{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 "" -1 259 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 260 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 262 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 263 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 261 264 "" 1 18 0 0 0 0 0 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 0 1 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 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Em phasis" -1 276 "Times" 1 12 96 52 84 1 0 1 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 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 1 0 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 "Maple 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 "Normal" -1 256 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 32 "The Newton interpolation formula " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada " }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "load interpolation and function approximation procedur es" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 276 10 "f cnapprx.m" }{TEXT -1 37 " contains the code for the procedure " } {TEXT 0 13 "newton_interp" }{TEXT -1 25 " used in this worksheet. " }} {PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives it s 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 "First order divided differences" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "Given a function " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" } {TEXT -1 23 ", and distinct numbers " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG 6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\" " }{TEXT -1 15 ", the quotient " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "g*[x[0], x[1]] = (g(x[1])-g(x[0]))/(x[1]-x[0]);" "6#/*& %\"gG\"\"\"7$&%\"xG6#\"\"!&F)6#F&F&*&,&-F%6#&F)6#F&F&-F%6#&F)6#F+!\"\" F&,&&F)6#F&F&&F)6#F+F8F8" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "is called a " }{TEXT 260 30 "first order divided difference" } {TEXT -1 4 " of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 " The first order divided difference " }{XPPEDIT 18 0 "g[x[0],x[1]]" "6# &%\"gG6$&%\"xG6#\"\"!&F'6#\"\"\"" }{TEXT -1 52 " is the gradient of th e secant line to the graph of " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"g G6#%\"xG" }{TEXT -1 20 " joining the points " }{XPPEDIT 18 0 "``(x[0], g(x[0])" "6#-%!G6$&%\"xG6#\"\"!-%\"gG6#&F'6#F)" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "``(x[1],g(x[1])" "6#-%!G6$&%\"xG6#\"\"\"-%\"gG6#&F'6#F) " }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 293 313 313 {PLOTDATA 2 "61-%'CURVESG6$7S7$$!3++++++++:!#<$\"3+++++++DEF*7 $$!3&*****\\P&3YV\"F*$\"3M6Kmlb)on#F*7$$!3!***\\ivGF*7$$!3'***\\7V0@&=\"F*$\"37MSGDB#p'GF*7$$!33+]i&exd7\"F*$\"3mMz&* *ot.\"HF*7$$!3'***\\i+#QU1\"F*$\"3MY0#\\g?Y&HF*7$$!3****\\i!3%f+5F*$\" 35@$p#3Te**HF*7$$!3;++D\"oS:P*!#=$\"3Uwr>\">(fVIF*7$$!3h*****\\<#)*=() FX$\"3QK?3T-.)3$F*7$$!3#*****\\(G3U9)FX$\"3f;[*pXhk7$F*7$$!3Y*****\\- \\r\\(FX$\"3yTpC0`$*oJF*7$$!3?+++vGVZoFX$\"35pBF>5u5KF*7$$!3_*****\\(4 J@iFX$\"3'[M-5uH-D$F*7$$!37++D1Bt_cFX$\"3Yx8;-+T&G$F*7$$!3')*****\\FPm (\\FX$\"3h!3$o<7SELF*7$$!3()*******4'*QS%FX$\"3s//9%*3TgLF*7$$!3?++Dc> mPPFX$\"3[,$)G&yY\"*R$F*7$$!3'3+++&=$z9$FX$\"3YbxcKRpKMF*7$$!3N***\\iX /4]#FX$\"3#)3#)3a-qoMF*7$$!3C***\\(o8y%)=FX$\"3A8XvI&3A]$F*7$$!33**** \\i:#>C\"FX$\"3y(f2Fbhj`$F*7$$!3O!***\\7ev:l!#>$\"3o=%*>qm*pc$F*7$$!3u F++](o2[\"!#?$\"3)3RdjRf#*f$F*7$$\"3i(***\\P>:mkFir$\"3(=ry%[E\">j$F*7 $$\"3d***\\iv&QA7FX$\"3G+++!*>=+5F*$\"37kzo$fa++%F*7$$\"3-++DE&4Q1 \"F*$\"3;v4J#pN(=SF*7$$\"3=+]P%>5p7\"F*$\"3Lq)o&QCYOSF*7$$\"39+++bJ*[= \"F*$\"3_9Mi)R\\?0%F*7$$\"33++Dr\"[8D\"F*$\"3KGf0ho3pSF*7$$\"3++++Ijy5 8F*$\"358&3Z3xN3%F*7$$\"31+]P/)fTP\"F*$\"391:R&Q[#)4%F*7$$\"31+]i0j\"[ V\"F*$\"33jPsp$Q:6%F*7$$\"3++++++++:F*$\"3+++++++DTF*-%'COLOURG6&%$RGB G$\"*++++\"!\")$\"\"!Fb[lFa[l-F$6%7$7$F(Fa[l7$FfzFa[l-%*LINESTYLEG6#\" \"\"-F[[l6&F][lFb[lFb[lFb[l-F$6%7$7$$!\"\"Fb[lFa[l7$Fb\\l$\"\"$Fb[l-Fi [l6#\"\"#F\\\\l-F$6%7$7$$F[\\lFb[lFa[l7$F^]l$\"\"%Fb[lFg\\lF\\\\l-F$6% 7$Fd\\lF_]l-F[[l6&F][lFa[lFa[lF^[l-%*THICKNESSGFh\\l-F$6&Fd]l-%'SYMBOL G6#%'CIRCLEGF\\\\l-%&STYLEG6#%&POINTG-F$6&Fd]l-F\\^l6#%(DIAMONDGF\\\\l F_^l-F$6&Fd]l-F\\^l6#%&CROSSGF\\\\lF_^l-%%TEXTG6$7$$!#8Fc\\l$\"#LFc\\l Q-(x0,~g(x0)~)6\"-F^_l6$7$$\"#6Fc\\l$\"#VFc\\lQ-(x1,~g(x1)~)Ff_l-F^_l6 %7$$!#;Fc\\l$\"#HFc\\lQ)y~=~g(x)Ff_lFjz-%*AXESSTYLEG6#%%NONEG-%*AXESTI CKSG6$Fb[lFb[l-%+AXESLABELSG6%Q\"xFf_lQ!Ff_l-%%FONTG6#%(DEFAULTG-%%VIE WG6$;Fb`l$\"#:Fc\\l;Fa[lF\\`l" 1 2 0 1 10 0 2 9 1 1 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" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "For examp le, if " }{XPPEDIT 18 0 "g(x) = sin(x),x[0] = 1.2;" "6$/-%\"gG6#%\"xG- %$sinG6#F'/&F'6#\"\"!-%&FloatG6$\"#7!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "x[1] = 1.3" "6#/&%\"xG6#\"\"\"-%&FloatG6$\"#8!\"\"" } {TEXT -1 6 " then " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "g*[x[0], x[1]] = (sin(1.3)-sin(1.2))/(1.3-1.2);" "6#/*&%\"gG\"\"\"7 $&%\"xG6#\"\"!&F)6#F&F&*&,&-%$sinG6#-%&FloatG6$\"#8!\"\"F&-F16#-F46$\" #7F7F7F&,&-F46$F6F7F&-F46$F " 0 "" {MPLTEXT 1 0 30 "(sin(1.3)-sin(1.2))/(1.3-1.2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*%*4>:$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "The first order dvided difference can be regarded as being a " }{TEXT 260 36 "numerical anal ogue of the derivative" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "g(x);" "6#- %\"gG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 3 "If " } {XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 34 " is a differenti able function and " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 26 " \+ are close together, then " }{XPPEDIT 18 0 "g*[x[0], x[1]];" "6#*&%\"gG \"\"\"7$&%\"xG6#\"\"!&F(6#F%F%" }{TEXT -1 1 " " }{TEXT 259 1 "~" } {TEXT -1 2 " " }{XPPEDIT 18 0 "g*`'`((x[0]+x[1])/2);" "6#*&%\"gG\"\" \"-%\"'G6#*&,&&%\"xG6#\"\"!F%&F,6#F%F%F%\"\"#!\"\"F%" }{TEXT -1 62 ", \+ that is, the gradient of the secant line joining the points " } {XPPEDIT 18 0 "``(x[0],g(x[0]));" "6#-%!G6$&%\"xG6#\"\"!-%\"gG6#&F'6#F )" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``(x[1],g(x[1]));" "6#-%!G6$&% \"xG6#\"\"\"-%\"gG6#&F'6#F)" }{TEXT -1 14 " on the curve " }{XPPEDIT 18 0 "y = g(x);" "6#/%\"yG-%\"gG6#%\"xG" }{TEXT -1 97 " is approximate ly equal to the gradient of the tangent line at the mid-pont of the in terval from " }{XPPEDIT 18 0 "x[0];" "6#&%\"xG6#\"\"!" }{TEXT -1 4 " t o " }{XPPEDIT 18 0 "x[1];" "6#&%\"xG6#\"\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "For the example above where " }{XPPEDIT 18 0 "g(x) = sin(x),x[0] = 1.2,x[1] = 1.3;" " 6%/-%\"gG6#%\"xG-%$sinG6#F'/&F'6#\"\"!-%&FloatG6$\"#7!\"\"/&F'6#\"\"\" -F06$\"#8F3" }{TEXT -1 9 " we have " }{XPPEDIT 18 0 "g*`'`(x) = cos(x) ;" "6#/*&%\"gG\"\"\"-%\"'G6#%\"xGF&-%$cosG6#F*" }{TEXT -1 5 " and " }} {PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "g*`'`((x[0]+x[1])/2); " "6#*&%\"gG\"\"\"-%\"'G6#*&,&&%\"xG6#\"\"!F%&F,6#F%F%F%\"\"#!\"\"F%" }{TEXT -1 49 " = .3153223624, which is approximately equal to " } {XPPEDIT 18 0 "g*[x[0], x[1]];" "6#*&%\"gG\"\"\"7$&%\"xG6#\"\"!