{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 10 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 1 20 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 1 16 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times " 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } {PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "AC - Title" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 1 2 258 1 }{PSTYLE "AC - Note" -1 259 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "AC - Normal Text" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "AC - Se ction Heading" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 12 0 1 0 1 0 2 2 260 1 }{PSTYLE "AC - Author" -1 262 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 4 4 1 0 1 0 2 2 259 1 }{PSTYLE "Normal" -1 263 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 258 "" 0 "" {TEXT 260 43 "Spline Method of Interpolation --Simulation " }{TEXT 257 0 "" }}{PARA 262 "" 0 "" {TEXT 256 7 "\251 2 003 " }{TEXT -1 144 "Nathan Collier, Autar Kaw, Jai Paul , Michael Ket eltas, University of South Florida , kaw@eng.usf.edu , http://numerica lmethods.eng.usf.edu/mws " }}{PARA 259 "" 0 "" {TEXT -1 171 "NOTE: Thi s worksheet demonstrates the use of Maple to illustrate the spline met hod of interpolation. We limit this worksheet to linear and quadratic spline interpolation." }}{SECT 0 {PARA 261 "" 0 "" {TEXT 258 12 "Intr oduction" }{TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT 281 13 "The Spline \+ me" }{URLLINK 17 "" 4 "" "" }{TEXT 282 66 "thod of interpolation (for \+ detailed explanation, you can read the " }{URLLINK 17 "textbook notes \+ and examples" 4 "http://numericalmethods.eng.usf.edu/mws/gen/05inp/mws _gen_inp_txt_spline.pdf" "" }{TEXT 284 12 ", and see a " }{URLLINK 17 "Power Point Presentation)" 4 "http://numericalmethods.eng.usf.edu/mws /gen/05inp/mws_gen_inp_ppt_spline.ppt" "" }{TEXT 283 15 " is illustrat ed" }{TEXT -1 145 ". Given 'n' data points of 'y' versus 'x', it is \+ required to find the value of y at a particular value of x using linea r and quadratic splines." }}{PARA 260 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 34 "with(LinearAlgebra):\nwith(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 104 "Warning, the previous binding of the name GramSchmi dt has been removed and it now has an assigned value\n" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have bee n redefined and unprotected\n" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 262 17 "Section I : Data." }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "The following is the array of x-y data which is used to interpo late. It is obtained from the " }{URLLINK 17 "physical problem" 4 "htt p://numericalmethods.eng.usf.edu/mws/gen/05inp/mws_gen_inp_phy_problem .pdf" "" }{TEXT -1 131 " of velocity of rocket (y-values) vs. time ( x-values) data. We are asked to find the velocity at an intermediate point of x=16." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "xy:=[[10 ,227.04],[0,0],[20,517.35],[15,362.78],[30,901.67],[22.5,602.97]]:" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "Value of x at which y is desired " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "xdesired:=16:" }}}} {PARA 3 "" 1 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 263 33 " Section II : Big scary functions." }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "This function will s ort the data matrix into ascending order and puts them into a new matr ix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "n:=rowdim(xy):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "for passnum from 1 to n-1 do\n for i from 1 to n-passnum do \n if xy[i,1]>xy[i+1,1] then\n temp1:=xy[i,1];\n temp2:=x y[i,2];\n xy[i,1]:=xy[i+1,1];\n xy[i,2]:=xy[i+1,2];\n x y[i+1,1]:=temp1;\n xy[i+1,2]:=temp2;\n end if:\n end do:\nend do:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "x:=matrix(n,1):\n y:=matrix(n,1):\nfor i from 1 to n do\nx[i,1]:=xy[i,1];\ny[i,1]:=xy[i, 2];\nend do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "ranger:=pr oc(ar,n)\nlocal i,xmin,xmax,xrange;\nxmin:=ar[1,1]:\nxmax:=ar[1,1]:\nf or i from 1 to n do\nif ar[i,1] > xmax then xmax:=ar[i,1] end if;\nif \+ ar[i,1] < xmin then xmin:=ar[i,1] end if;\nend do;\nxrange:=xmin..xmax ;\nend proc: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Plotting the given values of X and Y." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot(xy,ranger(x,n) ,style=POINT,color=BLUE,symbol=CIRCLE,symbolsize=30);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 743 492 492 {PLOTDATA 2 "6(-%'CURVESG6#7(7$$\"\"!F)F(7$$\"#5F)$\"3#***********RqA! #:7$$\"#:F)$\"3t**********zFOF/7$$\"#?F)$\"3A+++++]t^F/7$$\"3+++++++]A !#;$\"3F+++++qHgF/7$$\"#IF)$\"3f**********p;!*F/-%'COLOURG6&%$RGBGF(F( $\"*++++\"!\")-%'SYMBOLG6$%'CIRCLEGFB-%&STYLEG6#%&POINTG-%+AXESLABELSG 6$Q!6\"FW-%%VIEWG6$;F(FA%(DEFAULTG" 1 5 4 1 30 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 264 40 "Section III: Linear spline interpolation" }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{OLE 1 5122 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B >N:F:nyyyyy]::yyyyyy:::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::fyyyyya:nYf::G:I:K:wAyA:::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm =;:::::::n:;`:Z@[::JBTqutnBil[Aj;JZ`:<:=ja^GE=;:::::: :::N;?R:yyyyyy:>:<::::::JDJ:j:VBYmp>HYLkNG>::::::::N:;::::::::_Z:vyyuy:>:<::::::AJ:n:>:nYN::wyyyqR<:TnEj``pkDqqHqqTPtWdZfbk;>ZEZ?GHZhV@cYH_WV>Z::jysy:::J:V:Zy=::::::::^=N:_YjmR:B: >tei:AR:=r:=b<=r:=B:@JCHRvVZIF:QR:?B\\[[Kb:=::RFF::J:dZP::::::;C:?B:yay=J:>Z: :::::J<>:eS:>Z:J;vCS=[LsfFaMR>@>Z::::::::kJ;@j;>:C:yayA:<::::::MZ:Nx@J :<:=B:Y<>Z:>:::::::::J?:<::::::wqy[:::::::::::::vYxI:;Z::::: :::j:D[r;ZhC:]J?NJ:::;bZ:vYxY;:nsRjwD:[g;F:MZ =>e=J;^=>:E:Mb:>Z:f:NZ;F:f:FZ=:Gc;Y JGvyyuy;B:[B::G;Sj`@Pt\\Pd`QrP@;MHB:qi:EbyAa:> :[V:>Z<>ZaTXUeRYEUXQZB:GY;OrvgrJKxWyrJKxW;CjOxW>@C:US:>;N`Dni;V@=Z: >_f_=VYZ:>:[>DZjq:>:C:Uk:^:>x;J>JSd:EB;=Z:F_f_=VYZ:> :[>DZjq:F:>:C:Uk:^:>X;J>JSd:_E<=Z:F?jw;Lb:DZJ:Y=B:Gaoq\\=?Sy:;;J :C:[q:>;N`Djvij:B:K;VYZ:>:[>DZjq::[>DZjq::[> DZjq:tOF:L=J:^ZcTTPp s:C:[q:>;N@;Y:ES?=Z:ngf?;N@r>=B:nGJSJ>`GF:nGs:qQ:uI;B:>L=J:^Z<:VH:nGjIJJ:B:;jPF:C:[Q:>;N@;V:_E?=Z:Fi f?;N@>iLF:;JvlP>JSjdxj:>:m=N@VjFF:>IJSJ[Xj:>:m=N@nh?F:>IJSjQEj:jv:_;uo :<:kMHjw;<:[N:B:Cb:^D:::y:;;Z:jP>:C:[q:>;N@nqMF:JSjJuj:<:Sk?JSJb ]J>JSJAUj:JP>;N@Vf>F:^?JSjwG:;JP^]:jw?>B:>LB:Cb::::jIJJ: B:f?=JJSjHtj:jR:_;sx;=B:F@JSJ?Tj:jR:_;wt:=:[cEK:_ ;GU:=J:>@sB:VYZ:JB[X=J>JSd:GF>=Z:f_ h>;N@noGF:f?O:G;Ojysy=:;JHjw?:sg:B:=J;Dlc`qsLqlp@X;j;::::WTJWTL>Z:F:C:[a::;Jlj:fBE]:^[:>uXnu;^y yyyy]:F:;JVUrD>c>y;[GFYB:;N:?Q^yyyY;Z::Jyk=:F[:>:;:M:yyYZ:vYxYyK:B:yayqY>::<:1:" } {TEXT -1 191 " , fit linear splines to the data. 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The splines are given by\n " }{OLE 1 4098 1 "[x m]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy]::yyyyyy:::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::fyyyyya:nYf::G:jy ;::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::JcvGYMt>^:fBWMtNHm=;:::::::n:;`:Z@[::JBQv>u nBil[Aj;JZPZ:B:F:YLpfF>:::::::::J?NZ;vyyyyy=J:B:::::::c:;:=j[vGUMrvC?M oJ::::::::JCNZ;F:f:vYxY:B::::::F:;JmJ:j;vCj^nGGmq>:;::::::::_Z:vyyuy:> :<::::::AJ:n:>:nYN::wyyyqj>>:OJ:V;^;f;;JAjA>:[Z:F:AR:=r:=b<=b>AfSDJ;]\\LVjrc:?VDDJ=q]LVj[s:YR:hj:DJ >A=]J?Dk;]a=v[:r<]R:?B\\_;Ob:GvDDk;E`=v[;:KVF:b:GvDVj]s:Y:>KZAZ:N:s:;jDj\\FHemj^HMmqn G;KaFFJufF;J::::::>^:N:yay=J:>Z::::::JD_mlVH[KRJ:<::::::: >=?R:AB:;JZ:>:::::::::J?ryyYiwyyyy;R=vx;B:QB:n>^;UTRcETcTX[US>DsZ: Vy<>jxM:<:[V:>Z:^ZcTTUUSaEBWTSiEB_tUUURWMEHN^YqIHN^YQJ;N`D>n;noFF:;N@>m;N@>c;F:>_:^=VY;sy>Z:JB[:^Z>:<:U k:^:>x;J>JS>Z:ZD>:KT;=:]KHjw?JB:>L= J:^Z<::::y:;;Z:jP>:C:[q:>;N@;OEij:B:AM>JSJCaj:Jv>;N@^hBF:NFK:_;SX;=:k= 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "Y:=matrix(3*(n-1),1,0):\nfor i fro m 0 to n-2 do\nfor j from 0 to 1 do\nY[2*(i+1)+j,1]:=y[i+j+1,1];\nend \+ do;\nend do;\nevalm(Y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6 #717#\"\"!F'7#$\"&/F#!