Is g Since the eigenvectors of RGs and a do not coincide, there is little we can say about the smoothing properties of Gauss-Seidel by looking at the eigenvalues of the iteration matri Iterations required to reduce an A error mode by a factor of 100 By looking at the number of iterations required to reduce the amplitude of each mode, of the A matrit, by a factor of 100, we can determ ine the smoothing operties of the Gauss-Seidel scheme. It turns out that based on the above graph, the high frequency modes do in fact decay at a much faster rate than the 2.3 Restriction SLIDE 10 We shall require procedures for transferring information between grids. The process of transferring a vector from a fine to a coarse mesh is called restriction Given wh we obtain w 2h by restriction 2=功hhn=19 mode k v 2 (mode k=2) v 15 (mode k=15) λ(RGS) 0 2 4 6 8 10 12 14 16 18 20 -1 -0.8 -0.6 -0.4 0 0.2 0.4 0.6 0.8 1 ➊❀➋❬➌➎➍P➏✥➋❧➋✸➐➒➑✗➓❈➔✄→✗➓❈➣✂➍↕↔✣➙✓➙✗→✝➋✸➛✈➙✓➙P➜✏➝➃➓❈➞❊➟ ➠❈➠❈➠ ➡✥➢✬➤➦➥✐➧❷➨➫➩➃➧✈➧❈➢➯➭❨➧❈➤✜➲❥➧✐➥⑧➨➒➳❥➵✐➸↕➳✸➺✲➻❃➼✂➽❯➾P➤✩➚⑦➪➶➚❨➳➹➤✩➳❥➨✺➥✐➳❥➢✬➤➦➥⑧➢❙➚❨➧❊➘q➨✬➩✜➧⑧➵✏➧❾➢✄➸❷➴➯➢✬➨✕➨❿➴t➧⑦➷❽➧▼➥✐➾❥➤❑➸❈➾P➬ ➾✣➮✐➳P➱✗➨✶➨➫➩➃➧❾➸❊✃▼➳❇➳P➨✬➩✗➢✬➤☞➭❾❐✩➵✏➳✏❐✜➧⑧➵❊➨❿➢❙➧❊➸✈➳❀➺❮❒❬➾P➱✓➸✏➸❊❰❡➡✚➧⑧➢❙➚❨➧⑧➴Ï➮⑧➬✑➴t➳❇➳✍Ð❥➢✬➤✓➭❑➾❥➨q➨➫➩➃➧✈➧❈➢➯➭❨➧❈➤✜➲❥➾❥➴Ñ➱✜➧⑧➸↕➳✸➺ ➨➫➩➃➧❾➢✬➨❡➧❈➵❧➾❥➨❿➢❙➳❥➤❯✃✈➾P➨✕➵❊➢➯Ò✓Ó