Solution methods Iterative Techniques Lecture 6
✂✁☎✄✝✆✟✞✡✠☛✁☎☞✍✌✏✎✑✞✡✒✓✁✕✔✗✖✙✘ ✚ ✞✛✎✢✜✛✣✙✞✡✠✥✤✦✎★✧✩✎✫✪✛✒✓☞✗✠✭✬✦✆✮✎✫✖ ✯✎✫✪✑✞✡✆✮✜✰✎★✱
1 Motivation SLIDE 1 Consider a standard second order finite difference discretization of V-u= on a regular g 1.2. and 3 dimensions 1.1 1D Finite differences △m=h bandwidth b= 1 n points Cost of Gaussian elimination O(bin)=O(n) 1.2 2D Finite differences slide 3 i-1,},i+1,j 1 7× points bandwidth b= n Cost of Gaussian elimination 0(b2n2)=0(n")
✲ ✳✴☎✵✷✶✝✸☎✹✺✵✻✶✭✴✽✼ ✾❀✿❂❁❄❃❆❅❈❇ ❉✫❊●❋■❍❑❏▼▲❖◆✭P❘◗✕❍❑❙✥◗●❋❚▲❯◗❱P❲▲✟❍❳◆❩❨✭❊●❋❚▲✗❊●P❲▲❖◆❩P✫❬■❋❯❏❭❙✥◆❪▲❖❏❴❫❀◆❩P❳◆❩❋❚❨☛◆☎▲❖❏▼❍✥❨☛P✥◆☛❙❳❏❛❵❩◗❱❙❳❏❛❊●❋✓❊❱❜ ❝❡❞✦❢☛❣✮❤❥✐✛❦ ❊●❋✮◗✦P❳◆❩❧●♠❯♥▼◗❱P♦❧♣P❳❏▼▲ ❦ ❏❛❋rq ❦❚s❖❦ ◗●❋❚▲✮t✦▲❯❏❴✉✈◆✭❋■❍❑❏❛❊●❋❚❍❩✇ ① ②④③ ❤⑥⑤ ⑦✷⑧❑⑦ ⑦✰⑨✍⑩✕❶❑❷❸❶❺❹❖❻❼⑨❥❶❾❽❿❻✰➀❯❻❆❷❸➁➂❻❆➃ ✾❀✿❂❁❄❃❆❅r➄ ➅➇➆ ❊●❏❛❋♣❙❲❍ ➈ 0 1 2 3 4 5 6 0 1 2 3 4 5 6 nz = 13 ➅➊➉✗➅ ✉✈◗❱❙❳P✥❏❭➋ ➌◗❱❋■▲❖➍❘❏❛▲❖❙❳➎④➏ ❤ q ❉✫❊♣❍❑❙❘❊❱❜✑➐➑◗❱♠❚❍✥❍❑❏▼◗❱❋✗◆✭♥❛❏❴✉✈❏❛❋❚◗➒❙✥❏❴❊♣❋➔➓✈→➣➏❢ ➅❆↔ ❤⑥↕➇➙❾➛✙➜ ⑦✷⑧➞➝ ➝✷⑨✍⑩✕❶❑❷❸❶❺❹❖❻❼⑨❥❶❾❽❿❻✰➀❯❻❆❷❸➁➂❻❆➃ ✾❀✿❂❁❄❃❆❅r➟ ➅➊➉✗➅➠➆❊♣❏❴❋❂❙❲❍ ➈ 0 5 10 15 20 25 0 5 10 15 20 25 nz = 105 ➅❢ ➉✓➅❢ ✉✕◗➒❙❳P✥❏❴➋ ➌◗❱❋❚▲❯➍❘❏❛▲➡❙✥➎④➏ ❤ ➅ ❉✫❊♣❍❑❙✙❊❱❜✷➐➑◗❱♠❚❍✥❍❑❏▼◗❱❋✓◆✭♥❛❏❛✉✦❏❛❋❚◗❱❙❳❏❛❊●❋★➓✈→➣➏❢ ➅❢ ↔ ❤⑥↕➠➙❺➛♦➢➡➜ q
