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" }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be re ad into a Maple session by a command similar to the one that follows, \+ where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procdrs/fcnapprx.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "The Lagrang e basis polynomials " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{PARA 0 "" 0 "" {TEXT -1 27 "Suppose we have a sequence " }{XPPEDIT 18 0 "x[1],x[2],x[3],` . . . `,x[n]" "6'&%\"xG6#\"\"\"&F$6#\"\"#&F$6# \"\"$%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 4 " of " }{TEXT 272 1 "x" }{TEXT -1 8 " values." }}{PARA 0 "" 0 "" {TEXT -1 14 "We define the " }{TEXT 260 30 "Lagrange \"L\" basis polynomials" }{TEXT -1 1 " " }{XPPEDIT 18 0 "L[j](x);" "6#-&%\"LG6#%\"jG6#%\"xG" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "j = 1,2,` . . .`,n" "6&/%\"jG\"\"\"\"\"#%'~.~.~.G%\"nG " }{TEXT -1 16 " by: " }}{PARA 0 "" 0 "" {TEXT -1 83 " \+ \+ " }{XPPEDIT 18 0 "L[j](x) = product((x-x[i])/(x[j]-x[i]),i = 1 .. n)" "6#/-&%\"LG6#%\"jG6#%\"xG-%(productG6$*&,&F*\"\"\"&F*6#%\"iG! \"\"F0,&&F*6#F(F0&F*6#F3F4F4/F3;F0%\"nG" }}{PARA 0 "" 0 "" {TEXT -1 96 " \+ " }{XPPEDIT 18 0 "i <> j;" "6#0%\"iG%\"j G" }{TEXT 269 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "These are polynomial s in the variable " }{TEXT 273 1 "x" }{TEXT -1 11 " of degree " } {XPPEDIT 18 0 "n-1;" "6#,&%\"nG\"\"\"F%!\"\"" }{TEXT -1 26 ", having t he property that" }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "L[j](x[i]) = PIECEWISE([0, i <> j],[1, i = j]);" "6#/-&%\"LG6#%\"jG6# &%\"xG6#%\"iG-%*PIECEWISEG6$7$\"\"!0F-F(7$\"\"\"/F-F(" }{TEXT -1 3 " \+ ." }{TEXT 270 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 271 11 "_ __________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 18 "For example , when " }{XPPEDIT 18 0 "n = 4" "6#/%\"nG\"\"%" }{TEXT -1 31 " we have 4 cubic polynomials: " }}{PARA 256 "" 0 "" {TEXT -1 3 "\n " } {XPPEDIT 18 0 "L[1](x);" "6#-&%\"LG6#\"\"\"6#%\"xG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(x-x[2])*(x-x[3])*(x-x[4])/((x[1]-x[2])*(x[1]-x[3])*(x [1]-x[4]));" "6#**,&%\"xG\"\"\"&F%6#\"\"#!\"\"F&,&F%F&&F%6#\"\"$F*F&,& F%F&&F%6#\"\"%F*F&*(,&&F%6#F&F&&F%6#F)F*F&,&&F%6#F&F&&F%6#F.F*F&,&&F%6 #F&F&&F%6#F2F*F&F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L[2](x);" "6#-&%\"LG 6#\"\"#6#%\"xG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(x-x[1])*(x-x[3])*(x -x[4])/((x[2]-x[1])*(x[2]-x[3])*(x[2]-x[4]));" "6#**,&%\"xG\"\"\"&F%6# F&!\"\"F&,&F%F&&F%6#\"\"$F)F&,&F%F&&F%6#\"\"%F)F&*(,&&F%6#\"\"#F&&F%6# F&F)F&,&&F%6#F6F&&F%6#F-F)F&,&&F%6#F6F&&F%6#F1F)F&F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "L[3](x);" "6#-&%\"LG6#\"\"$6#%\"xG" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "(x-x[1])*(x-x[2])*(x-x[4])/((x[3]-x[1])*(x[3]-x[2])*(x[ 3]-x[4]));" "6#**,&%\"xG\"\"\"&F%6#F&!\"\"F&,&F%F&&F%6#\"\"#F)F&,&F%F& &F%6#\"\"%F)F&*(,&&F%6#\"\"$F&&F%6#F&F)F&,&&F%6#F6F&&F%6#F-F)F&,&&F%6# F6F&&F%6#F1F)F&F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "L[4](x);" "6#-&%\"LG6 #\"\"%6#%\"xG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(x-x[1])*(x-x[2])*(x- x[3])/((x[4]-x[1])*(x[4]-x[2])*(x[4]-x[3]));" "6#**,&%\"xG\"\"\"&F%6#F &!\"\"F&,&F%F&&F%6#\"\"#F)F&,&F%F&&F%6#\"\"$F)F&*(,&&F%6#\"\"%F&&F%6#F &F)F&,&&F%6#F6F&&F%6#F-F)F&,&&F%6#F6F&&F%6#F1F)F&F)" }{TEXT -1 1 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The fol lowing Maple procedure can be used to construct these Lagrange \"L\" b asis polynomials for a given list of " }{TEXT 276 1 "x" }{TEXT -1 10 " values: " }{XPPEDIT 18 0 "[x[1],x[2],x[3],` . . . `,x[n]]" "6#7'&%\" xG6#\"\"\"&F%6#\"\"#&F%6#\"\"$%(~.~.~.~G&F%6#%\"nG" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proc edure " }{TEXT 0 1 "L" }{TEXT -1 10 " requires " }{TEXT 260 20 "two in put parameters" }{TEXT -1 14 ": the list of " }{TEXT 274 1 "x" }{TEXT -1 143 " values is the first input parameter, and the symbol to be use d as the \"unknown\" in the Lagrange \"L\" basis polynomials is the se cond parameter." }}{PARA 0 "" 0 "" {TEXT -1 88 "For this reason the Ma ple notation is slightly different from the mathematical notation." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 376 "L := proc(xvals::list,j::posint,x)\nlocal z,xvalsdelete;\n if j > nops(xvals) then\n error \"2nd argument must no geater than th e list length\"\n end if;\n xvalsdelete := [op(1..j-1,xvals),op(j+ 1..nops(xvals),xvals)];\n if member(xvals[j],xvalsdelete) then\n \+ error \"the same point has been entered twice\"\n end if;\n mul( (x-z)/(xvals[j]-z),z=xvalsdelete);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 89 "\nWe can use Maple's ability to perform algebraic manipul ations to check how the function " }{TEXT 0 1 "L" }{TEXT -1 40 " works .\nFirst set up a list of symbolic " }{TEXT 275 1 "x" }{TEXT -1 13 " v alues . . ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 46 "n := 4:\nx := 'x':\nxvals := [seq(x[i],i=1..n)];" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7&&%\"xG6#\"\"\"&F'6#\"\"#&F '6#\"\"$&F'6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 30 "The Lagrange basis polynomial " }{XPPEDIT 18 0 "L[2](x) ;" "6#-&%\"LG6#\"\"#6#%\"xG" }{TEXT -1 40 " with index 2 for this symb olic data is:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "L[2 ](x) = product((x-x[i])/(x[2]-x[i]),i = 1 .. 4);" "6#/-&%\"LG6#\"\"#6# %\"xG-%(productG6$*&,&F*\"\"\"&F*6#%\"iG!\"\"F0,&&F*6#F(F0&F*6#F3F4F4/ F3;F0\"\"%" }{XPPEDIT 18 0 "`` = (x-x[1])/(x[2]-x[1])*(x-x[3])/(x[2]-x [3])*(x-x[4])/(x[2]-x[4]);" "6#/%!G*.,&%\"xG\"\"\"&F'6#F(!\"\"F(,&&F'6 #\"\"#F(&F'6#F(F+F+,&F'F(&F'6#\"\"$F+F(,&&F'6#F/F(&F'6#F5F+F+,&F'F(&F' 6#\"\"%F+F(,&&F'6#F/F(&F'6#F>F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 92 " \+ " }{XPPEDIT 18 0 "i <> 2;" "6#0%\"i G\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "L(xvals,2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*.,& %\"xG\"\"\"&F%6#F&!\"\"F&,&&F%6#\"\"#F&F'F)F),&F%F&&F%6#\"\"$F)F&,&F+F &F/F)F),&F%F&&F%6#\"\"%F)F&,&F+F&F4F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "If we let " }{XPPEDIT 18 0 "x = x[ 2];" "6#/%\"xG&F$6#\"\"#" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "L[2](x );" "6#-&%\"LG6#\"\"#6#%\"xG" }{TEXT -1 17 " simplifies to 1." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "L(xvals,2,x[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "If we let " }{XPPEDIT 18 0 "z = x[1],x[3];" "6$/%\"zG&%\"xG6#\"\"\"&F&6#\"\"$" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x[4];" "6#&%\"xG6#\"\"%" }{TEXT -1 6 " then " }{XPPEDIT 18 0 "L[2](x);" "6#-&%\"LG6#\"\"#6#%\"xG" }{TEXT -1 17 " simplifies to 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "L(xvals,2,x[3]);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "The basis polynomial " }{XPPEDIT 18 0 "L[j](x)" "6#- &%\"LG6#%\"jG6#%\"xG" }{TEXT -1 17 " for the points " }{XPPEDIT 18 0 "x[1],x[2],x[3],` . . . `,x[n]" "6'&%\"xG6#\"\"\"&F$6#\"\"#&F$6#\"\"$% (~.~.~.~G&F$6#%\"nG" }{TEXT -1 47 " is the interpolating polynomial fo r the points" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "``(x [1],0),` . . . `,``(x[j-1],0),``(x[j],1),``(x[j+1],0),` . . . `,``(x[n ],0);" "6)-%!G6$&%\"xG6#\"\"\"\"\"!%(~.~.~.~G-F$6$&F'6#,&%\"jGF)F)!\" \"F*-F$6$&F'6#F1F)-F$6$&F'6#,&F1F)F)F)F*F+-F$6$&F'6#%\"nGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "The following picture shows the graphs of the Lagrange \"L\" basis polynomials for the " }{TEXT 280 1 "x" }{TEXT -1 8 " values " } {XPPEDIT 18 0 "x[1]=1,x[2]=2,` . . . `,x[n]=n" "6&/&%\"xG6#\"\"\"F'/&F %6#\"\"#F+%(~.~.~.