\"#F)7#$\"&yi$F,F-7#$\"&N<&F,F07#$\"&(HgF,F37#$ \"&n,*F,F'F'F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Solving for t he coefficients, we get" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 26 "C:=evalm(inverse(A) &* Y);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"CG-%'matrixG6#717#$\"\"!F+7#$\"+++SqA!\")7#$\"+(R MV0\"!#B7#$\")++))))!\"'7#$\")++G\\!\"(7#$F6F/7#$!*++hT\"F77#$\"*++gc$ F;7#$!)++c8F/7#$\")+]Xb!\"&7#$!)+g&R$F77#$\"*++[g\"F/7#$!&8_\"!\"#7#$ \"*++g)GF;7#$\"))))))3#F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 282 "fquad:=proc(z)\nlocal d,i,j:\nif z <= x[2,1] then\n d:=C[1,1]+C[ 2,1]*z+C[3,1]*z^2:\nelse\n for i from 2 to n-1 do\n if z <= x[i+1, 1] and z > x[i,1] then\n d:=0:\n for j from 0 to 2 do\n \+ d:=d+C[3*(i-1)+j+1,1]*z^j:\n end do:\n end if:\n end do:\n end if:\nd;\nend proc:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Value o f function at desired value of X is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "fquad(xdesired);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++SOUR!\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Plotting the quadratic spline interpolant and the value o f Y for the desired X" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 198 "plot([xy,[[xdesired,fquad(xdesired)]],fquad],ra nger(x,n),style=[POINT,POINT,LINE],color=[BLUE,RED,BLACK],symbol=[CIRC LE,CROSS],symbolsize=[30,40],thickness=2,title=\"Quadratic spline inte rpolation\");" }}{PARA 13 "" 1 "" {GLPLOT2D 683 518 518 {PLOTDATA 2 "6 )-%'CURVESG6&7(7$$\"\"!F)F(7$$\"#5F)$\"3#***********RqA!#:7$$\"#:F)$\" 3t**********zFOF/7$$\"#?F)$\"3A+++++]t^F/7$$\"3+++++++]A!#;$\"3F+++++q HgF/7$$\"#IF)$\"3f**********p;!*F/-%'COLOURG6&%$RGBGF(F($\"*++++\"!\") -%&STYLEG6#%&POINTG-%'SYMBOLG6$%'CIRCLEGFB-F$6&7#7$$\"#;F)$\"3.++++SOU RF/-FF6&FHFIF(F(FL-FQ6$%&CROSSG\"#S-F$6&7SF'7$$\"3s******\\i9Rl!#=$\"3 e++gkxk%[\"F=7$$\"3*****\\PC#)GA\"!#<$\"3Y,+@Y=VwFF=7$$\"3))****\\Peui =Fho$\"3x.+Y\\\"y\"HUF=7$$\"3L++]i3&o]#Fho$\"3_2+A)>a:p&F=7$$\"3'**** \\(oX*y9$Fho$\"3S4+*)G)zp9(F=7$$\"3o***\\P9CAu$Fho$\"3)Q,qfpXj\\)F=7$$ \"3#****\\P*zhdVFho$\"3K>+,$*e`$*)*F=7$$\"31++v$>fS*\\Fho$\"3u-!\\$*>^ Q8\"F/7$$\"30++v=$f%GcFho$\"3U.!HPS&)yF\"F/7$$\"3;+++Dy,\"G'Fho$\"3E/! ))pGUgU\"F/7$$\"3I++]7F/7$$\"35++v$pnsM*Fho$\"3A4!*=dO?A@F/7$$\"3,++]siL-5F=$ \"3!*zZIC\"4dF#F/7$$\"3%********Q5'f5F=$\"3;#3G,p(*)3CF/7$$\"3*)**\\P/ QBE6F=$\"3iw:yLU;rDF/7$$\"3#******\\\"o?&=\"F=$\"3?)yXJy!Q@FF/7$$\"3-+ ]Pa&4*\\7F=$\"3a**z[lWI$*GF/7$$\"3.