1.3 3D Finite differences +1,k 7×n× n points bandwidth b=n This means that we we halve the grid spacing, we will have 8 times(23)more unknowns and the cost of soluing the problem will increase by a factor of 128 (2). It is apparent that, at least for practical three dimensional problems, faster methods are needed 2 Basic iterative methods 2.1J aconI 2.1.1 Intuiti d of sol at starting from an arbitrary u(a, 0) That is, we ecpect the solution af the time dependent problem to converge to the lution of Solution by relaxation By adding the time dependent term at, we have now a parabolic equation. If kept fixed, we expect the solution to"ce to the steady state solution (i.e 0). Recall that the time dependent heat transfer problem is modeled by this equation. In this case, we expect the temperature to settle to a steady state distribution provided that the heat source,f, and the boundary conditions, do not depend on t
➤✷➥➞➦ ➦✷➧✍➨✕➩❑➫❸➩❺➭❖➯❼➧❥➩❾➲❿➯✰➳❯➯❆➫❸➵➂➯❆➸ ➺❀➻❂➼❄➽❆➾➇➚ ➪r➶✗➪r➶✗➪r➹❀➘●➴❛➷♣➬❲➮ ➱ 0 20 40 60 80 100 120 0 20 40 60 80 100 120 nz = 725 ➪✰✃☎➶✓➪✰✃➑❐✕❒➒➬❳❮✥➴❴❰ Ï ❒❱➷■Ð❖Ñ❘➴❛Ð❖➬❳Ò④Ó❘ÔÕ➪✰Ö ×➘❂➮❾➬❘➘●Ø✷Ù➑❒●Ú❚➮❳➮❳➴▼❒❱➷✟Û❩Ü❴➴❛❐✈➴❴➷■❒➒➬❳➴❛➘●➷ÞÝ✈ß➞ÓÖ ➪✰✃❩àáÔ⑥â➇ã❾ä❘å➡æáç è➂é➡ê▼ë✦ì✈í❲î➒ï❚ë✽ðñé❯î❱ð❘òáí✈òáí❪é❯î❱óõô➒í✕ðñé❯í❿ö●÷✝êùø✗ë➣ú❯î♣û☛ê❄ï➡ö➒ü✙òáí✈ò✷ê❄ó❄ó❆é❚î➒ô➒í✂ý✮ð➣ê❄ì✈í☛ë✗þñÿ ✃✁ ì✄✂➒÷❳í ☎ï✝✆❱ï✞✂❱ò✷ï❯ë✂î➒ï➂ø ðñé❯í✓û✟✂➒ë✝ð✠✂☛✡✕ë☞✂❱óõôê❄ï❂ö➇ð❄é❚í➑ú➂÷✌✂✎✍✭ó❴í✭ì➔ò✷ê❄ó❄ó✫ê❄ï❀û✭÷❳í❲îë✭í✏✍☞✑➇î✒✡☛î●û✭ð✓✂❱÷✔✂☛✡✖✕✘✗♣ý þñÿ✘✙ ✎✚✜✛ð✛ê▼ë✙î❲ú●ú❚î➒÷❳í✭ï❚ð✻ðñé❯î❱ðùü✷î➒ð✛ó❴í❲îë✝ð✢✡☞✂❱÷✻ú❀÷✥î●û✭ð➞êùû❲î➒ó■ð❄é❖÷❳í❲í❸ø❱ê❄ì✈í✭ï❯ë✝ê✣✂❱ï❀î➒ó●ú❀÷✟✂✎✍✭ó❴í☛ì✽ë❲ü✢✡☛î➒ë❲ð❺í✭÷ ì✕í☛ðñé✤✂øë❪î➒÷✥í❿ï➂í✥í❲ø♣í✥ø✚ ✥ ✦★✧✪✩✬✫☞✭✯✮✞✰✬✱✳✲✴✧✠✰✬✫☞✵✶✱✸✷✱✹✰✬✺✼✻✾✽✿✩ ❀♦➥❑➤ ❁❃❂✻➵❅❄❇❆❸➩ ❈❊❉✣❋✞❉✣❋ ●☞❍❅■❑❏▼▲✣■◆▲✣❖◗P❘●❙❍❅■◆P✢❚❱❯▼❚❲P✢■◆❳❨■❑▲✣❩✞❍ ➺❀➻❂➼❄➽❆➾❭❬ ❪➷❚➮❑➬❳Û❒●Ð✗➘❱Ø✷➮❑➘♣Ü❴❫➡➴❛➷✤❵ ❛❝❜❅❞◆❞ Ô❢❡❤❣ Ñ✢Û➑➮❳➘●Ü✐❫●Û ❥ ❥✞❦❜ Ô ❜❞❑❞✳❧ ❡❤❣ ♠✒♥ ➮❑➬✥❒❱❮❳➬❳➴❛➷✤❵✽Øñ❮❳➘♣❐ ❒●➷✓❒●❮Ï ➴❴➬❳❮❲❒❱❮✟♦ ❜ ßq♣sr✟t♣à☞✉ ✈Û☎Û✭❰➡➹❀Û◆✇☛➬ ❜ ß✣♣sr ❦ ➱②①à ➱ ❜ ßq♣➂à✁✉ è➂é❯î❱ð✑ê▼ë❲ü✛òáí☎í✟③☛ú❯í❲û✭ð✷ð❄é❚í❸ë☞✂❱ó☎ð➣ê✣✂➒ï④✂☛✡❸ð❄é❚í✺ð➞ê❄ì✕í☎ø●í❺ú❚í☛ï➂ø●í✭ï❚ð❀ú❀÷✟✂✎✍✭ó❴í☛ì ð✓✂✈û⑤✂➒ï■ô í✭÷➞ö❂í➑ð✓✂✽ðñé❯í ë❙✂➒ó☎ð➣ê✣✂➒ï⑥✂☛✡➑ðñé❯í❿ë✝ðí❲î●ø✘✑✕ë✝ðî❱ðí♦ú❀÷✌✂⑦✍☛ó❛í☛ì✚ ⑧❭⑨❶⑩✁❷❘❸ ❹✬⑨❨❺✌❻▼⑩❽❼☛⑨✝❾➀❿⑦➁➃➂⑦❷✘❺➅➄❨➆s➄❶⑩❽❼☛⑨✝❾ ➇♦✓❒♣Ð❯Ð❖➴❛➷✤❵✂➬✥Ò❯Û❿➬✥➴❴❐✈Û✈Ð❖Û✭➹❀Û✭➷■Ð❖Û✭➷❂➬❡➬✥Û✭❮✥❐➉➈◆➊ ➈◆➋ ❣❀Ñ✢Û❪Ò❚❒❱❫♣Û❿➷❯➘➒Ñ ❒✂➹❚❒●❮✥❒Ï ➘♣Ü❴➴➌✇➑Û◆➍❂Ú❚❒❱➬❳➴❛➘●➷❊✉ ❪Ø ➬❳Ò❚Û Ï ➘♣Ú❯➷❚Ð❯❒●❮✌♦➎✇☛➘●➷■Ð❖➴❭➬✥➴❴➘♣➷❚➮❿❒●❮❳Û➐➏●Û❩➹❖➬➒➑❯❰❖ÛÐ❶❣✡Ñ✢Û✕Û☛❰❖➹❀Û◆✇☛➬☎➬❳Ò❯Û✟➮❳➘●Ü❛Ú❖➬✥➴❴➘♣➷➠➬✥➘➃➓✌✇✭➘●➷✝❫●Û❩❮✌❵♣Û◆➔ ➬❳➘➊➬✥Ò❯Û④➮❑➬❳Û❩❒♣Ð❨♦❈➮❑➬✥❒❱➬❳Û④➮❳➘●Ü❛Ú❖➬❳➴❛➘●➷ ßñ➴→✉ Û✎✉✐❣ ➈❑➊ ➈◆➋ Ô➣t❂à✁✉↕↔❘Û❑✇✭❒●Ü❴Ü❘➬✥Ò❚❒➒➬✕➬✥Ò❯Û ➬✥➴❴❐✈Û④Ð❖Û❩➹■Û❩➷❚Ð❖Û✭➷❂➬ Ò❯Û❒➒➬✂➬✥❮✥❒●➷❚➮❾ØñÛ❩❮✂➹❯❮✥➘Ï