{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 "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 262 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 264 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis " -1 265 "Times" 1 12 128 0 0 1 0 1 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 "Grey Em phasis" -1 271 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "N ormal" -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 "Time s" 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 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 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 46 "Local Taylor series approximation of functions" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 20 "Version: 25.3.2007\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 56 "l oad interpolation and function approximation procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 271 10 "fcnapprx.m" } {TEXT -1 37 " contains the code for the procedure " }{TEXT 0 9 "loctay lor" }{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 th e one that follows, where the file path gives its location. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procdrs/f cnapprx.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 46 "load numerical integration procedures and data" }} {PARA 0 "" 0 "" {TEXT -1 18 "The Maple m-files " }{TEXT 271 6 "intg.m " }{TEXT -1 5 " and " }{TEXT 271 8 "gkdata.m" }{TEXT -1 36 " contain t he code for the procedure " }{TEXT 0 5 "GKint" }{TEXT -1 39 " and the \+ associated nodes and weights. " }}{PARA 0 "" 0 "" {TEXT -1 122 "They c an be read into a Maple session by commands similar to those that foll ow, where the file paths give their location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "read \"K:\\\\Maple/procdrs/intg.m\";\nread \"K: \\\\Maple/procdrs/gkdata.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Defining procedures and viewing Maple lib rary code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 32 " A Maple procedure has the form:" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 0 23 "procedur e_name := proc(" }{TEXT -1 59 " argument or arguments (input parameter s) of the procedure " }{TEXT 0 1 ")" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 24 " BODY OF PROCEDURE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 " The result of last line of code i s the value returned by the procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 0 10 "end proc; " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "fn := proc(x)\n \+ if x<1 then x+1 else 2 end if\nend proc; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fnGf*6#%\"xG6\"F(F(@%29$\"\"\",&F+F,F,F,\"\"#F(F(F( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "It m ay be necessary to use quotes when using the procedure in a plot comma nd." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot('fn(x)',x=0..3,y=0..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7U7$$\"\"!F)$\"\"\"F) 7$$\"3s******\\i9Rl!#>$\"31++]i9Rl5!#<7$$\"3/++vVA)GA\"!#=$\"34+]PC#)G A6F27$$\"3+++]Peui=F6$\"3%****\\Peui=\"F27$$\"3A++]i3&o]#F6$\"33++D'3& o]7F27$$\"3%)***\\(oX*y9$F6$\"3/+](oX*y98F27$$\"3z***\\P9CAu$F6$\"3#** *\\P9CAu8F27$$\"3!)***\\P*zhdVF6$\"3.+]P*zhdV\"F27$$\"31++v$>fS*\\F6$ \"3++]P>fS*\\\"F27$$\"3$)***\\(=$f%GcF6$\"35+](=$f%Gc\"F27$$\"3Q+++Dy, \"G'F6$\"3$*****\\#y,\"G;F27$$\"33++]7 F27$$\"3e+]P4_J&o*F6$\"31+v$4_J&o>F27$$\"3,++]siL-5F2$\"\"#F)7$$\"3,++ DJL(4.\"F2F\\q7$$\"3-+++!R5'f5F2F\\q7$$\"3)***\\P/QBE6F2F\\q7$$\"3!*** ***\\\"o?&=\"F2F\\q7$$\"31+]Pa&4*\\7F2F\\q7$$\"33+]7j=_68F2F\\q7$$\"33 ++vVy!eP\"F2F\\q7$$\"34+](=WU[V\"F2F\\q7$$\"3)****\\7B>&)\\\"F2F\\q7$$ \"3)***\\P>:mk:F2F\\q7$$\"3'***\\iv&QAi\"F2F\\q7$$\"31++vtLU%o\"F2F\\q 7$$\"3!******\\Nm'[F2F\\q7$$\"3z*****\\@80+#F2F\\q7$$\"31++]7,Hl ?F2F\\q7$$\"3()**\\P4w)R7#F2F\\q7$$\"3;++]x%f\")=#F2F\\q7$$\"3!)**\\P/ -a[AF2F\\q7$$\"3/+](=Yb;J#F2F\\q7$$\"3')****\\i@OtBF2F\\q7$$\"3')**\\P fL'zV#F2F\\q7$$\"3>+++!*>=+DF2F\\q7$$\"3-++DE&4Qc#F2F\\q7$$\"3=+]P%>5p i#F2F\\q7$$\"39+++bJ*[o#F2F\\q7$$\"33++Dr\"[8v#F2F\\q7$$\"3++++Ijy5GF2 F\\q7$$\"31+]P/)fT(GF2F\\q7$$\"31+]i0j\"[$HF2F\\q7$$\"\"$F)F\\q-%'COLO URG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q\"yF\\x-%%VIEWG6$;F(F _wFax" 1 2 0 1 10 0 2 9 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 19 "plot(fn,0..3,0..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7U7$\"\"!$\"\"\"F(7$$\"1+++]i9R l!#<$\"1++]i9Rl5!#:7$$\"1++vVA)GA\"!#;$\"1+]PC#)GA6F17$$\"1++]Peui=F5$ \"1++v$eui=\"F17$$\"1++]i3&o]#F5$\"1++D'3&o]7F17$$\"1++voX*y9$F5$\"1+] (oX*y98F17$$\"1++vVTAUPF5$\"1+]P9CAu8F17$$\"1++v$*zhdVF5$\"1+]P*zhdV\" F17$$\"1++v$>fS*\\F5$\"1+]P>fS*\\\"F17$$\"1++v=$f%GcF5$\"1+](=$f%Gc\"F 17$$\"1+++Dy,\"G'F5$\"1++]#y,\"G;F17$$\"1++]7F17$$\"1,]P4_J&o *F5$\"1+v$4_J&o>F17$$\"1++]siL-5F1$\"\"#F(7$$\"1++DJL(4.\"F1F[q7$$\"1+ ++!R5'f5F1F[q7$$\"1+]P/QBE6F1F[q7$$\"1+++:o?&=\"F1F[q7$$\"1+]Pa&4*\\7F 1F[q7$$\"1+]7j=_68F1F[q7$$\"1++vVy!eP\"F1F[q7$$\"1+](=WU[V\"F1F[q7$$\" 1++DJ#>&)\\\"F1F[q7$$\"1+]P>:mk:F1F[q7$$\"1+]iv&QAi\"F1F[q7$$\"1++vtLU %o\"F1F[q7$$\"1+++bjm[F1F[q7$$\"1+++:K^+?F1F[q7$$\"1++]7,Hl?F1F[q7$$\"1+]P4w)R7# F1F[q7$$\"1++]x%f\")=#F1F[q7$$\"1+]P/-a[AF1F[q7$$\"1+](=Yb;J#F1F[q7$$ \"1++]i@OtBF1F[q7$$\"1+]PfL'zV#F1F[q7$$\"1+++!*>=+DF1F[q7$$\"1++DE&4Qc #F1F[q7$$\"1+]P%>5pi#F1F[q7$$\"1+++bJ*[o#F1F[q7$$\"1++Dr\"[8v#F1F[q7$$ \"1+++Ijy5GF1F[q7$$\"1+]P/)fT(GF1F[q7$$\"1+]i0j\"[$HF1F[q7$$\"\"$F(F[q -%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$%!GFjw-%%VIEWG6$;F(F^w F^x" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 136 "It is possible to view the code of a procedure which has been previously defined in the current Maple session by means of the procedure " } {TEXT 0 4 "eval" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "eval(fn);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#f*6#%\"xG6\"F&F&@%29$\"\"\",&F)F*F*F*\"\"#F&F&F&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 9 "interface" }{TEXT -1 71 " can be used to reset the \+ value of certain interface variables such as " }{TEXT 0 11 "verbosepro c" }{TEXT -1 20 " which controls how " }{TEXT 0 4 "eval" }{TEXT -1 8 " works. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "To " }{TEXT 259 41 "view the code for a Maple library routine" } {TEXT -1 19 " (procedure) reset " }{TEXT 0 11 "verboseproc" }{TEXT -1 4 " to " }{TEXT 265 1 "2" }{TEXT -1 27 " from the default value of " } {TEXT 265 1 "1" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "For exa mple, we can see how the procedure " }{TEXT 0 9 "nextprime" }{TEXT -1 59 " is defined in terms of the prime number testing procedure " } {TEXT 0 7 "isprime" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "interface(verboseproc=2);\ne val(nextprime);\ninterface(verboseproc=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#%\"nG6$%\"iG%#t1G6#%aoCopyright~(c)~1990~by~the~Uni versity~of~Waterloo.