{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Tim es" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "System" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 280 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{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 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 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 "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 "Map le Output" -1 12 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 "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 "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Tim es" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 43 "An introduction to polynomial int erpolation" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B. C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 20 "Version: 24.3.2007\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "lo ad root-finding procedures including: " }{TEXT 0 15 "findmax,findmin" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 280 7 "roots. m" }{TEXT -1 38 " contains the code for the procedures " }{TEXT 0 7 "f indmax" }{TEXT -1 5 " and " }{TEXT 0 7 "findmin" }{TEXT -1 25 " used i n this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be read int o 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 33 "read \"K:\\\\Maple/procdrs/roots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "Introductor y examples .. " }{TEXT 275 87 "solving a system of equations to obtain the coefficients of an interpolating polynomial" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 31 "Consider the following problem:" }}{PARA 0 "" 0 "" {TEXT -1 28 "Find the quadratic function " }{XPPEDIT 18 0 "p(x) = a*x^2+b*x+c;" "6#/-%\"pG6#%\"xG,(*& %\"aG\"\"\"*$F'\"\"#F+F+*&%\"bGF+F'F+F+%\"cGF+" }{TEXT -1 24 " such th at the graph of " }{XPPEDIT 18 0 "y = p(x);" "6#/%\"yG-%\"pG6#%\"xG" } {TEXT -1 24 " goes through the points" }{XPPEDIT 18 0 "``(1,2), ``(2,3 )" "6$-%!G6$\"\"\"\"\"#-F$6$F'\"\"$" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``(4,1)" "6#-%!G6$\"\"%\"\"\"" }{TEXT -1 3 ".\n " }}{PARA 0 "" 0 " " {TEXT -1 46 "We can set up three equations in the unknowns " } {XPPEDIT 18 0 "a,b" "6$%\"aG%\"bG" }{TEXT -1 5 " and " }{TEXT 264 1 "c " }{TEXT -1 53 " which can then be solved to find these coefficients. " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 259 4 "Note" }{TEXT -1 16 " : The procedure " }{TEXT 0 8 "unassign" }{TEXT -1 1 " " }{TEXT 259 6 " clears" }{TEXT -1 37 " any previously assigned values from " } {XPPEDIT 18 0 "a, b" "6$%\"aG%\"bG" }{TEXT -1 5 " and " }{TEXT 262 1 " c" }{TEXT -1 8 ", while " }{TEXT 0 6 "assign" }{TEXT -1 29 " uses the \+ equations given by " }{TEXT 0 5 "solve" }{TEXT -1 34 " to assign the c omputed values to " }{XPPEDIT 18 0 "a, b" "6$%\"aG%\"bG" }{TEXT -1 5 " and " }{TEXT 263 1 "c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "unassign('a','b','c'); \np := x->a*x^2+b*x+c;\neq1 := p(1)=2;\neq2 := p(2)=3;\neq3 := p(4)=1; \nsols := solve(\{eq1,eq2,eq3\},\{a,b,c\});\nassign(sols);\n'p(x)'=p(x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG% &arrowGF(,(*&%\"aG\"\"\")9$\"\"#F/F/*&%\"bGF/F1F/F/%\"cGF/F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,(%\"aG\"\"\"%\"bGF(%\"cGF(\" \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/,(*&\"\"%\"\"\"%\"aGF)F )*&\"\"#F)%\"bGF)F)%\"cGF)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ eq3G/,(*&\"#;\"\"\"%\"aGF)F)*&\"\"%F)%\"bGF)F)%\"cGF)F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%solsG<%/%\"cG#!