Ô✩Õ☞Ö✬×✮Ø✰Ù ➊❡➜✏➓❈➞✐➍❥➜✏➔t➙❨Ú✜➋✶➞❧➓✍Û❨➏✜➔t➞✏➓❇→✑➜❧➙✈➞✏➓❇→✗➏✜Ü❈➓❷➍PÚ❑Ý ➧❈➵❊➵❧➳❥➵❋✃▼➳❇➚❨➧✺Þ✓ß ➍⑦à❙➍✣Ü⑧➜❧➙❨➞✺➙Pàqá❇â✣â 0 2 4 6 8 10 12 14 16 18 20 0 20 40 60 80 100 120 140 160 180 200 n=19 Number of iterations mode k ➠❇➠❈➠✐➌❋➑✑➔✄➋✺➍ ➭☞➳❇➳✍➚✈➸❊✃▼➳❇➳P➨✬➩✜➧⑧➵ ➠ ã➬❮➴t➳✍➳❇ÐP➢✬➤☞➭❑➾P➨✶➨➫➩➃➧⑦➤✥➱✗✃✈➮✐➧❈➵❾➳❀➺❷➢✬➨➒➧❈➵❧➾❥➨❿➢❙➳❥➤✜➸❾➵❧➧✐ä⑧➱✗➢✬➵✏➧✏➚✑➨➒➳❮➵❧➧✐➚P➱✜➥✏➧⑦➨➫➩➃➧✈➾❥✃✲❐✩➴➯➢✬➨✕➱✜➚❨➧▼➳❀➺⑦➧✐➾❨➥✏➩ ✃▼➳❇➚❨➧❊➘▼➳❀➺①➨➫➩➃➧ Ý ✃✈➾P➨✕➵❊➢➯Ò✣➘↕➮❈➬å➾▼➺⑧➾✣➥❈➨➒➳❥➵✻➳❀➺❑æ❨ç✣ç❥➘↕➷❽➧❯➥✏➾P➤è➚✣➧❈➨➒➧⑧➵❊✃↕➢✬➤✩➧✻➨✬➩✜➧①➸❊✃▼➳❇➳P➨✬➩✗➢✬➤☞➭ ❐✩➵✏➳✏❐✜➧⑧➵❊➨❿➢❙➧❊➸❯➳✸➺é➨➫➩➃➧å❒✺➾P➱✓➸✐➸✐❰❡➡✚➧⑧➢❙➚❨➧⑧➴✲➸❈➥✏➩✜➧⑧✃▼➧❇Óëê❊➨❾➨❿➱✗➵❊➤➃➸❑➳❥➱✗➨↕➨➫➩➃➾❥➨▼➮✐➾ì➸❈➧✐➚å➳P➤è➨➫➩➃➧✰➾✣➮✐➳❥➲❥➧ ➭✣➵❧➾✏❐✜➩☞➘í➨➫➩➃➧✾➩✗➢➯➭✍➩❷➺✐➵❧➧✐ä⑧➱✜➧❈➤✩➥❈➬✈✃✈➳✍➚✣➧⑧➸❾➚✣➳✈➢✬➤⑦➺⑧➾✣➥❈➨q➚❨➧✏➥✐➾P➬▼➾P➨✶➾✈✃↕➱✜➥✐➩➎➺⑧➾ì➸❊➨➒➧⑧➵➎➵❧➾P➨❡➧❷➨✬➩✜➾❥➤❑➨➫➩➃➧ ➴t➳P➷❑➺✐➵❧➧✐ä⑧➱✜➧❈➤✩➥❈➬✑➳❥➤➦➧❊➸❈Ó îqï✕ð ñ✿ò✮ó❨ô✗õ✜ö✸÷✩ô➃ö❀ø③ù Ô✩Õ☞Ö✬×✮ØûúPü ý➧❑➸❧➩✜➾❥➴✬➴❃➵❧➧✐ä⑧➱✗➢✬➵❧➧▼❐✩➵✏➳❇➥✐➧✐➚❥➱✗➵❧➧⑧➸❾➺⑧➳❥➵❯➨✕➵✏➾❥➤➃➸➫➺⑧➧❈➵❊➵❊➢✬➤☞➭û➢✬➤❇➺⑧➳P➵❊✃✈➾P➨✕➢❙➳P➤þ➮✏➧❈➨❿➷❽➧✏➧❈➤✙➭✣➵❊➢❙➚ì➸❈Ó ÿ➦➩➃➧ ❐✩➵✏➳❇➥✐➧⑧➸✏➸✲➳✸➺✺➨✕➵✏➾❥➤➃➸➫➺⑧➧❈➵❊➵❊➢✬➤☞➭↕➾❷➲❥➧✐➥⑧➨➒➳❥➵✂➺✐➵✏➳❥✃ ➾✂✁❽➤✩➧✺➨➒➳❾➾↕➥✐➳❇➾P➵✐➸❈➧❬✃▼➧❊➸❧➩❮➢✄➸✲➥✏➾P➴✬➴t➧✐➚❷➵✏➧❊➸❊➨❿➵❊➢❙➥⑧➨❿➢❙➳❥➤➦Ó ➌❃➔☎✄✣➓❈Ú✝✆✟✞✡✠✶➓➎➙Þ➜✐➍P➔♥Ú☛✆✌☞✍✞ Þ☞ß✟✎✑✏✓✒✕✔✖✎✑✗✙✘✚✔✕✗✙✛✢✜ ✆✝☞✍✞✤✣✦✥✞ ☞✧✞ ✆✝✞ ✥ ✞ ☞✧✞✩★ ➞✏➓❇➋✸➜❧➞✏➔♥Ü⑧➜❧➔♥➙✣Ú✝➙✫✪✩➓❈➞✐➍❥➜❧➙❨➞✭✬❙➛▼➍❥➜❧➞✏➔✯✮✩✰⑧➠ ✱