~All~rights~reserved.GE\\s/\"\"\"\"\"#F-\"\"$F.\" \"&F/\"\"(F0\"#6F1\"#8\"\"*F1F2\"#<\"#B\"#H\"#;F4F4\"#>F8F5F6\"#J\"#DF 6@'-%%typeG6$9$%(integerG@%2F?F-F-C%@%/-%%iremG6$F?F-\"\"!>8%,&F?F,F,F ,>FK,&F?F,F-F,?(8$FKF-6\"4-%(isprimeG6#FPFQFP-F=6$F?%(numericGYQ9argum ent~must~be~integerFQ.-%*nextprimeG6#F?FQFQFQ" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 174 "A lot of Maple code is a ccessible in this way, but often you will find that it is quite diffic ult to track down the actual name of the procedure which actually does the work." }}{PARA 0 "" 0 "" {TEXT -1 78 "Here is the routine which p erforms 9-point Newton-Cotes numerical integration." }}{PARA 0 "" 0 " " {TEXT -1 96 "It is usually interfaced through other procedures which sort out the necessary input parameters." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "interface(verboseproc =2);\neval(`evalf/int/quanc8`);\ninterface(verboseproc=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6)%$funG%\"aG%\"bG%'relerrG%'abserrG%&flag sG%(HF_ModeG6G%'resultG%'errestG%&nofunG%%flagG%#w0G%#w1G%#w2G%#w3G%#w 4G%%areaG%#x0G%#f0G%&stoneG%%stepG%&cor11G%%tempG%&qprevG%%qnowG%&qdif 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FTFavQ,integral~=~FTF`tC$-F]p6%FjqF_p-Ffp6$Q5integrand~evals~=~%dFTFht -F]p6%FhuF_p-Ffp6&Q:on~%a~..~%a~integral~=~%aFTF[uFavF`t?(FTFfnFfnFT0F ]v,$*&FjqFfn-%%iquoG6$F]vFjqFfnFfnC$>F]vF[]m>F[v,&F[vFfnFfnFfq>F]v,&F] vFfnFfnFfn@%1F[vF^t>F_\\lF^tC&>Fdv&FW6#F[v>F_vFav>FfvFj`l?(FgdlFfnFfnF eoFc[lC$>&Fin6#,$*&FjqFfnFgdlFfnFfnFjdl>&F_oFa^mF`el>F`t,&F`tFfnFbtFfn @$/FdtF^tC&>F_uFdt>FcuFht>FfuF]tOF`t?(FTFfnFfnFTFc[lC%>83,&-F`cl6#F`tF fnFdtFfn@$0Fb_mFd_mC&>F_uFdt>FcuFht>FfuF]tOF`t>Fdt,$*&FjqFfnFdtFfnFfnF TFTFT" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "If you can figure out what parameters it needs, you can use it direct ly." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "flags := array(1..3):\n`evalf/int/quanc8`(x->x^2,0,2 .,0,.5*10^(-Digits),flags,false);\nprint(`Number of function evaluatio ns -->`,flags[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+nmmmE!\"*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%CNumber~of~function~evaluations~-->G \"#L" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "flags := array(1..3):\n`evalf/int/quanc8`(x->sqrt(x+ 0.0001),0,2.,0,.5*10^(-Digits),flags,false);\nprint(`Number of functio n evaluations -->`,flags[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+S )ed)=!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%CNumber~of~function~eval uations~-->G\"$D#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "Inserting " }{TEXT 0 28 "option `Copyright whatever`;" } {TEXT -1 51 " at the start of a procedure hides the actual code." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "bs := proc(x)\noption `Copyright 2001 Bart Simpson`;\n if x<1 th en exp(x) else exp(1) fi\nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#bsGf*6#%\"xG6\"6#% " 0 "" {MPLTEXT 1 0 61 "interface(verboseproc=2);\n eval(bs);\ninterface(verboseproc=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#f*6#%\"xG6\"6#% " 0 "" {MPLTEXT 1 0 21 "plot('bs(x)',x=0..3);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7V7$$\"\"!F)$\"\"\"F) 7$$\"3s******\\i9Rl!#>$\"3G7fS*\\F6$\"3 oZEA)3Uxk\"F27$$\"3$)***\\(=$f%GcF6$\"3I:LL\"*=mb6@\\)>F27$$\"3`+++v4&G](F 6$\"3>(eoSl.w6#F27$$\"3!)*****\\7nD:)F6$\"3_v@V0dvfAF27$$\"3[+++D!*oy( )F6$\"3Rwf$[JndS#F27$$\"3=+]Pf$yH1*F6$\"3?6s(y;U^Z#F27$$\"3))***\\Ppns M*F6$\"35gj3wv^YDF27$$\"3e+]P4_J&o*F6$\"3Y'4Gg`tSj#F27$$\"3,++]siL-5F2 $\"34X!f%G=G=FF27$$\"3,++DJL(4.\"F2Faq7$$\"3-+++!R5'f5F2Faq7$$\"3)*** \\P/QBE6F2Faq7$$\"3!******\\\"o?&=\"F2Faq7$$\"31+]Pa&4*\\7F2Faq7$$\"33 +]7j=_68F2Faq7$$\"33++vVy!eP\"F2Faq7$$\"34+](=WU[V\"F2Faq7$$\"3)****\\ 7B>&)\\\"F2Faq7$$\"3)***\\P>:mk:F2Faq7$$\"3'***\\iv&QAi\"F2Faq7$$\"31+ +vtLU%o\"F2Faq7$$\"3!******\\Nm'[F2Faq7$$\"3z*****\\@80+#F2Faq7$$\" 31++]7,Hl?F2Faq7$$\"3()**\\P4w)R7#F2Faq7$$\"3;++]x%f\")=#F2Faq7$$\"3!) **\\P/-a[AF2Faq7$$\"3/+](=Yb;J#F2Faq7$$\"3')****\\i@OtBF2Faq7$$\"3')** \\PfL'zV#F2Faq7$$\"3>+++!*>=+DF2Faq7$$\"3-++DE&4Qc#F2Faq7$$\"3=+]P%>5p i#F2Faq7$$\"39+++bJ*[o#F2Faq7$$\"33++Dr\"[8v#F2Faq7$$\"3++++Ijy5GF2Faq 7$$\"31+]P/)fT(GF2Faq7$$\"31+]i0j\"[$HF2Faq7$$\"\"$F)Faq-%'COLOURG6&%$ RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Fax-%%VIEWG6$;F(Fdw%(DEFAUL TG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "fsolve('bs(x)'=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1=ZJ p!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 48 "Local Tayl or series approximations for functions" }}{PARA 0 "" 0 "" {TEXT -1 102 "In this section we construct a procedure which uses a collection \+ of Taylor polynomials to approximate " }{XPPEDIT 18 0 "arctan(x)" "6#- %'arctanG6#%\"xG" }{TEXT -1 5 " for " }{TEXT 266 1 "x" }{TEXT -1 17 " \+ between 0 and 1." }}{PARA 0 "" 0 "" {TEXT -1 73 "We start by construct ing a Taylor polynomial of degree 11 to approximate " }{XPPEDIT 18 0 " arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 5 " for " }{TEXT 267 1 "x" }{TEXT -1 8 " near 0." }}{PARA 0 "" 0 "" {TEXT -1 97 "Note that the po lynomial is converted to the nested or Horner form for more efficient \+ evaluation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 88 "series(arctan(x),x=0,12):\npx := convert(%,polynom) ;\np1 := unapply(convert(px,horner),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#pxG,.%\"xG\"\"\"*&#F'\"\"$F'*$)F&F*F'F'!\"\"*&#F'\"\"&F'*$)F& F0F'F'F'*&#F'\"\"(F'*$)F&F5F'F'F-*&#F'\"\"*F'*$)F&F:F'F'F'*&#F'\"#6F'* $)F&F?F'F'F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1Gf*6#%\"xG6\"6$%) operatorG%&arrowGF(*&,&\"\"\"F.*&,&#F.\"\"$!\"\"*&,&#F.\"\"&F.*&,&#F. \"\"(F3*&,&#F.\"\"*F.*&#F.\"#6F.*$)9$\"\"#F.F.F3F.FDF.F.F.FDF.F.F.FDF. F.F.FDF.F.F.FEF.F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot([arctan(x),p1(x)],x=0..1.2,col or=[red,green],thickness=[1,2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 319 197 197 {PLOTDATA 2 "6&-%'CURVESG6%7S7$$\"\"!F)F(7$$\"3h*******\\ech#! #>$\"32,01HA1:EF-7$$\"3-+++v*G:*[F-$\"3%poTnBLw)[F-7$$\"3u******\\L)4X (F-$\"37mWT>0CPuF-7$$\"3)******\\MSF+\"!