We see that P(R3)=A(R3)l= a (R3)l and, since n is the highest frequency mode, it is clear that jacobi is not a smoother 2.2.2 Under-Relaxed Jacobi SLIDE 6 RwJ=wRJ+(1-w)I 入(R)=u入(F3)+(1-)=1-(1-(F3) k=1 e obs erve that for w< 1, Jacobi can in fact be a good smoother. If we set the condition A(n+1/2(R,J)=An(R J)l, we obtain w =2/3. We also note that for w> l the method becomes unstable (does not converge) since for some smoother is a slow down in co nvergence of the low frequency mode o be a good to be paid for Jacobi to IDE 7 Iterations required to reduce an error mode by a factor of 100 The graph shows the number of iterations required to reduce the the amplitude of each error mode by a factor of 100. We see that the standard Jacobi(w=1) algorithm, requires many iterations to eliminate the highest frequency mode On the other hand, the under-relazed Jacobi scheme, eliminates the high modes very quickly, but on the other hand, the low frequency modes take longer, that with standard Jacobi, to dis appear. We shall see that this slow down in the convergence of the low frequency modes is not really a problem and that, by using hes we will be able to speed up the oo nvergence of these modes 2.2.3 Gauss-SeidelÜ➝Ý✰Þ✁Ý✦Ý❜ß●à■á✿ßÔâ❩ã❱ä✔å❵æ✹çéè ê✾ëìã❱ä✔å✿æ✼è❞çéè ê❩í⑤ã❱ä✔å➂æ✼è➌á✿î➋ï✿ð✮Þ❦ñ☛î❩ò✞Ý✔ó✙ñ◆Þ✰ß☛à★Ý✰à✲ñ✯ô➂à■Ý☞Þ✦ß❾õ✦ö❑Ý✦÷☞ø★Ý✁î➋ò✁ù ú➉ûï⑩Ý❦ð✥ñ☛ß✮ñ◆Þ✗ò☞ü✴Ý✦á❵ö✰ß☛à★á✿ß✍ý❞á⑤òû⑤þ ñ➌ñ◆Þ✰îû ß✹á❧Þú➉û✼û ß☛à★Ý☞ö✁ÿ ￾✂✁✄￾✂✁✄￾ ☎✝✆✟✞✡✠☞☛✍✌✏✎✑✠✓✒✕✔✗✖✂✠✓✞✙✘✚✔✜✛✣✢✥✤✡✦ ✧✩★☞✪✬✫✮✭✰✯ ä✲✱✲å❇ç✴✳✥ä✔å✶✵✠ã✸✷✺✹✻✳✮æ✂✼ -1 -0.5 0 0.5 1 mode k λ(RωJ) ω =1.1 (UNSTABLE) ω =1/2 ω =2/3 ω =1 ê✩✽❞ã❱ä✾✱✲å➂æ✥ç✿✳✮ê✥✽■ã❱ä✔å✿æ✂✵❆ã❀✷✲✹✻✳✮æ✢ç❁✷✺✹✻✳✪ã✸✷✺✹ ê✩✽❞ã❱ä✔å❵æ⑧æ❃❂ ❄ ç❅✷✣❂❇❆❈❆❇❆❈❂⑧ó Ü➝Ý û⑩þ Þ✁Ý✁ö❊❉➂Ý➦ß●à■á❵ßÔõû ö❋✳❍●■✷⑤ð❇ý❞á⑤òû⑤þ ñ❇ò✦á❵î✻ñ☛î➦õ☞á⑤ò✁ß