Ü❛Û✭❐ ➴▼➮✂❐✈➘❖Ð❖Û✭Ü❛Û❩Ð Ï ♦❈➬❳Ò❯➴▼➮✟Û❑➍❂Ú❚❒➒➬✥➴❴➘♣➷❊✉ ❪➷Õ➬❳Ò❯➴▼➮✔✇❩❒●➮❳Û✎❣♦Ñ✢Û④Û☛❰❖➹❀Û◆✇✝➬ ➬❳Ò❚Û☎➬❳Û❩❐✦➹❀Û✭❮❲❒➒➬✥Ú❯❮✥Û☎➬❳➘✂➮❳Û☛➬❳➬❳Ü❛Û☎➬❳➘✟❒✟➮❾➬✥Û❩❒●Ð✤♦✗➮❑➬✥❒➒➬✥Û❪Ð❖➴▼➮❾➬✥❮❳➴ÏÚ❖➬✥➴❴➘♣➷ ➹❯❮✥➘❲❫❂➴▼Ð❖ÛÐ✟➬✥Ò❚❒➒➬❡➬❳Ò❯Û❿Ò❚Û❩❒➒➬ ➮❳➘●Ú❯❮⑤✇☛Û⑦❣❨❡❤❣❯❒●➷❚Ð✗➬❳Ò❯Û Ï ➘●Ú❚➷❚Ð❯❒❱❮✟♦✖✇✭➘●➷❚Ð❖➴❴➬❳➴❛➘●➷■➮❙❣❯Ð❖➘✈➷❯➘●➬❡Ð❖Û✭➹❀Û✭➷■Ð✗➘♣➷ ❦ ✉ ÿ
we use an inexpensive(explicit)method. Thus avoiding the need to solve ystem o equ For instance 20+ }2=1,f={f} Here, we use the super indec r to denote iteration(or time level). u will denote the solution vector. An appro imation to u at iteration r will be denoted by l' SLIDE 7 2 possible, i.e.(At=h2/2) h2 1+12f) 2.1.2 Matrix form SLIDE 8 A=D-L-U L: Lower triangl triangular (D-L-Uu=f Iterative method (L+Uu+ D-(L +U)u+D- f D-(D-A)ur+ (I-D-1A)
➙✞➛✢➜➞➝s➟❭➠ ➡▼➢➐➤✌➢✎➥✐➦✎➧ ➨✞➩➨◗➫✿➭ ➩ ➯◆➯➳➲➸➵ ➺➧✏➻◗➤❽➧④➼✎➽➸➾✐➽✤➧☞➚❨➪✞➧❙➽➶➤✌➾✐➦✎➧➘➹q➧❙➚✝➪➶➥❴➾➌➴☞➾❴➷⑤➬➐➮➐➧❙➷✌➱✤➢❨✃❶❐❮❒❅❰❨Ï✝Ð✿Ñ✘Ò❱Ó✘Ô✣Õ❲Ô➞Ö✢×➸Øq❰✤Ù❘Ö✞Ù⑤Ù⑤Õ⑥ØÚÓ❭Ð❙Ó❲ÛÜÒ❲Ù➎Ñ ÛÜÔ➞Ö❅Ù✟Ñ✘ÝÞÐ✁ß❱Ð⑤ØÚÙ❙àáÓ❽â➒Ù✟ã❙Ï➶Ñ❲Ø→Ô✣Ó❲Ö➶Ð❙ä å✤➢⑦æ❝➾❴➽◗➤☛➷⑤➼✘➽➶➴❙➧✎çéè➩✞ê✟ë▼ì í î è➩ê í ï ➫ ➭ è➩ê íë▼ì î➘ð è➩ê í ➲ è➩ê íqñ ì ò➶ó ➲ ➵è í ô ➭öõ è➩ í✓÷❱øí❴ù ì❱úüû❭➭ýõ ➵èíÚ÷❑øí✐ù ì þ✒Ù❙Ý✌Ù☞ÿ✁❇ÙÞÏ✝Ð❙Ù Øq❰✤Ù Ð✁Ï✄✂✤Ù❙Ý➳Ô➞Ö❅Õ✎Ù✆☎✞✝✾ØÚÓ➐Õ✎Ù❙Ö✞Ó❲ØÚÙ✠Ô➞ØÚÙ☞Ý✌Ñ✘Ø➅Ô✣Ó✘Ö✠✟ÚÓ✘Ý Ø→Ô➞à✄Ù✠Û❴Ù☞Ò❲Ù❙Û✡✎ä ô ✴Ô➞Û➞Û❊Õ✎Ù❙Ö✞Ó❲ØÚÙ Øq❰✤Ù Ð❙Ó❲ÛÜÏ❨Ø→Ô✣Ó❲Ö✿Ò❲Ù☞☛☞ØÚÓ❲Ý❙ä✍✌ Ö④Ñ✆✂✎✂✞Ý✌Ó✏☎✎Ô➞à✄Ñ❲Ø→Ô✣Ó❲Ö➎Ø✓Ó ô Ñ❲Ø✴Ô➞ØÚÙ❙Ý✌Ñ❲Ø→Ô✣Ó❲Ö✑✝✒✴Ô➞Û➞Û✔✓⑤Ù✪Õ✎Ù❙Ö✞Ó❲ØÚÙ⑤Õ✕✓☞ß ô ê ä ô ê✌ë▼ì ➭ ô ê ➲ ï ➫ ➹ û î✠✖ô ê ➬ ➭ ➹✘✗ î ï ➫✖➬ ô ê ➲ ï ➫û✚✙ ➙✞➛✢➜➞➝s➟✜✛ ✢➷⑤➼✤✣✤➾✐➥❴➾❴➷✦✥✖✃❨➾➌➴✁➷⑤➼❲➷✟➧◆➤✹➷✟➱➶➼❲➷ ï ➫★✧ òó ð ➡✹➱✝➻➶➤◆ç ➺➧✠➷✟➼✪✩✎➧ ï ➫ ➼⑦➤✹➥✐➼✎æ✬✫⑦➧✠➼⑦➤✹➪◗➢✢➤✌➤✌➾✭✣✤➥❴➧⑦ç❨➾➅❐ ➧✎❐✳➹ï ➫ ➭ òó✄✮ ð ➬✁❐ ô ê✌ë❤ì ➭✰✯✗ î òó ð ✖✲✱✴✳ô ê ➲ òó ð û ✵ è➩ê✌ë▼ì í ➭✷✶ ð ➹ è➩ê íë❤ì ➲ è➩ê íqñ ì ➲ òó ➵è í ➬✹✸q➢✎æ✻✺ ➭ ✶ ú✏✙✼✙✼✙✾✽ ❐ ✿❁❀❃❂❄❀❅✿ ❆❈❇❊❉✼❋❍●❏■▲❑✔▼◆❋❍❖ ➙✞➛✢➜➞➝s➟◗P ✢➪✤➥✐➾❴➷ ✖ ✖ ➭❙❘ î✠❚⑥î✜❯ ❱❲❳ ❘❩❨❭❬➾✐➼✪✫✎➢✎➽◗➼✘➥ ❚ ❨❭❪➢➺➧❙æ❃➷✟æ✌➾➌➼✘➽❊✫⑦➻✤➥➌➼✘æ ❯ ❨❭❫➪➶➪◗➧◆æ❝➷✌æ✟➾✐➼✎➽❊✫✎➻➶➥✐➼✎æ ✖ ô ➭➀û ✣✞➧◆➴❙➢✎➮➐➧◆➤ ➹❘ î✠❚⑥î✜❯➬ ô ➭❢û ❴✓➷✌➧◆æ✟➼✘➷✌➾✐➦✎➧✠➮➐➧☞➷✌➱➶➢✝✃ ❘ ô ê✌ë▼ì ➭ ➹❚ ➲ ❯➬ ô ê ➲ û ➙✞➛✢➜➞➝s➟◗❵ ô ê✟ë▼ì ➭ ❘ ñ ì ➹❚ ➲ ❯➬ ô ê ➲ ❘ ñ ì û ➭ ❘ ñ ì ➹❘ î◗✖➬ ô ê ➲ ❘ ñ ì û ➭ ➹✘✗ î ❘ ñ ì ✖➬ ô ê ➲ ❘ ñ ì û ❛
RJ R=(-D-1 acobi iteration matrix D"=n2 Note that, in order to implement this method t for matrices, but update the unknowns, one component at a time, according to +hifi)for 2.1.3 Implementation Jacobi iterat 2.2 Gauss-Seidel SLIDE 11 Assuming we are updating the unknowns according to their index i, at the time u r +I needs to be calculated, we already know ur+i. The idea of Gauss-Seidel iteration is to always use the most recently computed value nknown values ost recent +1 12f)