#=$\"3_74.Tm*R***F-7$$\"3#)*** *\\Fy:f7F=$\"3-'yqY'ec_7F=7$$\"3')****\\d'*)o\\\"F=$\"3mi'HwVde[\"F=7$ $\"3w****\\(>ZIu\"F=$\"3*>$e))o'4ds\"F=7$$\"3u****\\xOi(*>F=$\"3'pMN'o 0nr>F=7$$\"3#)****\\FPQ^AF=$\"3_#H(>O9Y9AF=7$$\"3/+++IrS7DF=$\"3WL8!o^ g9Y#F=7$$\"3p*****\\o;Bu#F=$\"3m3K/!f)fm=OYF=7$$\"33++]_u3Y_F=$\"3\"zf+1O-9$[F =7$$\"3[*****\\PJK]&F=$\"3J)4nGn74.&F=7$$\"3%*****\\n(p$RdF=$\"3xo3`E- N5_F=7$$\"3A*****\\#p2%*fF=$\"3#H*3Ti'Q)*R&F=7$$\"3o****\\xgkeiF=$\"3G lam%*pF=$\"31h3^\"\\yO5'F=7$$\"3?*****\\J igC(F=$\"3+%zC!4/]qiF=7$$\"3G****\\P*eU 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"plot((p1(x)-arctan(x))/arctan(x),x=0..0.15,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7]o7$$\"3/ +++,;u@5!#@$\"\"!F,7$$\"33+++-K[V?F*F+7$$\"37+++.[AlIF*F+7$$\"3%****** \\Smp3%F*F+7$$\"3-+++2'\\/8'F*$!3#3em,o_&oc'yM;FB$\"36`eZDbTE8F;7$$\"3=+++V)z @X#FBF+7$$\"3%******\\7t&pKFB$!3gAe$*p!>kK\"F;7$$\"3/+++s@*>p%FBF+7$$ \"3r******=7T9hFBF+7$$\"3u+++)=HPJ*FBF+7$$\"3'******4VDMD\"!#>F+7$$\"3 5+++%GZRd\"FfnF+7$$\"3=+++s?6r=FfnF+7$$\"31+++(**3)y@FfnF+7$$\"31+++(f Hq\\#Ffn$!3'pHOiP=(*Q\"F;7$$\"34+++f'HU\"GFfnF+7$$\"3-+++7*309$FfnF+7$ $\"3I+++ce*yU$FfnF+7$$\"3/+++)[D9v$FfnF+7$$\"3++++iNGwSFfnF+7$$\"3#*** ***>^W$*Q%FfnF+7$$\"3'******p%QjtYFfnF+7$$\"3*)*****>O\"o6]FfnF+7$$\"3 M+++]>0)H&Ffn$!3)Q)p*4;J4J\"F;7$$\"3?+++A!p6j&FfnF+7$$\"38+++vS.EfFfn$ !3Uw\\MmjGs6F;7$$\"3y*****>xZ&\\iFfn$!3k-lQu?CNLF;7$$\"39+++;$4wb'Ffn$ !3ex=nRJ&zN'F;7$$\"3Q******=#R!zoFfn$!3-W8J[4K#3)F;7$$\"3!*******3A@ur Ffn$!3E-IHL/<]:!#K7$$\"3S+++chf#\\(Ffn$!3=#)oZ7$fBT#F^s7$$\"3i*****pf2 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\"F[v$!3%)G;)4u75e(Fgx7$$\"33+++9hbv9F[v$!3PmMUq@#45)Fgx7$$\"3/+++wSq$ [\"F[v$!3)f0mrz(3`')Fgx7$$\"3++++Q?&=\\\"F[v$!3w\"R?j2-)R#*Fgx7$$\"3%* *************\\\"F[v$!3![WzTQ\\F')*Fgx-%+AXESLABELSG6$Q\"x6\"Q!Fg]l-%' COLOURG6&%$RGBGF+F+$\"*++++\"!\")-%%VIEWG6$;F+$\"#:!\"#%(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 15 "The polyn omial " }{XPPEDIT 18 0 "p1(x)" "6#-%#p1G6#%\"xG" }{TEXT -1 18 " will a pproximate " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 48 " with a precision of about 10 digits as long as " }{XPPEDIT 18 0 "abs(x)" "6#-%$absG6#%\"xG" }{TEXT -1 25 " is less than about 0.15. " }}{PARA 0 "" 0 "" {TEXT -1 54 "Let's try using a second Taylor serie s to approximate " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" } {TEXT -1 16 " centred around " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 4 ".3. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 196 "Digits := 16:\nseries(arctan(x),x=0.3,12):\np x := collect(convert(%,polynom),x):\nDigits := 10:\np2 := unapply(conv ert(evalf(px),horner),x);\nplot((p2(x)-arctan(x))/arctan(x),x=0.15..0. 45,color=blue);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#p2Gf*6#%\"xG6 \"6$%)operatorG%&arrowGF(,&$\"+;jl1C!#=\"\"\"*&,&$\"+!H*******!#5F0*&, &$\"+6(e^^(!#;F0*&,&$\"+KKMLLF5!\"\"*&,&$\"+fgg-g!#9F?*&,&$\"+l2n2?F5F 0*&,&$\"+>j[#=&!#7F?*&,&$\"+3mV*>\"F5F?*&,&$\"+mS#**)p!#6F?*&,&$\"+9#Q &zDF5F0*&,&$\"+H^(*4?F5F?*&$\"+g>aZcFVF09$F0F0F0F\\oF0F0F0F\\oF0F0F0F \\oF0F0F0F\\oF0F0F0F\\oF0F0F0F\\oF0F0F0F\\oF0F0F0F\\oF0F0F0F\\oF0F0F0F \\oF0F0F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 439 223 223 {PLOTDATA 2 "6 &-%'CURVESG6#7[o7$$\"3%**************\\\"!#=$\"3)4,-,uS%3;!#G7$$\"37+] ilyM;:F*$\"3/oSu5zK%[\"F-7$$\"3.++DJdpK:F*$\"3=;5,63Cv8F-7$$\"3%***\\( ofV!\\:F*$\"3)43:QQ`&z7F-7$$\"3%)****\\i9Rl:F*$\"39G-46Eu&>\"F-7$$\"31 +vVV)RQf\"F*$\"31%>aW*=Lu5F-7$$\"3.+]PC#)GA;F*$\"3#G\"R;$z8()y*!#H7$$ \"3!)*\\iSS\"Ga;F*$\"3Sy#R`%edj*)FL7$$\"3%)***\\Peuio\"F*$\"3wHB57;:Y$ )FL7$$\"3&******\\$)z%=mF9HK(FL7$$\"3/+](oX*y9=F*$\"3e!)>!z Y-Q;(FL7$$\"3()**\\iNf]W=F*$\"3Or94K5+nqFL7$$\"3)***\\P9CAu=F*$\"3y!z* Q)Qvu+(FL7$$\"3.+](o5#*\\!>F*$\"33$p2N!y?wpFL7$$\"34+]P*zhd$>F*$\"3_o) **p&zQopFL7$$\"3%***\\P>fS**>F*$\"3Hbp8rdA.qFL7$$\"3/+](=$f%G1#F*$\"3m 5QB'zc43(FL7$$\"34++]#y,\"G@F*$\"3]FsU(4)*[=(FL7$$\"3-++Dr\"zb=#F*$\"3 +ITr[RD)G(FL7$$\"3%*****\\(4&G]AF*$\"3U[m)FL7$$\"35++vVy!e(GF*$\"34OE\"H4X2#))FL7$$\"3@+](=WU[$HF*$\"3 i>i2%)e\\p*)FL7$$\"3G++DJ#>&)*HF*$\"3Uokpy:nK\"*FL7$$\"3U+]P>:mkIF*$\" 3s#=$=I@B0$*FL7$$\"3S+]iv&QA7$F*$\"3SB:1nfod%*FL7$$\"3_++vtLU%=$F*$\"3 $)>SO:n%[i*FL7$$\"3c+++bjm[KF*$\"3=rZ[J&41!)*FL7$$\"3e++vyb^6LF*$\"3Ea ?,Y)[Z(**FL7$$\"3K+]PMaKsLF*$\"3E*)[8A/c95F-7$$\"3S++D6W%)RMF*$\"30#e' GC)*zL5F-7$$\"37+++:K^+NF*$\"3mW+h,uJ^5F-7$$\"3G++]7,HlNF*$\"3%)exoR#Q -2\"F-7$$\"33+]P4w)Ri$F*$\"3C*31\"[[d(3\"F-7$$\"3Q++]x%f\")o$F*$\"3%HW fJ'Gx16F-7$$\"3Y+]P/-a[PF*$\"3vX`'*=P2D6F-7$$\"3P+](=Yb;\"QF*$\"3#\\pF _R?X9\"F-7$$\"3K++]i@OtQF*$\"39TtRg)pS;\"F-7$$\"3?+]PfL'z$RF*$\"3A\\H> .3Y&=\"F-7$$\"3>+++!*>=+SF*$\"3Aw9bT'4x?\"F-7$$\"3q++DE&4Q1%F*$\"33/KC q1ZL7F-7$$\"3%3+vV>5p7%F*$\"3')Q)3pN@WE\"F-7$$\"39+++bJ*[=%F*$\"3)[+9k m:5I\"F-7$$\"33++Dr\"[8D%F*$\"3)4)zyMX?f8F-7$$\"3W+++Ijy5VF*$\"3\">uIp (4NN9F-7$$\"3f+v=nIZUVF*$\"3a#e*QXBv*[\"F-7$$\"3u+]P/)fTP%F*$\"3)4S@EV ]rb\"F-7$$\"3S+++b!)[/WF*$\"3\"GN#Q&z!*oj\"F-7$$\"32+]i0j\"[V%F*$\"3&* eJ$[xy^t\"F-7$$\"3%*\\(=#HA6^WF*$\"317&>oWUrz\"F-7$$\"3#)*\\7G:3uY%F*$ \"3[.ngR4[m=F-7$$\"3o\\iSwSq$[%F*$\"3Bz:k`C.W>F-7$$\"35+++++++XF*$\"3j >2()))evI?F--%+AXESLABELSG6$Q\"x6\"Q!Fc_l-%'COLOURG6&%$RGBG$\"\"!Fj_lF i_l$\"*++++\"!\")-%%VIEWG6$;$\"#:!\"#$\"#XFd`l%(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 69 "This second polynomi al provides an accurate enough approximation for " }{XPPEDIT 18 0 "arc tan(x)" "6#-%'arctanG6#%\"xG" }{TEXT -1 35 " for x between about 0.15 \+ and 0.45." }}{PARA 0 "" 0 "" {TEXT -1 37 "The next Taylor polynomial i s OK for " }{TEXT 268 1 "x" }{TEXT -1 29 " between about 0.45 and 0.89 ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 197 "Digits := 16:\nseries(arctan(x),x=0.65,12):\npx := c ollect(convert(%,polynom),x):\nDigits := 10:\np3 := unapply(convert(ev alf(px),horner),x);\nplot((p3(x)-arctan(x))/arctan(x),x=0.45..0.89,col or=blue);\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#p3Gf*6#%\"xG6\"6$%)o peratorG%&arrowGF(,&$\"+P#R$F@F6*&,&$\"+0i)>3(F@F0*&,&$\"+:s8GmF@F6*&,&$\"+!\\Af[$F@F 0*&,&$\"+rQQ>5F@F6*&$\"+Ly^18FEF69$F6F0F6F[oF6F6F6F[oF6F6F6F[oF6F6F6F[ oF6F6F6F[oF6F6F6F[oF6F6F6F[oF6F6F6F[oF6F6F6F[oF6F6F6F[oF6F6F(F(F(" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7gn7 $$\"35+++++++X!#=$\"3adG')fe/)z#!#H7$$\"3oLLLyu!ff%F*$\"3#zl6X/r*eD!#G 7$$\"3#pm;C1c$zYF*$\"3*)*Q3,$)[=u$F37$$\"3YLL$Gs-Kx%F*$\"3o=x!**=_8`%F 37$$\"3sLL$)f9nn[F*$\"3SBnzRb0\")\\F37$$\"3emmT.7ph\\F*$\"3Spq52VtL_F3 7$$\"3oLL3T&f)[]F*$\"3]h,ruSOr`F37$$\"3e++v0t6R^F*$\"3EEQ<-uggaF37$$\" 3ULL3:?YK_F*$\"3d,\"*p'euF_&F37$$\"3p***\\nO2bK&F*$\"3Gg\"e+fY$pbF37$$ \"3fmmmZf@@aF*$\"3kw4/:dR4cF37$$\"3RLL$y6;b]&F*$\"3N_,yaIJTcF37$$\"3'* *****H9=/g&F*$\"3a.!*=#yq]n&F37$$\"3))*****\\%)4dp&F*$\"3&[)Rj$o!G2dF3 7$$\"3w+++d5a(y&F*$\"3.N)fU\\**ot&F37$$\"3aLL3&fK4(eF*$\"3mt)f7)>\\idF 37$$\"3=mmm*>$4qfF*$\"33\"y&y9P7\"z&F37$$\"3)omm'Q_4agF*$\"3'Q<@U47P\" eF37$$\"3a++vz&4=:'F*$\"3GUlOMS!y$eF37$$\"3qmmmGLIQiF*$\"33zjd,&fp&eF3 7$$\"3c***\\(z1?LjF*$\"3A0XAftLveF37$$\"3q***\\#*RlNU'F*$\"3_$)[f6I.!* eF37$$\"3Onm;/:&y^'F*$\"39rIobX9-fF37$$\"3kmmT\"eNWg'F*$\"3A$=PGy!35fF 37$$\"3#QLLe?Gyp'F*$\"3\"HD6ngi[\"fF37$$\"3wLL3&*o$[z'F*$\"3Ub)R=p]`\" fF37$$\"3q++DWKGzoF*$\"3it'f)y1q6fF37$$\"3!RLL[h([qpF*$\"3V,K[<:7.fF37 $$\"3K*****Rl5Z1(F*$\"3-%y=F%Hz))eF37$$\"33++]:&*)o:(F*$\"3]viy@+$*oeF 37$$\"3O++v.t2YsF*$\"39UAU/uxVeF37$$\"3'3++l80^M(F*$\"3y4WX&\\X$3eF37$ $\"3+mmm[g3MuF*$\"3%)e#fpPI\"pdF37$$\"3y*****\\;#4HvF*$\"3nu*HlC%e=dF3 7$$\"3pKL3F==:wF*$\"3ACQ)3j+Pm&F37$$\"3L+++n0I4xF*$\"39R7[ytF\"f&F37$$ \"3UmmTm*eyz(F*$\"3T$HDU*[32bF37$$\"3?++v5!G/*yF*$\"3![RC#)p5VR&F37$$ \"3Ummmr6$4)zF*$\"3++$pa+QaC&F37$$\"3K++v$fzc2)F*$\"3)*fG`2/x@]F37$$\" 3eLLL&eLp;)F*$\"3'y#>1%3Ymp%F37$$\"3WLL$=(RDg#)F*$\"31?b-/uMxTF37$$\"3 enmT=;!GN)F*$\"3=-56APGaLF37$$\"3L+++%HVyV)F*$\"3Q9A/$fRS;#F37$$\"3lnm ;^1JN&)F*$!3oWD&\\iKRO\"!#I7$$\"3m++DM')*)y&)F*$!3s#*pH#)fJ\"R\"F37$$ \"3mLLL,JF37$$\"3%omT&)\\g*o')F*$!3]M=)4TuTP&F37$$\" 3-++vzVV:()F*$!34R]dK!G\"=#)F37$$\"3W**\\(o4vwt)F*$!3E`6B:(z%=)*F37$$ \"3'******R\"e\"*f()F*$!3u_do)Qn&f6!#F7$$\"3[+]7Jl:#y)F*$!3M:.g;9wc8Fd [l7$$\"3#****\\#[sR/))F*$!3!4aH[x'Qv:Fd[l7$$\"3D+v=OzHG))F*$!3Ab(e#>Cl O=Fd[l7$$\"3e+]7C')>_))F*$!3v]E68a,G@Fd[l7$$\"3?]P4o*[T'))F*$!3E&>KQjd fG#Fd[l7$$\"3!)*\\i?J*4w))F*$!3e0;5m9j_CFd[l7$$\"3U\\7.c'\\!))))F*$!3: +`MZyZGEFd[l7$$\"39+++++++*)F*$!3*\\h\"GDv&R\"GFd[l-%+AXESLABELSG6$Q\" x6\"Q!Fa^l-%'COLOURG6&%$RGBG$\"\"!Fh^lFg^l$\"*++++\"!\")-%%VIEWG6$;$\" #X!\"#$\"#*)Fb_l%(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 56 "The last Taylor polynomial fills the gap from 0.89 to 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "Digits := 16:\nseries(arctan(x),x=0.9,12):\npx := co llect(convert(%,polynom),x):\nDigits := 10:\np4 := unapply(convert(eva lf(px),horner),x);\nplot(p4(x)-arctan(x),x=0.75..1,color=blue);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#>%#p4Gf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&$\"+-$QAh)!#8!\"\"*&,&$\"+P2v65!\"*\"\"\"*&,&$\"+.QQ/t!#6F0*&,&$ \"+;XSBhF;F0*&,&$\"+,it-n!#5F0*&,&$\"+[#*\\I8F5F6*&,&$\"+%Q,BH\"F5F0*& ,&$\"+Q0yZyFDF6*&,&$\"+vbSBJFDF0*&,&$\"+,q!R#zF;F6*&,&$\"+E9\"H:\"F;F0 *&$\"+[rP[rF/F69$F6F6F6FjnF6F6F6FjnF6F6F6FjnF6F6F6FjnF6F6F6FjnF6F6F6Fj nF6F6F6FjnF6F6F6FjnF6F6F6FjnF6F6F6FjnF6F6F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7S7$$\"3++++++++v!#=$ \"31k_g<5Vh>!#F7$$\"33nmT&)G\\avF*$\"3;')p41**4y>F-7$$\"3TLek`o!>g(F*$ \"3!Q!yQ;&QB*>F-7$$\"3Cm;z>)G_l(F*$\"3*)[\\dxZ73?F-7$$\"3=m;aQU!*3xF*$ \"3'y>*e^b#Q-#F-7$$\"3aLeRZXKixF*$\"3T\\tV4lHR?F-7$$\"3zm\"z>,_=\"yF*$ \"3IAegw&>N0#F-7$$\"3[*\\7G$[8jyF*$\"35P![HxO\"o?F-7$$\"3/n\"z%*frh\"z F*$\"3CsnX1\"QJ3#F-7$$\"33+Dcw#Q!pzF*$\"3Kh*\\yczz4#F-7$$\"3kLL3_\"=M- )F*$\"3?*zf%*y?J6#F-7$$\"3Sm;/wfJr!)F*$\"3>dUC<8NE@F-7$$\"3&****\\7eP_ 7)F*$\"3xr(>#F-7$$\"3)HLL$e'3IQ)F*$\"3IMr%\\_s&4AF-7$$\" 3N+DJq\"G&Q%)F*$\"30(G;#f6yBAF-7$$\"3iLL$eMsw[)F*$\"3b^#4&4zveHQ6iAF-7$ $\"3yL$ek`1lk)F*$\"3!*Hmw\"*z'\\F#F-7$$\"3eLe*[.-dp)F*$\"3a>&f$Q)\\lG# F-7$$\"3km;/Egw[()F*$\"3)=,1s/!z)H#F-7$$\"3\"f;z%*f%)Q!))F*$\"3]!pmR** =7J#F-7$$\"3g*\\(oza'=&))F*$\"3]B!pi(Qy@BF-7$$\"34n;zWho.*)F*$\"3dk*fF >Ad*)F*$\"3-jIt*o-TM#F-7$$\"3>**\\i:jf4!*F*$\"3-y 0ui3saBF-7$$\"3E*\\7`>r-1*F*$\"3I$H5XCF-7$$\"3Km;aQz]O '*F*$\"3II*e(Qun]CF-7$$\"3uKekG=4*o*F*$\"3'3'z;()zabCF-7$$\"3Q****\\i4 TP(*F*$\"3;uPZUq]fCF-7$$\"3/L$3F9!z#z*F*$\"3gJ`Sw%4MY#F-7$$\"3ImmmT>KU )*F*$\"37![%Q2;ImCF-7$$\"31+DJqJ8&*)*F*$\"3%3j:y)HuoCF-7$$\"31+voa-oX* *F*$\"3q&3OqjJ/Z#F-7$$\"\"\"\"\"!$\"35pT:Nu^rCF--%+AXESLABELSG6$Q\"x6 \"Q!F][l-%'COLOURG6&%$RGBG$FfzFfzFc[l$\"*++++\"!\")-%%VIEWG6$;$\"#v!\" #Fdz%(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 99 "We now set up a procedure to use one of these four polynomials \+ depending on where the input number " }{TEXT 269 1 "x" }{TEXT -1 41 " \+ is located in the interval from 0 to 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1013 "atan := proc(x::rea lcons)\n if x<0 or x>1 then\n error \"argument, %1, is outside \+ the required interval 0..1\",x;\n end if;\n if x<0.15 then\n \+ (1+(-1/3+(1/5+(-1/7+(1/9-1/11*x^2)*x^2)*x^2)*x^2)*x^2)*x \n elif x<0 .45 then\n .2406656306e-8+(.9999999290+(.7515158710e-6+\n (- .3333343232+(-.6002606059e-4+(.2007670765+\n (-.5182486319e-2+(-. 1199436608+(-.6989924066e-1+\n (.2579538214+(-.2009975129+.564754 1960e-1*\n x)*x)*x)*x)*x)*x)*x)*x)*x)*x)*x ;\n elif x<0.89 then \n -.7231739237e-5+(1.000071690+(.2221065783e-4+\n (-.337340 7799+(.3014969853e-1+(.7778835846e-1+\n (.3198181352+(-.708198620 5+(.6628137215+\n (-.3485922490+(.1019383871-.1306517833e-1*\n \+ x)*x)*x)*x)*x)*x)*x)*x)*x)*x)*x;\n else\n -.8612238302e-3+(1 .011750737+(-.7304383803e-1+\n (-.6123404516e-1+(-.6702736201+(1. 330499248+\n (-1.292301384+(.7847780538+(-.3123405575+\n (.7 923907001e-1+(-.1152911426e-1+.7148377148e-3*\n x)*x)*x)*x)*x)*x) *x)*x)*x)*x)*x;\n end if;\nend proc: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 "The relative error curve shows \+ that our approximation is accurate to about 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 31 "The worst situation occurs for " }{TEXT 270 1 "x" } {TEXT -1 26 " between about 0.85 and 1." }}{PARA 0 "" 0 "" {TEXT -1 126 "Note that, because we have defined atan as a procedure, we need t o delay evaluation with quotes in the following plot command." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot(('atan(x)'-arctan(x))/arctan(x),x=0..1,colour=blue);" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7[q7 $$\"3`*****\\n5;\"o!#@$!31$z#zeup\"f\"!#L7$$\"3#******\\8ABO\"!#?$\"\" !F37$$\"33+++-K[V?F1F27$$\"3#)******pUkCFF1F27$$\"3s*****\\Smp3%F1F27$ $\"3k******R&)G\\aF1F27$$\"3Y******4G$R<)F1F27$$\"3%******zqd)*3\"!#>F 27$$\"3*)*****>c'yM;FFF27$$\"3')*****fT:(z@FFF27$$\"3#*******zZ*z7$FFF 27$$\"33+++XTFwSFFF27$$\"3&******4z_\"4iFF$!3d.%GIsAzB#F-7$$\"3o****** R&phN)FF$!3KXSc\"G,E#))!#K7$$\"3++++*=)H\\5!#=$!3o3L*)o\"eKO\"!#I7$$\" 31+++=JN[6F[o$!3KVJ^(yRw,%F^o7$$\"3%******z/3uC\"F[o$!3IY[Na!RH3\"!