þÝ✸á➦ôû❈ûï➳Þú➉û✼û ß☛à★Ý☞ö✁ÿ❑❏❱õ▼▲✥Ý✄Þ✁Ý☞ß ß●à■Ý➦òû î➋ï❵ñ☛ß❛ñ û î✏è ê✚◆④ëP❖❾í✸◗❙❘✏❚⑩ã❛ä✱✲å æ✁è➓ç è ê✾ë✛ã❱ä✱✲å æ✼è ð❃▲✥Ý û⑩þ ß❂á✿ñ☛î❯✳ ç❲❱❨❳P❩✾ÿ❆Ü✸Ý➦á✿üÞ û îû ßÝ ß●à■á✿ß❩õû ö❬✳✿❭❪✷✄ß●à■Ý úÝ☞ß●à ûï þÝ✦òû✿úÝ☞Þ✐ø❞î★Þ❦ßá þ ü✴Ý✑❫❼ïûÝ❦Þ✰î û ß✫òûî✜❉✿Ý✁ö❛ô⑩Ý❀❴➉Þ❦ñ☛î➋ò✦Ý✮õû ö✰Þ û❵úÝ ❄ ð✔è ê✥✽★ã❱ä✾✱✲å➂æ✼è✮❭❲✷➐ÿ✑❏☞î Þ û✿úÝ❧Þ✁Ý✁î■Þ✁Ý❦ð✔ß☛à★Ý❃❵➋ö❦ñ❱ò✦Ý✄ßû✃þ Ý❃❵★á✿ñ❱ï✶õû ö➟ý❞á⑤òû⑤þ ñ✫ßû➊þ Ý✄á✄ô û❈ûï Þú❧û❈û ß●à■Ý✁ö✩ñ◆Þ✗á❧Þ❦üû▲✙ïû▲➌î➳ñ☛î→òû î✜❉✿Ý☞ö❛ô➐Ý✁î➋ò✦Ý ûõ✩ß●à■Ý✗üû▲➊õ✦ö❑Ý✦÷✁ø★Ý☞î❩ò☞ù ú➉ûï⑤Ý☞Þ☞ÿ ✧✩★☞✪✬✫✮✭❜❛ ❝❡❞✏❢❈❣✐❤❥❞❧❦♥♠✣♦✥♣q❣❧❢✍r❨s✜❦t❣✏❢❇✉✑❞❧♠✈❣✏❢❇✉✗s✜✇❈❢❋❤P♦①❢❈❣✏❣❧♠❨❣③②✈♠✗✉✗❢❋④✓⑤✑❤⑦⑥❙❤✣✇⑧❞❧♠❨❣✺♠P⑥q✷❇⑨✣⑨ 0 2 4 6 8 10 12 14 16 18 20 0 100 200 300 400 500 600 n=19 Number of iterations mode k ω = 1 ω=2/3 ⑩❩à■Ý✶ô❵ö❑á✐❵■à✃Þ❑à û▲ÔÞ❧ß☛à★Ý❧î★øú➉þ Ý☞ö ûõ❜ñ☛ßÝ✁ö❑á❵ß❛ñ û î■Þ✶ö✞Ý✞÷✁ø❞ñ☛ö❑Ý✦ï➦ßû ö❑Ý✦ï✿ø★ò✦Ý❧ß●à■Ý❧ß●à■Ý➟áú❵❩ü✆ñ☛ß❯ø★ï⑤Ý ûõ✩Ý✦á⑤ò✦à➊Ý✁ö❦öû ö ú➉ûï⑤Ý þ ù➟á✫õ☞á⑤ò✁ßû ö ûõ❷❶❨❸✣❸⑤ÿ❭Ü➝Ý✐Þ✁Ý✦Ý✩ß●à■á❵ß✢ß☛à★Ý✐Þ❦ß❂á✿î➋ï⑩á✿ö❑ï➟ý✲á⑩òû⑤þ ñ✢ã✬✳ ç❅✷❈æ á✿üô û ö❦ñ☛ß☛àúð✗ö✞Ý✞÷✁ø❞ñ☛ö❑Ý☞Þ úá✿î✾ù✃ñ☛ß❂Ý☞ö✞á✿ß❯ñ ûî■Þ➦ßû Ý✁ü✆ñúñ☛î➋á❵ßÝ✸ß●à■Ý❭à✲ñ✯ô➂à■Ý☞Þ✦ß✹õ✦ö❑Ý✦÷☞ø★Ý✁î➋ò✁ù ú❧ûï⑤Ý☞Þ✁ÿ ❹î➝ß☛à★Ý û ß●à■Ý✁ö✪à■á❵î➋ï➂ð✢ß●à■Ý✰ø❞î➋ï⑩Ý☞ö❊❺❂ö❑Ý✁ü✴á❇❻➐Ý✦ï➉ý❞á⑩òû⑩þ ñ✷Þ✁ò✦à■ÝúÝ☞ð✮Ý☞ü✯ñúñ☛î❩á✿ß❂Ý❦Þ✩ß●à■Ý✪à❞ñ✯ô❈à ú❧ûï⑤Ý☞Þ ❉✿Ý☞ö❦ù✸÷☞ø❞ñ❱ò✐❼✿ü✯ù➂ð þ ø❞ß ûî ß☛à★Ý û ß●à■Ý✁ö✐à★á✿î➋ï✿ð✫ß☛à★Ý✗üû▲ õ✦ö❑Ý✦÷☞ø★Ý✁î➋ò✁ù ú➉ûï⑩Ý❦Þ✶ßá✍❼⑤Ý❜üû î➐ô➐Ý☞ö✦ð✍ß●à■á✿î ▲➌ñ☛ß●à➸Þ❦ßá❵î➋ï⑩á✿ö❑ï ý✲á⑩òû⑤þ ñ◆ð❜ßû ï✿ñ◆Þ✁á✐❵✣❵★Ý✞á❵ö✁ÿ Ü✸Ý➊Þ❑à■á✿ü☛ü✇Þ☞Ý✦Ý➳ß☛à★á✿ß✶ß☛à❞ñ◆Þ➦Þ❦üû▲ ï û▲➌î✠ñ☛î❆ß●à■Ý òûî✥❉➂Ý✁ö❛ô➐Ý☞î❩ò✞Ý ûõ✹ß●à■Ý✍üû▲➟õ✦ö❑Ý✦÷☞ø★Ý✁î➋ò✁ù ú❧ûï⑤Ý☞Þ✹ñ◆Þ✍îû ß✛ö❑Ý✦á✿ü☛ü✯ù✗á③❵➋öû⑩þ ü④Ýú á✿î➋ï✩ß☛à★á✿ß❛ð þ ù✐ø✲Þ❦ñ☛î➐ô òûá❵ö✦Þ✁Ý☞ö úÝ❦Þ❑à★Ý❦Þ✦ð❽▲✥Ý❾▲➌ñ☛ü☛ü þ Ý❜á þ ü✴Ý✶ßû Þ❿❵★Ý✦Ý✞ï➟ø✍❵→ß●à■Ý❜òûî✜❉✿Ý✁ö❛ô⑩Ý✁î➋ò✦Ý ûõ✩ß●à■Ý☞Þ✁Ý ú➉ûï⑤Ý☞Þ☞ÿ ￾✂✁✄￾✂✁♥➀ ➁✝✔➃➂✡➄✍➄⑧✌⑧➅✂✠✓✦❙✞✡✠✓✒ ✧✩★☞✪✬✫✮✭✰➆ ➇✺❢✍✇❈❤P➈♥➈❿➉ ❩