~G/&F%6#%\"nGF0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 15 "For each index " }{TEXT 282 1 "j" }{TEXT -1 11 " the poin t " }{XPPEDIT 18 0 "``(x[j],1)=``(j,1)" "6#/-%!G6$&%\"xG6#%\"jG\"\"\"- F%6$F*F+" }{TEXT -1 133 " is plotted as a black circle surrounded by a circle of the same colour as the graph of the associated Lagrange \"L \" basis polynomial." }}{PARA 0 "" 0 "" {TEXT -1 22 "The value assigne d to " }{TEXT 281 1 "n" }{TEXT -1 16 " may be changed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 405 "x := ' x':\nn := 4:\nxvals := [seq(i,i=1..n)];\np1 := plot([seq(L(xvals,i,x), i=1..n)],x=1..n,\n color=[seq(COLOR(HUE,(i-1)/n),i=1..n)],thi ckness=2):\np2 := plot([seq([[i,1]],i=1..n)],style=point,symbol=circle ,\n symbolsize=18,color=[seq(COLOR(HUE,(i-1)/n),i=1..n)]):\n p3 := plot([seq([[i,1]],i=1..n)],style=point,symbol=circle,\n \+ symbolsize=10,color=black):\nplots[display]([p1,p2,p3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7&\"\"\"\"\"#\"\"$\"\"%" }}{PARA 13 "" 1 "" {GLPLOT2D 394 220 220 {PLOTDATA 2 "60-%'CURVESG6%7U7$$\"\" \"\"\"!F(7$$\"3.++DJdpK5!#<$\"3Lm5N:,@6%*!#=7$$\"31++]i9Rl5F.$\"3qb'Hm #4XV))F17$$\"31+vVV)RQ4\"F.$\"3'ezx%*y%Gm$)F17$$\"34+]PC#)GA6F.$\"3sMO H/aa/zF17$$\"3%****\\Peui=\"F.$\"3o7cWUe<@pF17$$\"33++D'3&o]7F.$\"3/x. 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Fbp7$Fcr$!3'H)>D8^-0RFbp7$Fhr$!3Of6Jve;)o%Fbp7$F]s$!3YG*zeTl)y`Fbp7$Fb s$!3516>QH)p(eFbp7$Fgs$!3oh,Un`xViFbp7$F\\t$!3o#3%>W#*Q5kFbp7$Fat$!3Q$ >DajG`N'Fbp7$Fft$!3')3DD+yejgFbp7$F[u$!3wQ_l!pkR[&Fbp7$F`u$!32Q8F**z4= YFbp7$Feu$!33`r^LdYvMFbp7$Fju$!3Nd]VF'zy#=Fbp7$F`v$\"3c3'R,%Q.73Y#>VuEJF17$F^y$\"3AM,c%Ryuw$F17$Fcy$\"3#>9y7\"f UlWF17$Fhy$\"3C94?h#=Q;&F17$F]z$\"3#)>j'z#y.MgF17$Fbz$\"3m'*[7D0\"y(oF 17$Fgz$\"3g')QT/g'z%yF17$Fdcm$\"34(45o!3sQ$)F17$F\\[l$\"3K@ZRVJ*p%))F1 7$F\\dm$\"3$[o6mQZIT*F17$Fa[lF(-Fe[l6$Fg[l#\"\"$Fb[lFh[l-F$6&7#F'Fd[l- %'SYMBOLG6$%'CIRCLEG\"#=-%&STYLEG6#%&POINTG-F$6&7#7$$F[\\lF*F(F[hlFa^n Ff^n-F$6&7#7$$F]^nF*F(F`dmFa^nFf^n-F$6&7#Fi]nFj]nFa^nFf^n-F$6&F`^n-%'C OLOURG6&%$RGBGF*F*F*-Fb^n6$Fd^n\"#5Ff^n-F$6&F\\_nFi_nF]`nFf^n-F$6&Fa_n Fi_nF]`nFf^n-F$6&Ff_nFi_nF]`nFf^n-%+AXESLABELSG6%Q\"x6\"Q!Fj`n-%%FONTG 6#%(DEFAULTG-%%VIEWG6$;F(Fa[lF_an" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "The Lagran ge interpolating polynomial " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 24 "For a general sequence " } {XPPEDIT 18 0 "x[1],x[2],x[3],` . . . `,x[n]" "6'&%\"xG6#\"\"\"&F$6#\" \"#&F$6#\"\"$%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 4 " of " }{TEXT 266 1 "x " }{TEXT -1 35 " values and corresponding sequence " }{XPPEDIT 18 0 "y [1],y[2],y[3],` . . . `,y[n]" "6'&%\"yG6#\"\"\"&F$6#\"\"#&F$6#\"\"$%(~ .~.~.~G&F$6#%\"nG" }{TEXT -1 5 ", of " }{TEXT 267 1 "y" }{TEXT -1 13 " values, the " }{TEXT 260 33 "Lagrange interpolating polynomial" } {TEXT -1 17 " is defined by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "p(x) = sum(L[j](x)*y[j],j = 1 .. n);" "6#/-%\"pG6#%\"xG -%$sumG6$*&-&%\"LG6#%\"jG6#F'\"\"\"&%\"yG6#F0F2/F0;F2%\"nG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 268 11 "___________" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "It has degree less than or equal to " }{XPPEDIT 18 0 "n- 1;" "6#,&%\"nG\"\"\"F%!\"\"" }{TEXT -1 19 ", and the property " } {XPPEDIT 18 0 "L[j](x[i]) = PIECEWISE([0, i <> j],[1, i = j]);" "6#/-& %\"LG6#%\"jG6#&%\"xG6#%\"iG-%*PIECEWISEG6$7$\"\"!0F-F(7$\"\"\"/F-F(" } {TEXT -1 15 " , means that " }{XPPEDIT 18 0 "p(x[i]) = y[i];" "6#/-% \"pG6#&%\"xG6#%\"iG&%\"yG6#F*" }{TEXT -1 6 " for " }{XPPEDIT 18 0 "i \+ = 1,2,` . . . `,n;" "6&/%\"iG\"\"\"\"\"#%(~.~.~.~G%\"nG" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "For \+ example, when " }{XPPEDIT 18 0 "n = 4" "6#/%\"nG\"\"%" }{TEXT -1 74 ", the Lagrange interploating polynomial which fits through the data poi nts" }{XPPEDIT 18 0 "``(x[1],y[1]),``(x[2],y[2]),``(x[3],y[3]),``(x[4] ,y[4])" "6&-%!G6$&%\"xG6#\"\"\"&%\"yG6#F)-F$6$&F'6#\"\"#&F+6#F1-F$6$&F '6#\"\"$&F+6#F8-F$6$&F'6#\"\"%&F+6#F?" }{TEXT -1 6 ", is: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "p(x) = L[1](x)*y[1]+L[2](x )*y[2]+L[3](x)*y[3]+L[4](x)*y[4];" "6#/-%\"pG6#%\"xG,**&-&%\"LG6#\"\" \"6#F'F.&%\"yG6#F.F.F.*&-&F,6#\"\"#6#F'F.&F16#F7F.F.*&-&F,6#\"\"$6#F'F .&F16#F?F.F.*&-&F,6#\"\"%6#F'F.&F16#FGF.F." }{TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "that is, " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "p(x) = ``((x-x[2])*( x-x[3])*(x-x[4])/((x[1]-x[2])*(x[1]-x[3])*(x[1]-x[4])))*y[1]+``((x-x[1 ])*(x-x[3])*(x-x[4])/((x[2]-x[1])*(x[2]-x[3])*(x[2]-x[4])))*y[2]+``((x -x[1])*(x-x[2])*(x-x[4])/((x[3]-x[1])*(x[3]-x[2])*(x[3]-x[4])))*y[3]+` `((x-x[1])*(x-x[2])*(x-x[3])/((x[4]-x[1])*(x[4]-x[2])*(x[4]-x[3])))*y[ 4];" "6#/-%\"pG6#%\"xG,**&-%!G6#**,&F'\"\"\"&F'6#\"\"#!\"\"F/,&F'F/&F' 6#\"\"$F3F/,&F'F/&F'6#\"\"%F3F/*(,&&F'6#F/F/&F'6#F2F3F/,&&F'6#F/F/&F'6 #F7F3F/,&&F'6#F/F/&F'6#F;F3F/F3F/&%\"yG6#F/F/F/*&-F+6#**,&F'F/&F'6#F/F 3F/,&F'F/&F'6#F7F3F/,&F'F/&F'6#F;F3F/*(,&&F'6#F2F/&F'6#F/F3F/,&&F'6#F2 F/&F'6#F7F3F/,&&F'6#F2F/&F'6#F;F3F/F3F/&FM6#F2F/F/*&-F+6#**,&F'F/&F'6# F/F3F/,&F'F/&F'6#F2F3F/,&F'F/&F'6#F;F3F/*(,&&F'6#F7F/&F'6#F/F3F/,&&F'6 #F7F/&F'6#F2F3F/,&&F'6#F7F/&F'6#F;F3F/F3F/&FM6#F7F/F/*&-F+6#**,&F'F/&F '6#F/F3F/,&F'F/&F'6#F2F3F/,&F'F/&F'6#F7F3F/*(,&&F'6#F;F/&F'6#F/F3F/,&& F'6#F;F/&F'6#F2F3F/,&&F'6#F;F/&F'6#F7F3F/F3F/&FM6#F;F/F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 56 "This polynomial fits through the data p oints because p( " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "y[1];" "6#&%\"yG6#\"\"\"" }{TEXT -1 5 ", p ( " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"#" }{TEXT -1 4 ") = " } {XPPEDIT 18 0 "y[2];" "6#&%\"yG6#\"\"#" }{TEXT -1 5 ", p( " }{XPPEDIT 18 0 "x[3];" "6#&%\"xG6#\"\"$" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "y[3] ;" "6#&%\"yG6#\"\"$" }{TEXT -1 8 " and p( " }{XPPEDIT 18 0 "x[4];" "6# &%\"xG6#\"\"%" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "y[4];" "6#&%\"yG6#\" \"%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 195 "We would usually expect that this polynomial would hav e degree 3, but it is possible for the degree to be less than 3. For e xample, if the given 4 data points are in the same straight line, then " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 132 " will give t he equation of this line, and so have degree 1. In such a situation, a fter expanding and collecting terms together, the " }{XPPEDIT 18 0 "x^ 2;" "6#*$%\"xG\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x^3;" "6#*$% \"xG\"\"$" }{TEXT -1 20 " terms would vanish." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The following procedure \+ " }{TEXT 0 8 "lagrange" }{TEXT -1 173 " can be used to check how the L agrange interpolating polynomial works symbolically. It can also be us ed to find a specific interpolating polynomial for given numerical dat a." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 16 ": The procedure " }{TEXT 0 1 "L" } {TEXT -1 36 " introduced in the previous section " }{TEXT 260 14 "must be active" }{TEXT -1 19 " for the procedure " }{TEXT 0 8 "lagrange" } {TEXT -1 38 " to work, so it is also included here." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "lagrange" {MPLTEXT 1 0 641 "L := proc(xvals::list,j::posint,x)\nlocal v,xvalsdelete;\n if j > nop s(xvals) then\n error \"2nd argument must no geater than the list length\"\n end if;\n xvalsdelete := [op(1..j-1,xvals),op(j+1..nop s(xvals),xvals)];\n if member(xvals[j], xvalsdelete) then\n err or \"the same point has been entered twice\"\n end if;\n mul((x-v) /(xvals[j]-v),v=xvalsdelete);\nend proc: # of L \n\nlagrange := proc(x vals::list, yvals::list, x)\n local m,n,j;\n n := nops(xvals);\n \+ m := nops(yvals);\n if n<>m then\n error \"1st and 2nd argument s must have the same length\"\n end if;\n add(L(xvals,j,x)*yvals[j ],j=1..n);\nend proc: # of lagrange " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 110 "n := 4:\nx := 'x': y := 'y':\nxvals := [seq (x[i],i=1..n)];\nyvals := [seq(y[i],i=1..n)];\nlagrange(xvals,yvals,x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7&&%\"xG6#\"\"\"&F'6#\" \"#&F'6#\"\"$&F'6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7& &%\"yG6#\"\"\"&F'6#\"\"#&F'6#\"\"$&F'6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**0,&%\"xG\"\"\"&F&6#\"\"#!\"\"F',&&F&6#F'F'F(F+F+,&F& F'&F&6#\"\"$F+F',&F-F'F0F+F+,&F&F'&F&6#\"\"%F+F',&F-F'F5F+F+&%\"yGF.F' F'*0,&F&F'F-F+F',&F(F'F-F+F+F/F',&F(F'F0F+F+F4F',&F(F'F5F+F+&F:F)F'F'* 0F \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 66 "The Lagrange interpolating polynomial can be arran ged in the form " }{XPPEDIT 18 0 "a[n]*z^n+a[n-1]*z^(n-1)+` . . . `+a[ 1]*z+a[0];" "6#,,*&&%\"aG6#%\"nG\"\"\")%\"zGF(F)F)*&&F&6#,&F(F)F)!