Part 4 Capter 6 Sple ad Peewe Iterpolato PowerPot orgazed y Dr. Mael R. Gutao II Duke Uverty Reved y Pro. Jag CAU All mage opyrgt Te MGraw-Hll Compae I. Permo requred or reproduto or dplay.
Capter Ojetve Udertadg d tat t ple mmze ollato y ttg lower-order polyomal to data a peewe ao. Kowg ow to develop ode to perorm tale lookup. Reogzg wy u polyomal are preerale to quadrat ad ger-order ple. Udertadg d te odto tat t uderle a u t. Udertadg te deree etwee atural lamped ad ot-a-kot ed odto. Kowg ow to t a ple to data wt MATLAB ult- uto. Udertadg d ow multdmeoal l terpolato t mplemeted wt MATLAB.
Itroduto to Sple - t order polyomal l a lead to erroeou reult eaue o roud-o error ad ollato. A alteratve approa to apply lower- order polyomal a peewe ao to uet o data pot. Tee oetg polyomal are alled ple uto. Trd order urve are alled u ple. Sple mmze ollato ad redue roud-o error due to ter lower-order ature.
Hger Order v. Sple Sple elmate ollato y ug mall uet o pot or ea terval rater ta every pot. T epeally ueul we tere are jump te data: a rd order polyomal 5 t order polyomal l 7 t order polyomal d Lear ple eve t order polyomal geerated y ug par o pot at a tme A mu more aeptale auray
Lear Sple Lear Sple For data pot tere are - terval ad ea terval a t ow l t F l l t l t ple uto. For lear ple ea uto merely te tragt le oetg te two pot et ea ed o te terval. Lear ple uto a e ormulated a a a Iterept ad te lope are Kot Lear ple uto amout to Newto rt-order polyomal Lear ple uto amout to Newto rt-order polyomal.
Eample 6. Q. Ft te data elow wt rt-order ple ad Evaluate te uto at =5.. 4.5.5. 4 7. 9..5.5 Soluto or te eod terval rom 4.5 to 7. te uto.5.. 4.5. 7. 4.5 5
Sple Developmet a Frt-order ple d tragt-le equato etwee ea par o pot tat Go troug te pot Prmary dadvatage: tey are ot moot. - > at te pot kot were te two ple meet te lope age aruptly. rt dervatve are ot otuou at te kot Seod-order ple d quadrat equato etwee ea par o pot tat Go troug te pot Cotuou rt dervatve at te teror pot Trd-order d ple d u equato etwee ea par o pot tat Go troug te pot Cotuou rt ad eod dervatve at te teror pot Note tat te reult o u ple terpolato are deret rom te reult o a terpolatg u.
Quadrat Sple To eure tat te t dervatve are otuou at te kot a ple o at leat + order mut e ued. Te quadrat ple ould ave otuou rt dervatve at te kot. For data pot tere are - terval ad oequetly - ukow oeet tat eed to e oud. a
Quadrat Sple Quadrat Sple. Cotuty odto: Te uto mut pa troug all te t T t d d t pot. Te umer o oeet redued to -. a or l dj l l l a a. Te uto value o adjaet polyomal mut e equal at te kot. or equato o tere are - remag odto. o tere are remag odto.
Quadrat Sple. Te rt dervatve at te teror ode mut e otuou. -> - equato -> > = -> oly oe more equato eed e determed. or 4. Aume tat t te eod dervatve zero at te rt pot. -> te rt two pot are oeted wt a tragt le. -> Lear equato A = rgt ad de vetor were ot o ad.
Eample 6. / Q. Ft te quadrat ple to te ame data ued E. 6.. Ue te reult to etmate te value at =5.. 4.5 7. 4.5..5 4 9 9. 5.5
Eample 6. / Soluto we ave our pot ad = terval. Ater applyg te otuty equato ad te zero eod-dervatve odto -=5 oeet e eed to e olved. 4.5 4.5..5. 7. 4.5.5.5 9. 7.. 4.5
Eample 6. /.5.5.5 6.5.5 4.64.. 6 5.5. 4.5.64 4.5 5. 5 4.5.645 4.5.5. 7..6 7..66 Tere are two ortomg rom quadrat ple:. Stragt le oetg te rt two pot.. Te ple or te lat terval eem to wg too g. -> Cu ple do ot et tee prolem.