&F(6#F% F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 10 "cos(1.25);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+COA`J!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Gen eral divided differences" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 93 "Higher order divided differences are defined recursively in terms of lower order differences." }}{PARA 0 " " 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "x[0],x[1];" "6$&%\"xG6#\"\"!&F $6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x[2];" "6#&%\"xG6#\"\"# " }{TEXT -1 70 " are three distinct real numbers (usually taken in inc reasing order), " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " g*[x[0], x[1], x[2]] = (g*[x[1], x[2]]-g*[x[0], x[1]])/(x[2]-x[0]);" " 6#/*&%\"gG\"\"\"7%&%\"xG6#\"\"!&F)6#F&&F)6#\"\"#F&*&,&*&F%F&7$&F)6#F&& F)6#F0F&F&*&F%F&7$&F)6#F+&F)6#F&F&!\"\"F&,&&F)6#F0F&&F)6#F+F?F?" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "is called a " }{TEXT 260 31 "second order divided difference" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "x[0], x[1],x[2];" "6%&%\"xG6#\"\"!&F$6#\"\"\"&F$6#\"\"#" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "x[3];" "6#&%\"xG6#\"\"$" }{TEXT -1 69 " are four dis tinct real numbers (usually taken in increasing order), " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "g*[x[0], x[1], x[2], x[3]] = ( g*[x[1], x[2], x[3]]-g*[x[0], x[1], x[2]])/(x[3]-x[0]);" "6#/*&%\"gG\" \"\"7&&%\"xG6#\"\"!&F)6#F&&F)6#\"\"#&F)6#\"\"$F&*&,&*&F%F&7%&F)6#F&&F) 6#F0&F)6#F3F&F&*&F%F&7%&F)6#F+&F)6#F&&F)6#F0F&!\"\"F&,&&F)6#F3F&&F)6#F +FFFF" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "is called a " } {TEXT 260 30 "third order divided difference" }{TEXT -1 4 " of " } {XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "In general, if " } {XPPEDIT 18 0 "x[0],x[1],` . . . `,x[n];" "6&&%\"xG6#\"\"!&F$6#\"\"\"% (~.~.~.~G&F$6#%\"nG" }{TEXT -1 5 " are " }{XPPEDIT 18 0 "n+1;" "6#,&% \"nG\"\"\"F%F%" }{TEXT -1 60 " distinct real numbers (usually taken in increasing order), " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "g*[x[0], x[1], x[2], `. . .`, x[n]] = (g*[x[1], `. . .`, x[n]]-g *[x[0], `. . .`, x[n-1]])/(x[n]-x[0]);" "6#/*&%\"gG\"\"\"7'&%\"xG6#\" \"!&F)6#F&&F)6#\"\"#%&.~.~.G&F)6#%\"nGF&*&,&*&F%F&7%&F)6#F&F1&F)6#F4F& F&*&F%F&7%&F)6#F+F1&F)6#,&F4F&F&!\"\"F&FDF&,&&F)6#F4F&&F)6#F+FDFD" } {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 264 33 "____ _____________________________" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "is called an " }{TEXT 260 29 "n th order divided difference" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "There is a relationship \+ between a higher order divided difference and the derivative of the sa me order." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "For example, if " }{XPPEDIT 18 0 "g(x) = sin(x),x[0] = 1.2,x[1] = \+ 1.21,x[2] = 1.22;" "6&/-%\"gG6#%\"xG-%$sinG6#F'/&F'6#\"\"!-%&FloatG6$ \"#7!\"\"/&F'6#\"\"\"-F06$\"$@\"!\"#/&F'6#\"\"#-F06$\"$A\"F;" }{TEXT -1 5 " then" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "g*[x[ 0], x[1]] = (sin(1.21)-sin(1.2))/(1.21-1.2);" "6#/*&%\"gG\"\"\"7$&%\"x G6#\"\"!&F)6#F&F&*&,&-%$sinG6#-%&FloatG6$\"$@\"!\"#F&-F16#-F46$\"#7!\" \"F=F&,&-F46$F6F7F&-F46$F " 0 "" {MPLTEXT 1 0 128 "f01 := (sin(1.21)-sin(1.2))/(1.21-1.2);\nf12 := (sin(1.22)-sin(1.21))/(1.22-1.21);\nf012 := (f12 - f01)/(1.22-1.2);\n -sin(1.21)/2;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$f01G$\")c\"pd$! \")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$f12G$\")ZN$[$!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%f012G$!)]/yY!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+3+3yY!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Computing divided differences" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 74 "An efficient way \+ of performing polynomial interpolation is to make use of " }{TEXT 260 19 "divided differences" }{TEXT -1 24 " in connection with\nthe " } {TEXT 260 28 "Newton interpolation formula" }{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT -1 32 "Suppose we are given a function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 16 " and a sequence " }{XPPEDIT 18 0 "x[0],x[1],` . . . `,x[n];" "6&&%\"xG6#\"\"!&F$6#\"\"\"%(~.~.~.~G &F$6#%\"nG" }{TEXT -1 13 " of distinct " }{TEXT 265 1 "x" }{TEXT -1 8 " values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "The associated first order divided differences are:" }{TEXT 266 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " \+ " }{XPPEDIT 18 0 "g*[x[0], x[1]] = (g(x[1])-g(x[0]))/(x[1]-x[0]);" "6 #/*&%\"gG\"\"\"7$&%\"xG6#\"\"!&F)6#F&F&*&,&-F%6#&F)6#F&F&-F%6#&F)6#F+! \"\"F&,&&F)6#F&F&&F)6#F+F8F8" }{TEXT -1 5 " , " }{XPPEDIT 18 0 "g*[x [1], x[2]] = (g(x[2])-g(x[1]))/(x[2]-x[1]);" "6#/*&%\"gG\"\"\"7$&%\"xG 6#F&&F)6#\"\"#F&*&,&-F%6#&F)6#F-F&-F%6#&F)6#F&!\"\"F&,&&F)6#F-F&&F)6#F &F8F8" }{TEXT -1 15 ", . . . . , " }{XPPEDIT 18 0 "g*[x[n-1], x[n]] = (g(x[n])-g(x[n-1]))/(x[n]-x[n-1]);" "6#/*&%\"gG\"\"\"7$&%\"xG6#,&% \"nGF&F&!\"\"&F)6#F,F&*&,&-F%6#&F)6#F,F&-F%6#&F)6#,&F,F&F&F-F-F&,&&F)6 #F,F&&F)6#,&F,F&F&F-F-F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The second order divided differenc es are:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "g*[x[0], x[1], x[2]] = (g*[x[1], x[2]]-g*[x[ 0], x[1]])/(x[2]-x[0]);" "6#/*&%\"gG\"\"\"7%&%\"xG6#\"\"!&F)6#F&&F)6# \"\"#F&*&,&*&F%F&7$&F)6#F&&F)6#F0F&F&*&F%F&7$&F)6#F+&F)6#F&F&!\"\"F&,& &F)6#F0F&&F)6#F+F?F?" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "g*[x[1], x[2] , x[3]] = (g*[x[2], x[3]]-g*[x[1], x[2]])/(x[3]-x[1]);" "6#/*&%\"gG\" \"\"7%&%\"xG6#F&&F)6#\"\"#&F)6#\"\"$F&*&,&*&F%F&7$&F)6#F-&F)6#F0F&F&*& F%F&7$&F)6#F&&F)6#F-F&!\"\"F&,&&F)6#F0F&&F)6#F&F?F?" }{TEXT -1 3 ", \+ " }{TEXT 267 7 ". . . ." }{TEXT -1 3 " ," }{XPPEDIT 18 0 "g*[x[n-2], \+ x[n-1], x[n]] = (g*[x[n-1], x[n]]-g*[x[n-2], x[n-1]])/(x[n]-x[n-2]);" "6#/*&%\"gG\"\"\"7%&%\"xG6#,&%\"nGF&\"\"#!\"\"&F)6#,&F,F&F&F.&F)6#F,F& *&,&*&F%F&7$&F)6#,&F,F&F&F.&F)6#F,F&F&*&F%F&7$&F)6#,&F,F&F-F.&F)6#,&F, F&F&F.F&F.F&,&&F)6#F,F&&F)6#,&F,F&F-F.F.F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 87 "The number of divided differences decreases at each ste p until we arrive at the single " }{TEXT 271 2 "n " }{TEXT -1 27 "th o rder divided difference" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "g*[x[0], ` . . .`, x[n]] = ( g*[x[1], ` . . . `, x[n]]-g*[x[0], ` . . . `, x[n-1]])/(x[n]-x[0]);" "6#/*&%\"gG\"\"\"7%&%\"xG6#\"\"!%'~.~.~.G&F)6#%\"nGF&*&,&*&F%F&7%&F)6# F&%)~~.~.~.~G&F)6#F/F&F&*&F%F&7%&F)6#F+%(~.~.~.~G&F)6#,&F/F&F&!\"\"F&F AF&,&&F)6#F/F&&F)6#F+FAFA" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Given lists of " }{TEXT 289 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "[x[0], x[1], ` . . . `, x[n]]; " "6#7&&%\"xG6#\"\"!&F%6#\"\"\"%(~.~.~.~G&F%6#%\"nG" }{TEXT -1 5 " and " }{TEXT 290 1 "y" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "[y[0], y[1] , ` . . . `, y[n]];" "6#7&&%\"yG6#\"\"!&F%6#\"\"\"%(~.~.~.~G&F%6#%\"nG " }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "y[i]=g(x[i])" "6#/&%\"yG6#%\" iG-%\"gG6#&%\"xG6#F'" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "i=0,1,` . . \+ . `,n" "6&/%\"iG\"\"!\"\"\"%(~.~.~.~G%\"nG" }{TEXT -1 75 ", the follow ing code will generate all the associated divided differences. " }} {PARA 0 "" 0 "" {TEXT -1 42 "You can try it out with various values of " }{TEXT 291 1 "n" }{TEXT -1 18 " (not too large!)." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 496 "n := 4:\n x := 'x': y := 'y':\nxvals := [seq(x[i],i=0..n)];\nyvals := [seq(y[i], i=0..n)];\nd := [yvals]:\nm := nops(xvals):\nprint(``);\nfor i to n do \n dd := [];\n for j from m-i+1 by -1 to 2 do\n t := (d[i,j]- d[i,j-1])/(xvals[j+i-1]-xvals[j-1]);\n dd := [t,op(dd)];\n end \+ do;\n if i%&xvalsG7'&%\"x G6#\"\"!&F'6#\"\"\"&F'6#\"\"#&F'6#\"\"$&F'6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7'&%\"yG6#\"\"!&F'6#\"\"\"&F'6#\"\"#&F'6#\"\"$ &F'6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%?4~order~1~divided~differences:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&*&,&&%\"yG6#\"\"\"F(&F&6#\"\"!!\"\"F(,&&%\"xGF'F(&F/F*F ,F,*&,&&F&6#\"\"#F(F%F,F(,&F.F,&F/F4F(F,*&,&&F&6#\"\"$F(F3F,F(,&F7F,&F /F;F(F,*&,&&F&6#\"\"%F(F:F,F(,&&F/FBF(F>F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%?3~order~2~divi ded~differences:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%*&,&*&,&&%\"yG6# \"\"#\"\"\"&F(6#F+!\"\"F+,&&%\"xGF-F.&F1F)F+F.F+*&,&F,F+&F(6#\"\"!F.F+ ,&F0F+&F1F6F.F.F.F+,&F2F+F9F.F.*&,&*&,&&F(6#\"\"$F+F'F.F+,&F2F.&F1F@F+ F.F+F%F.F+,&F0F.FCF+F.*&,&*&,&&F(6#\"\"%F+F?F.F+,&&F1FJF+FCF.F.F+F=F.F +,&FMF+F2F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%?2~order~3~divided~differences:G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$*&,&*&,&*&,&&%\"yG6#\"\"$\"\"\"&F*6#\"\"#!\"\"F-,&&% \"xGF/F1&F4F+F-F1F-*&,&F.F-&F*6#F-F1F-,&&F4F9F1F3F-F1F1F-,&F;F1F5F-F1F -*&,&F6F-*&,&F8F-&F*6#\"\"!F1F-,&F;F-&F4FBF1F1F1F-,&F3F-FEF1F1F1F-,&F5 F-FEF1F1*&,&*&,&*&,&&F*6#\"\"%F-F)F1F-,&&F4FOF-F5F1F1F-F'F1F-,&FRF-F3F 1F1F-F%F1F-,&FRF-F;F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%Csingle~order~4~divided~difference:G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&,&*&,&*&,&&%\"yG6#\"\"%\"\" \"&F,6#\"\"$!\"\"F/,&&%\"xGF-F/&F6F1F3F3F/*&,&F0F/&F,6#\"\"#F3F/,&&F6F ;F3F7F/F3F3F/,&F5F/F>F3F3F/*&,&F8F/*&,&F:F/&F,6#F/F3F/,&&F6FEF3F>F/F3F 3F/,&FGF3F7F/F3F3F/,&F5F/FGF3F3F/*&,&F@F/*&,&FBF/*&,&FDF/&F,6#\"\"!F3F /,&FGF/&F6FQF3F3F3F/,&F>F/FTF3F3F3F/,&F7F/FTF3F3F3F/,&F5F/FTF3F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Given lists of " }{TEXT 268 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "[x[0], x[1], ` . . . `, x[n]];" "6#7&&%\"xG6#\"\"!&F%6 #\"\"\"%(~.~.~.~G&F%6#%\"nG" }{TEXT -1 5 " and " }{TEXT 269 1 "y" } {TEXT -1 8 " values " }{XPPEDIT 18 0 "[y[0], y[1], ` . . . `, y[n]];" "6#7&&%\"yG6#\"\"!&F%6#\"\"\"%(~.~.~.~G&F%6#%\"nG" }{TEXT -1 8 ", wher e " }{XPPEDIT 18 0 "y[i]=g(x[i])" "6#/&%\"yG6#%\"iG-%\"gG6#&%\"xG6#F' " }{TEXT -1 5 " for " }{XPPEDIT 18 0 "i=0,1,` . . . `,n" "6&/%\"iG\"\" !\"\"\"%(~.~.~.~G%\"nG" }{TEXT -1 26 ", the following procedure " } {TEXT 0 12 "divideddiffs" }{TEXT -1 24 " constructs the list of " } {TEXT 260 74 "divided differences which are required by the Newton int erpolation formula" }{TEXT -1 65 ". These are the first members of the sequences above, along with " }{XPPEDIT 18 0 "g(x[0]);" "6#-%\"gG6#&% \"xG6#\"\"!" }{TEXT -1 162 ", which could be considered as a 0 th orde r divided difference. The other divided differences given above are ne eded during the computation, but are not retained." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "divideddiffs" {MPLTEXT 1 0 358 "divideddiffs := proc(xvals::list,yvals::list)\n local i,j,n,m,d;\n \+ n := nops(xvals);\n m := nops(yvals);\n if n<>m then \n err or \"the two lists must have the same length\"\n end if;\n d := yv als;\n\011 for i from 2 to n do\n\011 for j from n by -1 to i do \n\011 d[j] := (d[j]-d[j-1])/(xvals[j]-xvals[j-i+1]);\n\011 \+ end do;\n end do;\n d;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "You can check that it produces the c orrect results with various values of " }{TEXT 270 1 "n" }{TEXT -1 18 " (not too large!)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "n := 3:\nx := array(0..n):\ny := array(0.. n):\nxvals := [seq(x[i],i=0..n)];\nyvals := [seq(y[i],i=0..n)];\ndivid eddiffs(xvals,yvals);" }}{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#7&&%\"yG6#\"\"!*&,&&F%6#\"\"\"F,F$ !\"\"F,,&&%\"xGF+F,&F0F&F-F-*&,&*&,&&F%6#\"\"#F,F*F-F,,&&F0F7F,F/F-F-F ,F(F-F,,&F:F,F1F-F-*&,&*&,&*&,&&F%6#\"\"$F,F6F-F,,&&F0FCF,F:F-F-F,F4F- F,,&FFF,F/F-F-F,F2F-F,,&FFF,F1F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 136 " \nFor a numerical example, we take the set of equally spaced data poin ts used previously to find a degree 6 polynomial approximation for " } {XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 17 " on the interv al " }{XPPEDIT 18 0 "[0, Pi/2];" "6#7$\"\"!*&%#PiG\"\"\"\"\"#!\"\"" } {TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "h := eval f(Pi/12):\nxvals := [seq(h*(i-1),i=1..7)];\nyvals := [seq(sin(xvals[i] ),i=1..7)];\ndivdiffs := divideddiffs(xvals,yvals);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%&xvalsG7)$\"\"!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*$\"\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)divdiff sG7)$\"\"!F'$\"+&Hfh))*!#5$!+p2s'G\"F*$!+mglE:F*$\"+%)plf?!#6$\"+yU\\< l!#7$!+Lx*Rl*!#8" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 32 "Th e Newton interpolation formula" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 15 "For a function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 6 ", the " }{TEXT 260 28 "Newton \+ interpolation formula" }{TEXT -1 51 " involving the divided difference s associated with " }{TEXT 273 1 "x" }{TEXT -1 8 " values " }{XPPEDIT 18 0 "x[0],x[1],` . . . `,x[n];" "6&&%\"xG6#\"\"!&F$6#\"\"\"%(~.~.~.~G &F$6#%\"nG" }{TEXT -1 5 " and " }{TEXT 272 1 "y" }{TEXT -1 8 " values \+ " }{XPPEDIT 18 0 "y[0],y[1],` . . . `,y[n];" "6&&%\"yG6#\"\"!&F$6#\"\" \"%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "y[i]=g( x[i])" "6#/&%\"yG6#%\"iG-%\"gG6#&%\"xG6#F'" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "i=0,1,` . . . `,n" "6&/%\"iG\"\"!\"\"\"%(~.~.~.~G%\"nG " }{TEXT -1 5 ", is:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(x) = g(x[0])+g*[x[0], x[1]]*(x -x[0])+g*[x[0], x[1], x[2]]*(x-x[0])*(x-x[1])+g*[x[0], x[1], x[2], x[3 ]]*(x-x[0])*(x-x[1])*(x-x[2])+` . . . `+g*[x[0], `. . `,x[n]](x-x[0])* (x-x[1])*` . . . `*(x-x[n-1])" "6#/-%\"pG6#%\"xG,.-%\"gG6#&F'6#\"\"!\" \"\"*(F*F/7$&F'6#F.&F'6#F/F/,&F'F/&F'6#F.!\"\"F/F/**F*F/7%&F'6#F.&F'6# F/&F'6#\"\"#F/,&F'F/&F'6#F.F9F/,&F'F/&F'6#F/F9F/F/*,F*F/7&&F'6#F.&F'6# F/&F'6#FB&F'6#\"\"$F/,&F'F/&F'6#F.F9F/,&F'F/&F'6#F/F9F/,&F'F/&F'6#FBF9 F/F/%(~.~.~.~GF/*,F*F/-7%&F'6#F.%%.~.~G&F'6#%\"nG6#,&F'F/&F'6#F.F9F/,& F'F/&F'6#F/F9F/FgnF/,&F'F/&F'6#,&F`oF/F/F9F9F/F/" }{TEXT -1 2 ". " }} {PARA 256 "" 0 "" {TEXT -1 3 " " }{TEXT 262 40 "____________________ ____________________" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 73 ": This formula ma y be compared with the Taylor polynomial for a function " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x=x[0 ]" "6#/%\"xG&F$6#\"\"!" }{TEXT -1 2 ": " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "q(x) = g(x[0])+g*`'`(x[0])*(x-x[0])+``(g*`''`(x[ 0])/2!)*(x-x[0])^2+``(g*`'''`(x[0])/3!)*(x-x[0])^3+` . . . `+``(`@@`(g ,n)*``(x[0])/n!)*(x-x[0])^n;" "6#/-%\"qG6#%\"xG,.-%\"gG6#&F'6#\"\"!\" \"\"*(F*F/-%\"'G6#&F'6#F.F/,&F'F/&F'6#F.!\"\"F/F/*&-%!G6#*(F*F/-%#''G6 #&F'6#F.F/-%*factorialG6#\"\"#F9F/*$,&F'F/&F'6#F.F9FGF/F/*&-F<6#*(F*F/ -%$'''G6#&F'6#F.F/-FE6#\"\"$F9F/*$,&F'F/&F'6#F.F9FWF/F/%(~.~.~.~GF/*&- F<6#*(-%#@@G6$F*%\"nGF/-F<6#&F'6#F.F/-FE6#F^oF9F/),&F'F/&F'6#F.F9F^oF/ F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 15 "The polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG " }{TEXT -1 34 " has degree less than or equal to " }{XPPEDIT 18 0 "n; " "6#%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "For " } {TEXT 263 7 "example" }{TEXT -1 5 ", if " }{XPPEDIT 18 0 "n = 1" "6#/% \"nG\"\"\"" }{TEXT -1 20 ", we have two points" }{XPPEDIT 18 0 "``(x[0 ],g(x[0]));" "6#-%!G6$&%\"xG6#\"\"!-%\"gG6#&F'6#F)" }{TEXT -1 4 " and " }{XPPEDIT 18 0 "``(x[1],g(x[1]));" "6#-%!G6$&%\"xG6#\"\"\"-%\"gG6#&F '6#F)" }{TEXT -1 5 ", and" }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "p(x) = g(x[0])+(g(x[1])-g(x[0]))/(x[1]-x[0])*(x-x[0]); " "6#/-%\"pG6#%\"xG,&-%\"gG6#&F'6#\"\"!\"\"\"*(,&-F*6#&F'6#F/F/-F*6#&F '6#F.!\"\"F/,&&F'6#F/F/&F'6#F.F:F:,&F'F/&F'6#F.F:F/F/" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 72 "This is the equation of the straight \+ line passing through the two points" }{XPPEDIT 18 0 "``(x[0],g(x[0])); " "6#-%!G6$&%\"xG6#\"\"!-%\"gG6#&F'6#F)" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(x[1],g(x[1]));" "6#-%!G6$&%\"xG6#\"\"\"-%\"gG6#&F'6#F)" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The following procedure implements this formula." }} {PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 16 ": The procedure " } {TEXT 0 12 "divideddiffs" }{TEXT -1 36 " introduced in the previous se ction " }{TEXT 260 14 "must be active" }{TEXT -1 19 " for the procedur e " }{TEXT 0 10 "newtinterp" }{TEXT -1 38 " to work, so it is also inc luded here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " newtoninterp" {MPLTEXT 1 0 788 "divideddiffs := proc(xvals::list,yvals ::list)\n local i,j,n,m,d;\n n := nops(xvals);\n m := nops(yvals );\n if n<>m then \n error \"the two lists must have the same l ength\"\n end if;\n d := yvals;\n\n\011 for i from 2 to n do\n \011 for j from n by -1 to i do\n\011 d[j] := (d[j]-d[j-1]) /(xvals[j]-xvals[j-i+1]);\n\011 end do;\n end do;\n d;\nend pr oc: # of divideddiffs\n\nnewtinterp := proc(xvals::list,divdiffs::list ,x)\n local sum,term,prod,n,m,i;\n \n n := nops(xvals);\n m := nops(divdiffs);\n if n<>m then\n error \"the two lists must ha ve the same length\"\n end if; \n\011 prod := 1;\n\011 sum := divd iffs[1];\n\011\011 for i from 1 to n-1 do\n\011 prod := prod*(x-x vals[i]);\n\011\011 term := prod*divdiffs[i+1];\n\011\011 sum := sum+term;\n end do;\n sum;\nend proc: # of newtinterp " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The procedure c an be checked below. Since the " }{TEXT 274 1 "x" }{TEXT -1 68 " is us ed as the subscripted variable for the members of the list of " } {TEXT 275 1 "x" }{TEXT -1 21 " values, the symbol \"" }{TEXT 276 1 "z " }{TEXT -1 55 "\" is used to designate the intermediate input variabl e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 144 "z := 'z':\nn := 4:\nx := array(0..n):\nd := array(0. .n):\nxvals := [seq(x[i],i=0..n)];\ndivdiffs := [seq(d[i],i=0..n)];\nn ewtinterp(xvals,divdiffs,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xva lsG7'&%\"xG6#\"\"!&F'6#\"\"\"&F'6#\"\"#&F'6#\"\"$&F'6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)divdiffsG7'&%\"dG6#\"\"!&F'6#\"\"\"&F'6# \"\"#&F'6#\"\"$&F'6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,&%\"dG6 #\"\"!\"\"\"*&,&%\"zGF(&%\"xGF&!\"\"F(&F%6#F(F(F(*(F*F(,&F+F(&F-F0F.F( &F%6#\"\"#F(F(**F*F(F2F(,&F+F(&F-F5F.F(&F%6#\"\"$F(F(*,F*F(F2F(F8F(,&F +F(&F-F;F.F(&F%6#\"\"%F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "We can let Maple do the algebra to check that \+ the Newton formula does indeed provide the unique interpolating polyno mial." }}{PARA 0 "" 0 "" {TEXT -1 11 "Don't take " }{TEXT 277 1 "n" } {TEXT -1 47 " to be too large, or you may run out of memory." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "n := 4:\nx := array(0..n):\ny := array(0..n):\nxvals := [seq(x[i],i=0.. n)];\nyvals := [seq(y[i],i=0..n)];\ndivdiffs := divideddiffs(xvals,yva ls);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7'&%\"xG6#\"\"!&F'6# \"\"\"&F'6#\"\"#&F'6#\"\"$&F'6#\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%&yvalsG7'&%\"yG6#\"\"!&F'6#\"\"\"&F'6#\"\"#&F'6#\"\"$&F'6#\"\"%" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)divdiffsG7'&%\"yG6#\"\"!*&,&&F'6# \"\"\"F.F&!\"\"F.,&&%\"xGF-F.&F2F(F/F/*&,&*&,&&F'6#\"\"#F.F,F/F.,&&F2F 9F.F1F/F/F.F*F/F.,&FF/F .,&FWF.F3F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 34 "\nYou can check that ev aluation at " }{XPPEDIT 18 0 "x[i];" "6#&%\"xG6#%\"iG" }{TEXT -1 7 " g ives " }{XPPEDIT 18 0 "y[i];" "6#&%\"yG6#%\"iG" }{TEXT -1 20 " for i f rom 1 to n.\n" }{TEXT 260 4 "Note" }{TEXT -1 29 ": Unlike the Maple pr ocedure " }{TEXT 0 6 "interp" }{TEXT -1 20 ", and the procedure " } {TEXT 0 8 "lagrange" }{TEXT -1 16 ", the procedure " }{TEXT 0 10 "newt interp" }{TEXT -1 88 " requires the second argument to be the list of \+ divided differences and not the list of " }{TEXT 281 1 "y" }{TEXT -1 8 " values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 29 "newtinterp(xvals,divdiffs,z);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,&%\"yG6#\"\"!\"\"\"*(,&%\"zGF(&%\"xGF&!\"\"F(,&&F%6#F (F(F$F.F(,&&F-F1F(F,F.F.F(**F*F(,&F+F(F3F.F(,&*&,&&F%6#\"\"#F(F0F.F(,& &F-F:F(F3F.F.F(*&F/F(F2F.F.F(,&F=F(F,F.F.F(*,F*F(F5F(,&F+F(F=F.F(,&*&, &*&,&&F%6#\"\"$F(F9F.F(,&&F-FHF(F=F.F.F(F7F.F(,&FKF(F3F.F.F(*&F6F(F?F. F.F(,&FKF(F,F.F.F(*.F*F(F5F(FAF(,&F+F(FKF.F(,&*&,&*&,&*&,&&F%6#\"\"%F( FGF.F(,&&F-FYF(FKF.F.F(FEF.F(,&FfnF(F=F.F.F(FCF.F(,&FfnF(F3F.F.F(*&FBF (FNF.F.F(,&FfnF(F,F.F.F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 102 "We can check that the Newton interpolating polyno mial does indeed go through the interpolation points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 233 "simplify (newtinterp(xvals,divdiffs,x[0]));\nsimplify(newtinterp(xvals,divdiffs ,x[1]));\nsimplify(newtinterp(xvals,divdiffs,x[2]));\nsimplify(newtint erp(xvals,divdiffs,x[3]));\nsimplify(newtinterp(xvals,divdiffs,x[4])); \nunassign('x','y'):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"yG6#\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"yG6#\"\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#&%\"yG6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\" yG6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#&%\"yG6#\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 64 "A procedure for constructing \+ a Newton interpolating polynomial: " }{TEXT 0 13 "newton_interp" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 162 "It is co nvenient to have a procedure which will construct a Newton interpolati ng polynomial without first having to construct the required divided d ifferences. " }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 13 "newton_interp" }{TEXT -1 44 " presented formally here, uses the \+ lists of " }{TEXT 282 1 "x" }{TEXT -1 5 " and " }{TEXT 283 1 "y" } {TEXT -1 104 " values as the first two parameters. This procedure may \+ be used in a similar way to the Maple procedure " }{TEXT 0 6 "interp" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "newt on_interp: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 16 "Calling Sequence" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 59 " newton_interp( x , y, z ) or newtoninterp( x, y, z ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 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 280 11 "Descript ion" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " } {TEXT 0 13 "newton_interp" }{TEXT -1 134 " constructs the Lagrange int erpolating polynomial for the data given in the lists x and y, using z 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 t o activate:" }{TEXT 256 1 "\n" }{TEXT -1 155 "To make the procedures a ctive open the subsection, place the cursor anywhere after the prompt \+ [ > and press [Enter].\nYou can then close up the subsection." }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "newton_interp: implementation" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 722 "newtoninterp := proc() newton_interp(args[1..nargs]) end:\n\nnewt on_interp := proc(xvals::list,yvals::list,x::algebraic)\n local d,su m,term,prod,n,m,i,j;\n \n n := nops(xvals);\n m := nops(yvals); \n if n<>m then\n error \"the two lists must have the same leng th\"\n end if;\n d := yvals;\n\n # construct the required divide d differences\n\011 for i from 2 to n do\n\011 for j from n by -1 to i do\n\011 d[j] := (d[j]-d[j-1])/(xvals[j]-xvals[j-i+1]);\n \011 end do;\n end do;\n \n # construct the Newton interpolati ng polynomial\n\011 prod := 1;\n\011 sum := d[1];\n\011\011 for i f rom 1 to n-1 do\n\011 prod := prod*(x-xvals[i]);\n\011\011 term := prod*d[i+1];\n\011\011 sum := sum+term;\n end do;\n sum;\ne nd proc: # of newton_interp" }}}{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 39 "Examples are given in the next \+ section." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 13 "newton_interp" }{TEXT -1 11 ": examples " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 59 "We find the Newton interpolating polynomial for the point s " }{XPPEDIT 18 0 "``(1,5),``(2,3),``(4,2),``(5,4),``(6,3);" "6'-%!G6 $\"\"\"\"\"&-F$6$\"\"#\"\"$-F$6$\"\"%F*-F$6$F'F.-F$6$\"\"'F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "xvals := [1,2,4,5,6];\nyvals := [5,3,2,4,3];\npx := n ewton_interp(xvals,yvals,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xva lsG7'\"\"\"\"\"#\"\"%\"\"&\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%& yvalsG7'\"\"&\"\"$\"\"#\"\"%F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p xG,,\"\"(\"\"\"*&\"\"#F'%\"xGF'!\"\"*(F)F+,&F*F'F'F+F',&F*F'F)F+F'F'** \"#7F+F-F'F.F',&F*F'\"\"%F+F'F'*.F)F'\"#:F+F-F'F.F'F1F',&F*F'\"\"&F+F' F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "T he picture below shows the graph of the interpolating polynomial along with the interpolation points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "pts := [[1,5],[2,3],[4,2],[ 5,4],[6,3]];\nplot([pts,px],x=0..7,y=0..7,style=[point,line],symbol=ci rcle,\n color=[black,coral]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ptsG7'7$\"\"\"\"\"&7$\"\"#\"\"$7$\"\"%F*7$F(F-7$\"\" 'F+" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVES G6%7'7$$\"\"\"\"\"!$\"\"&F*7$$\"\"#F*$\"\"$F*7$$\"\"%F*F.7$F+F37$$\"\" 'F*F0-%'COLOURG6&%$RGBGF*F*F*-%&STYLEG6#%&POINTG-F$6%7[o7$$F*F*F.7$$\" 3gmmm\"z+e_\"!#=$\"3_@l<`$zE3$!#<7$$\"3sLL3->R`GFI$\"3')oz%e\\S7!QFL7$ $\"3mmm;apSYVFI$\"37T8$G*))e*Q%FL7$$\"3Onm;z'=$\\eFI$\"3!Q(y&e%*oex%FL 7$$\"3i+]7.I?(f'FI$\"3fCglHf/+\\FL7$$\"3!RL$3Ft3XtFI$\"3S['p9jqN)\\FL7 $$\"3J++DJ=ZQ!)FI$\"3/&FL7$$ \"3MML3Fe\")\\%*FI$\"3>4]*QE0%G]FL7$$\"33+](=`xn,\"FL$\"3Qqa@UQV))\\FL 7$$\"3#omT&y/Gl6FL$\"3'3\\W9y3I$[FL7$$\"3++]PurI88FL$\"3]/Y`p\"=\")f%F L7$$\"3aLL$e#3dl9FL$\"3)*4&*eNv<%H%FL7$$\"3ymm\"Ht%o*f\"FL$\"3'RNL5)\\ ')*)RFL7$$\"3K++]F_m]FL$\"3<49*Gd'>VK FL7$$\"3;++]s2O[?FL$\"3s0U#GwT9)GFL7$$\"3um;aG\"H5=#FL$\"3SsD)o*>EmDFL 7$$\"3^LL$ej%yQBFL$\"3&y\"=>fG2>AFL7$$\"3mLLLVUUsCFL$\"3EpP#z+,j&>FL7$ $\"35+](o()yyi#FL$\"3cY(\\^0_[p\"FL7$$\"3GLLLoD[lFFL$\"3:#pQ%=\\l3:FL7 $$\"3P+](oibk\"HFL$\"3'=:!)fM&)zN\"FL7$$\"3a+]i!o<-1$FL$\"3i5e7bVsp7FL 7$$\"3qLL3-$=-@$FL$\"3ZT*)=T(3oB\"FL7$$\"3kL$3xplzM$FL$\"3w*)\\mOk,g7F L7$$\"3gmm\"H([a'\\$FL$\"36pZX!3)=T8FL7$$\"3wm;ayo(3l$FL$\"3!*G2CPZA%[ \"FL7$$\"3?+]7VLA&y$FL$\"3Q]UHQ5r`;FL7$$\"3'pm;a?@.$RFL$\"3u)\\e.XI&y= FL7$$\"3)******\\\\@-3%FL$\"3a+BQ#)>?\\@FL7$$\"3Q++v$opoA%FL$\"3YO%Gfj vIW#FL7$$\"3c+](oMf(oVFL$\"3!zvC\"QjDXFFL7$$\"3#)***\\ii.j_%FL$\"39U.r cM\\)3$FL7$$\"3%GLL$oT'ym%FL$\"3[H<$)y^(4R$FL7$$\"3'3++DE5!>[FL$\"3%fb A![/X#p$FL7$$\"3Mm;a)3rf&\\FL$\"3.8JaJX7KRFL7$$\"3*4++vW0d5&FL$\"3;1W) RpZ.9%FL7$$\"3;L$3-\"QfY_FL$\"3Kln^]xVnUFL7$$\"39n;Hx#G-K&FL$\"3+Pkk/f ;,VFL7$$\"3C+]PWF'QR&FL$\"3hBqksA$*3VFL7$$\"3'o;/^*Q&eY&FL$\"3?[p)4eG( )G%FL7$$\"3[LL$e/Xy`&FL$\"3.`rn$32$QUFL7$$\"3emT&)3J@8cFL$\"3'G*>@kX4] TFL7$$\"3m**\\(=<\"e)o&FL$\"3FnggZ$yB-%FL7$$\"3%ymmm(zvLeFL$\"3+FU6&RD3$FL7$$\"3LM$3-7d%HhFL$\"3Uqc\"*Qt :)H#FL7$$\"31oTg2R5(>'FL$\"3=8!y^kFL$!3\")\\DbvP rV_!#?7$$\"3S+]7=p:*['FL$!3_w*\\zk!zZsFI7$$\"3'pmmmV,&elFL$!3chOw-^x;: FL7$$\"3,M3xcrVKmFL$!37n_K=22XCFL7$$\"3<+](o(GP1nFL$!39::k-PFkMFL7$$\" 3#3++]zQrx'FL$!37%z?Nr>\"HXFL7$$\"3g+]78Z!z%oFL$!3e%*G5ZKZ&o&FL7$$\"3W ]P%[`Gf)oFL$!31%4]o@UgM'FL7$$\"3I+DccB&R#pFL$!3N-/A-*y[.(FL7$$\"39]7Gy h(>'pFL$!3yP1'fxUEv(FL7$$\"\"(F*$!3++++++++&)FL-F:6&F<$\"*++++\"!\")$ \")AR!)\\F]alFE-F>6#%%LINEG-%'SYMBOLG6#%'CIRCLEG-%+AXESLABELSG6$Q\"x6 \"Q\"yF[bl-%%VIEWG6$;FEFe`lF`bl" 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 -1 117 "Expanding the terms of the New ton interpolating polynomial and collecting together terms involving t he same power of " }{TEXT 284 1 "x" }{TEXT -1 52 " gives the same poly nomial as that given by Maple's " }{TEXT 0 6 "interp" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "collect(px,x);\ninterp(xvals,yvals,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,\"\"#\"\"\"*&#F$\"#:F%*$)%\"xG\"\"%F%F%!\"\"*&#\"$,\" \"#gF%*$)F+\"\"$F%F%F%*&#\"$(RF1F%*$)F+F$F%F%F-*&#\"$@\"F(F%F+F%F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,\"\"#\"\"\"*&#F$\"#:F%*$)%\"xG\"\"%F %F%!\"\"*&#\"$,\"\"#gF%*$)F+\"\"$F%F%F%*&#\"$(RF1F%*$)F+F$F%F%F-*&#\"$ @\"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 9 "Exampl e 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 72 "We find the degree 6 Newton interpolating polynomial whic h approximates " }{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 30 " and agrees with the value of " } {XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 47 " at 7 equally \+ spaced data points between 0 and " }{XPPEDIT 18 0 "Pi/2" "6#*&%#PiG\" \"\"\"\"#!\"\"" }{TEXT -1 11 " inclusive." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "Digits := 15:\nh := \+ evalf(Pi/12);\nxvals := [seq(h*(i-1),i=1..7)];\nyvals := map(sin,xvals );\npx := newton_interp(xvals,yvals,x):\nDigits := 10:\np := evalf(una pply(px,x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"0\\\"*z(Q*zh# !#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7)$\"\"!F'$\"0\\\"*z(Q *zh#!#:$\"0)H)fv()fB&F*$\"0ZuRj\")R&yF*$\"0g'>^v>Z5!#9$\"0u&**Qp**38F1 $\"0*[zEjzq:F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&yvalsG7)$\"\"!F'$ \"0?D5X!>)e#!#:$\"0*************\\F*$\"0Zl=\"y1rqF*$\"0SWy.a-m)F*$\"0m !*GEe#f'*F*$\"0+++++++\"!#9" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf *6#%\"xG6\"6$%)operatorG%&arrowGF(,.9$$\"0o`YHfh))*!#:*($\"0(3Fn2s'G\" F0\"\"\"F-F4,&F-F4$\"0\\\"*z(Q*zh#F0!\"\"F4F8**$\"0+N)pglE:F0F4F-F4F5F 4,&F-F4$\"0)H)fv()fB&F0F8F4F8*,$\"0*[,aqlf?!#;F4F-F4F5F4F^v> Z5!#9F8F4F4*0$\"0]]5]()Rl*!#=F4F-F4F5F4F " 0 "" {MPLTEXT 1 0 40 "p(1.047197551);\nevalf(sin(1.047197551)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+MSDg')!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+PSDg')!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 34 "Elsewhere we get an approximation." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "p(1);\nevalf(sin(1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+.!4ZT )!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[)4ZT)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "We can obtain the us ual picture to illustrate the interpolation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "x := 'x':\npoints := evalf(zip((x,y)->[x,y],xvals,yvals));\nplot([points,p(x)],x=0..Pi/ 2,symbol=circle,\n color=[blue,red],style=[point,line]); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%'pointsG7)7$$\"\"!F(F'7$$\"+yQ*z h#!#5$\"+^/>)e#F,7$$\"+cx)fB&F,$\"+++++]F,7$$\"+M;)R&yF,$\"+7y1rqF,7$$ \"+^v>Z5!\"*$\"+QSDg')F,7$$\"+Rp**38F<$\"+j#e#f'*F,7$$\"+Fjzq:F<$\"+++ ++5F<" }}{PARA 13 "" 1 "" {GLPLOT2D 357 267 267 {PLOTDATA 2 "6'-%'CURV ESG6%7)7$$\"\"!F)F(7$$\"30+++yQ*zh#!#=$\"3y*****4X!>)e#F-7$$\"35+++cx) fB&F-$\"3++++++++]F-7$$\"3;+++M;)R&yF-$\"3M+++7y1rqF-7$$\"3++++^v>Z5!# <$\"3Q+++QSDg')F-7$$\"3.+++Rp**38F=$\"37+++j#e#f'*F-7$$\"3/+++Fjzq:F=$ \"\"\"F)-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%&STYLEG6#%&POINTG-F$6%7S F'7$$\"3NGK5j*))QU$!#>$\"35K*[[W0LU$Fen7$$\"3DXXYUk*HS'Fen$\"3I&*ov\"= Q()R'Fen7$$\"3=N68yVJ`(*Fen$\"35M\"y?Axzt*Fen7$$\"3%3h%eRSe78F-$\"3YYg p>%G)38F-7$$\"3k+,hQPB[;F-$\"3$H?(=4&)yS;F-7$$\"3'H#*ed(RUf>F-$\"3/2k, hY\"p%>F-7$$\"3kHX[TMk\"G#F-$\"3!Q!)yZX+>E#F-7$$\"3Saq%HF-$\"3(z(fwm%yX!HF-7$$\"3)pq'4OKt)G$F-$\"3** o'H9*owHKF-7$$\"3#f/N#RTo*e$F-$\"3]A:?cI38NF-7$$\"3Ut79wN[GRF-$\"3*Qns gq5#GQF-7$$\"3/-Ss3'F -7$$\"3oHyF.C6noF-$\"3T\\Eq@V(*RjF-7$$\"3(*zx.5Ir.sF-$\"3[M_X#pQmf'F-7 $$\"3'pz!QUu\"G^(F-$\"3uKZ]:4wDoF-7$$\"3A\"3@7LGi%yF-$\"3wiLc'G$elqF-7 $$\"3)3\"HIT&[D>)F-$\"3(\\\"R1'>sjI(F-7$$\"3gv)\\AI@S\\)F-$\"3\"*Q[*=$ f&)3vF-7$$\"3vn()=V,i>))F-$\"37mP\"[ot)>xF-7$$\"3'G:[dg&*f:*F-$\"3TF#) *o4z#HzF-7$$\"3sSt6rL2&[*F-$\"3_A:3n?YD\")F-7$$\"3mka(*HIZ.)*F-$\"3CuT lhm!pI)F-7$$\"3Em\"[l:+d,\"F=$\"33\"ydG+(\\)\\)F-7$$\"3c!3XyEmu/\"F=$ \"3Gdk&RJ(fh')F-7$$\"3sJ_*>P$Q\"3\"F=$\"32g%pTf!4E))F-7$$\"3.M\"G%4t67 6F=$\"3))\\V7YIQm*)F-7$$\"3co*Q4iL+(eB*F-7$$\"3Hk&>q'*z.@\"F=$\"35JwQQK]d$*F-7$$\"34Q$[) >&*oU7F=$\"3=C-3cMam%*F-7$$\"3I^lOFY^w7F=$\"3)>^%3+(3,d*F-7$$\"3`!)f6E A448F=$\"3KFx]N[]f'*F-7$$\"3=;T7EvSU8F=$\"3Uk)y>dA.u*F-7$$\"3a_wiepWv8 F=$\"3zGk0`\\z4)*F-7$$\"3U$obdw1eS\"F=$\"33c\\y'H%>k)*F-7$$\"3s$)o#3`- 1W\"F=$\"3O)R?(Q)e`\"**F-7$$\"3-#*)4zFCpT***F-7$$\"3 +++lBjzq:F=$\"2i3*************F=-FK6&FMFNF(F(-FR6#%%LINEG-%+AXESLABELS G6$Q\"x6\"Q!F\\^l-%'SYMBOLG6#%'CIRCLEG-%%VIEWG6$;F($\"+Fjzq:!\"*%(DEFA ULTG" 1 2 4 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 45 "\nAs usual we can plot th e absolute error for " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 25 " as an approximation for " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6# %\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[0,Pi/2]" "6#7 $\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "plot(p(x)-si n(x),x=0..Pi/2,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 357 267 267 {PLOTDATA 2 "6&-%'CURVESG6#7hq7$$\"\"!F)F(7$$\"3WN!zQ?h)zU!#?$\"3>wf]# HkkV\"!#C7$$\"3)32exSA(f&)F-$\"3i[\\^UENdFF07$$\"3s5P;h$eRG\"!#>$\"35/ 'QE0oz'RF07$$\"3=9;b\"[W>r\"F9$\"3#Rd/2\">Xt]F07$$\"3W@uKAn\"zc#F9$\"3 S4x4\">-)))pF07$$\"3NGK5j*))QU$F9$\"3E)*=(fbE9a)F07$$\"3X')Qy-FW8\\F9$ \"3o<%)\\ThJ[5!#B7$$\"3DXXYUk*HS'F9$\"3k\"QEfdSsF9$\"3W\"*z`'3IL>\"FP7$$\"3+<&R$ocOfw F9$\"3dbaG>%>E?\"FP7$$\"3!4%yH5a:y!)F9$\"3I\"FP7$$\"3=N68yVJ`(*F9$\"3KCey&>f!*=\"FP7$$\"3Di))pQx &R9\"!#=$\"3t(zIWM%z?6FP7$$\"3%3h%eRSe78F[q$\"3/\"[Hh;5p,\"FP7$$\"3ubt 4*))3/[\"F[q$\"3eagg9Su+*)F07$$\"3k+,hQPB[;F[q$\"3G?D\\+%\\v\\(F07$$\" 3!=^%=d)GQ!=F[q$\"3YS(or\\cs9'F07$$\"3'H#*ed(RUf>F[q$\"3!G&)H$3vt.[F07 $$\"3IET3cokMF07$$\"3kHX[TMk\"G#F[q$\"31yz^&*[a9AF07 $$\"3E&4\"odLE[CF[q$\"3@\"*QO!fKQ/\"F07$$\"3Saq%HF[q$!3_(f6U t4Qb\"F07$$\"3on$***=Q*y6$F[q$!3'*)G17&el5@F07$$\"3)pq'4OKt)G$F[q$!3=? \"oljYB]#F07$$\"3'=H\")='4(RO$F[q$!3Wi'G;)o'[i#F07$$\"3twem(o3#RMF[q$! 3<`n%)**fF=FF07$$\"3gh/X8kW9NF[q$!3;'\\o(zYu$y#F07$$\"3#f/N#RTo*e$F[q$ !3!QiS3TSD#GF07$$\"3!Ggh%)*RQuOF[q$!322Cksn)f$GF07$$\"3of\")odQ3fPF[q$ !3**Q\"\\%RsP>GF07$$\"3a;Z\"pr$yVQF[q$!3Sfc-.w$[x#F07$$\"3Ut79wN[GRF[q $!3$[(eaM_c/FF07$$\"3]XR6m)y&)4%F[q$!3cn%RV*)H[\\#F07$$\"3/^ P'F[q$\"3)3*p$=')[.Y\"F07$$\"3.n3p46^WlF[q$\"3;iuVA)o\"f9F07$$\"3!yM%[ c<\"eq'F[q$\"3YDCZ)f,fS\"F07$$\"3oHyF.C6noF[q$\"3n%3$z[QQ08F07$$\"3(*z x.5Ir.sF[q$\"3#o#Q+^N9^'*Fat7$$\"3'pz!QUu\"G^(F[q$\"3s'[5\"zfMr`Fat7$$ \"3A\"3@7LGi%yF[q$\"3m$*)R-gRUC\"F[t7$$\"3)3\"HIT&[D>)F[q$!3golsB[6(G& Fat7$$\"3gv)\\AI@S\\)F[q$!3k))F[q$!33'=@[-X !e7F07$$\"3'3YoW(y!y)*)F[q$!3-eqvi]hf8F07$$\"3'G:[dg&*f:*F[q$!3AaMlfn) 4T\"F07$$\"3zYFV)[M0K*F[q$!3yve#f`J%49F07$$\"3sSt6rL2&[*F[q$!3YF+3tF\" [N\"F07$$\"3p-ka+KFW'*F[q$!3?uDb\"R)4^7F07$$\"3mka(*HIZ.)*F[q$!3s.-g;# e$)4\"F07$$\"3Em\"[l:+d,\"!#<$!3CW?*GRp'yfFat7$$\"3c!3XyEmu/\"Fi_l$\"3 [u7slfReb!#F7$$\"3sJ_*>P$Q\"3\"Fi_l$\"3(y$\\TEr&*=uFat7$$\"3.M\"G%4t67 6Fi_l$\"3#*p/D0h@89F07$$\"3co*Q4i\"Fi_l$\"3vDju?z0TDF07$$\" 3Bi*[\"*>=@?\"Fi_l$\"3kE$Rp5vPb#F07$$\"3Hk&>q'*z.@\"Fi_l$\"3$HjPOC')4a #F07$$\"3pdnAbtX=7Fi_l$\"3=jT!=qQB]#F07$$\"3I^RVVZ`E7Fi_l$\"3B&oPduzkV #F07$$\"3qW6kJ@hM7Fi_l$\"3!z$>EGo;UBF07$$\"34Q$[)>&*oU7Fi_l$\"3fGd$o<# G=AF07$$\"3qWugt?gf7Fi_l$\"3qN*=5f&)*e=F07$$\"3I^lOFY^w7Fi_l$\"37J&ff. 'Qf8F07$$\"3`!)f6EA448Fi_l$!3KO6N+Z+YZFa`l7$$\"3O[+7w)\\dK\"Fi_l$!3a7n &)R9C')*)Fat7$$\"3=;T7EvSU8Fi_l$!3pj$=oJ23\">F07$$\"3C%)ePUs#*e8Fi_l$! 3Z>R1Y`Y7IF07$$\"3a_wiepWv8Fi_l$!3_nu.Qq8&=%F07$$\"3'zm\">ioi!R\"Fi_l$ !3c#4hsFObH&F07$$\"3U$obdw1eS\"Fi_l$!39Hw$[[xBS'F07$$\"3Y$G\"H[Y?B9Fi_ l$!3\"[CNkB8Ch(F07$$\"3s$)o#3`-1W\"Fi_l$!3!oyy\"\\w['o)F07$$\"3w(QoVSj hX\"Fi_l$!3?g%*fmR0d%*F07$$\"3-#*)4zFC+[\"Fi_l$!3O\\9\"yh@.,\"FP7$ $\"37R0[Iw;%[\"Fi_l$!3$[\\ow94H,\"FP7$$\"3!\\vqYT:$)[\"Fi_l$!37'Rj$35W 75FP7$$\"3oq4'))>jC\\\"Fi_l$!3Zm2U*o5(35FP7$$\"3A'=^I)4h'\\\"Fi_l$!3#f 8T>V/:+\"FP7$$\"3+-9Cn(e2]\"Fi_l$!3nnEI33,1**F07$$\"3y<;V^l!\\]\"Fi_l$ !3W@6__$>xv*F07$$\"3D8JH'Q'y?:Fi_l$!3%G-Hp=nOy)F07$$\"3\\3Y:@imO:Fi_l$ !3e&)3Ef3kaqF07$$\"3[c%ynu)>X:Fi_l$!3=^yE@p0edF07$$\"3C/BSs7t`:Fi_l$!3 [5c\"o!yvpTF07$$\"37GU@Nv*zb\"Fi_l$!3E)[wc\\*>dKF07$$\"3+_h-)zjAc\"Fi_ l$!33M'f'zungAF07$$\"37w!Q31Ilc\"Fi_l$!3Y*\\z^s%Gw6F07$$\"3+++lBjzq:Fi _l$!3LQ]m#\\Nr8*!#J-%'COLOURG6&%$RGBGF(F($\"*++++\"!\")-%+AXESLABELSG6 $Q\"x6\"Q!F`^m-%%VIEWG6$;F($\"+Fjzq:!\"*%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "The coefficients of the i nterpolating polynomial can be obtained by using the Maple procedure \+ " }{TEXT 0 7 "collect" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "collect(p(x),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$)%\"xG\"\"'\"\"\"$!0]]5]()Rl*!#=*& $\"+Q0'3.\"!#6F()F&\"\"&F(F(*&$\"*3:/4#F/F()F&\"\"%F(!\"\"*&$\"+`Z'[l \"!#5F()F&\"\"$F(F7*&$\"(>PI$F;F()F&\"\"#F(F7*&$\"+c\\.+5!\"*F(F&F(F( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 172 "When each successive term of the Newton interpolating po lynomial is added we increase the degree by one and obtain a polynomia l which passes through one more of the points " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(x[0],g(x[0])),``(x[1],g(x[1])),``(x[ 3],g(x[3])),` . . . `,``(x[n],g(x[n]));" "6'-%!G6$&%\"xG6#\"\"!-%\"gG6 #&F'6#F)-F$6$&F'6#\"\"\"-F+6#&F'6#F3-F$6$&F'6#\"\"$-F+6#&F'6#F<%(~.~.~ .~G-F$6$&F'6#%\"nG-F+6#&F'6#FF" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 127 "The following commands construct Newton interpolating po lynomials of steadily increasing degree which interpolate the function " }{XPPEDIT 18 0 "g(x) = exp(-x^2);" "6#/-%\"gG6#%\"xG-%$expG6#,$*$F' \"\"#!\"\"" }{TEXT -1 8 " at the " }{TEXT 286 1 "x" }{TEXT -1 7 " valu es" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[0, 1/4];" "6#7$ \"\"!*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[0, 1/4, \+ 1/2];" "6#7%\"\"!*&\"\"\"F&\"\"%!\"\"*&F&F&\"\"#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[0, 1/4, 1/2, 3/4];" "6#7&\"\"!*&\"\"\"F&\"\"%!\"\"*&F &F&\"\"#F(*&\"\"$F&F'F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[0, 1/4, 1/ 2, 3/4, 1];" "6#7'\"\"!*&\"\"\"F&\"\"%!\"\"*&F&F&\"\"#F(*&\"\"$F&F'F(F &" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[0, 1/4, 1/2, 3/4, 1, 5/4];" "6#7 (\"\"!*&\"\"\"F&\"\"%!\"\"*&F&F&\"\"#F(*&\"\"$F&F'F(F&*&\"\"&F&F'F(" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "[0,1/4,1/2,3/4,1,5/4,3/2]" "6#7)\"\"! *&\"\"\"F&\"\"%!\"\"*&F&F&\"\"#F(*&\"\"$F&F'F(F&*&\"\"&F&F'F(*&F,F&F*F (" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[0,1/4,1/2,3/4,1,5/4,3/2,7/4]" "6 #7*\"\"!*&\"\"\"F&\"\"%!\"\"*&F&F&\"\"#F(*&\"\"$F&F'F(F&*&\"\"&F&F'F(* &F,F&F*F(*&\"\"(F&F'F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[0,1/4,1/2,3 /4,1,5/4,3/2,7/4,2]" "6#7+\"\"!*&\"\"\"F&\"\"%!\"\"*&F&F&\"\"#F(*&\"\" $F&F'F(F&*&\"\"&F&F'F(*&F,F&F*F(*&\"\"(F&F'F(F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 434 "f := x -> exp(-x^2);\np := 'p':\nleft := 0;\nright := 2;\ndeg := \+ 8;\nDigits := 15;\nh := (right-left)/deg:\nh := evalf(h);\nall_xvals : = [seq(h*(i-1),i=1..deg+1)];\nall_yvals := map(f,all_xvals);\nDigits : = 10;\nfor k from 2 to deg+1 do \n xvals := [seq(all_xvals[i],i=1..k )];\n yvals := [seq(all_yvals[i],i=1..k)];\n q[k] := unapply(evalf (newton_interp(xvals,yvals,x)),x);\n print(p[k](x)=q[k](x));print(`` );\nend do:\np := eval(q): q := 'q':" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%$expG6#,$*$)9$\"\"#\"\"\" !\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%leftG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rightG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$degG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"0++++++]#!#:" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%*all_xvalsG7,$\"\"!F'$\"0++++++]#!#:$\"0++++++ +&F*$\"0++++++](F*$\"0+++++++\"!#9$\"0++++++D\"F1$\"0++++++]\"F1$\"0++ ++++v\"F1$\"0+++++++#F1$\"0++++++D#F1" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*all_yvalsG7,$\"\"\"\"\"!$\"0wM\"G18%R*!#:$\"0092$y+)y(F+$\"0B4tC Gyp&F+$\"0U9%'Digit sG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"pG6#\"\"#6#%\"xG,&$\" \"\"\"\"!F-*&$\"+)[xMU#!#5F-F*F-!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"pG6#\"\"$6#%\"xG,($\" \"\"\"\"!F-*&$\"+)[xMU#!#5F-F*F-!\"\"*($\"++u--!)F2F-F*F-,&F*F-$\"++++ +DF2F3F-F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-&%\"pG6#\"\"%6#%\"xG,*$\"\"\"\"\"!F-*&$\"+)[xMU#!#5F- F*F-!\"\"*($\"++u--!)F2F-F*F-,&F*F-$\"+++++DF2F3F-F3**$\"+su41bF2F-F*F -F7F-,&F*F-$\"+++++]F2F3F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 12 "" 1 "" {XPPMATH 20 "6#/-&%\"pG6#\"\"&6#%\"xG,,$\"\"\"\"\"!F -*&$\"+)[xMU#!#5F-F*F-!\"\"*($\"++u--!)F2F-F*F-,&F*F-$\"+++++DF2F3F-F3 **$\"+su41bF2F-F*F-F7F-,&F*F-$\"+++++]F2F3F-F-*,$\"+?\"H1;%!#6F-F*F-F7 F-F=F-,&F*F-$\"+++++vF2F3F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-&%\"pG6#\"\"'6#%\"xG,.$\"\"\"\"\"! F-*&$\"+)[xMU#!#5F-F*F-!\"\"*($\"++u--!)F2F-F*F-,&F*F-$\"+++++DF2F3F-F 3**$\"+su41bF2F-F*F-F7F-,&F*F-$\"+++++]F2F3F-F-*,$\"+?\"H1;%!#6F-F*F-F 7F-F=F-,&F*F-$\"+++++vF2F3F-F-*.$\"+649a>F2F-F*F-F7F-F=F-FDF-,&F*F-$\" +++++5!\"*F3F-F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#/-&%\"pG6#\"\"(6#%\"xG,0$\"\"\"\"\"!F-*&$\"+)[xM U#!#5F-F*F-!\"\"*($\"++u--!)F2F-F*F-,&F*F-$\"+++++DF2F3F-F3**$\"+su41b F2F-F*F-F7F-,&F*F-$\"+++++]F2F3F-F-*,$\"+?\"H1;%!#6F-F*F-F7F-F=F-,&F*F -$\"+++++vF2F3F-F-*.$\"+649a>F2F-F*F-F7F-F=F-FDF-,&F*F-$\"+++++5!\"*F3 F-F3*0$\"+F.5))*)FCF-F*F-F7F-F=F-FDF-FJF-,&F*F-$\"++++]7FMF3F-F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 #/-&%\"pG6#\"\")6#%\"xG,2$\"\"\"\"\"!F-*&$\"+)[xMU#!#5F-F*F-!\"\"*($\" ++u--!)F2F-F*F-,&F*F-$\"+++++DF2F3F-F3**$\"+su41bF2F-F*F-F7F-,&F*F-$\" +++++]F2F3F-F-*,$\"+?\"H1;%!#6F-F*F-F7F-F=F-,&F*F-$\"+++++vF2F3F-F-*.$ \"+649a>F2F-F*F-F7F-F=F-FDF-,&F*F-$\"+++++5!\"*F3F-F3*0$\"+F.5))*)FCF- F*F-F7F-F=F-FDF-FJF-,&F*F-$\"++++]7FMF3F-F-*2$\"+6!R#[[!#7F-F*F-F7F-F= F-FDF-FJF-FQF-,&F*F-$\"+++++:FMF3F-F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-&%\"pG6#\"\"*6#%\"xG,4$\" \"\"\"\"!F-*&$\"+)[xMU#!#5F-F*F-!\"\"*($\"++u--!)F2F-F*F-,&F*F-$\"++++ +DF2F3F-F3**$\"+su41bF2F-F*F-F7F-,&F*F-$\"+++++]F2F3F-F-*,$\"+?\"H1;%! #6F-F*F-F7F-F=F-,&F*F-$\"+++++vF2F3F-F-*.$\"+649a>F2F-F*F-F7F-F=F-FDF- ,&F*F-$\"+++++5!\"*F3F-F3*0$\"+F.5))*)FCF-F*F-F7F-F=F-FDF-FJF-,&F*F-$ \"++++]7FMF3F-F-*2$\"+6!R#[[!#7F-F*F-F7F-F=F-FDF-FJF-FQF-,&F*F-$\"++++ +:FMF3F-F3*4$\"+!>p9J\"FCF-F*F-F7F-F=F-FDF-FJF-FQF-FXF-,&F*F-$\"++++]< FMF3F-F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "We can plot the various i nterpolating polynomials in the same picture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 207 "pts := eval f(zip((x,y)->[x,y],xvals,yvals)):\nplot([pts,seq(p[i](x),i=2..deg+1)], x=-0.5..2,y=-0.2..1.2,\n style=[point,line$deg],symbol =circle,\n color=[black,seq(COLOR(HUE,i/deg),i=0..deg-1)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 489 302 302 {PLOTDATA 2 "6.-%'CURVESG6%7+7$$\"\"! 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{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 580 "The origins of the theory of interpolating polynomials goes ba ck to Newton, who wanted to draw conclusions from the observed locatio n of comets at equally spaced time intervals as to their locations at \+ arbitrary times. He arrived at the problem of determining a 'geometric al curve' passing through any number of given points. He solved this p roblem by means of the interpolating polynomial which bears his name. \+ How highly he regarded his result is revealed by a letter to Oldenburg in 1676, in which he wrote that this was one of the most beautiful re sults he had ever achieved. " }}{PARA 0 "" 0 "" {TEXT -1 51 "Newton us ed his formula to give the exact value of " }{XPPEDIT 18 0 "Int(f(x),x =a..b)" "6#-%$IntG6$-%\"fG6#%\"xG/F);%\"aG%\"bG" }{TEXT -1 27 " in ter ms of the values of " }{XPPEDIT 18 0 "f(x[i])" "6#-%\"fG6#&%\"xG6#%\"i G" }{TEXT -1 17 " of the function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#% \"xG" }{TEXT -1 26 " at equally spaced points " }{XPPEDIT 18 0 "x[0]=a ,x[1],x[2],` . . . `,x[n]=b" "6'/&%\"xG6#\"\"!%\"aG&F%6#\"\"\"&F%6#\" \"#%(~.~.~.~G/&F%6#%\"nG%\"bG" }{TEXT -1 7 ", when " }{XPPEDIT 18 0 "f (x)" "6#-%\"fG6#%\"xG" }{TEXT -1 27 " is a polynomial of degree " } {TEXT 292 1 "n" }{TEXT -1 48 ". His student Cotes calculated the coeff icients " }{XPPEDIT 18 0 "c[0],c[1],` . . . `,c[n];" "6&&%\"cG6#\"\"!& F$6#\"\"\"%(~.~.~.~G&F$6#%\"nG" }{TEXT -1 37 " needed to calculate the integral as " }{XPPEDIT 18 0 "Sum(c[i]*f(x[i]),i=0..n)" "6#-%$SumG6$* &&%\"cG6#%\"iG\"\"\"-%\"fG6#&%\"xG6#F*F+/F*;\"\"!%\"nG" }{TEXT -1 5 " \+ for " }{XPPEDIT 18 0 "n<=10" "6#1%\"nG\"#5" }{TEXT -1 208 ". This work , based on Newton's interpolation formula, must have been awkward. Lag range's interpolation formula would have simplified the work, but this was only published in 1795 (more than 100 years later). " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Reference: P. Tura n, " }{TEXT 294 45 "On some open problems of approximation theory" } {TEXT -1 20 ", J. Approx. Theory " }{TEXT 293 2 "29" }{TEXT -1 16 " (1 980), 23-85. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 60 "(a) Find the Newton interpol ating polynomial for the points " }{XPPEDIT 18 0 "``(-2,1/4),``(-1,1/2 ),``(0,1),``(1,3/2),``(2,5/4)" "6'-%!G6$,$\"\"#!\"\"*&\"\"\"F*\"\"%F(- F$6$,$F*F(*&F*F*F'F(-F$6$\"\"!F*-F$6$F**&\"\"$F*F'F(-F$6$F'*&\"\"&F*F+ F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 86 "(b) Plot the graph of the polynomial found in (a) along with the interpolation points." }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________ " }}{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 40 "________________________________________" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q2 \+ " }}{PARA 0 "" 0 "" {TEXT -1 21 "(a) For the function " }{XPPEDIT 18 0 "g(x) = 1/(1+25*x^2);" "6#/-%\"gG6#%\"xG*&\"\"\"F),&F)F)*&\"#DF)*$F' \"\"#F)F)!\"\"" }{TEXT -1 76 ", show explicitly the Newton interpolati ng polynomials in the standard form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p(x) = g(x[0])+(x-x[0])*g*[x[0], x[1]]+(x-x[0])*(x- x[1])*g*[x[0], x[1], x[2]]+(x-x[0])*(x-x[1])*(x-x[2])*g*[x[0], x[1], x [2], x[3]]+` . . . `+(x-x[0])*(x-x[1])*` . . . `*(x-x[n-1])*g*[x[0], ` . . `, x[n]]" "6#/-%\"pG6#%\"xG,.-%\"gG6#&F'6#\"\"!\"\"\"*(,&F'F/&F'6# F.!\"\"F/F*F/7$&F'6#F.&F'6#F/F/F/**,&F'F/&F'6#F.F4F/,&F'F/&F'6#F/F4F/F *F/7%&F'6#F.&F'6#F/&F'6#\"\"#F/F/*,,&F'F/&F'6#F.F4F/,&F'F/&F'6#F/F4F/, &F'F/&F'6#FHF4F/F*F/7&&F'6#F.&F'6#F/&F'6#FH&F'6#\"\"$F/F/%(~.~.~.~GF/* .,&F'F/&F'6#F.F4F/,&F'F/&F'6#F/F4F/FgnF/,&F'F/&F'6#,&%\"nGF/F/F4F4F/F* F/7%&F'6#F.%%.~.~G&F'6#FcoF/F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "for the following lists of " }{TEXT 288 1 "x" }{TEXT -1 9 " values: " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[0, 1/4 ];" "6#7$\"\"!*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " [0, 1/4, 1/2];" "6#7%\"\"!*&\"\"\"F&\"\"%!\"\"*&F&F&\"\"#F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[0, 1/4, 1/2, 3/4];" "6#7&\"\"!*&\"\"\"F&\" \"%!\"\"*&F&F&\"\"#F(*&\"\"$F&F'F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " [0, 1/4, 1/2, 3/4, 1];" "6#7'\"\"!*&\"\"\"F&\"\"%!\"\"*&F&F&\"\"#F(*& \"\"$F&F'F(F&" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[0, 1/4, 1/2, 3/4, 1, 5/4];" "6#7(\"\"!*&\"\"\"F&\"\"%!\"\"*&F&F&\"\"#F(*&\"\"$F&F'F(F&*&\" \"&F&F'F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[0,1/4,1/2,3/4,1,5/4,3/2] " "6#7)\"\"!*&\"\"\"F&\"\"%!\"\"*&F&F&\"\"#F(*&\"\"$F&F'F(F&*&\"\"&F&F 'F(*&F,F&F*F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "[0,1/4,1/2,3/4,1,5/4, 3/2,7/4]" "6#7*\"\"!*&\"\"\"F&\"\"%!\"\"*&F&F&\"\"#F(*&\"\"$F&F'F(F&*& \"\"&F&F'F(*&F,F&F*F(*&\"\"(F&F'F(" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " [0,1/4,1/2,3/4,1,5/4,3/2,7/4,2]" "6#7+\"\"!*&\"\"\"F&\"\"%!\"\"*&F&F& \"\"#F(*&\"\"$F&F'F(F&*&\"\"&F&F'F(*&F,F&F*F(*&\"\"(F&F'F(F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "(b) Plot the graphs of the 8 polynomials from part (a) together in the same picture." }}{PARA 0 "" 0 "" {TEXT -1 39 "___________________ ____________________" }}{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 40 "_______________________________________ _" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 3 "Q3 " }}{PARA 0 "" 0 "" {TEXT -1 21 "(a) For the fun ction " }{XPPEDIT 18 0 "g(x) = ln(1+x);" "6#/-%\"gG6#%\"xG-%#lnG6#,&\" \"\"F,F'F," }{TEXT -1 76 ", show explicitly the Newton interpolating p olynomials in the standard form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "p(x) = g(x[0])+(x-x[0])*g*[x[0], x[1]]+(x-x[0])*(x-x[1] )*g*[x[0], x[1], x[2]]+(x-x[0])*(x-x[1])*(x-x[2])*g*[x[0], x[1], x[2], x[3]]+` . . . `+(x-x[0])*(x-x[1])*` . . . `*(x-x[n-1])*g*[x[0], `. . \+ `, x[n]]" "6#/-%\"pG6#%\"xG,.-%\"gG6#&F'6#\"\"!\"\"\"*(,&F'F/&F'6#F.! \"\"F/F*F/7$&F'6#F.&F'6#F/F/F/**,&F'F/&F'6#F.F4F/,&F'F/&F'6#F/F4F/F*F/ 7%&F'6#F.&F'6#F/&F'6#\"\"#F/F/*,,&F'F/&F'6#F.F4F/,&F'F/&F'6#F/F4F/,&F' F/&F'6#FHF4F/F*F/7&&F'6#F.&F'6#F/&F'6#FH&F'6#\"\"$F/F/%(~.~.~.~GF/*.,& F'F/&F'6#F.F4F/,&F'F/&F'6#F/F4F/FgnF/,&F'F/&F'6#,&%\"nGF/F/F4F4F/F*F/7 %&F'6#F.%%.~.~G&F'6#FcoF/F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "for the following lists of " }{TEXT 287 1 "x" }{TEXT -1 8 " val ues:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "[0,1/5],[0,1/ 5,2/5],[0,1/5,2/5,3/5],[0,1/5,2/5,3/5,4/5],[0,1/5,2/5,3/5,4/5,1]" "6'7 $\"\"!*&\"\"\"F&\"\"&!\"\"7%F$*&F&F&F'F(*&\"\"#F&F'F(7&F$*&F&F&F'F(*&F ,F&F'F(*&\"\"$F&F'F(7'F$*&F&F&F'F(*&F,F&F'F(*&F1F&F'F(*&\"\"%F&F'F(7(F $*&F&F&F'F(*&F,F&F'F(*&F1F&F'F(*&F7F&F'F(F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 84 "(b) Plot the g raphs of the 5 polynomials from part (a) together in the same picture. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "_____ __________________________________" }}{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 40 "_______________________________ _________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 17 "Code for picture " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 545 "p1 := plot(-1/10*x^2+1/2*x+18/5,x=-1.5..1.5,color=red):\np2 := \+ plot([[[-1.5,0],[1.5,0]],[[-1,0],[-1,3]],[[1,0],[1,4]]],\n color= black,linestyle=[1,2$2]):\np3 := plot([[-1,3],[1,4]],color=blue,thickn ess=2):\np4 := plot([[[-1,3],[1,4]]$3],color=black,style=point,\n \+ symbol=[circle,diamond,cross]):\nt1 := plots[textplot]([[-1.3,3.3 ,`(x0, g(x0) )`],\n [1.1,4.3,`(x1, g(x1) )`]]):\nt2 := plot s[textplot]([-1.6,2.9,`y = g(x)`],color=red):\nplots[display]([p1,p2,p 3,p4,t1,t2],tickmarks=[0,0],axes=none,\n view=[-1.6..1.5,0..4.3] );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 }