which is The same as m+=n+n两+ For Hhis 1o be an identify for Hhe(y+1) arbitary coefficien ai, we mus Ehave The(y+ 1)condiaions x=/ de for j=0,1,…,y 3.3.6 M//cing ah/ x bn/ss Cric/ria SLIDE 13 ExacHhess condilion will be sahs fied if ahd only if rdr= 3.3.7 M//cing ch/ xxa hn/ss cric/ria SLIDE 1 organi exact x…xv」xk Nonlin /ar, since ai's and i's are unknowns Note 12 Now whatis a practical way of computing Hhe evaluahon poins and weighEs? We can wriTe Hhe exac mess cri leria in6 a malix from. The sys hem of equations is no Eeasy 1o solve since s and are un❃❄P✑✾❆❂◗P♦✾❆❅✉●■P✑❀③❅❖❍✮❜❝❀❻❍❋❅ ✞ ✣ ✛✢ ✣ ✫✮Ï✱✞ ✢ ✛✢ ✣ Ï❵✫❋Ï✺✱ ✡☞✡✩✡ ✱✞ ❖ ✛✢ ✣ Ï❖ ✫❋Ï ✙✟✞✣ ✫✽ ✃✿ ✢ ✔ ✃ ✱✞ ✢ ✫✽ ✃✿ ✢ ✔ ✃ Ï✃ ✱ ✡☞✡✩✡✥✱✞ ❖ ✫✽ ✃✿ ✢ ✔ ✃ Ï✃ ❖ ✘✑❉❋◆✉●❖P✑✾❆❅⑨●❖❉❦♣✰❀③❍❏❊☞✾❆❣❲❀✓❊❑●❖✾❭●q❛❦❵r❉❋◆✉●❖P✑❀ ✕✫✒☛✱ ✴ ✗➍❍✮◆❖♣✺✾❰●■◆■❍✮◆❖❛❯❂✓❉❑❀✓✒❯❂✠✾❁❀✓❊❑●■❅✓✞✃ ➚✺❃➍❀❻❜➵❡✭❅q●❄P✺❍✛❨❋❀ ●❖P✺❀ ✕✚✒☛✱✾✴ ✗⑨❂✠❉❋❊✺❣❲✾❭●❖✾❁❉✮❊✺❅ ✫✽ ✃✿ ✢ ✔ ✃✸Ï✭✃ ✂ ✙✜✛✢ ✣ Ï ✂ ✫❋Ï ✒✺✹✼✻ ✠ ✙✥✲✳✦✵✴❜✦✵❫❳❫❳❫ ✦ ✒ ✍✴→ ✍✴→✁￾ ☛➶✖✙✖❑✔✗✢✸✜✞✶➤✔✵✚✞✖❢✌✌➣✍✑✘❏✔✵✜✞✖✙✏✵✏ ↕✬✛✢✥✔✗✖❑✬✻✢✥✍ ➙✰➛❑➜➞➝✴➟✕➠❜✓ ❪➃Ñ✑❍✮❂✠●❖❊✑❀✵❅❖❅❄❂✓❉✮❊✺❣✑✾❰●■✾❭❉❋❊♦❃❄✾❁❴❭❴✴♣✭❀③❅■❍✻●■✾❁❅❇Ð✺❀✗❣♦✾❭❵✳❍✮❊✺❣✆❉❋❊✑❴❁❛✆✾❭❵ ✛✏✢ ✣ ✫❋Ï ✙ ✫✽ ✃✿ ✢ ✔ ✃ ✡ ✴ ✛✢ ✣ Ï✘✫✮Ï ✙ ✫✽ ✃✿ ✢ ✔ ✃ ✡✠Ï✃ ✈ ✈ ✈ ✛✢ ✣ Ï❖ ✫✮Ï ✙ ✫✽ ✃✿ ✢ ✔ ✃ ✡✠Ï✃ ❖ ✍✴→ ✍✴→✄✂ ☛➶✖✙✖❑✔✗✢✸✜✞✶➤✔✵✚✞✖❢✌✌➣✍✑✘❏✔✵✜✞✖✙✏✵✏ ↕✬✛✢✥✔✗✖❑✬✻✢✥✍ ➙✰➛❑➜➞➝✴➟✕➠❊✏ ✝❄❀✗❉✮◆■❩❋❍❏❊✺✾❭➲✗✾❭❊✑❩➭❀✠Ñ✑❍✮❂✠●❖❊✑❀✵❅❖❅✉❀✵t❑❡✺❍✻●■✾❭❉❋❊✺❅ SMA-HPC ©2000 MIT Normalized 1-D Problem General Quadrature Scheme Meeting the exactness criteria 1 1 2 2 1 1 2 0 1 1 1 1 0 n l l l l n n w x x x w x x x w x dx # $ # $# $ % & % &% & % & % &% & − = % & % &% & % & % &% & % & ' (%' &( % & ' ( " % & % & & & ' & & % ✎✪✰✜✞✼✸✢✥✜✣✖❑✍✑✬❑➚❲❅❖✾❭❊✭❂✠❀❈Ï✭✃❖➻ ❅❄❍❏❊✺❣ ✔ ✃❇➻ ❅❄❍❏◆■❀❻❡✑❊✑➴✙❊✑❉✻❃❄❊✺❅ ➡➤➢✧➥➧➦➩➨ ✑ ❝▲❉✻❃ ❃❄P✭❍✻●❻✾❁❅❻❍❯▼✺◆■❍❋❂➧●❖✾❆❂✓❍✮❴✞❃✉❍✛❛♥❉❏❵➍❂✓❉✮❜❝▼✑❡❲●■✾❭❊✺❩✆●■P✑❀➭❀✗❨✻❍❏❴❁❡✺❍✻●■✾❭❉❋❊✩▼✭❉❋✾❭❊❑●◗❅✎❍✮❊✺❣✩❃⑨❀✓✾❁❩✮P❑●◗❅✜✛ ➸➤❀▲❂✓❍❏❊➫❃❄◆■✾❰●■❀✉●❖P✑❀❄❀✓Ñ❲❍❋❂➧●■❊✑❀✗❅■❅✳❂✠◆■✾❰●■❀✓◆■✾❁❍✎✾❁❊❋●■❉❈❍➉❜❦❍❏●❖◆■✾❰Ñ③❵r◆■❉✮❜♥✈✳➾✉P✑❀▲❅❖❛✙❅❇●❖❀✗❜➶❉❏❵✹❀✗t❑❡✺❍❏●❖✾❁❉✮❊✺❅ ✾❆❅✉❊✑❉❏●▲❀✵❍✮❅❖❛❝●❖❉❯❅❖❉✮❴❁❨✮❀❻❅❇✾❁❊✺❂✠❀❈Ï✃ ➻ ❅❄❍❏❊✭❣ ✔ ✃ ➻ ❅❄❍✮◆❖❀❻❡✑❊✺➴❑❊✺❉✻❃❄❊✺❅✓✈ Ò ￾