Cu Sple Wle data o a partular ze preet may opto or te order o ple uto u ple are preerred eaue tey provde te mplet repreetato tat et te dered appearae o moote. Lear ple ave dotuou rt dervatve Quadrat ple ave dotuou eod dervatve ad requre ettg te eod dervatve at ome pot to a pre-determed value *ut* Quadrat or ger-order ple ted to et te talte eret ger order polyomal ll-odtog or ollato
Cu Sple ot I geeral te t ple uto or a u ple a e wrtte a: a d For data pot tere are - terval ad tu 4- ukow to evaluate to olve all te ple uto oeet. Cotuty odto eed to e ated ad rt dervatve ad eod dervatve eed to e otuou at teror pot. -> greatly eae te moote.
Solvg Sple Coeet g p Oe odto requre tat te ple uto goe troug te rt ad lat pot o te terval yeldg - equato o te orm: equato o te orm: or or a Aoter requre tat te rt dervatve otuou at d a Aoter requre tat te rt dervatve otuou at ea teror pot yeldg - equato o te orm: or d A trd requre tat te eod dervatve otuou at ea teror pot yeldg - equato o te orm: or d ea teror pot yeldg equato o te orm: 6 or d Tee gve 4-6 total equato ad two addtoal equato are eeded!
Solvg Sple Coeet Solvg Sple Coeet Ug te two equato Ug te two equato d Te mpled equato are gve y -> - multaeou trdagoal equato or g q wt - ukow oeet. I we ave two addtoal odto we a ave two addtoal odto we a olve or te.
Two Addtoal Equato Tere are everal opto or te al two equato: Natural ed odto - aume te eod ddervatve at tte ed kot are zero. Clamped ed odto - aume te rt dervatve at te rt ad lat kot are kow. Not-a-kot ed odto - ore otuty o te trd dervatve at te eod ad te et-to-lat to kot. reult te rt two terval avg te ame ple uto ad te lat two terval avg te ame ple uto
Commoly Ued Ed Codto Codto Frt ad Lat Equato Natural ] [ ] [ Clamped were ' ad ' are te peed rt dervatve at te rt ad lat ode repetvely ] [ p y Not-a-kot
Natural Cu Sple Natural Cu Sple Te eod dervatve at te rt ode equal to zero a 6 d 6 d d q Te eod dervatve at te lat ode equal to zero a 6 d d Te al equato a ow e wrtte matr orm a ] [ ] [ ] [ ] [
Eample 6. / p / Q. Ft u ple to te ame data ued E.5.. Ue te reult to etmate te value at = 5. Sol. ] [ ] [ ] [ ] [ 4 4 5 5 4 7.5. 4.5.5 8 4 5 8 5.5. 7. 9..5.5 4.5 7.. 4 4.8 4.8 9.5.5 8.5 4.7665994.895476 4
.497786 d.645674 6 4 d Eample 6. /.5 d.895476.444487.7756654.5.497786.895476..645674 4.5.895476 4.5.444487 4.5.7756654 7..5.5 7..7665994 7. 5..6456745 4.5.8954765 4.5.4444875 4.5.8897488974
Peewe Iterpolato MATLAB opto MATLAB a everal ult- uto to mplemet peewe terpolato. Te rt ple: yy=ple y T perorm u ple terpolato geerally ug ot-a-kot odto. I y ota two more value ta a etre te te rt ad lat value y are ued a te dervatve at te ed pot.e. lamped
Not-a-kot Sple Eample opto Geerate data: = lpae- 9; y =./+5*.^; Calulate model pot ad determe ot-a-kot terpolato = lpae- ; yy = ple y ; Calulate atual uto value at model pot ad data pot te 9-pot ot-a-kot terpolato old ad te atual uto daed d yr =./+5*.^ plot y o yy - yr --
Clamped Sple Eample opto Geerate data w/ rt dervatve ormato: = lpae- 9; y =./+5*. ^; y = [ y -4] Calulate model pot ad determe ot-a-kot terpolato = lpae- ; yy = ple y ; Calulate atual uto value at model pot ad data pot te 9-pot lamped terpolato old ad te atual uto daed yr =./+5*.^ plot y o yy - yr --
MATLAB terp Futo opto Wle ple a oly perorm u ple MATLAB terp uto a perorm everal deret kd o terpolato: y = terp y metod & y ota te orgal data ota te pot at w to terpolate metod a trg otag te dered metod: earet - earet egor terpolato lear - oet te pot wt tragt le ple - ot-a-kot u ple terpolato pp or u - peewe u Hermte terpolato
Peewe Polyomal Comparo opto