+]7j=_68F=$\"3a*)eO3O$R1$F/7$$\"3-+ +vVy!eP\"F=$\"3Et[8<2;\\KF/7$$\"39+](=WU[V\"F=$\"3\"pcJ$\\wsDMF/7$$\"3 1++DJ#>&)\\\"F=$\"3:5b@/R7BOF/7$$\"3)***\\P>:mk:F=$\"3\">Q5Kr6:$QF/7$$ \"3()**\\iv&QAi\"F=$\"3b9yQO*\\>,%F/7$$\"3;++vtLU%o\"F=$\"3u3lv(H=e?%F /7$$\"3%******\\Nm'[+0WF/7$$\"34++vyb^6=F=$\"3i$)*zr7\"y) f%F/7$$\"3++]PMaKs=F=$\"3Y3Rh&z__y%F/7$$\"3$)***\\7TW)R>F=$\"3=tF$o'G7 \"*\\F/7$$\"3z*****\\@80+#F=$\"3A'oh&z@0v^F/7$$\"3)*****\\7,Hl?F=$\"3) >h`O7_xP&F/7$$\"3()**\\P4w)R7#F=$\"3%*HGS4%fId&F/7$$\"3O++]x%f\")=#F=$ \"3a()Hn<`B*z&F/7$$\"3)***\\P/-a[AF=$\"3'=zsyQmiF/7$$\"3y****\\i@OtBF=$\"3=BrP,D'[]'F/7$$\"3/+]PfL'zV#F=$ \"3#yQp9&yAcnF/7$$\"35+++!*>=+DF=$\"3aJ[\\33(***pF/7$$\"3?++DE&4Qc#F=$ \"3%QS-(=e!4D(F/7$$\"3=+]P%>5pi#F=$\"3SewKoLV,vF/7$$\"3K+++bJ*[o#F=$\" 3mY(R?t3Jt(F/7$$\"3E++Dr\"[8v#F=$\"3\">I+eDi.+)F/7$$\"3#)******Hjy5GF= $\"34=)R$z+'4C)F/7$$\"3E+]P/)fT(GF=$\"3o_\"zLK8\"*\\)F/7$$\"3;+]i0j\"[ $HF=$\"3gv%\\aVqxu)F/7$FA$\"3z******>**p;!*F/-FF6&FHF(F(F(-FM6#%%LINEG FP-%*THICKNESSG6#\"\"#-%&TITLEG6#Q?Quadratic~spline~interpolation6\"-% +AXESLABELSG6$Q!F\\_lF`_l-%%VIEWG6$;F(FA%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" } }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 266 37 "Section V: Cubic spline inte rpolation" }}{PARA 0 "" 0 "" {TEXT -1 143 "The algorithm of cubic spli ne interpolation is not shown. However, we are using the Maple functi on to conduct the cubic spline interpolation, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "fcubic:=t->s pline(convert(x,vector),convert(y,vector),t,cubic):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Computing the v alue of the function at the desired value of X," }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "fcubic(xdesired);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+;?a@R!\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Plotting the cub ic spline interpolant and the value of Y at the desired calue of X," } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "p lot([xy,[[xdesired,fcubic(xdesired)]],fcubic],ranger(x,n),style=[POINT ,POINT,LINE],color=[BLUE,RED,BLACK],symbol=[CIRCLE,CROSS],symbolsize=[ 30,40],thickness=2,title=\"Cubic spline interpolation\");" }}{PARA 13 "" 1 "" {GLPLOT2D 707 547 547 {PLOTDATA 2 "6)-%'CURVESG6&7(7$$\"\"!F)F (7$$\"#5F)$\"3#***********RqA!#:7$$\"#:F)$\"3t**********zFOF/7$$\"#?F) $\"3A+++++]t^F/7$$\"3+++++++]A!#;$\"3F+++++qHgF/7$$\"#IF)$\"3f******** **p;!*F/-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%&STYLEG6#%&POINTG-%'SYMB OLG6$%'CIRCLEGFB-F$6&7#7$$\"#;F)$\"3A+++;?a@RF/-FF6&FHFIF(F(FL-FQ6$%&C ROSSG\"#S-F$6&7S7$F)F(7$$\"+]i9Rl!