Cubic Spline Interpolation - College of the Redwoods

1 CubicSplineInterpolationSkyMcKinleyandMe ganLevineMath45 , ,aseriesofuniquecubicpolynomialsarefitte dbetweeneachofthedatapoints, ,wewillonlydiscusssplineswhichinterpolat eequallyspaceddatapoints, , ,inthiscase, bend xfi=s1 xfiifx1 x<x2s2 xfiifx2 x<x3_sn?1 xfiifxn?1 x<xn (1)wheresiisathirddegreepolynomialdefine dbysi xfi=ai x?xifi3+bi x?xifi2+ci x?xifi+di (2)fori=1,2,..,n? ,andtheyare12siv xfi=3ai x?xifi2+2bi x?xifi+cisivv xfi=6ai x?xifi+2bi (3) (4)fori=1,2,..,n? xfiwillbecontinuousontheintervalflx1,xn xfiwillbecontinuousontheintervalflx1,xn xfiwillbecontinuousontheintervalflx1,xn SincethepiecewiecefunctionS xfiwillinterpolateallofthedatapoints,wec anconcludethatS xifi=yi (5)fori=1,2,..,n? ,xi+1 ,S xifi=si xifiandwecanuseequation(2)toproduceyi=si xifiyi=ai xi?xifi3+bi xi?xifi2+ci xi?xifi+diyi=di (6)foreachi=1,2,..,n? xfimustbecontinuousacrossitsentireinterv al,itcanbeconcludedthateachsub-functionm ustjoinatthedatapoints,sosi xifi=si?

2 1 xifi (7)fori=2,3,.., (2),si xifi=diandsi?1 xifi=ai?1 xi?xi?1fi3+bi?1 xi?xi?1fi2+ci?1 xi?xi?1fi+di?1 (8)sodi=ai?1 xi?xi?1fi3+bi?1 xi?xi?1fi2+ci?1 xi?xi?1fi+di?1 (9)fori=2,3,..,n? (8),wehavedi=ai?1h3+bi?1h2+ci?1h+di?1 (10)fori=2,3,..,n? ,tomakethecurvesmoothacrosstheinterval,t hederivativesmustbeequalatthedatapoints; thatis,siv xifi=si?1v xifi (11)3 However,byequation(3),siv xifi=ciandsi?1v xifi=3ai?1 xi?xi?1fi2+2bi?1 xi?xi?1fi+ci?1soci=3ai?1 xi?xi?1fi2+2bi?1 xi?xi?1fi+ci?1. (12)Again,lettingh=xi?xi?1,wearriveatci= 3ai?1h2+2bi?1h+ci?1 (13)fori=2,3,u,n? (4),sivv xfi=6ai x?xifi+2bi,sosivv xfi=6ai x?xifi+2bisivv xifi=6ai xi?xifi+2bisivv xifi=2bi (14)fori=2,3,..,n? ,sincesivv xfihastobecontinuousacrosstheinterval,si vv xifi=si+1vv xififori=1,2,3,`,n? (14)leadustotheequationsivv xi+1fi=6ai xi+1?xifi+2bisi+1vv xi+1fi=6ai xi+1?xifi+2bi (15) (16)and,lettingh=xi+1?xiandusingtheconcl usionfromequations(14)and(16),si+1vv xi+1fi=6ai xi+1?

3 Xifi+2bi2bi+1=6aih+2bi (17) (18)Theseequationscanbemuchsimplifiedbys ubstitutingMiforsivv ,bi,ci, xifi=2biMi=2bibi=Mi2 (19)anddihasalreadybeendeterminedtobedi= yi. (20)Similarly,usingequationaicanbere-wri ttenas42bi+1=6aih+2bi6aih=2bi+1?2biai=2b i+1?2bi6hai=2 Mi+12fi?2 Mi2fi6hai=Mi+1?Mi6h (21)andcicanbere-writtenasdi+1=aih3+bih2 +cih+dicih=?aih3?bih2?di+di+1ci=?aih3?bi h2?di+di+1hci=?aih3?bih2h+?di+di+1hci= ?aih2?bihfi?di?di+1hci=? Mi+1?Mi6hh2+Mi2hfi?yi?yi+1hci=yi+1?yih? Mi+1?Mi6h+3Mi6hfici=yi+1?yih? Mi+1?Mi+3Mi6fihci=yi+1?yih? Mi+1+2Mi6fih. (22)Wenowhaveourequationsfordeterminingt heweightsforourn?1equationsai=Mi+1?Mi6hb i=Mi2ci=yi+1?yih? Mi+1+2Mi6fihdi=yi (23)Thesesystemscanbehandledmoreconvenie ntlybyputtingthemintomatrixformasfollows 5ci+1=3aih2+2bih+ci3 Mi+1?Mi6hfih2+2 Mi2fih+yi+1?yih? Mi+1+2Mi6fih=yi+2?yi+1h? Mi+2+2Mi+16fih3 Mi+1?Mi6hfih2+2 Mi2fih? Mi+1+2Mi6fih+ Mi+2+2Mi+16fih=? yi+1?

4 Yihfi+yi+2?yi+1hh 3Mi+1?3Mi6+6Mi6? Mi+1+2Mi6fi+ Mi+2+2Mi+16fifi=yi?2yi+1+yi+2hh6 Mi+4Mi+1+Mi+2fi=yi?2yi+1+yi+2hMi+4Mi+1+M i+2=6 yi?2yi+1+yi+2h2fi (24)fori=1,2,3,`,n?1whichleadstothematri xequation1410`00000141`00000014`0000____ b____0000`41000000`14100000`0141M1M2M3M4 _Mn?3Mn?2Mn?1Mn=6h2y1?2y2+y3y2?2y3+y4y3? 2y4+y5_yn?4?2yn?3+yn?2yn?3?2yn?2+yn?1yn? 2?2yn?1+yn (25)Notethatthissystemhasn?2rowsandncolu mns, , (26) `00000141`00000014`0000____b____0000`410 00000`14100000`00010M2M3M4_Mn?3Mn?2Mn?10 =6h2y1?2y2+y3y2?2y3+y4y3?2y4+y5_yn?4?2yn ?3+yn?2yn?3?2yn?2+yn?1yn?2?2yn?1+yn (27)Forreasonsofconvenience,thefirstandl astcolumnsofthismatrixcanbeeliminated,as theycorrespondtotheM1andMnvalues, `000141`000014`000___b___000`410000`1410 00`014M2M3M4_Mn?3Mn?2Mn?1=6h2y1?2y2+y3y2 ?2y3+y4y3?2y4+y5_yn?4?2yn?3+yn?2yn?3?2yn ?2+yn?1yn?2?2yn?1+yn (28)Thisresultsinann?2byn?2matrix,whichw illdeterminetheremainingsolutionsforM2th roughMn?

5 ,M1andMn,beequaltoM2andMn? (29) `000141`000014`000___b___000`410000`1410 00`015M2M3M4_Mn?3Mn?2Mn?1=6h2y1?2y2+y3y2 ?2y3+y4y3?2y4+y5_yn?4?2yn?3+yn?2yn?3?2yn ?2+yn?1yn?2?2yn?1+yn (30)WecannowdeterminethevaluesforM2throu ghMn?1, , `000141`000014`000___b___000`410000`1410 00`006M2M3M4_Mn?3Mn?2Mn?1=6h2y1?2y2+y3y2 ?2y3+y4y3?2y4+y5_yn?4?2yn?3+yn?2yn?3?2yn ?2+yn?1yn?2?2yn?1+yn (31) , ;theyarenotintrinsicallysuperiorto, ,ofcourse, , ,however, , , ,thefollowingfigurewasgeneratedusingthef unctiony=sin ,wecanseethedegreetowhichthecubiccurveim itatestheoriginalfunctiony=sin , , xfi+cos xfi , tfollowanyspecificpatternwithoutasinglep olynomial ,forinstance, ,however, ,whilethesplineisnotexactlypleasingtothe eye, ; , (ofwhichtherearenone) , , , :PrenticeHallNievergelt, :Module718; ,MA:COMAP15 Rorres, , :PrenticeHall