\"\"\"\"$/%\"bGF*/%\"aG#!\"#F* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,(*&#\"\"#\"\"$\"\" \"*$)F'F+F-F-!\"\"*&F,F-F'F-F-#F-F,F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 40 "We can illustrate the result in a gra ph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "points := [[1,2],[2,3],[4,1]];\nplot([p(x),points],x =0..4.5,y,symbol=circle,color=[red,black],\n \+ style=[line,point],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %'pointsG7%7$\"\"\"\"\"#7$F(\"\"$7$\"\"%F'" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6%7S7$$\"\"!F)$!3:LLLLL LLL!#=7$$\"3e*****\\P>(3)*!#>$!3'Rm;Nr\"e[XF07$$\"3?+]ilLKM=F,$\"3c7)H ih?`%>F,7$$\"3E++Dc(=Tz#F,$\"3DJ\"[ri\\&GXF,7$$\"31++v$Hw-w$F,$\"3G>T* 3O][+(F,7$$\"3\\**\\7`=%=s%F,$\"3)R!ok$>1eM*F,7$$\"3'***\\i:iL8cF,$\"3 O&=B#3RgS6!#<7$$\"3V**\\i!*pUOlF,$\"3)*fB;XAwU8FL7$$\"35+]i!z)3\"\\(F, $\"3^$)RR&*Q))R:FL7$$\"3>**\\7y*)oU%)F,$\"3!3\"=cX+GCFL7$$\"3'***\\(ovo$G5FL$\"396R@PZuY?FL7$$\"39++DYwUD6 FL$\"3k*y?T,e&)>#FL7$$\"3#****\\(o])GA\"FL$\"3,0A+7mNQBFL7$$\"38++v`L! oJ\"FL$\"35N=&H,'4hCFL7$$\"3$**\\iS:!4-9FL$\"3sH[xGfOiDFL7$$\"3-++v3W] .:FL$\"3suG;))G;qEFL7$$\"3-+++&e:%*e\"FL$\"3UQFx`@v]FFL7$$\"3'**\\ilq] $*o\"FL$\"3d(oyJ\"\\6KGFL7$$\"3))****\\A-\"yx\"FL$\"3y+IoUX-$*GFL7$$\" 3?+DcJV'[(=FL$\"3W*Hx,#)[y%HFL7$$\"3C+vo%z#Gn>FL$\"3?foaa1Q))HFL7$$\"3 O+]ilIFL7$$\"3()**\\ig]jEDFL$\"33p')[&p[1*HFL7$$ \"3t****\\K&**Hi#FL$\"33)=.DB9*[HFL7$$\"3()**\\7oLF@#3\\k*em#FL7$$\"3I++vo^$z4$FL$\"3cs7t*HP Bc#FL7$$\"3y*\\iST\")f=$FL$\"3KvjoheidCFL7$$\"3E++D;#RAG$FL$\"3Wdq&pY@ 8L#FL7$$\"3q*\\ilI5GP$FL$\"3(o,=.,)>,AFL7$$\"3%)*\\7G>$[nMFL$\"3[(QzMy *[`?FL7$$\"3/++vVK/gNFL$\"3Q_mLmW_(*=FL7$$\"3!)*\\i!R]%pl$FL$\"3?:;PrP +A4#F,7$$\"3))*\\P%eWA-WFL $!3;y#Rk'z+QYF07$$\"3++++++++XFLF*-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F( -%&STYLEG6#%%LINEG-F$6%7%7$$\"\"\"F)$\"\"#F)7$Fh[l$\"\"$F)7$$\"\"%F)Ff [l-Fhz6&FjzF)F)F)-F_[l6#%&POINTG-%*AXESTICKSG6$%(DEFAULTGF\\\\l-%'SYMB OLG6#%'CIRCLEG-%+AXESLABELSG6$Q\"x6\"Q\"yFa]l-%%VIEWG6$;F($\"#X!\"\"Fh \\l" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Given any " } {TEXT 259 19 "two distinct points" }{TEXT -1 39 ", which are not horiz ontal, there is a " }{TEXT 259 11 "unique line" }{TEXT -1 54 " (or deg ree 1 polynomial ) which passes through them. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Given any " }{TEXT 259 8 "3 points" }{TEXT -1 54 ", which are not in a straight line, there is \+ a unique " }{TEXT 259 8 "parabola" }{TEXT -1 58 " (or degree 2 polynom ial) which passes through the points." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 16 "If we are given " }{TEXT 259 8 "4 poi nts" }{TEXT -1 51 ", say the previous 3 points together with the point " }{XPPEDIT 18 0 "``(3,3)" "6#-%!G6$\"\"$F&" }{TEXT -1 43 ", then we c an fit a degree 3 polynomial or " }{TEXT 259 5 "cubic" }{TEXT -1 95 " \+ through them, unless the 4th point lies on the parabola passing throug h the previous 3 points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "unassign('a','b','c','d');\np := x ->a*x^3+b*x^2+c*x+d;\neq1 := p(1)=2;\neq2 := p(2)=3;\neq3 := p(3)=3;\n eq4 := p(4)=1;\nsols := solve(\{eq1,eq2,eq3,eq4\},\{a,b,c,d\});\nassig n(sols);\n'p(x)'=p(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#% \"xG6\"6$%)operatorG%&arrowGF(,**&%\"aG\"\"\")9$\"\"$F/F/*&%\"bGF/)F1 \"\"#F/F/*&%\"cGF/F1F/F/%\"dGF/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/,*%\"aG\"\"\"%\"bGF(%\"cGF(%\"dGF(\"\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$eq2G/,**&\"\")\"\"\"%\"aGF)F)*&\"\"%F)%\"bGF)F)*& \"\"#F)%\"cGF)F)%\"dGF)\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3 G/,**&\"#F\"\"\"%\"aGF)F)*&\"\"*F)%\"bGF)F)*&\"\"$F)%\"cGF)F)%\"dGF)F/ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq4G/,**&\"#k\"\"\"%\"aGF)F)*& \"#;F)%\"bGF)F)*&\"\"%F)%\"cGF)F)%\"dGF)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG<&/%\"aG#!\"\"\"\"'/%\"bG#\"\"\"\"\"#/%\"cG#F/ \"\"$/%\"dGF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,**&#\" \"\"\"\"'F+*$)F'\"\"$F+F+!\"\"*&#F+\"\"#F+*$)F'F3F+F+F+*&#F3F/F+F'F+F+ F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "points := [[1,2],[2,3],[3,3],[4,1]];\nplot([p(x),poi nts],x=0..4.5,y,symbol=circle,color=[red,black],\n \+ style=[line,point],ytickmarks=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'pointsG7&7$\"\"\"\"\"#7$F(\"\"$7$F*F*7$\"\"%F'" }}{PARA 13 " " 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6(-%'CURVESG6%7U7$$\"\"!F)$ \"\"\"F)7$$\"3e*****\\P>(3)*!#>$\"3%*3&yosW+2\"!#<7$$\"3?+]ilLKM=!#=$ \"3UOwakK3Q6F27$$\"3E++Dc(=Tz#F6$\"3mQ#)G@Wn@7F27$$\"31++v$Hw-w$F6$\"3 %\\C,%f>_78F27$$\"3\\**\\7`=%=s%F6$\"3Pe/')4As39F27$$\"3'***\\i:iL8cF6 $\"3]s09)=\"H-:F27$$\"3V**\\i!*pUOlF6$\"3FM#>V^TGg\"F27$$\"35+]i!z)3\" \\(F6$\"3w^b\"e!e#*4**\\7y*)oU%)F6$\"3(3j-Q=V*==F27$$\"39,+] Pn_@%*F6$\"3SGJD:PaK>F27$$\"3'***\\(ovo$G5F2$\"3Wr$*)y2$4L?F27$$\"39++ DYwUD6F2$\"3'\\b&p!Q.g9#F27$$\"3#****\\(o])GA\"F2$\"3GgrFvr=eAF27$$\"3 8++v`L!oJ\"F2$\"3c?'R\"=YIkBF27$$\"3$**\\iS:!4-9F2$\"3ICsZD/FeCF27$$\" 3-++v3W].:F2$\"372#)Q[t9mDF27$$\"3-+++&e:%*e\"F2$\"3'RX4#*HBNl#F27$$\" 3'**\\ilq]$*o\"F2$\"3'HA#*)4ak\\FF27$$\"3))****\\A-\"yx\"F2$\"3eJ*\\:_ F2$\"3y6S')e\" fw(HF27$$\"3O+]il8$F27$$\"3'**\\i!zA*pM#F2$\" 39y'ohPZUN4GF27$$\"3E ++D;#RAG$F2$\"3+H\"[,`%R\"o#F27$$\"3q*\\ilI5GP$F2$\"3D)o/HA,$[nMF2$\"3'=#o?*zi[P#F27$$\"3/++vVK/gNF2$\"3yC!Gvus.>#F27$$\" 3!)*\\i!R]%pl$F2$\"3y-q&=J:P(>F27$$\"3]+++&)HF]PF2$\"3rW#*\\bOZTz9F27$$\"3E+Dc\"Hl.%RF2$\"3='*GDgK[$>\"F2 7$$\"3x****\\K(Rt-%F2$\"3g+Sw4'Hu2*F67$$\"3p**\\(oDAq7%F2$\"3'[9Ay>*[? bF67$$\"3W+++&\\zh@%F2$\"3kAt0b^ " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 35 "The basic idea of interpolation . . " }{TEXT 276 77 "constructing an interpolating polynomial to approxi mate a continuous function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 36 "Suppose we have a set of data poin ts" }{XPPEDIT 18 0 "``(x,y)" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 188 " whic h perhaps have been obtained experimentally, but in any case can be co nsidered to be representative of a function of some kind. The general \+ idea of interpolation is to try to predict " }{TEXT 265 1 "y" }{TEXT -1 38 " values at intermediate input numbers " }{TEXT 266 1 "x" } {TEXT -1 17 ". In the case of " }{TEXT 259 24 "polynomial interpolatio n" }{TEXT -1 70 " we fit a polynomial through the data points, interpo lation points or " }{TEXT 259 5 "nodes" }{TEXT -1 63 ", and evaluate t he polynomial to predict intermediate values. \n" }}{PARA 0 "" 0 "" {TEXT -1 96 "If the data points are obtained by evaluating some non-po lynomial mathematical function such as " }{XPPEDIT 18 0 "sin(x)" "6#-% $sinG6#%\"xG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "ln(x)" "6#-%#lnG6#%\" xG" }{TEXT -1 48 ", then we can regard the polynomial as being an " } {TEXT 259 13 "approximation" }{TEXT -1 327 " for the function under co nsideration. The hope is that the values of the polynomial do not stra y too far from the values of the function in between the data points. \+ For sake of convenience we shall often take equally spaced data points , but, in terms of obtaining a good approximation, this is not essenti al or even desirable." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "As an " }{TEXT 261 7 "example" }{TEXT -1 42 ", we find \+ an interpolating polynomial for " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6# %\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0, Pi/2];" "6 #7$\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 36 " using 7 equally spaced data points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "dist := evalf(Pi/2);\nh := dist/6;\nxvals := [s eq(h*(i-1),i=1..7)];\nyvals := [seq(sin(xvals[i]),i=1..7)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%distG$\"+Fjzq:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"+yQ*zh#!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%&xvalsG7)$\"\"!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 0 "" 0 "" {TEXT -1 97 "\nSince there are 7 data point s, we look for a polynomial of degree 6 to fit through these points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 191 "unassign('a','b','c','d','e','r','s');\np := x->a*x^6+b*x^5+c*x ^4+d*x^3+e*x^2+r*x+s;\neqns := \{seq(p(xvals[i])=yvals[i],i=1..7)\}:\n sols := solve(eqns,\{a,b,c,d,e,r,s\});\nassign(sols);\n'p(x)'=p(x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arro wGF(,0*&%\"aG\"\"\")9$\"\"'F/F/*&%\"bGF/)F1\"\"&F/F/*&%\"cGF/)F1\"\"%F /F/*&%\"dGF/)F1\"\"$F/F/*&%\"eGF/)F1\"\"#F/F/*&%\"rGF/F1F/F/%\"sGF/F(F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG<)/%\"bG$\"+K3'3.\"!#6/ %\"dG$!+.Z'[l\"!#5/%\"aG$!+?I*Rl*!#8/%\"eG$!+rxt.LF4/%\"cG$!+X2U!4#!#7 /%\"rG$\"+c\\.+5!\"*/%\"sG$\"\"!FF" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"pG6#%\"xG,.*&$\"+?I*Rl*!#8\"\"\")F'\"\"'F-!\"\"*&$\"+K3'3.\"!#6F- )F'\"\"&F-F-*&$\"+X2U!4#!#7F-)F'\"\"%F-F0*&$\"+.Z'[l\"!#5F-)F'\"\"$F-F 0*&$\"+rxt.LF,F-)F'\"\"#F-F0*&$\"+c\\.+5!\"*F-F'F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Why has s turned out t o be 0?" }}{PARA 0 "" 0 "" {TEXT -1 23 "Note that the function " } {XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 7 " is an " } {TEXT 259 12 "odd function" }{TEXT -1 11 ", that is, " }{XPPEDIT 18 0 "sin(-x)=-sin(x)" "6#/-%$sinG6#,$%\"xG!\"\",$-F%6#F(F)" }{TEXT -1 9 " \+ for all " }{TEXT 267 1 "x" }{TEXT -1 42 ", so its graph passes through the origin.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "points := [seq([xvals[i],yvals[i]],i=1..7)]:\nplot([p(x),points],x=0..Pi/2,symb ol=circle,color=[red,black],\n style= [line,point]);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 " 6'-%'CURVESG6%7S7$$\"\"!F)F(7$$\"3NGK5j*))QU$!#>$\"30&*)R_W0LU$F-7$$\" 3DXXYUk*HS'F-$\"3ON%HA=Q()R'F-7$$\"3=N68yVJ`(*F-$\"3S*QMCAxzt*F-7$$\"3 %3h%eRSe78!#=$\"3,0[q>%G)38F=7$$\"3k+,hQPB[;F=$\"3;o,;4&)yS;F=7$$\"3'H #*ed(RUf>F=$\"3[lB&4m9p%>F=7$$\"3kHX[TMk\"G#F=$\"3M3fna/!>E#F=7$$\"3Saq%HF=$\"3O48fm%yX!HF=7$$\"3)p q'4OKt)G$F=$\"3&z3E7*owHKF=7$$\"3#f/N#RTo*e$F=$\"35uz(f0$38NF=7$$\"3Ut 79wN[GRF=$\"3zZI$eq5#GQF=7$$\"3/)F=$\"3Mv.(e>sjI(F=7$$\"3gv)\\AI@S \\)F=$\"39EFpJf&)3vF=7$$\"3vn()=V,i>))F=$\"3I0\")f%ot)>xF=7$$\"3'G:[dg &*f:*F=$\"3C5Tm'4z#HzF=7$$\"3sSt6rL2&[*F=$\"30Ca#o1ia7)F=7$$\"3mka(*HI Z.)*F=$\"3bePPhm!pI)F=7$$\"3Em\"[l:+d,\"!#<$\"3)*oya-q\\)\\)F=7$$\"3c! 3XyEmu/\"F_u$\"3]4*=OJ(fh')F=7$$\"3sJ_*>P$Q\"3\"F_u$\"31HE!Qf!4E))F=7$ $\"3.M\"G%4t676F_u$\"3')GDtXIQm*)F=7$$\"3co*Q4iq'*z.@\"F_u$\"3G1A%zB.v N*F=7$$\"34Q$[)>&*oU7F_u$\"3[ek)*F=7$$\"3s$)o#3`-1W\"F_u$\"3#GCM&Q)e`\"**F=7$$\"3-#*)4zFCpT***F=7$$\"3+++lBjzq:F_u$\"3m)\\K+++++\"F_u-%'COLOURG6&%$ RGBG$\"*++++\"!\")F(F(-%&STYLEG6#%%LINEG-F$6%7)F'7$$\"30+++yQ*zh#F=$\" 3y*****4X!>)e#F=7$$\"35+++cx)fB&F=$\"3++++++++]F=7$$\"3;+++M;)R&yF=$\" 3M+++7y1rqF=7$$\"3++++^v>Z5F_u$\"3I+++PSDg')F=7$$\"3.+++Rp**38F_u$\"37 +++j#e#f'*F=7$$\"3/+++Fjzq:F_u$\"\"\"F)-Fhz6&FjzF)F)F)-F_[l6#%&POINTG- %'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q\"x6\"Q!F`^l-%%VIEWG6$;F($\"+Fjzq :!\"*%(DEFAULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "If t he graphs of the interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6#- %\"pG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6# %\"xG" }{TEXT -1 31 " are plotted over the interval " }{XPPEDIT 18 0 " [0, Pi/2];" "6#7$\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 81 " in the s ame picture, there is no discernable difference between the two graphs . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "plot([p(x),sin(x)],x=0..Pi/2,color=[red,green],thickn ess=[1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 398 254 254 {PLOTDATA 2 "6&- %'CURVESG6%7S7$$\"\"!F)F(7$$\"3NGK5j*))QU$!#>$\"30&*)R_W0LU$F-7$$\"3DX XYUk*HS'F-$\"3ON%HA=Q()R'F-7$$\"3=N68yVJ`(*F-$\"3S*QMCAxzt*F-7$$\"3%3h %eRSe78!#=$\"3,0[q>%G)38F=7$$\"3k+,hQPB[;F=$\"3;o,;4&)yS;F=7$$\"3'H#*e d(RUf>F=$\"3[lB&4m9p%>F=7$$\"3kHX[TMk\"G#F=$\"3M3fna/!>E#F=7$$\"3Saq%HF=$\"3O48fm%yX!HF=7$$\"3)pq'4 OKt)G$F=$\"3&z3E7*owHKF=7$$\"3#f/N#RTo*e$F=$\"35uz(f0$38NF=7$$\"3Ut79w N[GRF=$\"3zZI$eq5#GQF=7$$\"3/)F=$\"3Mv.(e>sjI(F=7$$\"3gv)\\AI@S\\) F=$\"39EFpJf&)3vF=7$$\"3vn()=V,i>))F=$\"3I0\")f%ot)>xF=7$$\"3'G:[dg&*f :*F=$\"3C5Tm'4z#HzF=7$$\"3sSt6rL2&[*F=$\"30Ca#o1ia7)F=7$$\"3mka(*HIZ.) *F=$\"3bePPhm!pI)F=7$$\"3Em\"[l:+d,\"!#<$\"3)*oya-q\\)\\)F=7$$\"3c!3Xy Emu/\"F_u$\"3]4*=OJ(fh')F=7$$\"3sJ_*>P$Q\"3\"F_u$\"31HE!Qf!4E))F=7$$\" 3.M\"G%4t676F_u$\"3')GDtXIQm*)F=7$$\"3co*Q4iq'*z.@\"F_u$\"3G1A%zB.vN*F =7$$\"34Q$[)>&*oU7F_u$\"3[ek) *F=7$$\"3s$)o#3`-1W\"F_u$\"3#GCM&Q)e`\"**F=7$$\"3-#*)4zFCpT***F=7$$\"3+++lBjzq:F_u$\"3m)\\K+++++\"F_u-%'COLOURG6&%$RGB G$\"*++++\"!\")F(F(-%*THICKNESSG6#\"\"\"-F$6%7SF'7$F+$\"3QsLeI+ABMF-7$ F1$\"3]DS+L@i)R'F-7$F6$\"3_b&ehJeyt*F-7$F;$\"3<&Q%f]#=)38F=7$FA$\"3V-y j65yS;F=7$FF$\"3q)*)ys&)4p%>F=7$FK$\"3G3RBS#)*=E#F=7$FP$\"3GM*4Ir&=&e# F=7$FU$\"3=7dd?+e/HF=7$FZ$\"3c0jx$Rp(HKF=7$Fin$\"3ML>uye38NF=7$F^o$\"3 V3zj5M@GQF=7$Fco$\"3T$oQx\\8-9%F=7$Fho$\"3UOK72VNOWF=7$F]p$\"3=(>VrOc6 q%F=7$Fbp$\"3SMGBr+f5]F=7$Fgp$\"37xKegS#yE&F=7$F\\q$\"3AFv)ygs5c&F=7$F aq$\"3M>X(*RG,:eF=7$Ffq$\"33g0zi&Qs3'F=7$F[r$\"3h+)=j,t*RjF=7$F`r$\"3Q z3MFxj'f'F=7$Fer$\"3$[8q$y.wDoF=7$Fjr$\"3wm47uKelqF=7$F_s$\"3M(RvZssjI (F=7$Fds$\"3C>#p\"po&)3vF=7$Fis$\"3%4zeG%\\()>xF=7$F^t$\"31()\\)y]!GHz F=7$Fct$\"3g&H%*=Uja7)F=7$Fht$\"3%3R7+w2pI)F=7$F]u$\"3Q?Zs+w\\)\\)F=7$ Fcu$\"3Ih!)R3tfh')F=7$Fhu$\"3VZPF_)*3E))F=7$F]v$\"3kW#3Hj\"Qm*)F=7$Fbv $\"3OvDs!yi+6*F=7$Fgv$\"3Uh#yueneB*F=7$F\\w$\"3Y(Q,up+vN*F=7$Faw$\"3OZ !)zP7am%*F=7$Ffw$\"3-w%)pSt5q&*F=7$F[x$\"3-KPDS[]f'*F=7$F`x$\"3B\"='y# [C.u*F=7$Fex$\"3$oY$>Q\"*z4)*F=7$Fjx$\"3#4Wi\"*p+U')*F=7$F_y$\"3aZ!3__ n`\"**F=7$Fdy$\"3-jN*Rxj4&**F=7$Fiy$\"3%zPL)R0Iy**F=7$F^z$\"3Qg9![CwT* **F=7$Fcz$Fa[lF)-Fhz6&FjzF(F[[lF(-F_[l6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q! F]el-%%VIEWG6$;F($\"+Fjzq:!\"*%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 24 "Outside of the interval " }{XPPEDIT 18 0 "[0, Pi/2];" "6#7$\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 15 " the gra phs of " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 41 " eventually di verge away from each other." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "plot([p(x),sin(x)],x=-3..6,color=[r ed,green],thickness=[1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 366 356 356 {PLOTDATA 2 "6&-%'CURVESG6%7jn7$$!\"$\"\"!$!3^**o:Fb.8>!#<7$$!3%)**** \\7c#Q!GF-$!3'\\M%3T)*QV:F-7$$!3=+](oKNJj#F-$!3/L$G]ZF9M\"F-7$$!3&**\\ 7yybr`#F-$!3+&o13>GXE\"F-7$$!3s***\\([i;SG!37F-7$$!3j***** \\\\gXM#F-$!3Iz*=$oE]n6F-7$$!3+++DTZ%zC#F-$!3rmJ3gtaR6F-7$$!35+]PH;jb? 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Between these " }{TEXT 271 1 "x" } {TEXT -1 154 " values the values differ by a greater amount than would be expected if only rounding errors were involved. Typically, for int erpolation points which are " }{TEXT 259 14 "equally spaced" }{TEXT -1 81 " (horizontally), that is, interpolation points which are based \+ on equally spaced " }{TEXT 273 1 "x" }{TEXT -1 59 " values, the maximu m (absolute) error occurs in either the " }{TEXT 259 26 "first or last sub-interval" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "We may \+ call such a curve an " }{TEXT 259 20 "absolute error curve" }{TEXT -1 139 " even though, technically, it shows a signed error, because it is easy to mentally take the magnitude of any error read off from the gr aph." }}{PARA 0 "" 0 "" {TEXT -1 36 "For the current example the maxim um " }{TEXT 259 14 "absolute error" }{TEXT -1 54 " occurs in the first sub-interval and is about 1.2e-6" }{XPPEDIT 18 0 "`` = 1.2;" "6#/%!G -%&FloatG6$\"#7!\"\"" }{TEXT -1 1 " " }{TEXT 278 1 "x" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "10^(-6)" "6#)\"#5,$\"\"'!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 7 "infnorm" }{TEXT -1 8 " in the " }{TEXT 0 9 "numappr ox" }{TEXT -1 69 " package can be used to obtain the maximum absolute \+ error as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 2 ": " }{TEXT 260 20 "infnorm(f(x),x=a..b )" }{TEXT -1 52 " computes an estimate for the value: max \+ " }{XPPEDIT 18 0 "abs(f(x))" "6#-%$absG6#-%\"fG6#%\"xG" }{TEXT -1 2 " \+ ." }}{PARA 0 "" 0 "" {TEXT -1 98 " \+ " } {TEXT 268 1 "x" }{TEXT -1 11 " in [a,b]" }{TEXT 23 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "numap prox[infnorm](p(x)-sin(x),x=0..1,'xmax');\nxmax;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+K9b37!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,ia w$)!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The special procedures " }{TEXT 0 7 "findmax" }{TEXT -1 5 " and " }{TEXT 0 7 "findmin" }{TEXT -1 18 " may also be used." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "findmax(p (x)-sin(x),x=0..xvals[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+^x Jw$)!#6$\"+L9b37!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "findmin(p( x)-sin(x),x=xvals[6]..Pi/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"+ lHj&[\"!\"*$!+1u585!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "We c ould also plot the relative error " }{XPPEDIT 18 0 "(p(x)-sin(x))/sin (x);" "6#*&,&-%\"pG6#%\"xG\"\"\"-%$sinG6#F(!\"\"F)-F+6#F(F-" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot((p(x)-sin(x))/sin(x),x=0..Pi/2,color=blue);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7eo7 $$\"\"!F)%%FAILG7$$\"35+++,`'*p5!#?$\"3_+;gomQgM!#A7$$\"3;+++-1$*R@F.$ \"3RY@7ALWDMF17$$\"3!)*****H!f*)4KF.$\"3O&4eRRo2R$F17$$\"3))*****\\?h) zUF.$\"3[l!*GC.OcLF17$$\"3.+++2=z>kF.$\"3]7S3b&Q$)G$F17$$\"3u******4Cs f&)F.$\"3z&3R3nl8A$F17$$\"35+++i$eRG\"!#>$\"3G))fH=x^!4$F17$$\"3&***** *>[W>r\"FN$\"3]#3>/&)>P'HF17$$\"35+++Bn\"zc#FN$\"3'y=D4H'*=s#F17$$\"3! 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However, the procedure " }{TEXT 0 7 "infnorm" } {TEXT -1 44 " gives the value of the limit just computed." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "numap prox[infnorm]((p(x)-sin(x))/sin(x),x=0..1,'xmax');\nxmax;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++g&\\$!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"!F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 49 "Comparis on with a Taylor polynomial approximation" }}{PARA 0 "" 0 "" {TEXT -1 91 "Another way to obtain a polynomial approximation for the sine func tion is to construct the " }{TEXT 259 24 "Taylor series expansion " } {TEXT -1 3 "of " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 47 " about the mid-point of the interval, that is, " }{XPPEDIT 18 0 "x = Pi/4;" "6#/%\"xG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 88 "This series can be truncated to obtain an alter native degree 6 polynomial approximation " }{XPPEDIT 18 0 "q(x)" "6#-% \"qG6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#% \"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,Pi/2]" "6#7$ \"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 3 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "taylor(sin(x ),x=Pi/4,7);\ncollect(convert(evalf(%), polynom),x):\nq := unapply(%,x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3,&%\"xG\"\"\"*&\"\"%!\"\"%#PiG F&F),$*&\"\"#F)F-#F&F-F&\"\"!F+F&,$*&F(F)F-F.F)F-,$*&\"#7F)F-F.F)\"\"$ ,$*&\"#[F)F-F.F&F(,$*&\"$S#F)F-F.F&\"\"&,$*&\"%S9F)F-F.F)\"\"'-%\"OG6# F&\"\"(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"qGf*6#%\"xG6\"6$%)opera torG%&arrowGF(,0$\"(![;G!#6!\"\"*&$\"+8v#4#)*!#8\"\"\")9$\"\"'F5F0*&$ \"+df0_5F/F5)F7\"\"&F5F5*&$\"+EbHkF!#7F5)F7\"\"%F5F0*&$\"+0*pak\"!#5F5 )F7\"\"$F5F0*&$\"*h0m\")*FAF5)F7\"\"#F5F0*&$\"+KOD+5!\"*F5F7F5F5F(F(F( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 35 "Note the following facts about the " } {TEXT 259 17 "Taylor polynomial" }{TEXT -1 22 " when compared to the \+ " }{TEXT 259 24 "interpolating polynomial" }{TEXT -1 1 ":" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 102 "The absolute err or behaves rather differently than the absolute error of the interpola ting polynomial." }}{PARA 15 "" 0 "" {TEXT -1 75 "The approximation is better than that of the interpolating polynomial when " }{TEXT 270 1 "x" }{TEXT -1 13 " is close to " }{XPPEDIT 18 0 "Pi/4;" "6#*&%#PiG\"\" \"\"\"%!\"\"" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 156 "The max imum absolute error occurs at the ends of the interval, and is greater in magnitude than the maximum absolute error for the interpolating po lynomial." }}{PARA 15 "" 0 "" {TEXT -1 10 "The graph " }{XPPEDIT 18 0 "y = q(x);" "6#/%\"yG-%\"qG6#%\"xG" }{TEXT -1 35 " of the Taylor polyn omial only has " }{TEXT 259 19 "one point in common" }{TEXT -1 16 " wi th the graph " }{XPPEDIT 18 0 "y = sin(x);" "6#/%\"yG-%$sinG6#%\"xG" } {TEXT -1 18 ", namely the point" }{XPPEDIT 18 0 "``(Pi/4,1/sqrt(2));" "6#-%!G6$*&%#PiG\"\"\"\"\"%!\"\"*&F(F(-%%sqrtG6#\"\"#F*" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(q(x)-sin(x),x=0..Pi/2,color=blue);" }}{PARA 13 " " 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7X7$$\"\"!F)$! 3%)*********zk\"G!#A7$$\"3=9;b\"[W>r\"!#>$!3%R9$>ug+5CF,7$$\"3NGK5j*)) QU$F0$!3eP6([:9]0#F,7$$\"3X')Qy-FW8\\F0$!3WX#>umrOy\"F,7$$\"3DXXYUk*HS 'F0$!3$HSol`8Pa\"F,7$$\"3!4%yH5a:y!)F0$!3ply#3<1vI\"F,7$$\"3=N68yVJ`(* F0$!3o4zmIa0.6F,7$$\"3%3h%eRSe78!#=$!3!fe&ot/*Ht(!#B7$$\"3k+,hQPB[;FO$ !3Or#>$4TkI`FR7$$\"3'H#*ed(RUf>FO$!3YeW#)e`52PFR7$$\"3kHX[TMk\"G#FO$!3 %H&\\gkB0$\\#FR7$$\"3Saq%HFO$!3# R&QIfdp;5FR7$$\"3)pq'4OKt)G$FO$!3$RY]mB')H6'!#C7$$\"3#f/N#RTo*e$FO$!3g Bk&=V!*4y$F[p7$$\"3Ut79wN[GRFO$!3JqhqU[(46#F[p7$$\"3/\"F`q7$$\"3uvW/^FF[s7$$\"3oHyF.C6noFO$!3cWL Eo8b\"*>F[s7$$\"3(*zx.5Ir.sFO$!3`#HV0/lI#>F[s7$$\"3'pz!QUu\"G^(FO$!3y' QS>O%el>F[s7$$\"3A\"3@7LGi%yFO$!3Q)>k+\"3k=?F[s7$$\"3)3\"HIT&[D>)FO$!3 -*3-ub%ot?F[s7$$\"3gv)\\AI@S\\)FO$!3b`Lg_ai:@F[s7$$\"3vn()=V,i>))FO$!3 %\\[c#f8=l?F[s7$$\"3'G:[dg&*f:*FO$!3=LRX[QF_8F[s7$$\"3sSt6rL2&[*FO$\"3 ?u\"p*e`LR>F[s7$$\"3mka(*HIZ.)*FO$\"3&*p9;7$[3B\"F`r7$$\"3Em\"[l:+d,\" !#<$\"3oih]8@wRWF`r7$$\"3c!3XyEmu/\"Ffv$\"3R\\9ZIYcE6F`q7$$\"3sJ_*>P$Q \"3\"Ffv$\"33W9D7MfgEF`q7$$\"3.M\"G%4t676Ffv$\"3UcC)Qa.UJ&F`q7$$\"3co* Q4i0\"F[p7$$\"3[cr`&*GLx6Ffv$\"39EYukM!)))=F[p7$ $\"3Hk&>q'*z.@\"Ffv$\"3ga)ztB#*\\J$F[p7$$\"34Q$[)>&*oU7Ffv$\"3W9))4'=& [7bF[p7$$\"3I^lOFY^w7Ffv$\"3p9/2=N5T!*F[p7$$\"3`!)f6EA448Ffv$\"3B4\"*z B=o59FR7$$\"3=;T7EvSU8Ffv$\"3c-SXn-oh@FR7$$\"3a_wiepWv8Ffv$\"3m!QY*4=v >KFR7$$\"3U$obdw1eS\"Ffv$\"3')*QN?(3e`XFR7$$\"3s$)o#3`-1W\"Ffv$\"3`&QN \"y]KOmFR7$$\"3-#*)4zFCF,7$$\"3+++lBjzq:Ffv$ \"3&evM^4z?J#F,-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%+AXESLABELSG6$Q\" x6\"Q!Fc]l-%%VIEWG6$;F($\"+Fjzq:!\"*%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 " Q1" }}{PARA 0 "" 0 "" {TEXT -1 30 "(a) Find a quartic polynomial " } {XPPEDIT 18 0 "p(x) = a*x^4+b*x^3+c*x^2+d*x+e;" "6#/-%\"pG6#%\"xG,,*&% \"aG\"\"\"*$F'\"\"%F+F+*&%\"bGF+*$F'\"\"$F+F+*&%\"cGF+*$F'\"\"#F+F+*&% \"dGF+F'F+F+%\"eGF+" }{TEXT -1 24 " such that the graph of " } {XPPEDIT 18 0 "y = p(x);" "6#/%\"yG-%\"pG6#%\"xG" }{TEXT -1 25 " goes \+ through the points " }{XPPEDIT 18 0 "``(1,2), ``(2,3), ``(3,3), ``(4,1 ), ``(6,-1/2)" "6'-%!G6$\"\"\"\"\"#-F$6$F'\"\"$-F$6$F*F*-F$6$\"\"%F&-F $6$\"\"',$*&F&F&F'!\"\"F5" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 99 "(b) Plot the graph of the polynomial from part (a) to show that it passes through the given points." }}{PARA 0 "" 0 "" {TEXT -1 37 "_ ___________________________________\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 36 "____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q 2" }}{PARA 0 "" 0 "" {TEXT -1 37 "(a) Find an interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 45 " of degree 5 \+ which interpolates the function " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arc tanG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,1]" " 6#7$\"\"!\"\"\"" }{TEXT -1 24 " based on the values of " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 31 " at 6 equally spaced values of " }{TEXT 277 1 "x" }{TEXT -1 28 " between 0 and 1 inclusive . " }}{PARA 0 "" 0 "" {TEXT -1 44 "(b) Illustrate the interpolating po lynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 108 " and the interpolation points in a graph.\n(c) Plot an absolute error curv e for the interpolating polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6# %\"xG" }{TEXT -1 30 " used as an approximation for " }{XPPEDIT 18 0 "a rctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 "(d) Find the maximum absolute error in using " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 16 " to approximate \+ " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 17 " on t he interval " }{XPPEDIT 18 0 "[0,1]" "6#7$\"\"!\"\"\"" }{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 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "3 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }