3/20/2013. Splines There are cases where polynomial interpolation is bad overshoot oscillations. Examplef x. Interpolation at -4,-3,-2,-1,0,1,2,3,4
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1 // Sples There re ses where polyoml terpolto s d overshoot oslltos Emple l s Iterpolto t -,-,-,-,,,,, Ide ehd sples use lower order polyomls to oet susets o dt pots mke oetos etwee djet sples smooth euse lower order polyomls, vod oslltos d overshoots We wll over ler sples qudrt sples u sples - most ommo Autod uses sples etesvely Ler sples - oet eh two pots wth strght le utos oetg eh pr o pots re m m m m s slope etwee pots m Ler sples re etly the sme s ler terpolto! Emple: ler sples m
2 // Prolem wth ler sples - ot smooth t dt pots (or kots) rst dervtve s ot otuous use hgher order sples to get otuous dervtves - equte dervtves t eghorg sples Qudrt sples - otuous rst dervtves hve dsotuous seod d hgher dervtves Derve seod order polyoml etwee eh pr o pots For + pots (=,,,) - tervls - ukow prmeters ( s, s, d s) Need equtos Equtos or teror kots (dt pots), equtos o ether sde must equl kot t kot - teror kots - teror equtos the rst d lst utos must pss through the ed pots more equtos: totl s so r rst dervtves t teror odes must e equl Aother - equtos or totl o - equtos We eed equtos. Wht s the lst equto? Lst equto s rtrry We use ssume seod dervtve s t rst pot Coet rst two pots y strght le
3 // Emple: Set up teror equtos ( o them) = 9 ukows Set up ed-pot equtos ( o them) Set up rst dervtve equtos t teror kots ( o them) Fl rtrry equto The 9 equtos re Rerrge, d use = to elmte Put mtr orm
4 // Solve y y mtr method Get 9 / () Result Cu sples vod the strght le d the over-swg d C develop method lke we dd or qudrt - ukows - equtos teror kot equlty ed pot ed teror kot rst dervtve equlty ssume dervtve vlue eeded C do set o lgrer mpultos (whh ook shows) to redue equtos to - equtos Ukows - seod dervtves wth ed odtos Seod dervtves re evluted usg Beomes trdgol system
5 // Method outle Set up equtos or seod dervtves get seod dertvtves d plug to sple equto or eh seto Emple: Get seod dervtves Note tht s - we wll ot hve true trdgol setup wth ed odtos Wrte the equtos s Now plug seod dervtve vlues to u sple equto Ths gets messy, so let s go to Mtl.. Beuse u sples re used so wdely, Mtl hs ult- ode or t >> help sple SPLINE Cu sple dt terpolto. PP = SPLINE(X,Y) provdes the peewse polyoml orm o the u sple terpolt to the dt vlues Y t the dt stes X, or use wth the evlutor PPVAL d the sple utlty UNMKPP. X must e vetor. I Y s vetor, the Y(j) s tke s the vlue to e mthed t X(j), hee Y must e o the sme legth s X -- see elow or eepto to ths. I Y s mtr or ND rry, the Y(:,...,:,j) s tke s the vlue to e mthed t X(j), hee the lst dmeso o Y must equl legth(x) -- see elow or eepto to ths. YY = SPLINE(X,Y,XX) s the sme s YY = PPVAL(SPLINE(X,Y),XX), thus provdg, YY, the vlues o the terpolt t XX. For ormto regrdg the sze o YY see PPVAL. I-lss emple to try: Use sple ommd to t set o u sples to l s gve vlues o y t ={ -,-,,,}. Use these u sples to geerte terpolted vlues t =-:.: Ht: = -:; y = log(s()+); = -:.:; I you wt hllege, ompre wth 8 th order terpoltg polyoml
6 // Should get somethg lke ths
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