\"\" F))F+,&F(F)F)F0F)F)%(~.~.~.~GF)*&&F&6#F)F)F+F)F)&F&6#\"\"!F)" }{TEXT -1 24 " by using the procedure " }{TEXT 0 7 "collect" }{TEXT -1 59 ". \+ A technical difficulty is the fact that the same symbol \"" }{TEXT 265 1 "x" }{TEXT -1 40 "\" is used for the subscripted variables " } {XPPEDIT 18 0 "x[1],x[2],x[3],` . . . `;" "6&&%\"xG6#\"\"\"&F$6#\"\"#& F$6#\"\"$%(~.~.~.~G" }{TEXT -1 68 ". The \"unknown\" in the Lagrange p olynomial can easily be changed to " }{TEXT 283 1 "z" }{TEXT -1 7 ", s ay. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "n := 3:\nx := 'x': y := 'y': z := 'z':\nxvals := [se q(x[i],i=1..n)];\nyvals := [seq(y[i],i=1..n)];\nlagrange(xvals,yvals,z );\ncollect(%,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7%&%\"xG 6#\"\"\"&F'6#\"\"#&F'6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yva lsG7%&%\"yG6#\"\"\"&F'6#\"\"#&F'6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*,,&%\"zG\"\"\"&%\"xG6#\"\"#!\"\"F',&&F)6#F'F'F(F,F,,&F&F'&F)6 #\"\"$F,F',&F.F'F1F,F,&%\"yGF/F'F'*,,&F&F'F.F,F',&F.F,F(F'F,F0F',&F(F' F1F,F,&F6F*F'F'*,F8F',&F.F,F1F'F,F%F',&F(F,F1F'F,&F6F2F'F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,*,,&&%\"xG6#\"\"\"F)&F'6#\"\"#!\"\"F-,&F&F )&F'6#\"\"$F-F-&%\"yGF(F)F*F)F/F)F)*,,&F&F-F*F)F-,&F*F)F/F-F-&F3F+F)F& F)F/F)F)*,,&F&F-F/F)F-,&F*F-F/F)F-&F3F0F)F&F)F*F)F)*&,(*(F%F-F.F-F2F)F )*(F5F-F6F-F7F)F)*(F9F-F:F-F;F)F)F))%\"zGF,F)F)*&,(*(,&*&F%F-F*F)F-*&F %F-F/F)F-F)F.F-F2F)F)*(,&*&F5F-F&F)F-*&F5F-F/F)F-F)F6F-F7F)F)*(,&*&F9F -F&F)F-*&F9F-F*F)F-F)F:F-F;F)F)F)FBF)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "For a numerical example, consider the problem of finding \+ the quadratic function " }{XPPEDIT 18 0 "f(x) = a*x^2+b*x+c;" "6#/-%\" fG6#%\"xG,(*&%\"aG\"\"\"*$F'\"\"#F+F+*&%\"bGF+F'F+F+%\"cGF+" }{TEXT -1 24 " such that the graph of " }{XPPEDIT 18 0 "y = f(x);" "6#/%\"yG- %\"fG6#%\"xG" }{TEXT -1 26 " passes through the points" }{XPPEDIT 18 0 "``(1,2), ``(2,3)" "6$-%!G6$\"\"\"\"\"#-F$6$F'\"\"$" }{TEXT -1 5 " a nd " }{XPPEDIT 18 0 "``(4,1)" "6#-%!G6$\"\"%\"\"\"" }{TEXT -1 75 ".Thi s example was considered in the intoductory worksheet on interpolation ." }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "collect " }{TEXT -1 62 " can be used to see the actual coefficients of the pol ynomial." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "x := 'x':\nlagrange([1,2,4],[2,3,1],x);\ncollect(%,x) ;\nsort(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\"#\"\"\",&%\"xG! \"\"F%F&F&,&*&\"\"$F)F(F&F)#\"\"%F,F&F&F&*(F,F&,&F(F&F&F)F&,&*&F%F)F(F &F)F%F&F&F&*&,&*&F,F)F(F&F&#F&F,F)F&,&*&F%F)F(F&F&F&F)F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#\"\"\"\"\"$!\"\"*&#\"\"#F&F%*$)%\"xGF*F%F %F'*&F&F%F-F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"#\"\"$\"\" \"*$)%\"xGF&F(F(!\"\"*&F'F(F+F(F(#F(F'F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "In the following example the data \+ points" }{XPPEDIT 18 0 "``(1,2), ``(2,3 ), ``(3,4) ,``(4,5)" "6&-%!G6$ \"\"\"\"\"#-F$6$F'\"\"$-F$6$F*\"\"%-F$6$F-\"\"&" }{TEXT -1 34 " clearl y lie on the straight line " }{XPPEDIT 18 0 "y = x+1;" "6#/%\"yG,&%\"x G\"\"\"F'F'" }{TEXT -1 43 ", so the Lagrange interpolating polynomial \+ " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 15 " simplifies to " }{XPPEDIT 18 0 "x+1;" "6#,&%\"xG\"\"\"F%F%" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "x := 'x':\nlagrange([1,2,3,4],[2,3,4,5],x);\ncollect(%,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,***\"\"#\"\"\",&%\"xG!\"\"F%F&F&,&*&F %F)F(F&F)#\"\"$F%F&F&,&*&F-F)F(F&F)#\"\"%F-F&F&F&**F-F&,&F(F&F&F)F&,&F (F)F-F&F&,&*&F%F)F(F&F)F%F&F&F&**F1F&,&*&F%F)F(F&F&#F&F%F)F&,&F(F&F%F) F&,&F(F)F1F&F&F&**\"\"&F&,&*&F-F)F(F&F&#F&F-F)F&,&*&F%F)F(F&F&F&F)F&,& F(F&F-F)F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$%\"xGF$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 66 "A procedure for constru cting a lagrange interpolating polynomial: " }{TEXT 0 8 "lagrange" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 8 "lagrange" }{TEXT -1 58 " discussed in the last sect ion is presented formally here." }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 11 ": The name " }{TEXT 0 15 "lagrange_interp" }{TEXT -1 38 " may also be used for this procedure. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 15 "lagrange: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 262 16 "Calling Sequence" }{TEXT -1 1 ":" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 57 " \+ lagrange( x, y, z ) or lagrange_interp( x, y, z )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 10 "Parameters" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{TEXT 23 4 "x - " }{TEXT -1 28 "list of independent values, " }{XPPEDIT 18 0 "[x[0], x[1], ` . \+ . . `, x[n]];" "6#7&&%\"xG6#\"\"!&F%6#\"\"\"%(~.~.~.~G&F%6#%\"nG" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{TEXT 23 5 " y \+ - " }{TEXT -1 27 "list of dependent values, " }{XPPEDIT 18 0 "[y[0], \+ y[1], ` . . . `, y[n]];" "6#7&&%\"yG6#\"\"!&F%6#\"\"\"%(~.~.~.~G&F%6#% \"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{TEXT 23 4 "z - " }{TEXT -1 57 "the variable to be used in the interpolating polynomial. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 11 "Description" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 8 "lagrange" }{TEXT -1 134 " constructs t he Lagrange interpolating polynomial for the data given in the lists x and y, using v as the variable in this polynomial." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 155 "To ma ke the procedures active open the subsection, place the cursor anywher e after the prompt [ > and press [Enter].\nYou can then close up the \+ subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "lagrange: implemen tation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 614 "lagrange_interp := proc() lagrange(args[1..nargs]) e nd:\n\nlagrange := proc(xvals::list,yvals::list,x::algebraic)\n loca l m,n,j,L;\n\nL := proc(xvals::list,j::posint,x)\nlocal z,xvalsdelete; \n xvalsdelete := [op(1..j-1,xvals),op(j+1..nops(xvals),xvals)];\n \+ if member(xvals[j], xvalsdelete) then\n error \"the same point h as been entered twice\"\n end if;\n mul((x-z)/(xvals[j]-z),z=xvals delete);\nend proc: # of L\n\n n := nops(xvals);\n m := nops(yvals );\n if n<>m then\n error \"1st and 2nd arguments lists must ha ve the same length\"\n end if;\n add(L(xvals,j,x)*yvals[j],j=1..n) ;\nend proc: # of lagrange" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 43 "Examples are given in a following sec tion. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Converting expressions to functions in Ma ple . . " }{TEXT 0 7 "unapply" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 243 "It takes a while to get used \+ to the difference between expressions and functions when using Maple. \nSome Maple procedures require functions as input and some require ex pressions. Some procedures can be used with either expressions or func tions." }}{PARA 0 "" 0 "" {TEXT -1 17 "For example, the " }{TEXT 0 4 " plot" }{TEXT -1 80 " command can be used with either a function or exp ression as the first argument." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 203 "The only difference in the graphs produc ed by the following commands is that, in the first case, both axes rem ain unlabled because the independent variable is not used when the1st \+ argument is a function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot(sin,0..Pi);\nplot(sin(x),x=0.. Pi);" }}}{PARA 0 "" 0 "" {TEXT -1 19 "\nThe Maple command " }{TEXT 0 7 "unapply" }{TEXT -1 54 " can be used to convert an expression into a function." }}{PARA 0 "" 0 "" {TEXT -1 51 "The following command defin es the function f where " }{XPPEDIT 18 0 "f(x) = sqrt(x)+2;" "6#/-%\"f G6#%\"xG,&-%%sqrtG6#F'\"\"\"\"\"#F," }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 58 "It is essentially equivalent to using the arrow notatio n: " }{TEXT 261 19 "f := x -> sqrt(x)+2" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "f := \+ unapply(sqrt(x)+2,x);\nf(x);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 23 "Here is another use of " }{TEXT 0 7 "unap ply" }{TEXT -1 67 " to convert a derivative obtained as an expression \+ into a function." }}{PARA 0 "" 0 "" {TEXT -1 77 "The end result is exa ctly the same as that obtained by applying the operator " }{TEXT 0 1 " D" }{TEXT -1 17 " to the function " }{TEXT 261 19 "f := x -> sqrt(x)+2 " }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 70 "f := x -> sqrt(x)+2;\nDiff(f(x),x);\nvalue(% );\ndf := unapply(%,x);\nD(f);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%%sqrtG6#9$\"\"\"\"\"#F1F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,&*$-%%sqrtG6#%\"xG\"\" \"F,\"\"#F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*$-%%sqrt G6#%\"xGF%!\"\"#F%\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dfGf*6#% \"xG6\"6$%)operatorG%&arrowGF(,$*&\"\"\"F.*$-%%sqrtG6#9$F.!\"\"#F.\"\" #F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)operatorG%& arrowGF&,$*&\"\"\"F,-%%sqrtG6#9$!\"\"#F,\"\"#F&F&F&" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 49 ": A procedure similar to, but much simpler than, " }{TEXT 0 7 "unapply" } {TEXT -1 75 " for functions of one variable, can easily be constructed .by making use of " }{TEXT 0 4 "subs" }{TEXT -1 34 " to substitute for the expression " }{TEXT 265 3 "_FX" }{TEXT -1 18 " and the variable \+ " }{TEXT 265 2 "_X" }{TEXT -1 17 " in the template " }{TEXT 265 8 "_X \+ ->_FX" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "simple_unapply := (fx,x) -> subs(\{'_FX'= fx, '_X'=x\},_X->_FX);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/simple_ unapplyGf*6$%#fxG%\"xG6\"6$%)operatorG%&arrowGF)-%%subsG6$<$/.%$_FXG9$ /.%#_XG9%f*6#F7F)F*F)F3F)F)F)F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f := simple_unapply(sqrt(x )+2,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)opera torG%&arrowGF(,&*$-%%sqrtG6#9$\"\"\"F2\"\"#F2F(F(F(" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Diff(f(x), x);\nvalue(%);\nsimple_unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%%DiffG6$,&*$-%%sqrtG6#%\"xG\"\"\"F,\"\"#F,F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*$-%%sqrtG6#%\"xGF%!\"\"#F%\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"xG6\"6$%)operatorG%&arrowGF&,$*&\"\" \"F,*$-%%sqrtG6#9$F,!\"\"#F,\"\"#F&F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 8 "lagrange" }{TEXT -1 11 ": examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }{TEXT 284 43 ".. fitting a quartic curve through 5 points" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 61 "We find the La grange interpolating polynomial for the points " }{XPPEDIT 18 0 "``(1, 5),``(2,2),``(4,1),``(5,3),``(6,3);" "6'-%!G6$\"\"\"\"\"&-F$6$\"\"#F*- F$6$\"\"%F&-F$6$F'\"\"$-F$6$\"\"'F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "xvals := [ 1,2,4,5,6];\nyvals := [5,2,1,3,3];\nlagrange(xvals,yvals,x):\np := una pply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7'\"\"\"\"\"#\" \"%\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7'\"\"&\"\"# \"\"\"\"\"$F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$% )operatorG%&arrowGF(,,*,\"\"&\"\"\",&9$!\"\"\"\"#F/F/,&*&#F/\"\"$F/F1F /F2#\"\"%F7F/F/,&*&#F/F9F/F1F/F2#F.F9F/F/,&*&#F/F.F/F1F/F2#\"\"'F.F/F/ F/*,F3F/,&F1F/F/F2F/,&*&#F/F3F/F1F/F2F3F/F/,&*&#F/F7F/F1F/F2#F.F7F/F/, &*&#F/F9F/F1F/F2#F7F3F/F/F/**,&*&#F/F7F/F1F/F/#F/F7F2F/,&*&#F/F3F/F1F/ F/F/F2F/,&F1F2F.F/F/,&*&#F/F3F/F1F/F2F7F/F/F/*,F7F/,&*&#F/F9F/F1F/F/#F /F9F2F/,&*&FSF/F1F/F/#F3F7F2F/,&F1F/F9F2F/,&F1F2FBF/F/F/*,F7F/,&*&#F/F .F/F1F/F/#F/F.F2F/,&*&FinF/F1F/F/#F/F3F2F/,&*&FWF/F1F/F/F3F2F/,&F1F/F. F2F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "The picture below shows the graph of the interpolating p olynomial along with the interpolation points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "pts := [[1, 5],[2,2],[4,1],[5,3],[6,3]];\nplot([pts,p(x)],x=0..7,y=0..7,style=[poi nt,line],symbol=circle,\n color=[black,coral]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7'7$\"\"\"\"\"&7$\"\"#F*7$\"\"%F'7$F( \"\"$7$\"\"'F." }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 " 6'-%'CURVESG6%7'7$$\"\"\"\"\"!$\"\"&F*7$$\"\"#F*F.7$$\"\"%F*F(7$F+$\" \"$F*7$$\"\"'F*F4-%'COLOURG6&%$RGBGF*F*F*-%&STYLEG6#%&POINTG-F$6%7hn7$ $F*F*$\"3W'*************f!#<7$$\"3+LLLeR+Hw!#>$\"3Zh[%3wr]5'FH7$$\"3gm mm\"z+e_\"!#=$\"3S.9QVjWqhFH7$$\"3;+](oM'f*=#FR$\"3WZV^huA(>'FH7$$\"3s LL3->R`GFR$\"3c-ViH7!z>'FH7$$\"3=+]7G%**)*f$FR$\"33(y2)=xmphFH7$$\"3mm m;apSYVFR$\"3QM7_9258hFH7$$\"3Gnmm;G'y4&FR$\"3ko#=y(Q&*HgFH7$$\"3Onm;z '=$\\eFR$\"3U'Q4G5qG#fFH7$$\"3!RL$3Ft3XtFR$\"3O(f+5=tuk&FH7$$\"3tmmTNj &=t)FR$\"3!z[dk%*)\\I`FH7$$\"33+](=`xn,\"FH$\"31l[9'=*)Q&\\FH7$$\"3#om T&y/Gl6FH$\"3.<$Q][im_%FH7$$\"3++]PurI88FH$\"3Go#f:&R8wSFH7$$\"3aLL$e# 3dl9FH$\"3]=\"*Rm1D+OFH7$$\"3ymm\"Ht%o*f\"FH$\"3kmD**p1]!=$FH7$$\"3K++ ]F_m]FH$\"3%*)p6*[`CsAFH7$$\"3;++]s2 O[?FH$\"31Y79baRq=FH7$$\"3um;aG\"H5=#FH$\"3\">0HUg'eM:FH7$$\"3^LL$ej%y QBFH$\"3K]Ru0)4y<\"FH7$$\"3mLLLVUUsCFH$\"3[vE`>$R?;*FR7$$\"35+](o()yyi #FH$\"3'pAGtj_;j'FR7$$\"3GLLLoD[lFFH$\"3/5j2L2@!)[FR7$$\"3P+](oibk\"HF H$\"3TUvq!=Jy]$FR7$$\"3a+]i!o<-1$FH$\"3%3ea0cPdu#FR7$$\"3qLL3-$=-@$FH$ \"34F())pAGz^#FR7$$\"3kL$3xplzM$FH$\"3j3;r^Wo5GFR7$$\"3gmm\"H([a'\\$FH $\"3M%G\"HpLCYOFR7$$\"3wm;ayo(3l$FH$\"3!)35Fgf7`]FR7$$\"3?+]7VLA&y$FH$ \"3sFisJ4]*o'FR7$$\"3'pm;a?@.$RFH$\"3LVYF)4v5%))FR7$$\"3)******\\\\@-3 %FH$\"3o'yT,Y$HU6FH7$$\"3Q++v$opoA%FH$\"36$=8>r3JU\"FH7$$\"3c+](oMf(oV FH$\"3Ns&*fe'4Qr\"FH7$$\"3#)***\\ii.j_%FH$\"3)eynY#*Q%[?FH7$$\"3%GLL$o T'ym%FH$\"3%[dEj7\\)\\BFH7$$\"3'3++DE5!>[FH$\"3kq4>'Hz2m#FH7$$\"3Mm;a) 3rf&\\FH$\"3ym\\1AY)>#HFH7$$\"3*4++vW0d5&FH$\"3!QPl!*y&>sJFH7$$\"3;L$3 -\"QfY_FH$\"3a>*H!*ef3O$FH7$$\"3C+]PWF'QR&FH$\"3O;)[PpvZ\\$FH7$$\"3'o; /^*Q&eY&FH$\"35f&)>tn*=`$FH7$$\"3[LL$e/Xy`&FH$\"3s)3MS3Pxa$FH7$$\"3emT &)3J@8cFH$\"3q0F`mLGRNFH7$$\"3m**\\(=<\"e)o&FH$\"3K4E]2&GG]$FH7$$\"3?M 3Fu&p6w&FH$\"3JX$RCK;*QMFH7$$\"3%ymmm(zvLeFH$\"3qDwU9>UWLFH7$$\"3-nm\" zAAA)fFH$\"3;p(GT0hf/$FH7$$\"3LM$3-7d%HhFH$\"3['=BHz:Kf#FH7$$\"3#4++]p ]ZE'FH$\"3$3r@o7C6-#FH7$$\"3$QL$e*R7)>kFH$\"3EJ4[q&3&e6FH7$$\"3S+]7=p: *['FH$\"3kQJ>p9$R%pFR7$$\"3'pmmmV,&elFH$\"3?-Ep$****Hy\"FR7$$\"3,M3xcr VKmFH$!3\\r-.3\"z7K%FR7$$\"3<+](o(GP1nFH$!3[Id2/0y26FH7$$\"3#3++]zQrx' FH$!3g^9k]zc==FH7$$\"3g+]78Z!z%oFH$!39529iI,&f#FH7$$\"3I+DccB&R#pFH$!3 [!ol[n(*f]$FH7$$\"\"(F*$!3!H************\\%FH-F:6&F<$\"*++++\"!\")$\") AR!)\\F``lFE-F>6#%%LINEG-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q\"x6\"Q \"yF^al-%%VIEWG6$;FEFh_lFcal" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "Expanding the terms of the Lag range interpolating polynomial and collecting together terms involving the same power of " }{TEXT 277 1 "x" }{TEXT -1 41 " gives the polynom ial in a simpler form. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sort(collect(p(x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*&#\"#6\"$?\"\"\"\"*$)%\"xG\"\"%F(F(!\"\"*&#F &\"#5F(*$)F+\"\"$F(F(F(*&#\"$R%F'F(*$)F+\"\"#F(F(F-*&#\"#L\"#?F(F+F(F( \"\"'F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 \+ " }{TEXT 285 50 ".. constructing the Lagrange \"L\" basis polynomials " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The Lagrange \"L\" basis polynomials for the sequence of \+ " }{TEXT 278 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "x[1] = 1,x[ 2] = 2,x[3] = 3;" "6%/&%\"xG6#\"\"\"F'/&F%6#\"\"#F+/&F%6#\"\"$F/" } {TEXT -1 32 " can be constructed as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "lagrange([1 ,2,3],[1,0,0],x);\nL1 := unapply(%,x):\n'L1(x)'=L1(x);\nlagrange([1,2, 3],[0,1,0],x):\nL2 := unapply(%,x):\n'L2(x)'=L2(x);\nlagrange([1,2,3], [0,0,1],x):\nL3 := unapply(%,x):\n'L3(x)'=L3(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG!\"\"\"\"#\"\"\"F(,&*&F'F&F%F(F&#\"\"$F'F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#L1G6#%\"xG*&,&F'!\"\"\"\"#\"\"\" F,,&*&F+F*F'F,F*#\"\"$F+F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#L2 G6#%\"xG*&,&F'\"\"\"F*!\"\"F*,&F'F+\"\"$F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#L3G6#%\"xG*&,&*&\"\"#!\"\"F'\"\"\"F-#F-F+F,F-,&F'F- F+F,F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The Lagrange interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6#- %\"pG6#%\"xG" }{TEXT -1 15 " for the points" }{XPPEDIT 18 0 "``(1,4),` `(2,1),``(3,2)" "6%-%!G6$\"\"\"\"\"%-F$6$\"\"#F&-F$6$\"\"$F*" }{TEXT -1 56 " can be constructed from the Lagrange basis polynomials." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "4*L1(x)+L2(x)+2*L3(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\" %\"\"\",&%\"xG!\"\"\"\"#F&F&,&*&F*F)F(F&F)#\"\"$F*F&F&F&*&,&F(F&F&F)F& ,&F(F)F.F&F&F&*(F*F&,&*&F*F)F(F&F&#F&F*F)F&,&F(F&F*F)F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {TEXT 0 8 "lagrange" }{TEXT -1 33 " gives the same result directly. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "lagrange([1,2,3],[4,1,2],x);\np := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*(\"\"%\"\"\",&%\"xG!\"\"\"\"#F&F&,&*&F*F)F(F&F) #\"\"$F*F&F&F&*&,&F(F&F&F)F&,&F(F)F.F&F&F&*(F*F&,&*&F*F)F(F&F&#F&F*F)F &,&F(F&F*F)F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6 $%)operatorG%&arrowGF(,(*(\"\"%\"\"\",&9$!\"\"\"\"#F/F/,&*&#F/F3F/F1F/ F2#\"\"$F3F/F/F/*&,&F1F/F/F2F/,&F1F2F8F/F/F/*(F3F/,&*&#F/F3F/F1F/F/#F/ F3F2F/,&F1F/F3F2F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 153 "The following picture shows the graphs of the \+ 3 terms of the Lagrange polynomial obtained by multiplying the Lagrang e basis polynomial by the associated " }{TEXT 279 1 "y" }{TEXT -1 7 " \+ value." }}{PARA 0 "" 0 "" {TEXT -1 147 "Each component graph passes th rough just one of the interplation points, whereas the Lagrange interp olating polynomial passes through all 3 points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "pts := [[1, 4],[2,1],[3,2]]:\nplot([p(x),4*L1(x),L2(x),2*L3(x),pts],x=1..3,style=[ line$4,point],\n color=[red,blue,green,magenta,black],thickness=[2,1$3 ],symbol=circle);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6*-%'CURVESG6&7S7$$\"\"\"\"\"!$\"\"%F*7$$\"3ALLL3VfV5!#<$\"3cd9x&QH ey$F07$$\"3smm\"H[D:3\"F0$\"3(zDYTRlcg$F07$$\"3XLL$e0$=C6F0$\"3EUba$eF *4MF07$$\"3QLL$3RBr;\"F0$\"3B&fg8]V-A$F07$$\"3imm\"zjf)47F0$\"3X&ppc%R yQIF07$$\"3WLLe4;[\\7F0$\"3K9R7*4uq(GF07$$\"3-++Dmy]!H\"F0$\"3ybiX4.D; FF07$$\"3>LLezs$HL\"F0$\"31NgNm!3qb#F07$$\"31++D@1Bv8F0$\"3%=K(*>$HW0C F07$$\"3pmmm@Xt=9F0$\"3=\"\\P4f/qD#F07$$\"3MLL$3y_qX\"F0$\"3g*HoZeID8# F07$$\"3'******\\1!>+:F0$\"3M3%\\s()H%**>F07$$\"3*******\\Z/Na\"F0$\"3 +^+p`=Ft=F07$$\"35+++NfC&e\"F0$\"3M[!oQ$fzeb\"F07$$\"3gmmmEpS1veo)fY\"F07$$\"3%)***\\i`A3v\"F0$\"3ilF0$\"3#>-))*=Z]'4\"F07$$\"3mmm\"zi hl&>F0$\"3_L=Uc@@Z5F07$$\"3KLL$3#G,**>F0$\"33\"37#G\"*)4+\"F07$$\"3_^1,$*Fes7$$\"3'******\\)HxeCF0$\"3T/iJ&=B$\"3%=EF06?J0$F07$FC$\"31HIv2V#*GGF07$FH $\"3*3eS&*[#fFEF07$FM$\"3wbi?VCuDCF07$FR$\"3(=qsnyqSA#F07$FW$\"3y@tu5B @I?F07$Ffn$\"3EC3Fp+FQ=F07$F[o$\"3/m\\$R!yZv;F07$F`o$\"3g3%\\A\")R#*\\ \"F07$Feo$\"3C^+pytwH8F07$Fjo$\"3+[!o))**\\N<\"F07$F_p$\"3e*>KxqFx.\"F 07$Fdp$\"3(4t\"GR--P))Fes7$Fip$\"3a4H&3K*z&f(Fes7$F^q$\"3kbw\"QWP`A'Fe s7$Fcq$\"3Kwh'Go%3y]Fes7$Fhq$\"3Ave#Q)o\\!*QFes7$F]r$\"3Cp*4dy5)GGFes7 $Fbr$\"3S_N@)*[*Hz\"Fes7$Fgr$\"3k\"p;0&G0l!*!#>7$F\\s$\"3ervuytIw>!#?7 $Fas$!3-qZ&)3:))\\#)F``l7$Fgs$!39\\[\"G]Fq\\\"Fes7$F\\t$!3_B[d)f\\m:#F es7$Fat$!33,zpXt!fw#Fes7$Fft$!3AU#)H()e%4H$Fes7$F[u$!335(*fxf5KPFes7$F `u$!3UuN\")4i'\\Fes7$Fcw$!3clxyxlFFes7$Ffy$!3&3G7jf5Y?#Fes7$F[z$!3$za+)4q5P:Fes7$F`z$!3 8*obW+yMJ)F``l7$FezF`[l-Fjz6&F\\[lF`[lF`[lF][lFa[l-Ff[l6#F)-F$6&7S7$F( F`[l7$F.$\"3!4ZgZH:)G&)F``l7$F4$\"3Mx`o9c/k:Fes7$F9$\"3\"4i0J!oWHBFes7 $F>$\"3%pNI!Rb;jIFes7$FC$\"3)z=o?3#ycPFes7$FH$\"3qgPYc9AnVFes7$FM$\"3_ @(o974i'\\Fes7$FR$\"3kdJIqKF]bFes7$FW$\"3e\"RjPBKm4'Fes7$Ffn$\"3y6#zpV /8i'Fes7$F[o$\"3WO=*><$3_qFes7$F`o$\"3MdHv)G+>](Fes7$Feo$\"3%Ru\\lN=h \"zFes7$Fjo$\"3#*e(fcl!zz#)Fes7$F_p$\"3')oc]l'\\)z&)Fes7$Fdp$\"3ooCpj! 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"6#7$\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "dist := evalf(Pi/2):\nh := dist/6:\nxvals := [seq(h*(i-1),i=1..7) ];\nyvals := [seq(sin(xvals[i]),i=1..7)];\nx := 'x':\nlagrange_interp( xvals,yvals,x):\np := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%&xvalsG7)$\"\"!F'$\"+yQ*zh#!#5$\"+cx)fB&F*$\"+M;)R&yF*$\"+^v>Z5!\"* $\"+Rp**38F1$\"+Fjzq:F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7) $\"\"!F'$\"+^/>)e#!#5$\"+++++]F*$\"+7y1rqF*$\"+PSDg')F*$\"+j#e#f'*F*$ \"\"\"F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)oper atorG%&arrowGF(,.*0$\"+%Hfh))*!#5\"\"\"9$F1,&*&$\"+M'=(>Q!\"*F1F2F1!\" \"$\"+++++?F7F1F1,&*&$\"+<$f)4>F7F1F2F1F8$\"+++++:F7F1F1,&*&$\"+X&RKF \"F7F1F2F1F8$\"+LLLL8F7F1F1,&*&$\"+(e'H\\&*F0F1F2F1F8$\"++++]7F7F1F1,& *&$\"+osVRwF0F1F2F1F8$\"+++++7F7F1F1F1*0$\"+&e'H\\&*F0F1F2F1,&*&$\"+M' =(>QF7F1F2F1F1$\"+**********F0F8F1,&*&$\"+M'=(>QF7F1F2F1F8$\"+++++IF7F 1F1,&*&$\"+=$f)4>F7F1F2F1F8$\"+,+++?F7F1F1,&*&$\"+X&RKF\"F7F1F2F1F8$\" +nmmm;F7F1F1,&*&$\"+(e'H\\&*F0F1F2F1F8F?F1F1F1*0$\"+kJ;.!*F0F1F2F1,&*& $\"+<$f)4>F7F1F2F1F1$\"+++++]F0F8F1,&*&FXF1F2F1F1$\"+++++?F7F8F1,&*&$ \"+P'=(>QF7F1F2F1F8$\"+-+++SF7F1F1,&*&$\"+<$f)4>F7F1F2F1F8$\"+++++DF7F 1F1,&*&$\"+W&RKF\"F7F1F2F1F8$\"+********>F7F1F1F1*0$\"+KM$*p#)F0F1F2F1 ,&*&$\"+X&RKF\"F7F1F2F1F1$\"+MLLLLF0F8F1,&*&$\"+=$f)4>F7F1F2F1F1$\"+++ ++5F7F8F1,&*&$\"+P'=(>QF7F1F2F1F1$\"+-+++IF7F8F1,&*&$\"+J'=(>QF7F1F2F1 F8$\"+'*******\\F7F1F1,&*&$\"+;$f)4>F7F1F2F1F8$\"+********HF7F1F1F1*0$ \"+c(H\"ztF0F1F2F1,&*&$\"+(e'H\\&*F0F1F2F1F1$\"+++++DF0F8F1,&*&F`rF1F2 F1F1$\"+ommmmF0F8F1,&*&FapF1F2F1F1$\"+++++:F7F8F1,&*&$\"+J'=(>QF7F1F2F 1F1$\"+'*******RF7F8F1,&*&$\"+J'=(>QF7F1F2F1F8$\"+'*******fF7F1F1F1*0$ \"+Bx>mjF0F1F2F1,&*&$\"+osVRwF0F1F2F1F1$\"+++++?F0F8F1,&*&FatF1F2F1F1$ \"+,+++]F0F8F1,&*&$\"+W&RKF\"F7F1F2F1F1$\"+%*********F0F8F1,&*&$\"+;$f )4>F7F1F2F1F1$\"+)*******>F7F8F1,&*&F_uF1F2F1F1$\"+'*******\\F7F8F1F1F (F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "Here is a numerical example which compares the value of the inter polating polynomial with the value given by the sine function.\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "p(1);\nevalf(sin(1));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3!4ZT)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[)4ZT)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 54 "Now we plot the graph of the interpolatin g polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 37 " \+ along with the interpolation points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "pts := [seq([xvals[i],yva ls[i]],i=1..7)]:\nplot([p(x),pts],x=0..Pi/2,y,style=[line,point],symbo l=circle,\n color=[coral,black]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)F(7$$\"3N GK5j*))QU$!#>$\"3U\\U%fU0LU$F-7$$\"3DXXYUk*HS'F-$\"3+&H:`:Q()R'F-7$$\" 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Q31Ilc\"Fi_l$!3a60>B9x%=\"F07$$\"3+++lBjzq:Fi_l$!3@Pt&>cC=J)!#F-%'COLO URG6&%$RGBGF(F($\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!F`^m-%%VIEWG6$;F ($\"+Fjzq:!\"*%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note " }{TEXT -1 64 ": If the interpolating polynomial is to be arranged in the form " }{XPPEDIT 18 0 "a[n]*x^n+a[n-1]*x^(n-1)+` . . . `+a[1]*x+a [0];" "6#,,*&&%\"aG6#%\"nG\"\"\")%\"xGF(F)F)*&&F&6#,&F(F)F)!\"\"F))F+, &F(F)F)F0F)F)%(~.~.~.~GF)*&&F&6#F)F)F+F)F)&F&6#\"\"!F)" }{TEXT -1 24 " by using the procedure " }{TEXT 0 7 "collect" }{TEXT -1 32 ", it is p robably a good idea to " }{TEXT 260 22 "increase the precision" } {TEXT -1 22 " for the computation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 226 "Digits:= 20:\ndist := eval f(Pi/2):\nh := dist/6:\nxvals := [seq(h*(i-1),i=1..7)];\nyvals := [seq (sin(xvals[i]),i=1..7)];\nx := 'x':\npx := collect(lagrange_interp(xva ls,yvals,x),x):\nDigits := 10:\npx := evalf(px):\np := unapply(px,x); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7)$\"\"!F'$\"5`O%\\\"*z(Q *zh#!#?$\"51t))H)fv()fB&F*$\"5f4$[uRj\")R&yF*$\"5hu(f'>^v>Z5!#>$\"5E=Z d**Qp**38F1$\"5#>m*[zEjzq:F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&yva lsG7)$\"\"!F'$\"5Mi2_-^/>)e#!#?$\"5******************\\F*$\"5QCva'=\"y 1rqF*$\"5uY'QWy.a-m)F*$\"5t'Go!*GEe#f'*F*$\"5+++++++++5!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,.* &$\"+,v)Rl*!#8\"\"\")9$\"\"'F1!\"\"*&$\"+Q0'3.\"!#6F1)F3\"\"&F1F1*&$\" +#3:/4#!#7F1)F3\"\"%F1F5*&$\"+_Z'[l\"!#5F1)F3\"\"$F1F5*&$\"+K)>PI$F0F1 )F3\"\"#F1F5*&$\"+c\\.+5!\"*F1F3F1F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(p(x)-sin(x),x =0..Pi/2,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 482 250 250 {PLOTDATA 2 "6&-%'CURVESG6#7hq7$$\"\"!F)F(7$$\"3WN!zQ?h)zU!#?$\"3l9eBt :ZO9!#C7$$\"3)32exSA(f&)F-$\"3mZe$Q?ntv#F07$$\"3s5P;h$eRG\"!#>$\"3+ldQ %*)*)z'RF07$$\"3=9;b\"[W>r\"F9$\"3I5!zC.\"[t]F07$$\"3W@uKAn\"zc#F9$\"3 cXaOre%)))pF07$$\"3NGK5j*))QU$F9$\"3r()z)=z%[T&)F07$$\"3X')Qy-FW8\\F9$ \"3q\"QEfdSsF9$\"3/?XB%RUL>\"FP7$$\"3+<&R$ocOf wF9$\"3m')zgPCj-7FP7$$\"3!4%yH5a:y!)F9$\"3%=mgiuev?\"FP7$$\"3#[;cA:Xp \\)F9$\"3?YQIm8S37FP7$$\"3t)[9U*[t:*)F9$\"3E9:UI,V07FP7$$\"3E6G\"FP7$$\"3=N68yVJ`(*F9$\"3Maa$Rwv!*=\"FP7$$\"3Di))pQx &R9\"!#=$\"3#fM]kw837\"FP7$$\"3%3h%eRSe78F[q$\"3F*zDpVKp,\"FP7$$\"3ubt 4*))3/[\"F[q$\"3:v0)o,&*4!*)F07$$\"3k+,hQPB[;F[q$\"3A4,-J'Gy\\(F07$$\" 3!=^%=d)GQ!=F[q$\"3='3eP$=cZhF07$$\"3'H#*ed(RUf>F[q$\"3,\\$f2))oS![F07 $$\"3IESaq%HF[q$!3 ch%>:8:Lb\"F07$$\"3on$***=Q*y6$F[q$!3zNQkWM85@F07$$\"3)pq'4OKt)G$F[q$! 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" }}{PARA 0 "" 0 "" {TEXT -1 113 " (b) Use the Lagrange \"L\" basis polynomials found in part (a) to con struct the Lagrange interpolating polynomial " }{XPPEDIT 18 0 "p(x)" " 6#-%\"pG6#%\"xG" }{TEXT -1 15 " for the points" }{XPPEDIT 18 0 "``(0,3 ),``(2,-1),``(3,2),``(6,1);" "6&-%!G6$\"\"!\"\"$-F$6$\"\"#,$\"\"\"!\" \"-F$6$F'F*-F$6$\"\"'F," }{TEXT -1 24 ", and plot the graph of " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 35 " along with the f our given points. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 295 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 " (a) " }}{PARA 0 "" 0 "" {TEXT -1 40 "The four Lagrange \"L\" basis pol ynomials " }{XPPEDIT 18 0 "L[1](x)" "6#-&%\"LG6#\"\"\"6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "L[2](x)" "6#-&%\"LG6#\"\"#6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "L[3](x)" "6#-&%\"LG6#\"\"$6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "L[4](x)" "6#-&%\"LG6#\"\"%6#%\"xG" }{TEXT -1 18 " associated with " }{XPPEDIT 18 0 "x[1]=0" "6#/&%\"xG6#\"\"\" \"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[2]=2" "6#/&%\"xG6#\"\"#F'" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "x[3]=3" "6#/&%\"xG6#\"\"$F'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[4]=6" "6#/&%\"xG6#\"\"%\"\"'" }{TEXT -1 46 " respectively, can be constructed as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 273 "lagrange([0 ,2,3,6],[1,0,0,0],x):\nL1 := unapply(%,x): \n'L1(x)'=L1(x);\nlagrange( [0,2,3,6],[0,1,0,0],x):\nL2 := unapply(%,x):\n'L2(x)'=L2(x);\nlagrange ([0,2,3,6],[0,0,1,0],x):\nL3 := unapply(%,x):\n'L3(x)'=L3(x);\nlagrang e([0,2,3,6],[0,0,0,1],x):\nL4 := unapply(%,x):\n'L4(x)'=L4(x);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#L1G6#%\"xG*(,&*&\"\"#!\"\"F'\"\"\" F,F-F-F-,&*&\"\"$F,F'F-F,F-F-F-,&*&\"\"'F,F'F-F,F-F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#L2G6#%\"xG,$**\"\"#!\"\"F'\"\"\",&F'F+\"\"$F,F ,,&*&\"\"%F+F'F,F+#F.F*F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#L 3G6#%\"xG,$**\"\"$!\"\"F'\"\"\",&F'F,\"\"#F+F,,&*&F*F+F'F,F+F.F,F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#L4G6#%\"xG,$**\"\"'!\"\"F'\"\"\" ,&*&\"\"%F+F'F,F,#F,\"\"#F+F,,&*&\"\"$F+F'F,F,F,F+F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "The following pictu re shows the graphs of the four Lagrange \"L\" basis polynomials toget her with the four points" }{XPPEDIT 18 0 " ``(0,1),``(2,1),``(3,1),``( 6,1)" "6&-%!G6$\"\"!\"\"\"-F$6$\"\"#F'-F$6$\"\"$F'-F$6$\"\"'F'" } {TEXT -1 73 " to indicate where the each of the polynomial functions h as the value 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 161 "pts := [[0,1],[2,1],[3,1],[6,1]]:\nplot([L1(x ),L2(x),L3(x),L4(x),pts],x=-1..7,y=-2..2,color=[red,blue,green,magenta ,black],\n style=[line$4,point],symbol=circle);" }}{PARA 13 "" 1 "" {GLPLOT2D 682 255 255 {PLOTDATA 2 "6*-%'CURVESG6%7U7$$!\"\"\"\"!$\"3[L LLLLLLB!#<7$$!3PLLLLQ6G\"*!#=$\"3;X>7K\\`)=#F-7$$!3ummmmwAc#)F1$\"3KLZ $)='Q&\\?F-7$$!3)******\\F-7$$!3ALLL$o!)*QnF1$ \"3CA$*eSQ;@=F-7$$!3mmmmmxnK]F1$\"35Py]i\"*>%e\"F-7$$!3#ommmOk]J$F1$\" 37\"fb:\"z4m8F-7$$!37MLL$[9cg\"F1$\"37plyVObo6F-7$$!3QBnmm;ct?!#?$\"3, 2()pv[2-5F-7$$\"3v)*****\\YJ?;F1$\"3IF>5N[se%)F17$$\"3/KLL$=\"\\SaT\\T\"F17$$\"3ILLL=Zg#\\\"F-$\"3EbLsjuFw&*!#>7$$\"3 cmmmEn*Gn\"F-$\"3R\\zSdKw<_F_q7$$\"3Tmmm1xiD=F-$\"3_A2Y*z)\\uBF_q7$$\" 3!)*****\\9!H.?F-$!3'e#)3Ew\"pSO!#@7$$\"3Immm1:bg@F-$!3NF)Rua!RP9F_q7$ $\"3<+++X@4LBF-$!39v;hrQqiAF_q7$$\"31+++N;R(\\#F-$!3.\"=W]s+BV#F_q7$$ \"3wmmm;4#)oEF-$!3'4p:&>ef\\?F_q7$$\"3jmmm6lCEGF-$!3;K[b%p]cE\"F_q7$$ \"3ELLL$G^g*HF-$!37:-V&p<>G$F_r7$$\"3oKLL=2VsJF-$\"3+BtCf8'ye\"F_q7$$ \"3f*****\\`pfK$F-$\"3gd57CG^5KF_q7$$\"3!HLLLm&z\"\\$F-$\"3;$>AFgr:6&F _q7$$\"3s******z-6jOF-$\"3=woXdy$)erF_q7$$\"3<******4#32$QF-$\"3IfTU(G eR;*F_q7$$\"3O*****\\#y'G*RF-$\"3Ca=-VC<.6F17$$\"3G******H%=H<%F-$\"3' zWmbq+NH\"F17$$\"35mmm1>qMVF-$\"3O)3@Ylo9W\"F17$$\"3%)*******HSu]%F-$ \"3Qe'))*H;6n:F17$$\"3'HLL$ep'Rm%F-$\"3gsAyuX3X;F17$$\"3')******R>4N[F -$\"3Azb#3*Q]$o\"F17$$\"3#emm;@2h*\\F-$\"3#\\%))ev6tn;F17$$\"3]*****\\ c9W;&F-$\"3)4JP`lD(*e\"F17$$\"3Lmmmmd'*G`F-$\"3ux\"f\"*fb^W\"F17$$\"3j *****\\iN7]&F-$\"3S[9)p(=I87F17$$\"3aLLLt>:ncF-$\"3_Aa7#yeJ/*F_q7$$\"3 5LLL.a#o$eF-$\"39:+_()\\\\L\\F_q7$$\"3ammm^Q40gF-$!3o12:S&**Hq\"FU7$$ \"3y******z]rfhF-$!3dv+tB'f6$eF_q7$$\"3gmmmc%GpL'F-$!35j1!fEaWN\"F17$$ \"3/LLL8-V&\\'F-$!3C\"y\\I#*yC;#F17$$\"3=+++XhUkmF-$!3+))GGQ)HY:$F17$$ \"3=+++:oV l?F-7$FD$!3?9v-?TAL9F-7$FI$!3JLI7,*p2u)F17$FN$!3[n$ohVJy!RF17$FS$!3[GX oz>NqYFU7$FY$\"3GWkmodmbLF17$Fhn$\"3-X8Pzy%=F'F17$F]o$\"3[BY**3[)\\g)F 17$Fbo$\"3r3WVy$fX/\"F-7$Fgo$\"3ax'o4*=\"G;\"F-7$F\\p$\"3]x%eSm%G]7F-7 $Fap$\"3)*\\#3mu.KH\"F-7$Ffp$\"3%)>r#zBAcH\"F-7$F[q$\"3a\"=!=4Tnn7F-7$ Faq$\"3^a'RH))G3?\"F-7$Ffq$\"3+;:'[M9(=6F-7$F[r$\"3cWq1jLGv**F17$Far$ \"3\"\\$puyQP/()F17$Ffr$\"3\"R+xB(3%>8(F17$F[s$\"3)R\")*=&)fj&\\&F17$F `s$\"3wEe(o?g.o$F17$Fes$\"3Az'>=mp\"[>F17$Fjs$\"3xr>x`&)HUWFU7$F_t$!3i <*)fF17$Fdt$!3j$=%)*Q>'Qi$F17$Fit$!3k,8/gs,%Q&F17$F^u$!3G(R^s-;b4 (F17$Fcu$!3=ft$*yC!*G')F17$Fhu$!3mmxH'Q=j%**F17$F]v$!3P&[Y\"H(Hy6\"F-7 $Fbv$!3\"4ek'H+L/7F-7$Fgv$!3%Q()yRR'on7F-7$F\\w$!3O^dznn1'H\"F-7$Faw$! 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3i9K)oj%H)>(FU7$F[s$!3afY6WFEr')FU7$F`s$!3l(32/T7.@)FU7$Fes$!3TO,niwMN cFU7$Fjs$!3R%o7%QZkO;F_r7$F_t$\"3l+-i$y:w!*)FU7$Fdt$\"3)[wtT>Cm*>F_q7$ Fit$\"3g#Gi.=O!eNF_q7$F^u$\"3(R()=mw$z5cF_q7$Fcu$\"3Kmi'4pB74)F_q7$Fhu $\"3)GE*e(R#H(4\"F17$F]v$\"3)R\\6%3(HrZ\"F17$Fbv$\"3gs8'zdTg(=F17$Fgv$ \"3!\\uDnJ\"HmBF17$F\\w$\"3ww3CV&=9(GF17$Faw$\"3yyYYhLz$\\$F17$Ffw$\"3 g\">%)*)3F*\\TF17$F[x$\"3[D\"=3lCF\"\\F17$F`x$\"3$)3q*y&\\HQdF17$Fex$ \"3mcN[D^>\"p'F17$Fjx$\"3c=90Yb2DQ+\"F-7$Fiy$\"3Tn-Nq'[W7\"F-7$F^z$\"3gqlUyVst7F-7$Fcz$\"3W#*Q&>;zv T\"F-7$Fhz$\"3i2TM=>5#e\"F-7$F][l$\"3C[NfGjq]F- -Fg[l6&Fi[lFj[lF]\\lFj[lF^\\l-F$6%7&7$F]\\l$\"\"\"F*7$$\"\"#F*F_[n7$$ \"\"$F*F_[n7$$\"\"'F*F_[n-Fg[l6&Fi[lF*F*F*-F_\\l6#%&POINTG-%+AXESLABEL SG6$Q\"x6\"Q\"yFc\\n-%'SYMBOLG6#%'CIRCLEG-%%VIEWG6$;F(Fb[l;$!\"#F*Fb[n " 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" {TEXT -1 38 "T he Lagrange interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG 6#%\"xG" }{TEXT -1 15 " for the points" }{XPPEDIT 18 0 "``(0,3),``(2,- 1),``(3,2),``(6,1);" "6&-%!G6$\"\"!\"\"$-F$6$\"\"#,$\"\"\"!\"\"-F$6$F' F*-F$6$\"\"'F," }{TEXT -1 4 " is " }{XPPEDIT 18 0 "p(x)=3*L[1](x)-L[2] (x)+2*L[3](x)+L[4](x)" "6#/-%\"pG6#%\"xG,**&\"\"$\"\"\"-&%\"LG6#F+6#F' F+F+-&F.6#\"\"#6#F'!\"\"*&F4F+-&F.6#F*6#F'F+F+-&F.6#\"\"%6#F'F+" } {TEXT -1 81 " and is obtained by multiplying each Lagrange basis polyn omial by the associated " }{TEXT 296 1 "y" }{TEXT -1 38 " value and ad ding the resulting terms." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "3*L1(x)-L2(x)+2*L3(x)+L4(x);\np := \+ unapply(%,x):\n'p(x)'=p(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,***\" \"$\"\"\",&*&\"\"#!\"\"%\"xGF&F*F&F&F&,&*&F%F*F+F&F*F&F&F&,&*&\"\"'F*F +F&F*F&F&F&F&**F)F*F+F&,&F+F*F%F&F&,&*&\"\"%F*F+F&F*#F%F)F&F&F**,F)F&F %F*F+F&,&F+F&F)F*F&,&*&F%F*F+F&F*F)F&F&F&**F0F*F+F&,&*&F5F*F+F&F&#F&F) F*F&,&*&F%F*F+F&F&F&F*F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6 #%\"xG,***\"\"$\"\"\",&*&\"\"#!\"\"F'F+F/F+F+F+,&*&F*F/F'F+F/F+F+F+,&* &\"\"'F/F'F+F/F+F+F+F+**F.F/F'F+,&F'F/F*F+F+,&*&\"\"%F/F'F+F/#F*F.F+F+ F/*,F.F+F*F/F'F+,&F'F+F.F/F+,&*&F*F/F'F+F/F.F+F+F+**F4F/F'F+,&*&F9F/F' F+F+#F+F.F/F+,&*&F*F/F'F+F+F+F/F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 16 ": The procedure " } {TEXT 0 8 "lagrange" }{TEXT -1 33 " gives the same result directly. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "lagrange([0,2,3,6],[3,-1,2,1],x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,***\"\"$\"\"\",&*&\"\"#!\"\"%\"xGF&F*F&F&F&,&*&F%F*F+F&F*F&F&F& ,&*&\"\"'F*F+F&F*F&F&F&F&**F)F*F+F&,&F+F*F%F&F&,&*&\"\"%F*F+F&F*#F%F)F &F&F**,F)F&F%F*F+F&,&F+F&F)F*F&,&*&F%F*F+F&F*F)F&F&F&**F0F*F+F&,&*&F5F *F+F&F&#F&F)F*F&,&*&F%F*F+F&F&F&F*F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "The following picture shows the gra phs of the Lagrange polynomial together with the given interpoloation \+ points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "pts := [[0,3],[2,-1],[3,2],[6,1]]:\nplot([p(x),pts], x=-1..7,style=[line,point],color=[red,black],symbol=circle);" }}{PARA 13 "" 1 "" {GLPLOT2D 581 348 348 {PLOTDATA 2 "6'-%'CURVESG6%7W7$$!\"\" \"\"!$\"#:F*7$$!3PLLLLQ6G\"*!#=$\"3)HXl;_&=f8!#;7$$!3ummmmwAc#)F0$\"3P iWS*>2eA\"F37$$!3)******\\7$$\"3qnmmm3Q\\nF0$!3yVe. q&R%oqF07$$\"3aLLLL76#G)F0$!3S-n+6+6_6FH7$$\"3&)******f-w+5FH$!3%[#\\> +B?,:FH7$$\"3%*********y,u6FH$!32L3Ob%=?q\"FH7$$\"3)*******RP)4M\"FH$! 3_*R!4t`tl-D: #H1O))F07$$\"3jmmm6lCEGFH$\"3-=7FH7$$\"3oKLL=2VsJFH$\"3AG&)3L@+(e#FH7$$\"3f*****\\`pfK$FH$\"3#ys%*z Y(H*4$FH7$$\"3!HLLLm&z\"\\$FH$\"3\")*)fBK0uIOFH7$$\"3s******z-6jOFH$\" 3F*z$3\\_8WTFH7$$\"3<******4#32$QFH$\"3m'yX'*3)R*f%FH7$$\"3O*****\\#y' G*RFH$\"3Y*eK/_$[%)\\FH7$$\"3G******H%=H<%FH$\"32_\\:(GE^L&FH7$$\"35mm m1>qMVFH$\"3!RrfmzK&pbFH7$$\"3%)*******HSu]%FH$\"398HE879BdFH7$$\"3'HL L$ep'Rm%FH$\"3g%RTl!pcldFH7$$\"3')******R>4N[FH$\"3U**3O\\j)\\p&FH7$$ \"3#emm;@2h*\\FH$\"3v\\Yg@d71bFH7$$\"3]*****\\c9W;&FH$\"33&=>&*pW-<&FH 7$$\"3Lmmmmd'*G`FH$\"3gm@ETzv$p%FH7$$\"3j*****\\iN7]&FH$\"3V*)y\\EO\"e -%FH7$$\"3aLLLt>:ncFH$\"3#Q7h@0?s?$FH7$$\"35LLL.a#o$eFH$\"3Y%y&=\")R;! =#FH7$$\"3ammm^Q40gFH$\"3<*\\[S/3+g*F07$$\"3y******z]rfhFH$!39DoveUd%[ $F07$$\"3gmmmc%GpL'FH$!3Aoj;!o743#FH7$$\"3/LLL8-V&\\'FH$!394()e-*z>&QF H7$$\"3=+++XhUkmFH$!3smRs'eqB)fFH7$$\"3=+++![,`u'FH$!3S-dC$z@P4(FH7$$ \"3=+++:o " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }{TEXT 286 80 ".. approximating a continuous function using a Lagrange interpolat ing polynomial" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 288 8 "Questio n" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 45 "This question is co ncerned with the function " }{XPPEDIT 18 0 "f(x) = Int(arctan(t)^2,t = 0 .. x);" "6#/-%\"fG6#%\"xG-%$IntG6$*$-%'arctanG6#%\"tG\"\"#/F/;\"\"! F'" }{TEXT -1 74 ", which can be evaluated by numerical integration vi a the Maple procedure " }{TEXT 0 5 "evalf" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) Find the degree 8 " }{TEXT 265 33 "Lagrange i nterpolating polynomial" }{TEXT -1 1 " " }{XPPEDIT 18 0 "p(x)" "6#-%\" pG6#%\"xG" }{TEXT -1 33 " which agrees with the values of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 21 " at 9 equally spaced " } {TEXT 289 1 "x" }{TEXT -1 35 " values between 0 and 1 inclusive. " }} {PARA 0 "" 0 "" {TEXT -1 49 "(b) Plot a graph of the interpolating pol ynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 32 " found in (a) over the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" } {XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 64 " along with the int erpolation points used for its construction. " }}{PARA 0 "" 0 "" {TEXT -1 84 "(c) Plot a graph which shows the (absolute) error when th e interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 37 " found in (a) is used to approximate " }{XPPEDIT 18 0 "f( x)" "6#-%\"fG6#%\"xG" }{TEXT -1 18 " on the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 17 "(d) Estimate the " }{TEXT 265 22 "maximum absolute error" }{TEXT -1 35 " when the interpolating poly nomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 37 " found \+ in (a) is used to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 18 " on the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\" xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 290 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 46 ": Since w e are specifically asked to find the " }{TEXT 260 8 "Lagrange" }{TEXT -1 37 " interpolating polynomial, we should " }{TEXT 260 7 "not use" } {TEXT -1 1 " " }{TEXT 0 7 "collect" }{TEXT -1 40 " to arrange the poly nomial in the form " }{XPPEDIT 18 0 "a[8]*x^8+a[7]*x^7+` . . . `+a[1] *x+a[0];" "6#,,*&&%\"aG6#\"\")\"\"\"*$%\"xGF(F)F)*&&F&6#\"\"(F)*$F+F/F )F)%(~.~.~.~GF)*&&F&6#F)F)F+F)F)&F&6#\"\"!F)" }{TEXT -1 3 " . " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "f := x -> Int(arctan(t)^2,t =0..x);\nleft := 0;\nright := 1;\ndeg := 8;\nh := evalf(abs(right-left )/deg);\nxvals := [seq(left+h*i,i=0..deg)];\nyvals := [seq(evalf(f(xva ls[i+1])),i=0..deg)];\nlagrange_interp(xvals,yvals,x):\np := unapply(% ,x):\n'p(x)'=p(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(-%$IntG6$*$)-%'arctanG6#%\"tG\"\"#\"\"\"/F4; \"\"!9$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%leftG\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rightG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$degG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$ \"++++]7!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7+$\"\"!F'$\"+ +++]7!#5$\"+++++DF*$\"++++]PF*$\"+++++]F*$\"++++]iF*$\"+++++vF*$\"++++ ]()F*$\"+++++5!\"*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7+$\"\" !F'$\"+)Hr+Z'!#8$\"+D8T#3&!#7$\"+dy#fm\"!#6$\"+O**G*z$F0$\"+)oj]3(F0$ \"+mQ2i6!#5$\"+O`&Ru\"F7$\"+N?\"GX#F7" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,2*4$\"+Qq0w^!#7\"\"\"F'F-,&*&$\"+++++!)!\"*F-F'F-!\" \"$\"+++++?F2F-F-,&*&$\"+++++SF2F-F'F-F3$\"+++++:F2F-F-,&*&$\"+nmmmEF2 F-F'F-F3$\"+MLLL8F2F-F-,&*&$\"+++++?F2F-F'F-F3$\"++++]7F2F-F-,&*&$\"++ +++;F2F-F'F-F3$\"+++++7F2F-F-,&*&$\"+LLLL8F2F-F'F-F3$\"+mmmm6F2F-F-,&* &$\"+Vr&G9\"F2F-F'F-F3$\"+Vr&G9\"F2F-F-F-*4$\"+IX'H.#!#6F-F'F-,&*&$\"+ ++++!)F2F-F'F-F-$\"+++++5F2F3F-,&*&$\"+++++!)F2F-F'F-F3$\"+++++IF2F-F- ,&*&$\"+++++SF2F-F'F-F3F4F-F-,&*&$\"+nmmmEF2F-F'F-F3$\"+nmmm;F2F-F-,&* &$\"+++++?F2F-F'F-F3F:F-F-,&*&$\"+++++;F2F-F'F-F3$\"+++++9F2F-F-,&*&$ \"+LLLL8F2F-F'F-F3$\"+LLLL8F2F-F-F-*4$\"+'GuCW%FgnF-F'F-,&*&$\"+++++SF 2F-F'F-F-$\"+++++]!#5F3F-,&*&FjnF-F'F-F-$\"+++++?F2F3F-,&*&$\"+++++!)F 2F-F'F-F3FcqF-F-,&*&$\"+++++SF2F-F'F-F3$\"+++++DF2F-F-,&*&$\"+nmmmEF2F -F'F-F3F4F-F-,&*&$\"+++++?F2F-F'F-F3$\"++++]5O8\"FgqF-F'F-,&*&F4F-F'F-F-$\"+++++DFgqF3F-,&*&F[tF-F'F-F-$\"+ommm mFgqF3F-,&*&FcqF-F'F-F-$\"+++++:F2F3F-,&*&FjnF-F'F-F-$\"+++++SF2F3F-,& *&$\"+++++!)F2F-F'F-F3$\"+++++gF2F-F-,&*&$\"+++++SF2F-F'F-F3$\"+++++NF 2F-F-,&*&$\"+nmmmEF2F-F'F-F3F[tF-F-F-*4$\"+@=V\\:FgqF-F'F-,&*&FdsF-F'F -F-$\"+++++?FgqF3F-,&*&F4F-F'F-F-$\"+++++]FgqF3F-,&*&F[tF-F'F-F-$F]oF2 F3F-,&*&FcqF-F'F-F-$\"+++++?F2F3F-,&*&FjnF-F'F-F-$\"+++++]F2F3F-,&*&$ \"+++++!)F2F-F'F-F3$\"+++++qF2F-F-,&*&$\"+++++SF2F-F'F-F3FcqF-F-F-*4$ \"+8=4$*>FgqF-F'F-,&*&F\\qF-F'F-F-$\"+mmmm;FgqF3F-,&*&FdsF-F'F-F-$\"++ +++SFgqF3F-,&*&F4F-F'F-F-$\"+++++vFgqF3F-,&*&F[tF-F'F-F-$\"+MLLL8F2F3F -,&*&FcqF-F'F-F-$\"+++++DF2F3F-,&*&FjnF-F'F-F-$\"+++++gF2F3F-,&*&$\"++ +++!)F2F-F'F-F3FjnF-F-F-*4$\"+N?\"GX#FgqF-F'F-,&*&FXF-F'F-F-$\"+H9dG9F gqF3F-,&*&F\\qF-F'F-F-$\"+KLLLLFgqF3F-,&*&FdsF-F'F-F-$\"+++++gFgqF3F-, &*&F4F-F'F-F-$F]oF2F3F-,&*&F[tF-F'F-F-$\"+nmmm;F2F3F-,&*&FcqF-F'F-F-$ \"+++++IF2F3F-,&*&FjnF-F'F-F-$\"+++++qF2F3F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 136 "pts := [seq([xvals[i+1],yvals[i+1]],i=0..deg) ]:\nplot([p(x),pts],x=0..1,y,style=[line,point],symbol=circle,\n \+ color=[coral,black]);" }}{PARA 13 "" 1 "" {GLPLOT2D 435 348 348 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$\"\"!F)F(7$$\"3emmm;arz@!#>$\"3D>')oq9 w,H!#B7$$\"3[LL$e9ui2%F-$\"3qfxe#\\?K>#!#A7$$\"3nmmm\"z_\"4iF-$\"3bA^b RVJ;zF67$$\"3[mmmT&phN)F-$\"3fIDZ)H\"QO>!#@7$$\"3CLLe*=)H\\5!#=$\"3%3Q +lD\")G$QFA7$$\"3gmm\"z/3uC\"FE$\"3AJV,My0IkFA7$$\"3%)***\\7LRDX\"FE$ \"3uG%))ya4J,\"!#?7$$\"3]mm\"zR'ok;FE$\"3VmD\\<*H5_\"FR7$$\"3w***\\i5` h(=FE$\"33@>=C#45<#FR7$$\"3WLLL3En$4#FE$\"3,k7D5.'o+$FR7$$\"3qmm;/RE&G #FE$\"3?[#=rRTu*QFR7$$\"3\")*****\\K]4]#FE$\"3%=aM;g;\")3&FR7$$\"3$*** ***\\PAvr#FE$\"37Xx'yXq&*\\'FR7$$\"3)******\\nHi#HFE$\"3Q)p())z>()y!)F R7$$\"3jmm\"z*ev:JFE$\"3O#)y-M%o0r*FR7$$\"3?LLL347TLFE$\"37xF4\"F -7$$\"3,LLLLY.KNFE$\"3B!45\\J:,S\"F-7$$\"3w***\\7o7Tv$FE$\"3'QyY1!pAr; 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "numapprox[infnorm](p(x)-f(x),x=0..1 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*]S+L'!#:" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The maximum absolute er ror when " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 24 " is u sed to approximate " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 18 " on the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" } {XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 16 " is about 6.3e-7" } {XPPEDIT 18 0 "`` = 6.3;" "6#/%!G-%&FloatG6$\"#j!\"\"" }{TEXT -1 1 " \+ " }{TEXT 291 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-7)" "6#)\"#5,$ \"\"(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 61 "(a) Find the Lagrange interpolating polynomial for the po ints" }{XPPEDIT 18 0 "``(1,2), ``(2,3), ``(3,3), ``(4,1)" "6&-%!G6$\" \"\"\"\"#-F$6$F'\"\"$-F$6$F*F*-F$6$\"\"%F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 99 "(b) Plot the graph of the polynomial from part (a ) to show that it passes through the given points." }}{PARA 0 "" 0 "" {TEXT -1 37 "____________________________________\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 36 "____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) Use the procedure \+ " }{TEXT 0 8 "lagrange" }{TEXT -1 62 " to construct the five Lagrange \+ \"L\" basis polynomials for the " }{TEXT 297 1 "x" }{TEXT -1 8 " value s " }{XPPEDIT 18 0 "x[1] = -3;" "6#/&%\"xG6#\"\"\",$\"\"$!\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "x[2] = -2;" "6#/&%\"xG6#\"\"#,$F'!\"\" " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[3] = 0;" "6#/&%\"xG6#\"\"$\"\"!" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "x[4] = 2;" "6#/&%\"xG6#\"\"%\"\"#" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "x[5]=3" "6#/&%\"xG6#\"\"&\"\"$" } {TEXT -1 45 ", and plot their graphs in the same picture. " }}{PARA 0 "" 0 "" {TEXT -1 113 "(b) Use the Lagrange \"L\" basis polynomials fo und in part (a) to construct the Lagrange interpolating polynomial " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 15 " for the points" }{XPPEDIT 18 0 "``(-3, -1),``(-2,2),``(0, 1/2),``(2,-2);" "6&-%!G6$,$ \"\"$!\"\",$\"\"\"F(-F$6$,$\"\"#F(F.-F$6$\"\"!*&F*F*F.F(-F$6$F.,$F.F( " }{TEXT -1 1 "," }{XPPEDIT 18 0 " ``(3,1)" "6#-%!G6$\"\"$\"\"\"" } {TEXT -1 24 ", and plot the graph of " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG 6#%\"xG" }{TEXT -1 57 " along with the five given points. What is the \+ degree of " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 2 "? " } }{PARA 0 "" 0 "" {TEXT -1 52 "(c) Construct the Lagrange interpolating polynomial " }{XPPEDIT 18 0 "q(x);" "6#-%\"qG6#%\"xG" }{TEXT -1 15 " \+ for the points" }{XPPEDIT 18 0 "``(-3,-1),``(-2,2),``(0,0),``(2,-2);" "6&-%!G6$,$\"\"$!\"\",$\"\"\"F(-F$6$,$\"\"#F(F.-F$6$\"\"!F1-F$6$F.,$F. F(" }{TEXT -1 1 "," }{XPPEDIT 18 0 " ``(3,1)" "6#-%!G6$\"\"$\"\"\"" } {TEXT -1 24 ". What is the degree of " }{XPPEDIT 18 0 "q(x)" "6#-%\"qG 6#%\"xG" }{TEXT -1 2 "? " }}{PARA 0 "" 0 "" {TEXT -1 36 "_____________ _______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 36 "____________________ ________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 73 "(a) Use the Lagrange formula to find a degree 5 polynomial approximat ion " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "arctan(x);" "6#-%'arctanG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, 1];" "6#7$\"\"!\"\"\"" }{TEXT -1 34 ", which agrees with the values of " }{XPPEDIT 18 0 "arctan(x)" "6#-%'ar ctanG6#%\"xG" }{TEXT -1 6 " at 6 " }{TEXT 265 15 "equally spaced " } {TEXT -1 10 "values of " }{TEXT 298 1 "x" }{TEXT -1 55 " between 0 and 1 inclusive. You may use the procedure " }{TEXT 0 8 "lagrange" } {TEXT -1 25 " given in this worksheet." }}{PARA 0 "" 0 "" {TEXT -1 17 "(b) Estimate the " }{TEXT 265 22 "maximum absolute error" }{TEXT -1 25 " in using the polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG " }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arc tanG6#%\"xG" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "[0, 1]; " "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "___ _________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 36 "____________________ ________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 74 "(a) Use the Lagrange formula to find a degree 12 polynomial approxima tion " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "f(x)=(ln(x^2+1))/exp(x)" "6#/-%\"fG6#%\"xG*&-%#lnG6#,&* $F'\"\"#\"\"\"F/F/F/-%$expG6#F'!\"\"" }{TEXT -1 17 " on the interval \+ " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 2;" "6#1 %!G\"\"#" }{TEXT -1 33 " which agrees with the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " at 13 " }{TEXT 265 14 "equ ally spaced" }{TEXT -1 11 " values of " }{TEXT 299 1 "x" }{TEXT -1 15 " between 0 and " }{XPPEDIT 18 0 "2;" "6#\"\"#" }{TEXT -1 11 " inclusi ve." }}{PARA 0 "" 0 "" {TEXT -1 51 "(b) Plot the graph of the interpol ating polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 83 " found in part (a) together with the interpolation points used in \+ its construction." }}{PARA 0 "" 0 "" {TEXT -1 17 "(c) Estimate the " } {TEXT 265 22 "maximum absolute error" }{TEXT -1 25 " in using the poly nomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 42 " found \+ in (a) to approximate the function " }{XPPEDIT 18 0 "f(x)=(ln(x^2+1))/ exp(x)" "6#/-%\"fG6#%\"xG*&-%#lnG6#,&*$F'\"\"#\"\"\"F/F/F/-%$expG6#F'! \"\"" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"! %\"xG" }{XPPEDIT 18 0 "`` <= 2;" "6#1%!G\"\"#" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 36 "____________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 36 "____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q 5" }}{PARA 0 "" 0 "" {TEXT -1 74 "(a) Use the Lagrange formula to find a degree 10 polynomial approximation " }{XPPEDIT 18 0 "p(x)" "6#-%\"p G6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "f(x) = x*exp(x)/(x^2-x+1 );" "6#/-%\"fG6#%\"xG*(F'\"\"\"-%$expG6#F'F),(*$F'\"\"#F)F'!\"\"F)F)F0 " }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"x G" }{XPPEDIT 18 0 "`` <= 1;" "6#1%!G\"\"\"" }{TEXT -1 33 " which agree s with the values of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 7 " at 11 " }{TEXT 265 14 "equally spaced" }{TEXT -1 11 " values of " }{TEXT 300 1 "x" }{TEXT -1 15 " between 0 and " }{XPPEDIT 18 0 "1; " "6#\"\"\"" }{TEXT -1 11 " inclusive." }}{PARA 0 "" 0 "" {TEXT -1 51 "(b) Plot the graph of the interpolating polynomial " }{XPPEDIT 18 0 " p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 83 " found in part (a) together with the interpolation points used in its construction." }}{PARA 0 "" 0 " " {TEXT -1 17 "(c) Estimate the " }{TEXT 265 22 "maximum absolute erro r" }{TEXT -1 25 " in using the polynomial " }{XPPEDIT 18 0 "p(x)" "6#- %\"pG6#%\"xG" }{TEXT -1 43 " found in (a) to approximate the function \+ " }{XPPEDIT 18 0 "f(x) = x*exp(x)/(x^2-x+1);" "6#/-%\"fG6#%\"xG*(F'\" \"\"-%$expG6#F'F),(*$F'\"\"#F)F'!\"\"F)F)F0" }{TEXT -1 17 " on the int erval " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 1; " "6#1%!G\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "______ ______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 36 "____________________ ________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }