Mtg 38: Wed, 30 Mar 11
page38-1
( 1 ) p .37 − 3 Z ′ = h z ∙ ( 1 ) {\displaystyle {\color {red}(1)}{\color {blue}p.37-3}\ Z^{\color {red}'}=h{\overset {\color {red}\bullet }{z}}\ {\color {red}(1)}}
z ( s ) = ∑ i = 1 4 N i ¯ ( s ) d i ¯ ( 2 ) {\displaystyle z(s)=\sum _{i=1}^{4}{\color {red}{\overline {\color {black}N_{i}}}}(s){\color {red}{\overline {\color {black}d_{i}}}}\ {\color {red}(2)}}
z ( t ) = ∑ i = 1 4 N i ( t ) d i ( 3 ) {\displaystyle z(t)=\sum _{i=1}^{4}N_{i}(t)d_{i}\ {\color {red}(3)}}
Recall:
C o l l o c a t i o n a t t i → ( 5 ) p .36 − 4 {\displaystyle Collocation\ at\ t_{i}\ \rightarrow \ {\color {red}(5)}{\color {blue}p.36-4}}
C o l l o c a t i o n a t t i + 1 → ( 6 ) p .36 − 4 {\displaystyle Collocation\ at\ t_{i+1}\ \rightarrow \ {\color {red}(6)}{\color {blue}p.36-4}}
Now:
C o l l o c a t i o n a t t i + 1 2 ⇒ {\displaystyle Collocation\ at\ t_{i+{\color {red}{\frac {1}{2}}}}\Rightarrow }
z ∙ i + 1 2 = f i + 1 2 = f ( z i + 1 2 , t i + 1 2 ) ( 4 ) {\displaystyle {\overset {\color {red}\bullet }{z}}_{i+{\color {red}{\frac {1}{2}}}}=f_{i+{\color {red}{\frac {1}{2}}}}=f(z_{i+{\color {red}{\frac {1}{2}}}},\ t_{i+{\color {red}{\frac {1}{2}}}})\ {\color {red}(4)}}
z i + 1 2 = z ( s = 1 2 ) {\displaystyle z_{i+{\color {red}{\frac {1}{2}}}}=z(s={\color {red}{\frac {1}{2}}})}
= H W ∗ 6.6 1 2 ( z i + z i + 1 ) + h 8 ( f i − f i + 1 ) ( 5 ) {\displaystyle {\overset {\color {blue}HW^{*}6.6}{=}}{\frac {1}{2}}(z_{i}+z_{i+1})+{\frac {h}{8}}(f_{i}-f_{i+1})\ {\color {red}(5)}}
page38-2
( 1 ) p .38 − 3 z ∙ i + 1 2 = z i + 1 2 ′ 1 h ( 1 ) {\displaystyle {{\color {red}(1)}{\color {blue}p.38-3}\ {\overset {\color {red}\bullet }{z}}}_{i+{\color {red}{\frac {1}{2}}}}={z}_{i+{\color {red}{\frac {1}{2}}}}^{\color {red}'}{\color {blue}{\frac {1}{h}}}\ {\color {red}(1)}}
z ∙ i + 1 2 = z ′ ( s = 1 2 ) = H W ∗ 6.6 { ( 1 ) p .37 − 2 ( 1 ) p .37 − 3 − 3 2 ( z i − z i + 1 ) − 1 4 ( z i ′ + z i + 1 ′ ) ( 2 ) {\displaystyle {\overset {\color {red}\bullet }{z}}_{i+{\color {red}{\frac {1}{2}}}}=z^{\color {red}'}(s={\color {red}{\frac {1}{2}}}){\color {blue}{\overset {HW^{*}6.6{\begin{cases}&{\color {red}(1)}{\color {blue}p.37-2}\\&{\color {red}(1)}{\color {blue}p.37-3}\end{cases}}}{\color {black}=}}}-{\frac {3}{2}}(z_{i}-z_{i+1})-{\frac {1}{4}}({z}_{i}^{\color {red}'}+{z}_{i+1}^{\color {red}'})\ {\color {red}(2)}}
( 1 ) ( 2 ) : {\displaystyle {\color {red}(1)\ _{\ }(2):}}
z ∙ i + 1 2 = − 3 2 h ( z i − z i + 1 ) − 1 4 ( f i + f i + 1 ) ( 3 ) {\displaystyle {\overset {\color {red}\bullet }{z}}_{i+{\color {red}{\frac {1}{2}}}}={\frac {-3}{2{\color {blue}h}}}(z_{i}-z_{i+1})-{\frac {1}{4}}(f_{i}+f_{i+1})\ {\color {red}(3)}}
z ∙ i + 1 2 ≠ f i + 1 2 i n g e n e r a l {\displaystyle {\overset {\color {red}\bullet }{z}}_{i+{\color {red}{\frac {1}{2}}}}{\color {red}\neq f_{i+{\frac {1}{2}}}}\ in\ general}
G a p = Δ z ∙ i + 1 2 − f i + 1 2 ( 4 ) {\displaystyle Gap\ =\ \Delta \ {\overset {\color {red}\bullet }{z}}_{i+{\color {red}{\frac {1}{2}}}}-f_{i+{\color {red}{\frac {1}{2}}}}\ {\color {red}(4)}}
C o l l o c a t i o n a t t i + 1 2 ⇒ Δ = 0 ( 5 ) {\displaystyle Collocation\ at\ t_{i+{\color {red}{\frac {1}{2}}}}\ \Rightarrow \ \Delta \ =\ 0\ {\color {red}(5)}}
G o a l : _ F i n d ( z i , z i + 1 ) s t Δ = 0 ( 6 ) {\displaystyle {\color {blue}{\underline {Goal:}}}\ Find\ (z_{i},z_{i+1})\ st\ \Delta \ =\ 0\ {\color {red}(6)}}
page38-3
Δ = 0 ⇒ z i + 1 = ( 1 ) z i + h / 2 3 [ f i + 4 f i + 1 2 + f i + 1 ] ⏟ S i m p s o n ′ s r u l e ! ( 2 ) p .7 − 4 {\displaystyle \Delta =0\Rightarrow \ z_{i+1}{\overset {\color {red}(1)}{=}}{\color {blue}{\underset {Simpson's\ rule\ {\color {red}!\ (2)}\ p.7-4}{\underbrace {\color {black}z_{i}+{\frac {h/2}{3}}[f_{i}+4f_{i+{\frac {1}{2}}}+f_{i+1}]} }}}}
z ∙ = f ( z , t ) ( 2 ) {\displaystyle {\overset {\color {red}\bullet }{z}}=f(z,t)\ {\color {red}(2)}}
∫ t i t i + 1 z ∙ d t ⏟ z i + 1 − z i = ∫ t i t i + 1 d t ⏟ a p p l y s i m p s o n ′ s r u l e ⇒ ( 1 ) ( 3 ) {\displaystyle {\color {blue}{\underset {\color {black}z_{i+1}-z_{i}}{\underbrace {\color {black}\int _{t_{i}}^{t_{i+1}}{\overset {\color {red}\bullet }{z}}dt} }}}={\color {blue}{\underset {apply\ simpson's\ rule\ \Rightarrow \ {\color {red}(1)}}{\underbrace {\color {black}\int _{t_{i}}^{t_{i+1}}dt} }}}\ {\color {red}(3)}}
Opt. control pb.:{zi, i=1,2,...,n} unknownsolved by NLP(nonlin. progr)
I V P ( I n i t i a l V a l u e ⏟ ( 4 ) z ( t o ) = z 0 p b ) : I n t . n o n l i n e a r O D E s ⏟ ( 2 ) {\displaystyle {\color {blue}IVP\ ({\underset {{\color {red}(4)}{\color {black}z(t_{o})=z_{0}}}{\underbrace {\color {blue}Initial\ Value} }}pb)}:\ Int.\ {\color {blue}{\underset {\color {blue}(2)}{\underbrace {\color {black}nonlinear\ ODEs} }}}}
page38-4
Hermite-Simpson time-stepping algo
Assume zi known, find zi+1 using(1)p.38-3
Need:
1 ) f i = f ( z i , t i ) c a n c o m p . {\displaystyle {\color {blue}1)}\ f_{i}=f(z_{i},t_{i})\ {\color {red}can\ comp.}}
2 ) f i + 1 = f ( z i + 1 ⏟ u n k n o w n , t i + 1 ⏟ k n o w n ) u n k n o w n {\displaystyle {\color {blue}2)}\ f_{i+1}=f({\color {red}{\underset {unknown}{\underbrace {\color {black}z_{i+1}} }}},{\color {red}{\underset {known}{\underbrace {\color {black}t_{i+1}} }}})\ {\color {red}unknown}}
3 ) f i + 1 2 = f ( z i + 1 2 ⏟ u n k n o w n , t i + 1 2 ⏟ k n o w n ) u n k n o w n {\displaystyle {\color {blue}3)}\ f_{i+{\color {red}{\frac {1}{2}}}}=f({\color {red}{\underset {unknown}{\underbrace {\color {black}z_{i+{\color {red}{\frac {1}{2}}}}} }}},{\color {red}{\underset {known}{\underbrace {\color {black}t_{i+{\color {red}{\frac {1}{2}}}}} }}})\ {\color {red}unknown}}
t i + 1 2 = t i + h 2 {\displaystyle t_{i+{\color {red}{\frac {1}{2}}}}=t_{i}+{\frac {h}{2}}}
(5) p.38-1:
z i + 1 2 = g ( z i , z i + 1 ) ( 1 ) {\displaystyle z_{i+{\color {red}{\frac {1}{2}}}}=g(z_{i},z_{i+1})\ {\color {red}(1)}}
(1) p.38-1:
z i + 1 = z i + h / 2 3 [ f i + 4 f ( g ( z i , z i + 1 ) , t i + 1 2 ) + f i + 1 ] ( 2 ) {\displaystyle z_{i+1}=z_{i}+{\frac {h/2}{3}}[f_{i}+4f(g(z_{i},z_{i+1}),t_{i+{\color {red}{\frac {1}{2}}}})+f_{i+1}]\ {\color {red}(2)}}
⇔ F ( z i + 1 ) = ( 3 ) 0 ⏟ n o n l i n e a r a l g . e q . {\displaystyle \Leftrightarrow \ F(z_{i+1}){\overset {\color {red}(3)}{=}}{\underset {\color {blue}nonlinear\ alg.\ eq.}{\underbrace {0} }}}
⇒ N e w t o n − R a p h s o n − S i m p s o n {\displaystyle {\color {blue}\Rightarrow \ Newton-Raphson-Simpson}}
![{\displaystyle \Delta =0\Rightarrow \ z_{i+1}{\overset {\color {red}(1)}{=}}{\color {blue}{\underset {Simpson's\ rule\ {\color {red}!\ (2)}\ p.7-4}{\underbrace {\color {black}z_{i}+{\frac {h/2}{3}}[f_{i}+4f_{i+{\frac {1}{2}}}+f_{i+1}]} }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c532dc2d696d0ba7334f8a0b52fd212605763fd)
![{\displaystyle z_{i+1}=z_{i}+{\frac {h/2}{3}}[f_{i}+4f(g(z_{i},z_{i+1}),t_{i+{\color {red}{\frac {1}{2}}}})+f_{i+1}]\ {\color {red}(2)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71d6e8ffbb7171452f77baec8a5c9986e119c3f4)
