we are looking into the space aan: 1HH.. I'H This basis always becomes ill nditioned with increasing The solution is to obt ain a polynomial basis that is " normalized"in some sense so that it is properly conditioned aa1F se Di=erent olynomials SLIDE 17 xactness criteria will be satisfied if and only if (x)ax=∑wa(x) j=(x)a=∑(x) Each c, polynomial must Contain an x' term j=(x)ak=∑v(x) e 15 only difference from the previous set of criteria is that these polynomials have better properties than the ones we chose before aall rthogonal olynomials SLIDE 18 ?ic(x)c(x)dx=0j≠i The above integral is often referred to as an inner product and ascribed the not at ion (c H)=? ic(a)c(a) The connection between polynomial inner products and vector inner products aa12 Exploiting the Di=eren 0 dlynomals SLIDE 19 er rewriting the exactness criteria a(x)∑啊《x)j(x)t:∑w(x) ()=∑()=-(=立w(x) Low order terms❃⑨❀➵❍❏◆■❀❈❴❭❉✙❉❋➴❑✾❁❊✑❩❦✾❁❊❑●❖❉❦●■P✑❀➵❅❖▼✺❍✮❂✓❀ ✽✍✞❑➷✁￾ ✴ ✦❇Ï❨✦✵❫❳❫✵❫ ✦❇Ï❖✄✂❋✈❄➾✉P✑✾❆❅▲♣✺❍❋❅❇✾❆❅❬❍✮❴❭❃✉❍✛❛❲❅✉♣✭❀✵❂✠❉✮❜❝❀✵❅▲✾❁❴❭❴ ❂✠❉❋❊✺❣❲✾❭●❖✾❁❉✮❊✑❀✵❣❦❃❄✾❭●❖P❯✾❁❊✺❂✠◆■❀✗❍❋❅❇✾❁❊✑❩✙✒❇✈✣➾✉P✺❀➉❅❖❉✮❴❁❡❲●❖✾❁❉✮❊❯✾❆❅❫●❖❉➵❉❋♣❲●■❍✮✾❭❊❯❍➵▼✭❉❋❴❭❛✙❊✑❉❋❜❝✾❁❍✮❴✭♣✺❍❋❅❇✾❆❅➃●■P✺❍✻● ✾❆❅✣✖➧❊✑❉❋◆❖❜❦❍❏❴❁✾❁➲✓❀✗❣ ✖▲✾❁❊☞❅❖❉✮❜❝❀❈❅❖❀✓❊✺❅❖❀③❅❇❉➭●❖P✭❍✻●▲✾❭●▲✾❁❅❄▼✺◆❖❉❋▼✭❀✗◆❖❴❁❛❯❂✠❉❋❊✺❣❲✾❭●❖✾❁❉✮❊✑❀✵❣✧✈ ✍✴→ ✍✴→✥↔✆☎ ✝✏✵✖✰✎☞✢✟✞❫✖❑✬✻✖✙✜✹✔ ✦➉✪✭✼✥✒➣✜✞✪✰✯➆✢✸✍✑✼✥✏ ➙✰➛❑➜➞➝✴➟✕➠ ✏ ❪➃Ñ✑❍✮❂✠●❖❊✑❀✵❅❖❅❄❂✓◆❖✾❭●❖❀✗◆❖✾❆❍➭❃❄✾❭❴❁❴➣♣✰❀③❅❖❍❏●❖✾❆❅qÐ✺❀✵❣✆✾❭❵➃❍❏❊✺❣☞❉✮❊✺❴❭❛❯✾❭❵ Normalized 1-D Problem General Quadrature Scheme Use different polynomials ( ) ( ) 1 0 0 0 1 n i i i c x dx wc x = " = ! ( ) ( ) 1 1 1 0 1 n i i i c x dx wc x = " = ! ( ) ( ) 1 0 1 n l i l i i c x dx wc x = " = ! Each polynomial must i c Contain an term i x Be linearly independent BUT ➡➤➢✧➥➧➦➩➨ ☛ ➾✉P✑❀❝❉✮❊✺❴❭❛✩❣❲✾❭➯✹❀✓◆■❀✓❊✺❂✓❀➵❵r◆■❉✮❜ ●■P✑❀❝▼✑◆■❀✓❨✙✾❭❉❋❡✺❅❻❅❇❀✓●✎❉✮❵⑨❂✓◆❖✾❭●❖❀✗◆❖✾❆❍✆✾❆❅➉●❖P✺❍❏●❻●❖P✑❀✵❅❇❀❝▼✰❉✮❴❁❛❑❊✺❉✮❜❝✾❁❍✮❴❁❅ P✺❍✛❨❋❀➉♣✰❀✠●❖●❖❀✓◆❬▼✑◆❖❉❋▼✭❀✗◆❇●■✾❭❀✵❅⑨●❖P✺❍✮❊✆●■P✑❀❈❉✮❊✺❀✗❅✉❃⑨❀❻❂◗P✺❉❋❅❖❀✎♣✰❀✠❵r❉❋◆❖❀❋✈ ✍✴→ ✍✴→✥↔✰↔ ✠✬✛✔✗✚✣✪✰✶✭✪✰✜✞✍✺✼✑✦➉✪✭✼✥✒➣✜✞✪✰✯➆✢✥✍✺✼✥✏ ➙✰➛❑➜➞➝✴➟✕➠ ✒ ✘✑❉❋◆✉●❖P✑❀❈❊✺❉✮◆■❜❝❍✮❴❭✾❁➲✓❀✵❣❯✾❁❊❑●❖❀✓❩❋◆■❍✮❴✸➚❑●q❃⑨❉❝▼✭❉✻❛✙❊✑❉❋❜➫✾❆❍❏❴❆❅✉❍❏◆■❀❈❅❖❍✮✾❁❣✆●❖❉❝♣✰❀③✪✺✬✛✔✗✚✣✪✰✶✭✪✰✜✞✍✺✼✞✾❭❵ ✛✢ ✣ ☛✡ ✃ ✕rÏ ✗ ✡ ✂ ✕✥Ï✘✗❚✫✮Ï ✙✥✲ ✒✺✹✼✻ ✠ ✛✙❺❐ ➾✉P✑❀♥❍❏♣✰❉✻❨✮❀♦✾❭❊❑●❖❀✗❩✮◆◗❍❏❴➍✾❁❅➫❉❏❵➞●■❀✓❊ ◆❖❀✓❵r❀✓◆■◆❖❀✵❣➤●❖❉⑥❍✮❅❝❍❏❊✕✾❭❊✺❊✑❀✓◆❝▼✑◆■❉❲❣❲❡✺❂➧●❦❍❏❊✭❣➆❍✮❅■❂✠◆■✾❭♣✰❀✗❣➆●❖P✑❀ ❊✑❉✮●■❍✻●■✾❭❉❋❊ ✕ ✡ ✃✺✦ ✡ ✂ ✗✚✙✜✛✏✢ ✣ ✡ ✃❚✕✥Ï✘✗ ✡ ✂❆✕rÏ ✗✬✫❋Ï ➾✉P✑❀❝❂✠❉❋❊✑❊✑❀✗❂✠●❖✾❁❉✮❊✷♣✭❀✓●q❃➍❀✗❀✓❊➩▼✰❉✮❴❁❛✙❊✑❉✮❜❝✾❆❍❏❴✣✾❭❊✺❊✑❀✓◆❻▼✑◆■❉❲❣❲❡✺❂➧●◗❅❻❍❏❊✺❣➩❨✮❀✗❂✠●❖❉❋◆➉✾❁❊✑❊✑❀✗◆✎▼✺◆❖❉❲❣❲❡✺❂✠●■❅ ❂✓❍✮❊♦♣✰❀③❅❖❀✓❀✓❊☞♣✙❛✆❅■❍❏❜❝▼✑❴❁✾❭❊✑❩✭✈ ✍✴→ ✍✴→✥↔✑➔ ✌✡✌✧✱✳✼✥✪✰✢r✔✵✢✥✜✣✶➩✔✵✚✞✖ ✎♦✢☞✞❫✖❑✬✛✖✙✜✹✔✩✦❬✪✰✼r✒➣✜✣✪✭✯✢✥✍✑✼✸✏ ➙✰➛❑➜➞➝✴➟✕➠ ☎ ❳❉✮❊✭❅❇✾❆❣❲❀✓◆✉◆■❀✓❃❄◆■✾❰●■✾❭❊✺❩➭●■P✑❀❈❀✠Ñ✑❍❋❂➧●❖❊✺❀✗❅■❅⑨❂✓◆❖✾❭●❖❀✗◆❖✾❆❍ Normalized 1-D Problem General Quadrature Scheme Exploiting the different polynomials ( ) ( ) 1 0 0 0 1 n i i i c x dx wc x = " = ! ( ) ( ) 1 1 1 0 1 Low order terms n n i n i i c x dx wc x − − = " = ! !"""""#"""""$ ( ) ( ) 1 0 1 n n i n i i c x dx wc x = " = ! ( ) ( ) 1 2 1 2 1 0 1 High Order Terms n n i n i i c x dx wc x − − = " = ! !"""""#"""""$ Ò ✥