Interpolants. Interpolation. Polynomials are the most common choice of interpolants because they are easy to: Evaluate Differentiate, and Integrate.

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1 /4/7 Iterpolto Gve set o dscrete vlues,,,,,, d ucto tht tches these vlues ectl The resultg ucto c the e used to estte the vlue o t vlue o tht s ot gve Iterpolts Polols re the ost coo choce o terpolts ecuse the re es to: Evlute Derette, d Itegrte

2 /4/7 Drect Method Gve + dt pots,,,,,, pss polol o order through the dt s gve elow: where,, re rel costts Set up + equtos to d + costts To d the vlue t gve vlue o, spl susttute the vlue o the ove polol Ler Iterpolto The upwrd veloct o rocket s gve s ucto o te Fd the veloct t t6 secods usg the drect ethod or ler terpolto t s 5 5 vt /s Veloct s ucto o te Veloct vs te dt or the rocket eple

3 /4/7 Ler Iterpolto t t v v v s 5735 rge desred Solvg the ove two equtos gves, Hece s s, rge, desred s + t t, 5 t / s v v 5735 t s vt /s Fd the veloct t t6 secods usg the drect ethod or qudrtc terpolto v v v t + t v + t s rge desred v / s s, rge, desred 3

4 /4/7 t s vt /s v v Cuc Iterpolto Fd the veloct t t6 secods usg the drect ethod or cuc terpolto 3 t + t + t v + 3t 3 t t + 364t t, t / s s 5 rge desred s, rge, desred 5 v 3 t t + 364t t, t 5 Fd the dstce covered the rocket ro ts to t6s? 6 s v t s 6 dt t + 365t t dt 4 65 Fd the ccelerto o the rocket t t6s d d 3 t v t t + 364t 5466t + dt dt t + 638t, t / s 4

5 /4/7 Lgrg Iterpolto Cosder le seget tht goes through, d, where s the slope 5

6 /4/7 Geerl or o Lgrge polols 6

7 /4/7 Copre wth Tlor pproto 7

8 /4/7 For N Error ouds or equll spced odes Lgrge polol pproto or o tervl [,] Iterpolto Etrpolto <,> 8

9 /4/7 Newto s Dvded Derece Polol Method It s soetes useul to d severl pprotg polols P, P,,P N d the choose the oe tht suts our eeds I the Lgrge polols re used, there s o costructve reltoshp etwee P N d P N Ech polol hs to e costructed dvdull, d the work requred to copute the hgher-degree polols volves coputtos 9

10 /4/7 Newto polols hve the recursve ptter: Newto s Dvded Derece Method Ler terpolto: Gve,,, ler terpolt through the dt +, pss where

11 /4/7 Qudrtc Iterpolto Gve,,,, d,, t qudrtc terpolt through the dt + + Geerl For + + where Rewrtg ],, [ ], [ ] [ + + ] [ ], [ ], [ ], [ ],, [

12 /4/7 Gve + Geerl For dt pots,,,,,,,,, s where [ ] [, ],, ] [ M [,,, ],,, ] [ Geerl or The thrd order polol, gve,,,,,, d,, s [ ] + + [, [, 3 ],, + [ ],, ] [, ] [,, ] 3, ],,, ] [ [ 3,, ] 3, ] 3 3 [ [ 3

13 /4/7 The upwrd veloct o rocket s gve s ucto o te Fd the veloct t t6 secods usg the Newto Dvded Derece ethod or cuc terpolto v t + t t + t t t t + 3 t t t t t t t s vt /s Tle : Veloct s ucto o te t s vt /s t k [t k ] [, ] [,, ] [,,, ] t t 5, t, t 3 5, ; 748; 3766; *

14 /4/7 74; 748; 3766; *

15 /4/7 Sple Iterpolto Method Pttlls o polol pproto + 5 Tle : S equdsttl spced pots [-, ] Fgure : 5 th order polol vs ect ucto

16 /4/7 Pttlls o polol pproto th Order Polol 5th Order Polol Fgure : Hgher order polol terpolto s d de Ler sple terpolto Applg ler Lgrge terpolto polol pecewse, tht s etwee two successve pots We get N ler polols or N+ pots Note: the terpolto ucto s cotuos over the do, whle the slope s dscotuous

17 /4/7 Qudrtc sple terpolto We c lso use qudrtc Lgrge terpolto polol pecewse usg three cosecutve pots Note: the terpolto ucto d ts slope re cotuos over the do, whle the curvture s dscotuous Most populr sples re the Cuc Sples Gve N+ pots costruct N cuc polols S k or ech tervl [ k, k+ ] so tht the resultg cuc sple S, ts rst dervtve S, d ts secod dervtve S re cotuous over [, N ] 3

18 /4/7 Costructo o Cuc Sples Costructo o Cuc Sples cot 4

19 /4/7 Costructo o Cuc Sples cot Costructo o Cuc Sples cot 5

20 /4/7 Ler Regresso Ler Curve Fttg 6

21 /4/7 Wht s Regresso? Gve dt pots,,,,,, est t to the dt The est t s geerll sed o zg the su o the squre o the resduls, r S Resdul t pot s ε S r Su o the squre o the resduls, Fgure Bsc odel or regresso, Ler Regresso-Crtero# Gve dt pots,,,,,, est t + to the dt, ε,,, 3 3, ε Fgure Ler regresso o vs dt showg resduls t tpcl pot, Does zg ε work s crtero, where ε + 7

22 /4/7 Eple or Crtero# Eple: Gve the dt pots,4, 3,6,,6 d 3,8, est t the dt to strght le usg Crtero# Tle Dt Pots Fgure Dt pots or vs dt Ler Regresso-Crtero# Usg 4-4 s the regresso curve Tle Resduls t ech pot or regresso odel 4 4 predcted ε - predcted ε 3 4 Fgure Regresso curve or 4-4, vs dt 8

23 /4/7 Ler Regresso-Crtero# Usg 6 s regresso curve Tle Resduls t ech pot or 6 predcted ε - predcted ε Fgure Regresso curve or 6, vs dt Ler Regresso Crtero # 4 ε or oth regresso odels o 4-4 d 6 The su o the resduls s s sll s possle, tht s zero, ut the regresso odel s ot uque Hece the ove crtero o zg the su o the resduls s d crtero 9

24 /4/7 Ler Regresso-Crtero# Wll zg ε work etter?, ε,,, 3 3, ε Fgure Ler regresso o vs dt showg resduls t tpcl pot, Ler Regresso-Crtero Usg 4-4 s the regresso curve Tle The solute resduls eplog the 4-4 regresso odel predcted ε - predcted 4 ε Fgure Regresso curve or 4-4, vs dt

25 /4/7 Ler Regresso-Crtero# Usg 6 s regresso curve Tle Asolute resduls eplog the 6 odel predcted ε predcted 4 ε Fgure Regresso curve or 6, vs dt 4 ε 4 Ler Regresso-Crtero# or oth regresso odels o 4-4 d 6 The su o the errors hs ee de s sll s possle, tht s 4, ut the regresso odel s ot uque Hece the ove crtero o zg the su o the solute vlue o the resduls s lso d crtero

26 /4/7 Lest Squres Crtero The lest squres crtero zes the su o the squre o the resduls the odel, d lso produces uque le r S ε ε,, 3 3,,, ε Fgure Ler regresso o vs dt showg resduls t tpcl pot, Fdg Costts o Ler Model r Mze the su o the squre o the resduls: S ε To d r S r S gvg + d we ze wth respect to d r S + Solve or d

27 /4/7 Eple The torque, T eeded to tur the torso sprg o ousetrp through gle, s gve elow Fd the costts or the odel gve T k + k θ Tle: Torque vs Agle or torsol sprg 4 Agle, θ Rds Torque, T N Torque N θ rds Fgure Dt pots or Agle vs Torque dt Eple cot The ollowg tle shows the sutos eeded or the clcultos o the costts the regresso odel Tle Tulto o dt or clculto o portt sutos θ T θ Rds N Rds Tθ N--Rds Usg equtos descred or wth 5 d N N-/rd 3

28 /4/7 Eple Results Usg ler regresso, tred le s oud ro the dt 4 T θ Torque N θ rds Fgure Ler regresso o Torque versus Agle dt C ou d the eerg the sprg t s twsted ro to 8 degrees? 4

29 /4/7 Geerlzed Polol Model Gve,,,,,, to gve dt set est t + + +,,,, + + K+ Fgure Polol odel Geerlzed Polol Model cot The resdul t ech dt pot s gve E The su o the squre o the resduls the s S r E 5

30 /4/7 6 Geerlzed Polol Model cot To d the costts o the polol odel, we set the dervtves wth respect to where r r r S S S M M M M,, K equl to zero Geerlzed Polol Model cot These equtos tr or re gve + + The ove equtos re the solved or,,, K

31 /4/7 Noler Regresso 7

32 /4/7 Noler equtos C, A Newto s ethod c e used or the soluto serch cod MATLAB Lerzto o Dt For soe odels whch requre sulteous soluto o oler equtos, dt lerzto e thetcll coveet For eple, the dt or epoetl odel c e lerzed See Tle 56 the tetook or other odels 8

33 /4/7 9

34 /4/7