#H7 $$\"37+++p3p)H\"F[o$!35X'H')o(ya_)zl9F[o$!3AyA&H&fB#[(Fio7$$\"3)******Rrd!z9F[o $!3'4w;Zz'=,\"F\\s7$$\"37+++)R'ok;F[o$\"3'=-v'p\"yCu)Fio7$$\"31+++_(>/x\"F [o$\"3YDwxgXf-uFio7$$\"3-+++1J:w=F[o$\"3=')f`A'=Y+(Fio7$$\"33+++3En$4# F[o$\"3;Tds^GxFrFio7$$\"3-+++/RE&G#F[o$\"3Uv*eQT7A[(Fio7$$\"3\")***** \\K]4]#F[o$\"3tru,0.QQzFio7$$\"3$******\\PAvr#F[o$\"3<7JA-]6N%)Fio7$$ \"3)******\\nHi#HF[o$\"3eJqsaqoZ*)Fio7$$\"3*)*****p*ev:JF[o$\"3$zCn1$p MS%*Fio7$$\"3$)*****z!47TLF[o$\"3\"pu)fZIx05F\\s7$$\"3?+++LY.KNF[o$\"3 A%Hd))z%[g5F\\s7$$\"3u*****4o7Tv$F[o$\"3LsW(HUen7\"F\\s7$$\"31+++$Q*o] RF[o$\"39Ou)eza)*=\"F\\s7$$\"3?+++\"=lj;%F[o$\"3_i`&*)4-#)G\"F\\s7$$\" 3#)*****>O_!pUF[o$\"3Apm*)\\B'*y8F\\s7$$\"3++++V&RkT)o;?>F\\s7$$\"3/+++Xh-'e%F[o$\"39(H. w7OrP#F\\s7$$\"3E+++UrT%o%F[o$\"3MDp4*3Umz$F\\s7$$\"3!*******Q\"3Gy%F[ o$\"37\"R;@,V(*e%F\\s7$$\"3%******>7O*))[F[o$\"33nb!)>rI^]F\\s7$$\"3!) *****H5k]*\\F[o$\"33A`uF6a%H&F\\s7$$\"3Y******\\7I0^F[o$\"3*Q8Q*=.iJaF \\s7$$\"3A+++(RQb@&F[o$\"3b!\\O#G#oH^&F\\s7$$\"3`+++=>Y2aF[o$\"3g22D+3 &Rg&F\\s7$$\"3\\+++yXu9cF[o$\"3IeO2Gd,!o&F\\s7$$\"3d******[y))GeF[o$\" 3];6![-T(\\dF\\s7$$\"3.+++i_QQgF[o$\"3H4$G.'Hh4eF\\s7$$\"3A+++!y%3TiF[ o$\"3M3PT.UadeF\\s7$$\"35+++O![hY'F[o$\"3mbw3P&Hf*eF\\s7$$\"3I+++#Qx$o mF[o$\"3>l(>x9zP\"fF\\s7$$\"3s*****RP+V)oF[o$\"3s-RZJQO6fF\\s7$$\"3Y** ***ppe*zqF[o$\"3QX6,2a#f)eF\\s7$$\"3u*****R#\\'QH(F[o$\"30q\">T-k`F\\s7$$\"3\\******o e()=!)F[o$\"3CfvqB`Cm^F\\s7$$\"3a+++IXaE\")F[o$\"3#)f6$f8yw&[F\\s7$$\" 3H+++ZACI#)F[o$\"3)3vS+ea7P%F\\s7$$\"37+++l*RRL)F[o$\"3_ZH()3v)\\b$F\\ s7$$\"3y*****z&e)*R%)F[o$\"3^?/f)fIo7#F\\s7$$\"3i*****HvJga)F[o$!3eUe= +qN_KFio7$$\"3Q+++Uch)f)F[o$!3')y&=ZV&[>@F\\s7$$\"39+++K&*>^')F[o$!3g= [s[:YWWF\\s7$$\"3))*****>U$y.()F[o$!3l,p7qwwWuF\\s7$$\"3s*****HJnjv)F[ o$!3q$*=4/b&*H6!#F7$$\"3;+++zo_!y)F[o$!3S)yLT,5;M\"Fgbl7$$\"3h*****fW' o/))F[o$!3%R1obIz$y:Fgbl7$$\"3;+++8g%)G))F[o$!3f#)Q!*[e)H%=Fgbl7$$\"3g ******zb+`))F[o$!31#*3\"=y7%Q@Fgbl7$$\"3Y+++k`3l))F[o$!3%zn(>8\"*p)H#F gbl7$$\"39+++Z^;x))F[o$!3!fUnPCNzY#Fgbl7$$\"3c******R]?$)))F[o$!3a4M!R hcgb#Fgbl7$$\"3#******4$\\C*)))F[o$!3wy=%Qs)eYEFgbl7$$\"3M+++B[G&*))F[ o$!30t,[dRfRFFgbl7$$\"3e*****RrC8!*)F[o$\"3_<0tSDz1KFgbl7$$\"38+++\"G% [D*)F[o$\"3&>q4BXfy?$Fgbl7$$\"3k******[Qk\\*)F[o$\"3<\"[ " 0 "" {MPLTEXT 1 0 47 "evalf(Int(arctan, 0..1));\nevalf(Int(atan,0..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" +JdC)Q%!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+KdC)Q%!#5" }}}{PARA 0 "" 0 "" {TEXT -1 65 "Here is another example involving the evaluatio n of the integral " }{XPPEDIT 18 0 "Int(1/arctan(x),x=0..1)" "6#-%$Int G6$*&\"\"\"F'-%'arctanG6#%\"xG!\"\"/F+;\"\"!F'" }{TEXT -1 2 " ." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "evalf(Int(1/arctan,1/2..1)); \nevalf(Int(1/atan,1/2..1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+h8 ND!)!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+g8ND!)!#5" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 65 "A procedure for constructing loc al Taylor series approximations: " }{TEXT 0 9 "loctaylor" }}{PARA 0 " " 0 "" {TEXT -1 159 "The procedure in this section is designed to auto matically construct a procedure which use local taylor series to appro ximate a function over a given interval." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "loctaylor: \+ usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 260 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 261 2 " \+ " }{TEXT -1 22 " loctaylor( fx, rng) " }}{PARA 0 "" 0 "" {TEXT -1 27 " local_taylor( fx, rng) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 10 " fx - " }{TEXT -1 55 " \+ an expression involving a single variable, say x," }}{PARA 0 "" 0 "" {TEXT -1 85 " where f(x) evaluates to a real floating point number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 12 " rng - " }{TEXT 262 56 "the range x=a..b over which to approximate the function." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }}{PARA 256 "" 0 " " {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proced ure " }{TEXT 0 9 "loctaylor" }{TEXT -1 200 " attempts to construct a s equence of truncated local Taylor series approximations of fx to cover the given interval. A procedure which uses this family of Taylor seri es to approximate fx is returned. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 263 8 "Options:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 15 "info=true/false" }}{PARA 0 "" 0 "" {TEXT -1 128 "Wit h the option \"info=true\" the centres and radii of the Taylor series \+ approximations are printed as the computation progresses." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "degree=n" }}{PARA 0 "" 0 "" {TEXT -1 66 "The degree of the Taylor polynomials used as lo cal approximations." }}{PARA 0 "" 0 "" {TEXT -1 46 "The default is \"d egree=floor(2*Digits^(2/3))\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 25 "errtype=absolute/relative" }}{PARA 0 "" 0 "" {TEXT -1 45 "With the (default) option \"errtype=absolute\" " } {TEXT 0 9 "loctaylor" }{TEXT -1 55 " constructs a minimax approximatio n that minimises the " }{TEXT 259 14 "absolute error" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 39 " max \+ " }{XPPEDIT 18 0 "abs(f(x)-r(x));" "6#-%$absG6#,&-%\"fG6#%\"xG\"\"\" -%\"rG6#F*!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 27 " \+ " }{XPPEDIT 18 0 "a<=x" "6#1%\"aG%\"xG" } {XPPEDIT 18 0 "``<=b" "6#1%!G%\"bG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "and with the option \"errtype=relative\" " }{TEXT 0 9 "lo ctaylor" }{TEXT -1 48 " constructs an approximation that minimises the " }{TEXT 259 14 "relative error" }{TEXT -1 43 ", \n \+ max " }{XPPEDIT 18 0 "abs(f(x)-r(x))/abs(f(x));" "6# *&-%$absG6#,&-%\"fG6#%\"xG\"\"\"-%\"rG6#F+!\"\"F,-F%6#-F)6#F+F0" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 27 " \+ " }{XPPEDIT 18 0 "a<=x" "6#1%\"aG%\"xG" }{XPPEDIT 18 0 "``<=b" "6#1 %!G%\"bG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 73 "\"errtype=REL ATIVE\" and \"errtype=REL\" are equivalent to \"errtype=relative\"" }} {PARA 0 "" 0 "" {TEXT -1 75 "\"errtype=ABSOLUTE\" and \"errtype=ABS\" are equivalent to \"errtype=absolute\"." }}{PARA 0 "" 0 "" {TEXT -1 43 "\"errtype\" can also be typed as \"errortype\"." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "maxsteps=n" }}{PARA 0 " " 0 "" {TEXT -1 46 "The maximum number of Taylor polynomials used." }} {PARA 0 "" 0 "" {TEXT -1 106 "The default is \"min(4*max(iquo(100,degr ee),1)*trunc(max(evalf(abs(b-a)),1)),15*max(iquo(100,degree),1))\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "maxrad ius=r\nThe maximum width of the interval to be used for a particular l ocal Taylor series approximation." }}{PARA 0 "" 0 "" {TEXT -1 90 "The \+ default is \"maxradius=b-a\", which does not affect the estimated radi i of convergence. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "info=true/false/0/1/2" }}{PARA 0 "" 0 "" {TEXT -1 114 "Wi th the option \"info=1\" the centre of each Taylor series approximatio ns is printed as the computation progresses." }}{PARA 0 "" 0 "" {TEXT -1 109 "With the option \"info=2\" the radius and the maximum error of each Taylor series approximations is also given." }}{PARA 0 "" 0 "" {TEXT -1 36 "\"info=true\" is the same as \"info=1\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To ma ke the procedure active open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the s ubsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "loctaylor: implemen tation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10445 "# to allow for different names for the procedure\n local_taylor := proc() loctaylor(args[1..nargs]) end:\n\nloctaylor := \+ proc(fx::algebraic,rng::equation)\n local a,b,i,j,saveDigits,eps,c,d rv,d,derivs,first,last,fact,\n rd,h,maxr,x,Options,rs,aa,bb,n,prn tflg,f,r,mxstps,deg,\n vars,drv_proc,eps2,tiny,stp,k,t,ft,pval,er r,ertyp,w,\n prevc,prevr,rads;\n\n if nargs<2 then\n error \"at least 2 arguments are required; the basic syntax is: 'momentpoly (f(x),x=a..b)'.\"\n end if;\n\n vars := indets(fx,name) minus inde ts(fx,realcons);\n if nops(vars)>1 then \n if not has(indets(fx ),\{Int,Sum,RootOf\}) then\n error \"the 1st argument, %1, is \+ invalid .. it should be an expression which depends only on a single v ariable\",fx;\n end if;\n end if;\n if has(fx,\{Heaviside,abs ,signum,piecewise,min,max\}) then\n error \"cannot handle piecewi se functions\"\n end if;\n\n if not type(rng,name=realcons..realco ns) then\n error \"the 2nd argument, %1, is invalid .. it should \+ have the form 'name=a..b', to provide the interval for the approximati ng function\",rng;\n end if;\n\n x := op(1,rng);\n if nops(vars) >1 and not member(x,vars) then\n error \"the 1st argument, %1, is invalid .. it should depend on the variable %2\",fx,x;\n end if;\n \n rs := op(2,rng);\n aa := op(1,rs);\n bb := op(2,rs);\n a := evalf(aa);\n b := evalf(bb);\n\n # Get the options \"maxsteps\", \"degree\" and \"info\".\n # Set the default values to start with\n \+ h := evalf(abs(b-a));\n maxr := h;\n deg := trunc(2*(evalf(Digit s))^(2/3));\n rd := max(iquo(100,deg),1);\n mxstps := min(4*rd*tru nc(max(h,1)),15*rd);\n ertyp := 0;\n prntflg := 0;\n if nargs>2 \+ then\n Options:=[args[3..nargs]];\n if not type(Options,list (equation)) then\n error \"each optional argument must be an e quation\"\n end if;\n if hasoption(Options,'degree','deg','O ptions') then \n if not type(deg,posint) then\n err or \"\\\"degree\\\" must be a positive integer\"\n end if;\n \+ rd := max(iquo(100,deg),1);\n mxstps := min(4*rd*trunc( max(h,1)),15*rd);\n end if;\n if hasoption(Options,'maxsteps ','mxstps','Options') then \n if not type(mxstps,posint) then \n error \"\\\"maxsteps\\\" must be a positive integer\"\n \+ end if;\n end if;\n if hasoption(Options,'maxradius' ,'maxr','Options') then \n if not type(maxr,realcons) or not s ignum(0,maxr,0)=1 \n or signum(0,maxr-h,0)=1 then\n \+ error \"\\\"maxradius\\\" must be a positive real constant tha t is no greater than the width of the interval for the approximation\" \n end if;\n end if;\n if hasoption(Options,'errtype ','ertyp','Options') or \n hasoption(Options,'errortype','ertyp' ,'Options') then\n if not member(ertyp,\{'absolute','ABSOLUTE' ,'ABS','relative','RELATIVE','REL'\}) then \n error \"\\\"e rrtype\\\" option must be 'absolute' <-> 'ABSOLUTE' <-> 'ABS' or 'rela tive' <-> 'RELATIVE' <-> 'REL'\"\n end if;\n if member (ertyp,\{'absolute','ABSOLUTE','ABS'\})\n then ertyp := 0 else ertyp := 1 end if;\n end if;\n if hasoption(Options,'info', 'prntflg','Options') then\n if not member(prntflg,\{true,false ,0,1,2\}) then\n error \"\\\"info\\\" option must be 0 <-> \+ false,1 <-> true or 2\"\n end if;\n if prntflg=false t hen prntflg := 0\n elif prntflg=true then prntflg := 1 end if; \n end if; \n if nops(Options)>0 then\n error \"%1 i s not a valid option for %2\",op(1,Options),procname;\n end if;\n end if;\n\n if type(fx,polynom(anything,x)) and degree(fx)<=deg t hen\n error \"in order to obtain an approximation for a polynomia l, the \\\"degree\\\" option must be set to a value less than the degr ee of the polynomial\"\n end if;\n\n # Increase precision for the \+ computation by about 30%.\n saveDigits := Digits;\n Digits := Digi ts+max(5,trunc(Digits*0.3));\n \n a := evalf(aa);\n b := evalf(bb) ;\n if a>=b then\n error \"invalid range of values for %1\",x; \n end if;\n n := deg+1;\n drv := array(1..n);\n drv_proc := a rray(1..n);\n d := array(1..n+2);\n drv[1] := fx;\n drv_proc[1] \+ := traperror(codegen[optimize](codegen[makeproc](drv[1],x)));\n if d rv_proc[1]=lasterror then\n drv_proc[1] := proc(x) drv[1] end pro c:\n end if;\n\n for i from 2 to n do\n drv[i] := traperror(d iff(drv[i-1],x));\n if drv[i]=lasterror then\n error \"co uld not find the %-1 derivative\",i;\n end if; \n drv_proc[i ] := traperror(codegen[optimize](codegen[makeproc](drv[i],x)));\n \+ if drv_proc[1]=lasterror then\n drv_proc[1] := proc(x) drv[1] end proc:\n end if;\n end do;\n \n if prntflg>0 then\n \+ print(`finding degree `||deg||` local Taylor polynomial approximatio ns`);\n if ertyp=0 then \n print(`while attempting to con trol the absolute error`);\n else\n print(`while attempti ng to control the relative error`);\n end if;\n end if;\n\n e ps := Float(1,-saveDigits-2);\n eps2 := 2*eps;\n tiny := Float(1,- 3*saveDigits);\n c := evalf(a);\n r := 0;\n prevc := c;\n prev r := 0;\n derivs := NULL; \n last := false;\n first := true;\n for stp from 1 to mxstps do\n \n if prntflg>0 then\n \+ print(`step .. `||stp,` centre .. `,c);\n end if;\n\n d[1 ] := prevc+r;\n d[2] := c;\n d[3] := traperror(evalf(drv_pro c[1](c)));\n if d[3]=lasterror or not type(d[3],numeric) then\n \+ d[3] := traperror(evalf(limit(drv_proc[1](_u),_u=c)));\n \+ if d[3]=lasterror or not type(d[3],numeric) then\n error \+ \"failed to evaluate function at %1\",c;\n end if;\n end \+ if;\n \n fact := evalf(1);\n for j from 2 to n do\n \+ fact := evalf(fact/(j-1));\n d[j+2] := traperror(evalf(drv_pr oc[j](c)*fact));\n if d[j+2]=lasterror or not type(d[j+2],nume ric) then\n d[j+2] := traperror(evalf(limit(drv_proc[j](_u) ,_u=c)));\n if d[j+2]=lasterror or not type(d[j+2],numeric) then\n error \"failed to evaluate derivative of order % 1 at %2\",j-1,c;\n end if;\n end if;\n end do; \n \n \n prevr := r;\n if last then\n r := b-prevc-prevr;\n if prntflg>1 then\n print(`last r adius .. `,r);\n end if;\n else\n # estimate of u sable radius of convergence\n for j from n to 2 by -1 do \n \+ if d[j+2]<>0 then\n if ertyp=0 then\n \+ r := evalf(abs(eps/d[j+2])^(1/(j-1)));\n else\n \+ r := evalf(abs(eps*max(abs(d[3]),tiny)/d[j+2])^(1/(j- 1)));\n end if;\n break;\n end \+ if;\n end do;\n if j=1 then \n r := b-c;\n \+ if prntflg>0 then\n print(`all derivatives ar e zero .. function is constant`);\n end if;\n end i f;\n if prntflg>1 then\n print(`radius estimate .. \+ `,r);\n end if;\n end if;\n\n for k to 10 do \+ \n # check value given by previous series\n err := 0;\n if first then rads := [r] \n elif last then rads := [-r]\n else rads := [-r,r] end if;\n for t in rads do\n ft := traperror(evalf(drv_proc[1](c+t)));\n \+ if ft=lasterror or not type(ft,numeric) then\n error \+ \"failed to evaluate function at %1\",c+t;\n end if;\n \+ pval := d[n+2];\n for i from n+1 to 3 by -1 do pval \+ := t*pval+d[i] end do;\n if ertyp = 0 then\n \+ err := max(err,abs(pval-ft));\n else\n err := max(err,abs(pval-ft)/max(abs(ft),tiny));\n end if;\n \+ end do;\n if prntflg>1 then\n print(`error .. `, evalf[iquo(Digits,3)](err),`error bound .. `,eps2);\n end if; \n if err1 then\n print(`reducing radius to .. `,r);\n \+ end if;\n if not first then\n c := prevc+prevr+r ;\n # recalculate derivatives at reduced c\n d[1 ] := prevc+prevr;\n d[2] := c;\n d[3] := traperr or(evalf(drv_proc[1](c)));\n if d[3]=lasterror or not type( d[2],numeric) then\n error \"failed to evaluate function at %1\",c;\n end if;\n \n fact := evalf(1);\n \+ for j from 2 to n do\n fact := evalf(fact/(j- 1));\n d[j+2] := traperror(evalf(drv_proc[j](c)*fact)); \n if d[j+2]=lasterror or not type(d[j+2],numeric) then \n error \"failed to evaluate derivative of order %1 \+ at %2\",j-1,c;\n end if;\n end do;\n \+ end if; \n end do;\n if k>10 then \n error \"can not achieve a satisfactory Taylor series approximation at %1\",c;\n \+ end if;\n derivs := derivs,convert(d,list);\n\n prevc := \+ c;\n r := min(r,maxr);\n if c+r>=b then\n last := tr ue;\n break; \n end if;\n c := c+2*r;\n if c>=b then\n c := b;\n last := true;\n end if;\n end do;\n Digits := saveDigits;\n if not last then\n WARNING(`pr ocedure is not defined on all of the specified interval -- try increas ing \"maxsteps\" or \"degree\"`)\n end if;\n\n # Return the proced ure which evaluates the approximation.\n subs(\{dat=[derivs],a1=a,b1 =b\},\n proc(x_loctaylor::realcons)\n local aa,bb,xk,yk,jF,jM,jS,n ,h,j,data,fn,xx,saveDigits,pval,m;\n options `Copyright 2003 by Pete r Stone`;\n aa := a1;\n bb := b1;\n data := dat;\n\n saveDigit s := Digits;\n Digits := max(trunc(evalhf(Digits)),trunc(Digits*4/3) );\n xx := evalf(x_loctaylor);\n if xxbb then\n err or \"independent variable is outside of the interpolation interval: %1 \",evalf[saveDigits](aa..bb);\n end if;\n \n # Peform a binary s earch for the interval containing x.\n n := nops(data);\n jF := 0; \n jS := n+1;\n while jS-jF>1 do\n jM := trunc((jF+jS)/2);\n \+ if xx>=data[jM,1] then jF := jM else jS := jM end if;\n end do;\n \n # Get the data needed from the list.\n xk := data[jF,2];\n h \+ := xx-xk;\n m := nops(data[jF]);\n pval := data[jF,m];\n for j f rom m-1 to 3 by -1 do pval := h*pval+data[jF,j] end do;\n Digits := \+ saveDigits;\n return evalf(pval);\n end proc); # closing bracket \+ of \"subs\"\nend proc:" }}}{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 sectio n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "loctaylor " }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 " " {TEXT -1 53 "We construct a local Taylor series approximation for " }{XPPEDIT 18 0 "f(x) = sqrt(x)" "6#/-%\"fG6#%\"xG-%%sqrtG6#F'" }{TEXT -1 24 " over the interval from " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\" " }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "gn := loctaylor(sqrt(x),x=1..3,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Xfinding~degree~9~local~Taylor~polynomial~a pproximationsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Owhile~attempting~t o~control~the~absolute~errorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*ste p~..~1G%.~~~centre~..~G$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %%*step~..~2G%.~~~centre~..~G$\"02W8PhL:\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~3G%.~~~centre~..~G$\"0!*H=\"f%)G8!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~4G%.~~~centre~..~G$\"0PA;!fW H:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~5G%.~~~centre~..~G$ \"0gQ:sJ&e?!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~ 7G%.~~~centre~..~G$\"0/wjT/yJ#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%% *step~..~8G%.~~~centre~..~G$\"0pDSGZql#!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~9G%.~~~centre~..~G$\"\"$\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "The code for the pro cedure " }{TEXT 271 2 "gn" }{TEXT -1 27 " can be viewed as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "interface(verboseproc=2):\neval(gn);\ninterface(verboseproc=1): " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6#'%,x_loctaylorG%)realconsG62%# aaG%#bbG%#xkG%#ykG%#jFG%#jMG%#jSG%\"nG%\"hG%\"jG%%dataG%#fnG%#xxG%+sav eDigitsG%%pvalG%\"mG6#%>Copyright~2003~by~Peter~StoneG6\"C4>8$$\"\"\" \"\"!>8%$\"\"$FA>8.7+7.F?F?F?$\"0+++++++&!#:$!0++++++D\"FL$\"0,+++++D' !#;$!0,+++]i!RFQ$\"0,+++vVt#FQ$!0+++D\"y]?FQ$\"0++]7G8h\"FQ$!0^i:5/#48 FQ$\"0vozT.54\"FQ7.$\"0/sco!ow5!#9$\"02W8PhL:\"F[o$\"01yLjYR2\"F[o$\"0 kwL#[sbYFL$!0k(ftV;45FL$\"0f:Pq$)[P%FQ$!0&Gf?ZsqBFQ$\"0(*zn=W)Q9FQ$!0> z[E=kN*!#<$\"070#yK&RP'F\\p$!0#=xs>@!\\%F\\p$\"0snD%RHWKF\\p7.$\"0*peT O5T7F[o$\"0!*H=\"f%)G8F[o$\"0y&>!zbF:\"F[o$\"0X%)y7KuL%FL$!05Y+w[,;)FQ $\"0'\\_$Q*QqIFQ$!03f_D0TW\"FQ$\"0A:K(\\:2wF\\p$!0Fp_kuMH%F\\p$\"0)*)y !>F'QDF\\p$!0l\\g(**>_:F\\p$\"0a!o0.,M(*!#=7.$\"08En!f9H9F[o$\"0PA;!fW H:F[o$\"0q3ko2nB\"F[o$\"0Uy,e#*H/%FL$!0!=yl.f3mFQ$\"0\\wXZ_/;#FQ$!068` 2v&G))F\\p$\"0N$o52oSSF\\p$!0X0)HMW\")>F\\p$\"0>PUi;z,\"F\\p$!0\"Q\"zO hvS&F\\r$\"0%yv/>OYHF\\r7.$\"0\\!e6)))Rk\"F[o$\"0gQ:sJ&e*F\\r$\"0rU0QI(3TF\\r$!0)GYb(p$)*=F\\r$\"0-cQ1)*f **)!#>7.$\"0%)RA,9#*)=F[o$\"03THI'*)>?F[o$\"0(*)Qf0B@9F[o$\"0l!RD!y!=N FL$!0yz)*[!GaVFQ$\"0')f>`Zy2\"FQ$!0X]@]&4NLF\\p$\"0Qll\\&yb6F\\p$!0*4N *3.:H%F\\r$\"0kPsyS$p;F\\r$!0Z,U9'*[r'F_u$\"0h17*QJqFF_u7.$\"0ce'f.&)o @F[o$\"0/wjT/yJ#F[o$\"0tO\\rLC_\"F[o$\"0nEWB:UG$FL$!0b;[WxBa$FQ$\"0Fz* 3gmTwF\\p$!0Bygb)eg?F\\p$\"0.sV]#=BiF\\r$!0+n&3/r8?F\\r$\"0S(HiAHEoF_u $!0l&o&3QHR#F_u$\"0r9[mlMg)!#?7.