❜❞❝✆❡✁❢❤❣❙✐❦❥❧❜★❝❧♠♦♥❩♣q❢sr ✐t❥✉❣✇✈✘①③②✠♥④♣q❢⑥⑤❦⑦⑨⑧❶⑩❁❷❊❸✪❹◆❺❶❻❤❼❾❽✏❿➁➀❍❷➂❽✏❻✘❹◆➃❙➄✴❷➂❽✼➀❍❻❏➅ ♥ ♣q❢ ➆➇➆ ❣➉➈◆➊✄➋✤➌ ➍➏➎✤➐➒➑➓➐❏➔❊→✤➐❃➣✍↔➙↕➛➎❍➜✆➝✪➑✼➜➞➐➟➎✒↔➙➠✲➡➤➢➇➑⑥➠✕➑✼↕➥➐✲➐❏➔➦↔❅➧➓➠➨➑✼➐➙➔➥➎✏➝➫➩❧➑✕➩❧➎❍➭➂➢➇➝➫➐➲➯✆➡❄↔❃➳☞→❍➢➙➢➵➯➫↕➤➎❍➐❁➸⑥➎✤➜❾➠➺→❍↕◆➯ ➠✕→❍➐➲➜❾↔❃➳✆➑⑥➧☞➣❞➻⑥➭➂➐❞➭✄➡❊➝✎→❍➐➟➑③➐❏➔❊➑➏➭➂↕➦➼✤↕❄➎✤➩s↕❊➧☞➣❞➎❍↕➤➑➞➳☞➎✤➠✲➡➥➎❍↕➤➑⑥↕◆➐✍→✤➐★→✕➐➲↔➙➠✕➑❾➣★→✎➳☞➳✆➎✤➜✬➝❍↔➙↕➦➽✒➐➒➎ ➾ ➪➚ ❝✬❡➶❢ ➆ ❣➘➹ ➌ ✈➪➚ ❝ ➆❡✁❢ ♠ ➪➚ ❝ ➆ ♣q❢ ♠➴➈➊ ➷➚ ➆ ⑦➮➬❏➱✎✃➉❐❶❣ ➹✎❒✼❮✼❮✏❮✦❰ Ï❁Ð❃Ñ❄Ð✭Ò ❼⑥Ó♦Ô➶Õ✘❿➁Ó➛❿➁➃➤❽✄❷➂❽✏❻✘❹◆➃ Ö❄×➁Ø➙Ù✔Ú➴Û✤Ü known values unknown values Ý➁Þ✪ß➱✎à❊á➤á➵â✆ã✼✃Þâ✆á➇➱✎ä ➪❝✬❡➶❢ ➆ ❣ ❢ ➊ ✈➪❝ ➆❡✁❢ ♠ ➪❝ ➆ ♣❁❢ ♠➛➈◆➊ ➷ ➆ ⑦ å✍æ✘å ç❙è❶é❦ê➦ê✤ë✎ì★í✔î✬ï✲í✔ð Ö❄×➁Ø➙Ù✔Ú➴Û❊Û ñ ➧☞➧❾➭➂➠➓↔➙↕➁➽➨➩❧➑➓→❍➜✬➑➏➭✄➡➥➝✪→❍➐➲↔➙↕➁➽ò➐❏➔❊➑✉➭➂↕➂➼❍↕➤➎❍➩s↕➥➧✉→✎➳✆➳☞➎✤➜✬➝❍↔➙↕➦➽ò➐➟➎➨➐❏➔❊➑✼↔➙➜✉↔➙↕❄➝✎➑✬ó ❐ ➣❧→❍➐❧➐❏➔❊➑➏➐➲↔➙➠➨➑ ➪❝✬❡✁❢ ➆ ↕❄➑☞➑☞➝ô➧✑➐➟➎õ➻☞➑④➳☞→❍➢➇➳✼➭➂➢➇→✤➐➒➑☞➝ô➣✉➩❧➑④→❍➢➵➜✬➑☞→✪➝❍➯❩➼❍↕➤➎❍➩ ➪❝✆❡✁❢ ➆ ♣q❢➤ö✻÷➔➥➑➫↔❃➝✪➑☞→✠➎✦➸ùø❤→✤➭➦➧☞➧☞ú➒ûq➑⑥↔❃➝✎➑⑥➢ ↔➙➐➟➑⑥➜✬→✤➐✘↔❃➎✤↕ù↔❅➧✉➐➟➎ò→❍➢➵➩❧→❍➯ô➧③➭➦➧✼➑✉➐❏➔❊➑➏➠➨➎❍➧❾➐❞➜✬➑☞➳☞➑✼↕➥➐➲➢ü➯✑➳✆➎✤➠✲➡➤➭➂➐➒➑☞➝✑ýô→✤➢ü➭➥➑ ö known values unknown values þ❷❊ÿ✁✂☎✄✝✆✔❿➦❻✟✞➶❿➦Õ❧❻❏❽✄❿➁➀ô❷❊❽✏❻❃❹❄➃❙✈ ß➱✪ä✡✠✬á☞☛➂ã✼✃✍✌➨➱✎✠✾â❤✃✬ã ßã✏ä✎â❤á➇â✬ã✏✃Þâ✬ã✄⑦ ➪❝✬❡✁❢ ➆ ❣ ❢ ➊ ✈➪❝ ➆❡✁❢ ♠ ➪✑✏✓✒ Ñ ➆ ♣q❢ ♠➴➈➥➊ ➷ ➆ ⑦ ✔