#5$\"+DmT,9FK7$$\"+WA)GA\"!\"*$\"+k4 XAEFK7$$\"+Qeui=Fio$\"+\">H$**RFK7$$\"+i3&o]#Fio$\"+^jC\"R&FK7$$\"+pX* y9$Fio$\"+%fkWy'FK7$$\"+WTAUPFio$\"+ac(\\3)FK7$$\"+%*zhdVFio$\"+MJGU%* FK7$$\"+%>fS*\\Fio$\"+qZ$f3\"!\"(7$$\"+>$f%GcFio$\"+*eI(G7Fjq7$$\"+Dy, \"G'Fio$\"+q#GuP\"Fjq7$$\"+7Fjq7$$\"+%p nsM*Fio$\"+!\\9r5#Fjq7$$\"+siL-5FK$\"+ZOIwAFjq7$$\"+!R5'f5FK$\"+4$RBU# Fjq7$$\"+/QBE6FK$\"+I.O&f#Fjq7$$\"+:o?&=\"FK$\"+&o?8v#Fjq7$$\"+a&4*\\7 FK$\"+Y([a#HFjq7$$\"+j=_68FK$\"+YY<%4$Fjq7$$\"+Wy!eP\"FK$\"+'=NKF$Fjq7 $$\"+UC%[V\"FK$\"+DDOSMFjq7$$\"+J#>&)\\\"FK$\"+id]BOFjq7$$\"+>:mk:FK$ \"+v1*o\"QFjq7$$\"+w&QAi\"FK$\"+9#zy)RFjq7$$\"+uLU%o\"FK$\"+mJOvTFjq7$ $\"+bjm[FK$\"+g!eq(\\Fjq7$$\"+:K^+?FK$\"+r()=v^Fjq7$$\"+7,Hl ?FK$\"+)*3B!R&Fjq7$$\"+4w)R7#FK$\"+VFx)e&Fjq7$$\"+y%f\")=#FK$\"+y%o0\" eFjq7$$\"+/-a[AFK$\"+&ygW-'Fjq7$$\"+ial6BFK$\"+*yaRD'Fjq7$$\"+i@OtBFK$ \"+O/)Q['Fjq7$$\"+fL'zV#FK$\"+F]$*HnFjq7$$\"+!*>=+DFK$\"+SLcrpFjq7$$\" +E&4Qc#FK$\"+&Q[GA(Fjq7$$\"+%>5pi#FK$\"+A^qvuFjq7$$\"+bJ*[o#FK$\"+#[33 r(Fjq7$$\"+r\"[8v#FK$\"+Al'H)zFjq7$$\"+Ijy5GFK$\"+%y`$G#)Fjq7$$\"+/)fT (GFK$\"+L&=:\\)Fjq7$$\"+1j\"[$HFK$\"+d))RW()Fjq7$FB$\"+-+q;!*Fjq-FF6&F HF(F(F(-FM6#%%LINEGFP-%*THICKNESSG6#\"\"#-%&TITLEG6#Q;Cubic~spline~int erpolation6\"-%+AXESLABELSG6$Q!F^_lFb_l-%%VIEWG6$;F(FA%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2 " "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 274 0 "" }{TEXT 275 23 "Section VI: Conclusion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 259 "Maple helped us to apply our knowledge of numerical meth ods of interpolation to find the value of y at a particular value of x using linear and quadratic spline interpolation. Using Maple function s and plotting routines made it easy to illustrate this method." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 4 "" 0 "" {TEXT 277 10 "Referen ces" }}{PARA 0 "" 0 "" {TEXT -1 4 "[1] " }{TEXT 279 174 "Nathan Collie r, Autar Kaw, Jai Paul , Michael Keteltas, Holistic Numerical Methods \+ Institute, See http://numericalmethods.eng.usf.edu/mws/gen/05inp/mws_g en_inp_sim_spline.mws" }}{PARA 0 "" 0 "" {TEXT 280 76 "http://numerica lmethods.eng.usf.edu/mws/gen/05inp/mws_gen_inp_txt_spline.pdf" }{TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT 276 10 "Disclaimer" }{TEXT 278 1 ":" }{TEXT -1 248 " While every effor t has been made to validate the solutions in this worksheet, Universit y of South Florida and the contributors are not responsible for any er rors contained and are not liable for any damages resulting from the u se of this material." }}}{MARK "6 5 6 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }