3.3: Co, +Hting the Hoint] and 0 eight SLIDE 15 Could use Newton F Method F(=0=_ee+l-y=-F The nonlinear function for Newton is then Note 13 Newton's method is an iterative technique for finding a value y such that F(y 0. The method is based on linearizing the problem about a guess at y, and then updating the value of y by solving the linearized problem. In particular, the terate y+ is determined from y" by solving the linear system of equations y where J_(") obian(multidimensional derivative) of the nonlinear ction F(). The iteration is continued until the updated y is sufficiently close to the exact solution, a criterion that can be difficult to verify. Newtons method does not always converge, a phenomenon that is more likely when -(y) nearly singular. For more about Newton's method, see the f. 33fs1f9205 2.09f Co, +Hting the Hointj and 0 eight Newton method lcobian reveals problem 00 x1x2…xWV2 Note 14 Looking at th bian of the problem, we realize that the first n columns become incre learly dependent for large ffi This is bound to happen since✍✴→ ✍✴→✁￾ ↕✪✰✯✲✱✣✦✞✔✗✢✸✜✞✶✷✔✗✚✣✖✝✦➉✪✭✢✸✜✹✔✗✏❝✍✑✜✣❱★￾✖✙✢✥✶✰✚✹✔✗✏ ➙✰➛❑➜➞➝✴➟✕➠ ✞ ❳❉✮❡✺❴❁❣♦❡✺❅❖❀ ✎✖✄❝✔✵✪✭✜✄✂❭✏ ☛➶✖❑✔✗✚✣✪➣❱ ☎❩✕✝✆✳✗✚✙✾✲✟✞✡✠☞☛ ✖ ✆✍✌ ✘ ✖ ✆✍✌❬ ✢ ❄✎✆✍✌ ✘ ✙✟❄✏☎ ✖ ✆✍✌ ✘ ➾✉P✑❀❻❊✑❉❋❊✑❴❭✾❁❊✑❀✵❍❏◆✉❵r❡✑❊✺❂✠●❖✾❁❉✮❊☞❵r❉✮◆☞❝❬❀✓❃✉●■❉✮❊♥✾❁❅⑨●■P✑❀✓❊ SMA-HPC ©2000 MIT Normalized 1-D Problem General Quadrature Scheme Computing the points and weights 1 1 2 2 1 1 2 0 1 1 1 1 0 n l l l l n n w w x x x w F x x x x w x dx # $ # $# $ % & % &% & % & ) * + , = − % &% & % & = - . % &% & % & % &% & % & ' ('% (& % & ' ( " % & % & & & ' & & % ➡➤➢✧➥➧➦➩➨ ❯ ❝▲❀✗❃✉●❖❉❋❊➣➻ ❅✣❜❝❀✠●■P✑❉❲❣③✾❆❅✳❍❏❊③✾❭●❖❀✗◆■❍❏●❖✾❁❨✮❀✿●■❀✗❂◗P✑❊✺✾❁t❑❡✑❀➍❵r❉✮◆✞Ð✺❊✺❣❲✾❁❊✑❩❈❍▲❨✻❍✮❴❭❡✑❀✑✆❈❅❇❡✭❂◗P③●❖P✭❍✻●✒☎❩✕✓✆✳✗✚✙ ✲✑✈✳➾✉P✑❀▲❜❝❀✓●❖P✑❉❲❣❦✾❆❅✿♣✺❍❋❅❇❀✵❣➫❉✮❊❦❴❁✾❭❊✑❀✵❍❏◆■✾❭➲✗✾❭❊✺❩❻●❖P✑❀▲▼✺◆❖❉❋♣✑❴❭❀✗❜ ❍✮♣✭❉❋❡❲●❫❍❈❩❋❡✑❀✗❅■❅✿❍✻●✄✆✰➚✙❍❏❊✭❣➫●❖P✑❀✗❊ ❡✑▼✹❣✑❍✻●■✾❭❊✺❩❢●❖P✑❀♦❨✻❍❏❴❁❡✑❀❯❉❏❵✔✆⑥♣✙❛➤❅❖❉✮❴❁❨✙✾❭❊✺❩♥●❖P✑❀❯❴❁✾❁❊✑❀✗❍✮◆❖✾❁➲✓❀✵❣⑥▼✑◆❖❉❋♣✑❴❁❀✓❜♥✈❢❤✐❊✲▼✺❍❏◆❖●❖✾❆❂✠❡✺❴❁❍✮◆✗➚✞●❖P✑❀ ✾❭●❖❀✓◆◗❍✻●■❀✕✆ ✌ ❬ ✢ ✾❁❅❬❣❲❀✠●■❀✓◆■❜➫✾❁❊✑❀✵❣❯❵r◆■❉✮❜✖✆ ✌ ♣✙❛✆❅❖❉✮❴❁❨❑✾❁❊✑❩➭●❖P✑❀❈❴❁✾❁❊✑❀✗❍✮◆▲❅❇❛❲❅❇●❖❀✓❜❸❉❏❵✳❀✗t❑❡✺❍✻●■✾❭❉❋❊✺❅ ☎❩✕✝✆ ✌ ✗ ✱✗✠☛ ✕✝✆ ✌ ✗✞✖✘✆ ✌ ❬ ✢ ❄✎✆ ✌ ✘ ✙✾✲ ❃❄P✑❀✗◆❖❀✙✠✚☛ ✕✓✆ ✌ ✗❦✾❆❅❦●■P✑❀✜✛❑❍✮❂✓❉✮♣✑✾❆❍❏❊✡❡r❜➭❡✑❴❰●■✾❁❣✑✾❭❜❝❀✓❊✭❅❇✾❁❉✮❊✺❍✮❴✎❣✑❀✓◆■✾❭❨✻❍✻●■✾❭❨❋❀ ❢➫❉✮❵❈●❖P✑❀✩❊✺❉✮❊✑❴❁✾❭❊✺❀✗❍❏◆ ❵r❡✑❊✺❂✠●❖✾❁❉✮❊✢☎❩✕✓✆✳✗✠✈❺➾✉P✑❀❢✾❰●■❀✓◆◗❍✻●■✾❭❉❋❊✕✾❆❅❦❂✠❉❋❊❋●■✾❭❊✙❡✑❀✵❣ ❡✺❊❋●■✾❭❴▲●❖P✑❀❢❡✑▼✹❣✑❍❏●❖❀✗❣✣✆✕✾❁❅❦❅❖❡✒❯❂✠✾❁❀✓❊❑●❖❴❁❛ ❂✠❴❁❉❋❅❖❀❻●❖❉❯●❖P✑❀③❀✓Ñ✑❍✮❂➧●✎❅❇❉❋❴❭❡❲●■✾❭❉❋❊➣➚✺❍❯❂✓◆❖✾❭●❖❀✗◆❖✾❁❉✮❊☞●■P✺❍✻●➉❂✗❍❏❊❢♣✰❀➵❣❲✾✄✒❯❂✠❡✑❴❭●❬●■❉❯❨✮❀✗◆❖✾❭❵r❛✮✈❑❝▲❀✗❃✉●❖❉✮❊✴➻ ❅ ❜❝❀✠●■P✑❉❲❣➭❣❲❉✙❀✵❅✞❊✺❉❏●✿❍❏❴❁❃⑨❍✛❛❲❅✞❂✠❉✮❊✙❨❋❀✓◆■❩✮❀✮➚✻❍➉▼✑P✑❀✗❊✑❉✮❜❝❀✓❊✺❉✮❊③●■P✺❍✻●➃✾❆❅✳❜❝❉✮◆■❀➍❴❁✾❭➴❋❀✓❴❁❛❈❃❄P✑❀✓❊✤✠☛ ✕✝✆ ✗ ✾❆❅✴❊✑❀✵❍❏◆■❴❭❛❻❅❖✾❭❊✑❩❋❡✑❴❆❍❏◆✵✈ ✘✑❉❋◆✴❜❝❉✮◆■❀➍❍✮♣✭❉❋❡❲● ❝❬❀✓❃✉●❖❉❋❊➣➻ ❅✴❜❝❀✠●■P✑❉❲❣✧➚❏❅❖❀✓❀✿●■P✑❀ ★ ✈ ➼❋➼★✦✥✙Ò ★ ✈ ✔✤✥ ￾ ✥ ✥ ✈ ￾✔✤★ ❂✠❉❋❡✑◆◗❅❇❀✎❊✑❉❏●■❀✗❅✗✈ ✍✴→ ✍✴→✁✧ ↕✪✰✯✲✱✣✦✞✔✗✢✸✜✞✶✷✔✗✚✣✖✝✦➉✪✭✢✸✜✹✔✗✏❝✍✑✜✣❱★￾✖✙✢✥✶✰✚✹✔✗✏ ➙✰➛❑➜➞➝✴➟✕➠ ✪ ❝▲❀✗❃✉●❖❉❋❊☞❞❢❀✓●❖P✑❉❲❣★✛❑❍✮❂✠❉❋♣✑✾❆❍❏❊✆◆■❀✓❨❋❀✗❍✮❴❁❅⑨▼✑◆■❉✮♣✑❴❁❀✓❜ Normalized 1-D Problem General Quadrature Scheme Computing the points and weights 1 2 1 2 1 1 1 2 1 1 1 1 1 1 1 1 1 n n F l l l l l n n w x x x w w w J x x x x lw x lw x − − # $ % & ) * + , = - . ' ( % % % % & & ' & & & ' & % % % Columns become linearly dependent for high order 2n 0 0 0 ➡➤➢✧➥➧➦➩➨✶✚ ➺➣❉✙❉❋➴❑✾❁❊✑❩❢❍✻●❈●■P✑❀✩✛❑❍✮❂✓❉✮♣✑✾❆❍❏❊✷❉❏❵✉●❖P✑❀❦▼✺◆❖❉❋♣✑❴❭❀✗❜♥➚✞❃⑨❀❦◆❖❀✵❍❏❴❁✾❭➲✗❀➫●❖P✺❍❏●❈●❖P✑❀❦Ð✭◆■❅❇● ✙ ➷➃➻✞❂✠❉✮❴❁❡✑❜❝❊✺❅ ♣✰❀✗❂✠❉❋❜❝❀➃✾❭❊✺❂✓◆❖❀✵❍✮❅❖✾❭❊✺❩✮❴❁❛▲❴❭✾❁❊✑❀✵❍❏◆■❴❭❛✎❣❲❀✓▼✰❀✓❊✺❣✑❀✓❊❑●➣❵r❉✮◆✴❴❁❍✮◆❖❩❋❀ ✒❇✈✴➾✉P✑✾❆❅➣✾❆❅✧♣✰❉✮❡✑❊✺❣✎●■❉▲P✺❍✮▼✑▼✭❀✗❊❈❅❇✾❁❊✺❂✠❀ Ò✮Ò