛ ➪➃➶✒➹✠Ò☞➱❉➘❴Ð✔❐❃❒✏❐✜➱✙➘②❒✏➶❾❮✁❒ Ô❃❐✜➷ ü❉✡❡☞ þü ý✇✍ ❐❃✃➊❮✙➘❴Ð÷➱❉➘✒Ô②①÷❐❃✃ ✔ ØþÛ❦✛✢✜ ③