Temperature distribution Let u(a) and q(a) denote the temperature and heat flux in a homogeneous heat conducting bar of unit length. The bar is subjected to a distributed volumetric heat source p(a)and the temperature is maint ained at zero at the end points; the sides of the bar are assumed insulated so that the heat flow is one-dimensional cT(e) Acp da (q dq The stationary temperature distribution can be obt ained by considering the energy balance Ac(q+dq)-Acq= Acp d and the empirical relation between the temperature and the heat flux In the above equations, k is the heat conductivity and Ac is the bar cross ectional area. Defining f=p/k and eliminating q from the above equations 1. 1.1 Solution Properti SLIDE 2 The solution u(a) always exists ·u(x) is always“ smoother"than the data f(x) (see first lecture). In particular, if f has m continuous derivatives, u will have n+2 continuous derivatives. Thus, if Eco, thenuEC2 ·If(x)≥0 for all ax, then a(x)≥0 for all az Follows from the positivity of Greens function us≤(1/8川fls Given f(a) the solution u(a)is unique Continuous stability estimate ecall that for a function u: Q-R
➝r➞✾➟✁➠➢➡ ➤③➠✬➥r➦✉➠✬➧✹➨✾➟❂➩❤➧P➠✏➫❇➭❾➯P➟☞➧❱➭☞➲✱➩④➟✿➭⑤➞✈➳ ➵✛➸✢➺♦➻❦➼★➽❇➾③➚✬➪❤➶✐➹✡➼★➽❇➾❝➶✡➸❂➪✓➘✬➺❁➸❍➺☞➴✮➸✤➺☞➸❂➷●➬❤➸❂➮❁➚✬➺☞➱✓➮❁➸✝➚❃➪✮➶●➴✓➸❂➚✬➺❝✃✮➱✓❐●❒❆➪✰➚✷➴✮➘❃➷●➘❃❮P➸✢➪✓➸❂➘❃➱✮❰✦➴✮➸❂➚✹➺ Ï➘P➪✮➶✡➱Ï➺☞❒❆➪✓❮✗Ð✮➚✬➮✴➘✬Ñ❏➱✮➪✓❒⑨➺❍Ò❆➸✢➪✓❮❃➺☞➴✛Ó③Ô♠➴✮➸✝Ð✮➚❃➮♦❒✻❰♠❰✿➱✮Ð✡Õ⑤➸Ï➺☞➸❂➶✰➺☞➘●➚❶➶✓❒❆❰✿➺☞➮❁❒◆Ð✮➱✡➺☞➸❱➶✰ÖP➘❃Ò❆➱✓➷●➸❊➺☞➮❁❒ Ï ➴✓➸❱➚✹➺④❰☞➘❃➱✓➮ Ï➸✾×✱➼★➽❇➾✛➚✬➪❤➶❫➺☞➴✓➸❝➺☞➸✢➷●➬✑➸✢➮❸➚✹➺❁➱✓➮☞➸✦❒✻❰✛➷❧➚❃❒◆➪❅➺❁➚❃❒◆➪✮➸❂➶✷➚✹➺④Ø✢➸✢➮❁➘✉➚✬➺✛➺☞➴✓➸❝➸✢➪✮➶✷➬✑➘❃❒❆➪❅➺❁❰❂Ù❩➺☞➴✓➸ ❰☞❒❆➶✡➸❱❰✴➘✬Ñ✛➺❁➴✓➸✝Ð❤➚✬➮❍➚✬➮❁➸✝➚P❰☞❰☞➱✓➷●➸❂➶✰❒◆➪✮❰☞➱✓Ò✻➚✹➺❁➸❂➶✰❰☞➘⑥➺❁➴✮➚✹➺♠➺❁➴✓➸✵➴✓➸❂➚✬➺♦✃✮➘✹Ú➙❒✻❰✴➘❃➪✓➸✢ÛÜ➶✓❒◆➷●➸✢➪❤❰✿❒❆➘❃➪✮➚❃Ò❾Ó Ô♠➴✓➸➢❰✿➺❁➚✬➺☞❒❆➘❃➪✮➚❃➮☞ÝÞ➺☞➸✢➷●➬✑➸✢➮❸➚✹➺❁➱✓➮☞➸❢➶✓❒❆❰✿➺☞➮❁❒◆Ð✮➱✡➺☞❒❆➘❃➪ Ï➚✬➪➙Ð✑➸❢➘❃Ð✡➺❸➚✬❒❆➪✓➸❂➶↕Ð✈Ý Ï➘❃➪❤❰✿❒✻➶✡➸✢➮❁❒❆➪✓❮❼➺☞➴✓➸ ➸✢➪✮➸✢➮❁❮❃Ý❧Ð✮➚❃Ò❆➚❃➪Ï➸ ß✝à ➼❾➹❍á❛â❅➹✬➾❦ã ß❫à ➹✵ä ß✝à×❢âP➽❏å ➚✬➪❤➶✐➺❁➴✓➸✷➸✢➷●➬✓❒❆➮☞❒ Ï➚❃Ò✾➮☞➸❂Ò❆➚✬➺☞❒❆➘❃➪✄Ð❤➸✢➺⑤Ú♦➸❂➸✢➪⑩➺☞➴✓➸✵➺❁➸✢➷●➬✑➸✢➮❸➚✹➺☞➱✮➮☞➸⑥➚❃➪✮➶✐➺❁➴✓➸✷➴✓➸❱➚✹➺♠✃✮➱✡❐ ➹✷äæã✉ç✈è✝éëê ➼✲ì✮➘❃➱✓➮❁❒◆➸❂➮❂í ❰✴Ò❆➚❩Ú✉➾ î✍➪q➺❁➴✓➸❼➚✬Ð✑➘✹Ö❃➸➢➸❱ï❅➱✮➚✹➺❁❒◆➘P➪✮❰✢ð❫ç➙❒✻❰✰➺❁➴✓➸r➴✓➸❂➚✬➺ Ï➘❃➪✮➶✡➱Ï➺☞❒❆Ö✈❒⑨➺⑤Ýq➚❃➪✮➶ ßà ❒❆❰⑩➺☞➴✮➸ñÐ✮➚❃➮ Ï➮❁➘P❰❁❰ ❰☞➸Ï➺❁❒◆➘P➪✮➚✬Ò③➚❃➮☞➸❱➚✓Ó❧ò✝➸❊ó✮➪✓❒❆➪✓❮➢ôõäö×❇÷❃ç➢➚❃➪✮➶r➸✢Ò❆❒◆➷●❒❆➪✮➚✹➺❁❒◆➪✓❮❢➹✰Ñ★➮❁➘❃➷ø➺❁➴✓➸✐➚✬Ð✑➘✹Ö❃➸●➸❂ï❅➱✮➚✬➺☞❒❆➘❃➪✮❰ Ú✴➸❫➘PÐ✡➺❁➚❃❒◆➪⑩➺☞➴✮➸⑥ù❦➘❃❒✻❰☞❰☞❒◆➘P➪✰➸❱ïP➱❤➚✹➺☞❒❆➘❃➪❏Ó ú✑û✲ú✑û✲ú ü❏ý❤þ❾ÿ✁✄✂✲ý✆☎✞✝✠✟❩ý☛✡✌☞✍✟✎✏✂✑☞✍✒ ✓ ✔✍✕✗✖✁✘✚✙ ✛ Ô♠➴✓➸⑥❰☞➘❃Ò❆➱✡➺☞❒❆➘❃➪✄➻✱➼✲➽✑➾♠➚❃Ò◆Ú♠➚❩Ý✡❰ ☞✍✜✢✂✑✒✣✄✒ ✛ ➻✱➼✲➽✑➾✴❒✻❰✤➚✬Ò❆Ú♠➚❩Ý✈❰✥✤✒✄✦Þý✛ý☛✏✧★☞✩✟✫✪ ➺❁➴✮➚✬➪⑩➺☞➴✮➸⑥➶✓➚✹➺❸➚●ô❦➼✲➽✑➾ ✬✮✭✣✯✰✯✲✱✴✳✵✭✷✶✹✸✺✯✵✻✼✶✮✽✾✳✿✯❁❀❃❂❅❄✷❆❈❇❊❉✫✳✷✶✮❋●✻✣✽✾✸✺❉✫✳✵❍✹❋■ ô❑❏ ❉✫✭▼▲◆✻✵❖P❆❊✶✮❋✗❆❊✽❊❖✫✽◗✭✥❘✩✯✼✳✷❋✗❙✫❉✫✶✮❋✗❙✎✯✼✭✵❍ ➻❯❚❋✗✸✗✸ ❏ ❉✫❙✫✯ ▲ á❲❱ ✻✵❖✫❆☛✶✑❋✗❆☛✽❊❖✫✽◗✭❳❘❃✯✣✳✷❋✗❙✎❉P✶✑❋✗❙✫✯✷✭✣❂❩❨❏✽◗✭✰❍✲❋■ ô❭❬❫❪❵❴ ❍✴✶❏ ✯✣❆ ➻❛❬❜❪✢❝ ❂ ✛ îÜÑ✦ô❦➼✲➽✑➾❡❞❯❢♥Ñ★➘❃➮✤➚✬Ò❆Ò✛➽④ð✡➺☞➴✓➸❂➪❻➻✱➼✲➽✑➾❡❞❯❢✗Ñ★➘❃➮✤➚✬Ò❆Ò✾➽ ❣❖✫✸✗✸✺❖❚✭✌■✵✳✿❖✫❤✐✶❏ ✯❡❇❊❖✎✭✷❋✗✶✮❋✗❙✏❋✗✶✮❥❫❖❦■❈❧❡✳✿✯✵✯✼❆❵♠✭♥■✵✽✾❆✆✻✣✶✑❋●❖P❆✆❂ ✛❜♦✺♦ ➻ ♦✺♦ ♣rq ➼❁s✹÷✫t❅➾ ♦✉♦ ô ♦✉♦ ♣ ✈✠✇ ✛②①❒◆ÖP➸✢➪✏ô❦➼★➽❇➾✴➺☞➴✓➸⑥❰☞➘❃Ò❆➱✡➺❁❒◆➘P➪⑩➻❦➼★➽❇➾✴❒❆❰ ÿ★☎③✂✑④✾ÿ③☞ ✈t ➝r➞✾➟✁➠⑥⑤ ⑦❶➞✡➳✦➟✿➭❾➳③➩✾➞✛➩✮➯❧➯❅➟☞➨✛➲❤➭✑⑧✲➭✿➟●⑨➊➠✬➯❅➟✿➭❾➥❛➨✾➟✁➠ ⑩r➸✷➮☞➸ Ï➚✬Ò❆Ò✑➺☞➴✮➚✬➺❍Ñ★➘❃➮✤➚♥Ñ★➱✓➪Ï➺☞❒❆➘❃➪✏➻❛❶✩❷✞❸ î❹ ❺
where Q is the domain of definition. For example, the -norm of the functions z, z(1-x), eva and sin(T), in the interval Q=[0, 1] is 1, 1/4, e and 1, respectively. non-negative we have u()≤/G(x,y)f()y≤川fGx,y)4=12(1-) l=sup,(x)≤ll This estimate is a consequence of the fact that the solution u depends cor tinuously on the data f. In other words, we can say that if f is small so is Note 8 Solution uniqueness Uniqueness of the solution follows directly from the above estimate. If we have two solutions u, and u2 which satisfies the Poisson problem for a given f, we ave that uf-u2=(u1-u2"=0. This implies that u1-u2 sat isfy the Poisson problem for f =0. Thus, we can use the above stability estimate to show that J01-u2llo0=0. Therefo (We note that the be reached by integrating(u1 -u2)"=0 twice and imposing the appropriate boundary conditions.) 2 Numerical solution 2.1 Finite differences 2.1. 1 Discretization Subdivide interval(0, 1)into n+1 equal subintervals △ 0
❻✉❻ ❼♥❻✺❻ ❽❿❾➁➀❦➂➄➃ ➅❃➆❃➇ ❻ ❼★➈●➉✆➊✣❻➌➋ ➍❅➎➄➏✏➐✿➏❭➑➓➒ ➀❈➔➎➄➏❭→✾➣✩↔❈↕P➒✉➙❯➣❃➛✠→➄➏✼➜❊➙➄➒➔ ➒✉➣❃➙✁➝➟➞➄➣✩➐➠➏✼➡➄↕❃↔➃❊➢➏✩➤ ➔➎➄➏ ❻✉❻➄➥✢❻✉❻ ❽❜➦ ➙➄➣✩➐✿↔➧➣P➛ ➔➎➄➏ ➛➂ ➙❊➨➔ ➒✉➣❃➙➀✲➉ ➤ ➉★➈❁➩✴➦➫➉➭➊ ➤✾➯P➲➅ ↕P➙❊→ ➀ ➒✉➙ ➈➵➳❵➉➭➊ ➤✾➒✉➙ ➔➎➄➏➸➒✉➙➔ ➏✏➐✿➺✫↕➢ ➑❯➻➽➼➾ ➋✣➩✣➚ ➒ ➀➪➩ ➤ ➩✎➶✎➹ ➤✩➯✠↕P➙☛→ ➩ ➤✾➐✰➏➀✿➃ ➏✏➨➔ ➒✺➺✩➏➢✉➘➝ ➴ ➒✉➙❊➨✼➏❳➷➽➒ ➀ ➙➄➣❃➙✾➬➮➙➄➏✏➱✩↕➔ ➒✺➺✩➏➸➍✹➏✠➎❊↕✎➺❃➏ ❻ ❼★➈●➉✆➊✣❻✾✃❒❐❯❮ ❰ ➷➈➵➉③➋❦Ï➄➊✣❻ ÐÑ➈➵Ï➄➊✏❻ Ò✩Ï❫✃Ó❻✺❻ Ð♥❻✉❻ ❽ ❐❯❮ ❰ ➷➈➵➉③➋❦Ï➄➊❁Ò✩ÏÔ❾Õ❻✉❻ Ð♥❻✉❻ ❽ ➩ Ö ➉③➈❦➩❅➦×➉➭➊✷Ø Ù➎➄➏✏➐✿➏✣➛➵➣❃➐✰➏ ❻✉❻ ❼♥❻✺❻ ❽ ❾ ➀✿➂➄➃ ➅❃➆✾Ú❰✣Û ❮ÝÜ ❻ ❼★➈●➉✆➊✏❻◗✃ ➩ Þ ❻✺❻ Ð♥❻✺❻ ❽ Ø Ù➎➄➒ ➀ ➏ ➀❦➔ ➒✉↔❈↕➔ ➏❜➒ ➀ ↕⑥➨✣➣❃➙➀ ➏✏ß➂ ➏✏➙❊➨✼➏➫➣P➛ ➔➎❊➏➫➛●↕❃➨➔❈➔➎❊↕➔❈➔➎➄➏ ➀➣➢✉➂✾➔ ➒✉➣❃➙ ❼ →✾➏➃ ➏✏➙❊→➀ ➨✣➣❃➙✾➬ ➔ ➒✉➙➂ ➣➂☛➀❦➢✉➘ ➣❃➙ ➔➎➄➏❜→❊↕➔ ↕ Ð ➝✚àÝ➙á➣➔➎➄➏✣➐②➍✹➣❃➐✵→➀ ➤♥➍✹➏❫➨✏↕P➙ ➀ ↕➘✚➔➎❊↕➔ ➒✺➛ Ð ➒ ➀Ô➀↔❈↕➢✉➢❡➀➣×➒ ➀ ❼ ➝ â✚ã❵ä✷å❛æ ç♥ã✾è✿é③ä❦êÝã✾ëìé❊ë★êÝí③é✢åPë✴åPî✄î ï➙❊➒✉ß➂ ➏✏➙➄➏➀✰➀ ➣P➛ ➔➎➄➏ ➀➣➢✉➂✾➔ ➒✉➣❃➙❫➛➵➣➢✉➢➣✫➍➀ →✾➒✉➐✰➏✏➨➔✰➢✺➘ ➛➵➐✿➣✩↔ ➔➎➄➏ð↕❃ñ☛➣✫➺✩➏➸➏ ➀❁➔ ➒✺↔❈↕➔ ➏❃➝♥à➮➛★➍✲➏✠➎❊↕✎➺✩➏ ➔➍✹➣ ➀➣➢✉➂✾➔ ➒✉➣❃➙➀➪❼ ❮ ↕P➙❊→ ❼✆ò ➍❅➎➄➒ó➨✵➎ ➀ ↕➔ ➒ ➀➜❊➏➀➸➔➎➄➏❈ôÑ➣❃➒ ➀✿➀➣❃➙ ➃ ➐✿➣✩ñ➢➏✏↔õ➛➵➣❃➐ð↕❫➱❃➒✉➺❃➏✣➙ Ð ➤❵➍✲➏ ➎❊↕✎➺✩➏ ➔➎❊↕➔♥❼➭ö ö ❮ ➦÷❼✆öòö ❾➽➈➵❼ ❮ ➦✥❼➭ò✏➊❦ö ö✆❾ ➾❊➝ Ù➎➄➒ ➀ ➒✉↔➃❊➢ ➒✉➏➀★➔➎❊↕➔♥❼ ❮ ➦÷❼✆ò❡➀ ↕➔ ➒ ➀ ➛➘ð➔➎❊➏❡ôÑ➣✩➒ ➀✰➀➣❃➙ ➃ ➐✰➣❃ñ➢➏✣↔r➛➵➣❃➐ Ð➫❾ ➾❊➝ Ù➎➂❊➀ ➤➄➍✲➏❳➨✏↕P➙ ➂❊➀ ➏ ➔➎➄➏❳↕❃ñ☛➣✫➺✩➏ ➀❦➔ ↕❃ñ➄➒ ➢ ➒➔❁➘ ➏ ➀❁➔ ➒✺↔❈↕➔ ➏ ➔➣ ➀➎➄➣✫➍ ➔➎❊↕➔ ❻✉❻ ❼ ❮ ➦❲❼ò ❻✺❻ ❽ ❾ ➾❊➝ Ù➎➄➏✣➐✰➏✼➛➵➣✩➐✿➏✩➤ ❼ ❮ ❾Õ❼ò ➈➵ø➏❈➙➄➣➔ ➏ ➔➎❊↕➔ð➔➎➄➏ ➀ ↕P↔Ô➏❈➨✼➣❃➙☛➨➢✉➂❊➀ ➒✉➣❃➙✚➨✣↕❃➙ ñ✆➏②➐✰➏✏↕✩➨✵➎➄➏✏→ùñ➘ ➒✺➙➔ ➏✣➱❃➐✵↕➔ ➒✺➙❊➱ ➈●❼ ❮ ➦❑❼ò ➊❁ö ö✴❾ ➾ ➔➍❅➒✉➨✣➏❈↕P➙❊→×➒✺↔➃ ➣➀ ➒✺➙❊➱ ➔➎➄➏➠↕➃➄➃➐✿➣➃ ➐✿➒ó↕➔ ➏ ñ✆➣➂ ➙❊→❊↕P➐➘ ➨✼➣✩➙❊→✾➒➔ ➒✉➣❃➙➀ ➝ ➊ ú ûýü❛þÿ✁✄✂✆☎✞✝✠✟☛✡✌☞✍✟ü✏✎ ✂✆☞✒✑ ✓✕✔✗✖ ✘✚✙✗✛✜✙✣✢✥✤✧✦★✙✣✩✠✤✫✪✬✤✭✛✯✮✰✤✲✱ ✳✫✴✶✵✷✴✶✵ ✸✺✹✶✻✽✼✿✾❁❀❃❂❄✹❆❅❈❇✥❂✽✹✶❉✷❊ ❋✷●❃❍❏■✲❑▼▲ ➴➂ ñ➭→✾➒✉➺◗➒✉→✾➏ð➒✉➙➔ ➏✏➐✿➺✫↕➢✴➈ ➾ ➋✏➩✄➊ ➒✺➙➔ ➣✚◆✚❖ ➩ ➏✏ß➂ ↕➢✢➀✿➂ñ➄➒✉➙➔ ➏✏➐✿➺✫↕➢ó➀ P➉❜❾ ➩ ◆✚❖ ➩ ◗
du w J F yy n FR5se ut, W vel owe ke-cC w1 Wuy wl ew-IR5se We 51 yeg lodf W=1yz-W: e7Wgw Toss erWgt'ydomy, maw nffifn FI FoFVE mO/// 4 A 7e(Cqr-18Wn5171 W=lRze Tol= Wow f Edu. tIo cs ea w T r,2 1 y j
❘✬❙✯❚❱❯❳❲❨❘✭❩ ❭❪❙✜❫❴❭❪❙✯❵☛❭❜❛✶❘✥❙✽❝ ❬ ❞❢❡❈❣✐❤✒❥ ❯ ❥❧❦✚♠☛♥ ♦q♣✬rts✈✉❙①✇③②❁④⑤④r⑤⑥⑧⑦②s✷⑨❁⑥✒⑩② ⑩❢♣✬r❷❶❁❸❁❹❻❺❪r✆⑥ ②❽❼ ❸ ✇③②s❪⑩⑧❾❏s❪❺② ❺❳⑥ ❼❺✥s✇ ⑩⑧❾ ②s✈✉ ❛❢❘✷❝⑤❿ ❸❁⑩➀⑦② ❾❏s❪⑩ ❘✬❙❈❿ ➁r ➁❾❏❹❏❹➂s ② ⑩➀➃✚❸❄➄➅r❨❸❁s➇➆❷⑨✿❾➈⑥⑤⑩❆❾ ✇ ⑩❆❾ ② s➊➉③r✆⑩➁r③r✆s❷✉❙ ❸❁s✰⑨➋✉ ❛✶❘✥❙➌❝❄➍ ♦✈r ➁❾❏❹❏❹➂❺❳⑥tr ❭❪❙ ❬ ⑩② ⑨❈rts② ⑩✣r ⑩❢♣✬r❷❸➎⑦➅⑦④➏②❄➐❾❏➃✚❸❁⑩⑧❾ ②s➑⑩②✍❭❪❙❈➍ ♦✈r ➁❾❏❹❏❹➒❺❳⑥✆r✍⑩❢♣✬r✒❺✥s✰⑨❈r ④⑥ ✇③②✿④r✍⑩② ❾❏s✷⑨✿❾ ✇❸❁⑩✣r❷❶➌r✇ ⑩②✿④t➍✈➓♣✥❺❳⑥ ❿ ✉ ⑨❈rts② ⑩✣r✆⑥➋⑩❢♣✬r❨❶❁r✇ ⑩②❁④➋➔ ✉ ❙✿→✿➣③↔❪❙✆↔✷↕✫➍ ➙✫➛✶➜✷➛➈➙ ➝✌➞✭➞❜➟❁➠❳➡✫➢✶➤➦➥✥➧✽➢✶➠✷➨ ➩✷➫❃➭❏➯✲➲➵➳ ➸❡➅❣✞➺✆➻✬➼✿➽❷➾✬➚➪➺❷➶t➶❄➶ ✉❳➹ ➹ ❛❢❘✥❙❁❝ ❫ ♥ ❲✍❘ ❛✉❳➹ ❛❢❘❙➏➘✭➣✗➴➎➷ ❝➒➬ ✉❃➹ ❛❢❘❙✆➮➂➣➏➴➎➷ ❝➏❝ ❫ ♥ ❲✍❘ ❛ ✉ ❙➏➘❜➣ ➬ ✉ ❙ ❲✍❘ ➬ ✉ ❙ ➬ ✉ ❙✆➮✫➣ ❲✍❘ ❝ ❚ ✉ ❙➏➘✭➣ ➬✃➱✉ ❙ ♠ ✉ ❙✆➮✫➣ ❲✍❘➷ ❞❢❡➅❣ ❲❨❘➊❐➽✚➼✿➚➪➚ ❒ ➃②❁④r ❼✆②❁④➃✚❸❁❹✕⑨❈r ④❾❏❶➌❸✿⑩❆❾ ② s ②✗❼ ⑨❁❾❮✯r ④rts✇r✺❸➎⑦❈⑦ ④➏②t➐❾❏➃✚❸❁⑩⑧❾ ②s❪⑥q⑩②❨❼❺✥s✇ ⑩❆❾ ② s❱⑨❈r ④❾❏❶➌❸✿⑩❆❾❏❶❁r✆⑥ ➁❾❏❹❏❹❜➉③r ✇③②s✬⑥⑤❾✶⑨➅r ④ ❹❰❸✿⑩Ïr ④t➍ ➙✫➛✶➜✷➛➪Ð ÑÓÒ➂Ô✭➥✬➧❄➢✶➠✷➨❜Õ ➩✷➫❃➭❏➯✲➲▼Ö ➬✞❭✰×❄×Ø❚★Ù ❐➏Ú✬Û❈Û➺ ❐✗Ü➎❐ ➶t➶❄➶ ➬ ❭❬ ❙➏➘✭➣ ➬Ý➱✭❬❭❙ ♠ ❭❬ ❙✆➮➂➣ ❲✍❘➷ ❚★Ù➒❛❢❘❙ ❝ ♥➋❥ ❯ ❥❱❦ ❭✷Þß❚à❬ ❬ ❭✰↕✿➘✭➣✞❚ ❤ ❚✫á âã❬❭ ❚✧Ù ä
&du i b P aui) 2aU2) a WW盏 a(能 u aqui csthn2trint ∈Rl 66 for an≠,( n is SPD) arAxi Matrix Properties We give below the definition of some matrix classes and their main properties w sthn2trin Positiv2 D2finit2 (SPDt We sa that a matrix A sitive definite if M> for an non-zero vector For s mmetric matrices this condition is e Divalent to rehiring that all the elg es of the matrix be positive To show this we note that if n is s, mmetric and has real coet cients, it can bwwwritten as M=e6e, where 6 is the diagonal matrix of eigenvalues and e is an orthonormal transformation Iwe_A sinve e is non-Singtlar), impies br>, for an -#, or an. all the greater than zero bviousl, an matrix which is i s b is also non-sing ular and therefore invertiblE Ci=e 6 ele t Dieronel Dominent ft atv W that a matrix M diagonal dominant if
å✷æ❃ç❏è✲é▼ê ë★ì í î❨ï✷ð ñò ò ò ò ò ò òó ô õ í ö ÷❄÷t÷ ö õ í ô õ í ø ø ø ø ø ø ö ø ø ø ø ø ø ø ø ø ö ø ø ø ø ø ø õ í ô õ í ö ÷t÷t÷ ö õ í ô ùtúú ú ú ú ú ú ûýü ÿþ ì ñò ò ò ò òó ÿ✁ þÿþ ð ø ø ÿ✄✂✆☎ þ ø ÿ✄✂þ ùtúú ú ú ú û ü ✝ ì ñò ò ò ò òó ✝✟✞ï ✡✠ ✝✟✞ï ð ✠ ø ø ✝✟✞ï ✂✆☎ ø ✠ ✝✟✞ï ✂☛✠ ùtúú ú ú ú û ☞✍✌✏✎✁✑✒✑✔✓✖✕✘✗✚✙✜✛✆✢ ë✤✣✦✥✧✂✩★✪✂ ÿþ ü ✝ ✣✦✥✧✂ ✫✁✬✜✭✩✬✯✮ ✌✱✰✳✲✵✴✱✕✶✙✜✰✩✷ å✷æ❃ç❏è✲é✹✸ ✺✘✻ ë ✷✼✰✩✷✱✽ ✻ ✙✜✷✿✾✳✴✿✲✜❀✪✗❂❁ ❃❅❄❇❆❉❈❋❊✆● ❍ ì❏■ ❍ ü ❍ ð ü ❑✘❑✡❑ ü ❍ ✂✩▲✶▼ ❍▼ ë ❍ ì í î✍ïð ✞❍ ð ❖◆ ✂ P◗❘ ð ✞❍ ◗ õ ❍ ◗ ☎ ✘✠ ð ◆ ❍ ð ✂ ✠ ❙❉❚✡❊☛❯❱❚ ❍▼ ë ❍ ❲ ö❅❳✆❨❄❇❆❉❈❋❊✆●❩❍ ❭❬ ö ✞ë❫❪❵❴ ✌✱❛❝❜✠ ❞❢❡ ë ÿþ ì ✝ ❣ ÿþ ✓✖❤✁✙ ✻✕ ✻✐❈❥❊☛❦ ❪❵❴ ✴✿✷✿✙✜❧✏✴✿✓ ❞ í❄ö ♠♦♥✏♣rqts ✉✇✈✏♣②①④③⑥⑤⑧⑦❝①❇♥❥⑨❢q✚①✆♣⑩③❶q❋❷ ❸❖❹❅❚✐❺❻❈✚❼⑩❆ ❪✯❽✚ë ❹☛❈❴ ❈❾❊✖❿☛❺❾➀✩❚✘❆❖❄❨✱➁❆⑩❄➁ ❚✡❆②❼ ❪❚ ❴ ❼⑩❹✳❈✚❼❖➂❪✯➃❵➃ ➀✳❚➄❚❽ ➁ ➃❄❪❼⑩❚✶❦ ❪❊❩❼⑩❹☛❚✐❈❥❊☛❈➃●❴②❪➅❴ ➆♦❚❝➇ ø ❪❵➈❚✐➀✳❚➃❄✚➂➉❼⑩❹❅❚➊❦❅❚❱➋☛❊❪❼ ❪❄❥❊✦❄❨ ❴❄❥❺➌❚➄❺➌❈❋❼⑩❆ ❪➍❽ ❯➃❈❴✍❴❚ ❴ ❈❥❊☛❦❂❼⑩❹❅❚❪❆➎❺❻❈❪❊ ➁ ❆✍❄➁ ❚✘❆⑩❼ ❪❚❴ ø ✌✁✎✁✑✔✑✔✓✖✕✘✗✚✙✜✛➏❛➐✰✻ ✙✜✕✡✙⑥➑✩✓➒❜➓✓❇➔✟✷✿✙⑥✕✡✓✔☞✍✌✁❛❝❜→✢ ➆♦❚ ❴❈④●✔❼⑩❹☛❈❋❼❂❈➣❺❻❈✚❼⑩❆ ❪✯❽⑧↔ ❪❵❴ ➁ ❄❴⑩❪❼ ❪❵➈❚➒❦✪❚✘➋☛❊❪❼⑩❚ ❪❨ ❍▼↔❍ ❲ ö♦❨❄❥❆➏❈❥❊✆●✒❊❅❄❇❊✪↕➛➙✡❚✘❆✍❄ ➈❚✶❯r❼✍❄❥❆➎❍ ø ❃☛❄❥❆ ❴●✆❺➌❺➜❚✘❼⑩❆ ❪❯➄❺❻❈❋❼⑩❆ ❪❯✘❚❴ ❼✍❹❪❵❴ ❯❱❄❇❊☛❦❪❼ ❪❄❥❊ ❪➅❴ ❚✡➝✖❿❪❵➈❈➃❚✘❊✖❼❉❼⑩❄❻❆✍❚✡➝✖❿❪ ❆ ❪❊❅➇➌❼✍❹☛❈✚❼ ❈➃❵➃ ❼✍❹❅❚➊❚❪➇❥❚✡❊➈❈➃❿☛❚❴ ❄ ❨ ❼✍❹❅❚➊❺❻❈✚❼✍❆❪✯❽ ➀✩❚ ➁ ❄❴⑩❪❼ ❪✯➈❚ ø ❸✱❄ ❴❹❅❄✚➂➉❼⑩❹❪➅❴ ➂➞❚➊❊❅❄❋❼✍❚➄❼⑩❹☛❈❋❼ ❪❨ ë❫❪❵❴ ❴●✖❺➌❺➌❚❱❼✍❆❪ ❯❝❈❋❊✳❦❂❹✳❈❴ ❆✍❚✡❈➃ ❯❱❄✆❚❱➟❩❯❪❚✡❊❇❼ ❴ ❳ ❪❼❢❯✡❈❋❊✦➀✩❚➊➂➎❆❪❼②❼✍❚✘❊➒❈❴➐↔ ì➉➠▼✱➡➠ ❳ ➂➎❹❅❚✡❆⑩❚ ➡ ❪➅❴ ❼⑩❹☛❚➢❦❪❈❋➇❇❄❥❊☛❈➃ ❺❻❈✚❼⑩❆ ❪✯❽ ❄ ❨ ❚❪➇❥❚✘❊➈❈➃❿❅❚ ❴ ❈❋❊☛❦ ➠➤❪❵❴ ❈❋❊→❄❥❆⑩❼⑩❹☛❄❥❊❅❄❇❆⑩❺❻❈➃ ❼⑩❆➥❈❋❊❴ ❨❄❥❆✍❺❻❈✚❼ ❪❄❥❊ ❳ ❪ ø ❚ ø ➠ ☎ ì➦➠▼ ø ❸❖❹❅❚✘❊ ❳ ❍ ▼↔❍ ì ❍ ▼ ➠▼➡➠❍ ì➉➧▼ ➡➧ ❲ ö❻❨❄❥❆❉❈❋❊✆●❂❍ ❭❬ ö ✞❄❇❆➐❈❋❊✆● ➧ ì❫➠❍ ❭❬ ö ❴②❪❊☛❯❱❚ ➠➨❪❵❴ ❊❅❄❥❊❅↕ ❴⑩❪❊☛➇❥❿➃❈❋❆ ✠ ❳ ❪❺ ➁ ➃❵❪❚ ❴ ❼⑩❹✳❈✚❼✐❈➃✯➃ ❼⑩❹❅❚❾❚✘❊✖❼⑩❆ ❪❚ ❴❉❪❊ ➡ ❺➢❿❴❼✐➀✳❚ ➇❥❆✍❚✡❈❋❼⑩❚✡❆➩❼⑩❹☛❈❥❊➏➙✡❚✘❆✍❄ ø✟➫➀➈✆❪❄❥❿❴⑩➃● ❳ ❈❥❊✆●➢❺❻❈✚❼✍❆❪✯❽ ➂➎❹❪ ❯➥❹ ❪❵❴❖➭✪➯➳➲➉❪➅❴ ❈➃➅❴❄➊❊☛❄❥❊✪↕ ❴②❪❊❅➇❇❿➃❈❥❆➵❈❋❊✳❦ ❼⑩❹☛❚✘❆✍❚❨❄❥❆✍❚ ❪❊➈❚✡❆②❼ ❪➀➃❚ ❳ ↔☎ ì➉➠➡ ☎ ➠▼ ø ❜➓✙✜❀❅✾✩✰✳✷✿❀❅✲➞❜❂✰✩✑✔✙✵✷✼❀❅✷✄✕ ➆♦❚ ❴❈④●➌❼✍❹☛❈✚❼➐❈➜❺❻❈✚❼✍❆❪✯❽➓↔ ì✤■✶➸◗➻➺ ▲ r➼ ◗ ➽ ➺ ➽ ➼ ✂ ❪➅❴ ❦❪❈❥➇❥❄❇❊☛❈➃❵➃●❩❦❅❄❥❺❪❊✳❈❋❊✖❼ ❪❨ ➾
uP rat ictl b. Fire iww pint d s ira vidp ictl b- leys t we ra p pat m dix iw wpPidple di g/c. e dmic.cpl c t wafs c Ka pypdlidple di g/c Ie d/mic.cp Cidt we f wt dlt ra pfbdm pixf iwc pvpidple di gfc. Ie df m ofd ds wytxdt ppat ndyp cdF vpyrat tl b- He a/Edw,.pid wra p dt di gfc.He dmie·cp: cd vbda ra p w ofo· pwI.vp fct afs rat ict1面 e aiww pivntd ic Tt Ge Fa: pf iw. di g/c He fmic cpm pix ic iadtdbdi E ofdnl o I -matrix Cm函 xj iwC班d·c,- mix io ip w pint w up dc. t vafs c 101 1 pdm·x· FFpat tcpditw c/c-ctg pilt cbm tdl pAixyvic dt rat. fIt vidp ictI b Hire iwe p wpivntd fdt Ttde ds1H/ s dyip d c·Rf· t vafs c ra pio Fat F yp ictl b.ire… fft iwat fat d e. c tl b Fireycd j iw. c iadt dbd. F di g/c. Fe d/mic.cpm pixyratc.F c /c-ctg' pi Tire fo Fat dftffiditcpw fof -fi ur Nit ww tis ys t s iiF aflt pat rat dtffiditcpw dt cfc-ctg pill Thomas’ algorithn G bvi c tFimic pic dc. t tffiditcple cJc-vic gbFa pidi e ptm Jo pat bvicg pat ofE s icg. Igfdiram
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