$\"0'3?]eU([#F[o$\"0pDSGZql#F[o$\"0)G_ y^/I;F[o$\"0*G\"Qk*RnIFL$!0G*z'4)4')GFQ$\"0=-s4D5V&F\\p$!0=/+\"[]x7F\\ p$\"0#3g:2flLF\\r$!0/?F\"H***\\*F_u$\"0@/ItQ#4GF_u$!0W=t4(Q!f)Fbx$\"0U \"zBZ@%p#Fbx7.$\"0*)zZ;D+&GF[oFD$\"0))ov!30KF\\r$!0_%37m]s[F_u$ \"0PcMFbx$\"0)*3$R_`+'*!#@>81%'DigitsG>Ff[l-%$maxG 6$-%&truncG6#-%'evalhfG6#Ff[l-F\\\\l6#,$*(\"\"%F@FE!\"\"Ff[lF@F@>80-%& evalfG6#9$@$52Fh\\lF>2FCFh\\lY6$Q\\oindependent~variable~is~outside~of ~the~interpolation~interval:~%1F;-&Fj\\l6#Fe[l6#;F>FC>8+-%%nopsG6#FG>8 (FA>8*,&Fj]lF@F@F@?(F;F@F@F;2F@,&Fa^lF@F_^lFf\\lC$>8)-F\\\\l6#,&*&#F@ \"\"#F@F_^lF@F@*&F]_lF@Fa^lF@F@@%1&FG6$Fh^lF@Fh\\l>F_^lFh^l>Fa^lFh^l>8 &&FG6$F_^lF^_l>8,,&Fh\\lF@Fg_lFf\\l>83-F\\^l6#&FG6#F_^l>82&FG6$F_^lF^` l?(8-,&F^`lF@F@Ff\\lFf\\lFE%%trueG>Fd`l,&*&F[`lF@Fd`lF@F@&FG6$F_^lFh`l F@>Ff[lFe[lO-Fj\\l6#Fd`lF;F;F;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 38 "The values of the numerical procedure " } {TEXT 271 2 "gn" }{TEXT -1 36 " appear to agree well with those of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "gn(1.5); \nsqrt(1.5);\ngn(1.001);\nsqrt(1.001);\ngn(2.999);\nsqrt(2.999);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+r[uC7!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+r[uC7!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+v) *\\+5!\"*" }}{PARA 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" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x G" }{TEXT -1 58 " is an \"area function\" which gives areas under the \+ graph " }{XPPEDIT 18 0 "y = 1/(x+exp(x))" "6#/%\"yG*&\"\"\"F&,&%\"xGF &-%$expG6#F(F&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "t := 't':\nf := x->Int(1/( t+exp(t)),t=0..x);\nfn := evalf[15](loctaylor(f(x),x=-.55..5,errtype=r elative,info=true));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"x G6\"6$%)operatorG%&arrowGF(-%$IntG6$*&\"\"\"F0,&%\"tGF0-%$expG6#F2F0! \"\"/F2;\"\"!9$F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Yfinding~deg ree~12~local~Taylor~polynomial~approximationsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Owhile~attempting~to~control~the~relative~errorG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~1G%.~~~centre~..~G$!#b!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~2G%.~~~centre~..~G$!58H9*G wH4@[&!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~3G%.~~~centre~. .~G$!5qe(>J,;+CY&!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~4G%. ~~~centre~..~G$!5Jzx,4!H%pSa!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*s tep~..~5G%.~~~centre~..~G$!5JL0$))3#oz;a!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~6G%.~~~centre~..~G$!59NuGb7\\\\!R&!#?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~7G%.~~~centre~..~G$!5)H(3(zx% !#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~19G%.~~~centre~..~G$! 5b1%e!)[@3(*o%!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~20G%.~~ ~centre~..~G$!5h_g+DeF;$f%!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+ste p~..~21G%.~~~centre~..~G$!5I?3*z)pSi([%!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~22G%.~~~centre~..~G$!52@Zv5!QKBP%!#?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~23G%.~~~centre~..~G$!53L_8#G hxkC%!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~24G%.~~~centre~. .~G$!5Yj?4![7*>4T!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~25G% .~~~centre~..~G$!5!4Sjd5V'efR!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+ step~..~26G%.~~~centre~..~G$!5pnJLY\"H\"o'z$!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~27G%.~~~centre~..~G$!5+!3TM`n\"[>O!#?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~28G%.~~~centre~..~G$!5/=#!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~34G%.~~~centre~. .~G$!5\"y5pImip3*=!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~35G %.~~~centre~..~G$!5*phGCr@^Hc\"!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% %+step~..~36G%.~~~centre~..~G$!5&RH*>)\\V)!#@" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~38G%.~~~centre~..~G$!51\"H*o >NO$=c%!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~39G%.~~~centre ~..~G$!4!z'QH)=lE4g!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~40 G%.~~~centre~..~G$\"5?*\\vbI3_h)H!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6%%+step~..~41G%.~~~centre~..~G$\"5w-\"e>mFR_L(!#@" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%%+step~..~42G%.~~~centre~..~G$\"5L8f_c$f'pM7!#?" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~43G%.~~~centre~..~G$\"52VbA? ()R'pz\"!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~44G%.~~~centr e~..~G$\"5rL#Rr-%z'=U#!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~.. ~45G%.~~~centre~..~G$\"5tCo*y(=WE8J!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~46G%.~~~centre~..~G$\"5^0Fw+K3KwQ!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~47G%.~~~centre~..~G$\"53S0i2-q>&)fr%)z/8!*!#?" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%%+step~..~52G%.~~~centre~..~G$\"599iK(Qbgm.\"!#>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~53G%.~~~centre~..~G$\"5aflX; U#4a=\"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~54G%.~~~centre ~..~G$\"5//C$fGt])[8!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~5 5G%.~~~centre~..~G$\"5xO*Q/,'3TG:!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6%%+step~..~56G%.~~~centre~..~G$\"5]agNhB9lD" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~57G%.~~~centre~..~G$\"5!p>i&f'pzA%>!#>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~58G%.~~~centre~..~G$\"5)fy$) Rf" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~59G%.~~~centre ~..~G$\"5_%)4$\\.\"\\HTC!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~ ..~60G%.~~~centre~..~G$\"5\"=cs\"3)p2zs#!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~61G%.~~~centre~..~G$\"5lYk%=FzfD/$!#>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~62G%.~~~centre~..~G$\"5Erk5* e[g&)Q$!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~63G%.~~~centre ~..~G$\"5M#34N&4&p+x$!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~ 64G%.~~~centre~..~G$\"5e(>ym6>'))*=%!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~65G%.~~~centre~..~G$\"5\"Rd()e8%Q#[k%!#>" }}{PARA 11 " " 1 "" {XPPMATH 20 "6%%+step~..~66G%.~~~centre~..~G$\"\"&\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot(['fn'(x),f(x)],x=-.55..5,thickness=[1,2]);" }}{PARA 13 "" 1 " " {GLPLOT2D 477 477 477 {PLOTDATA 2 "6&-%'CURVESG6%7fn7$$!#b!\"#$!+g' \\k;#!\"*7$$!+A6RCa!#5$!+WV=M>F-7$$!+VAy[`F1$!+E$zYw\"F-7$$!+lL&F1$!+&pu6_\"F-7$$!+HnMY]F1$!+'>1iM\"F-7$$!+s*G^*[F1 $!+3Fs47F-7$$!+eMp#f%F1$!+J]K.5F-7$$!+WzD!H%F1$!+wQ!=\\)F17$$!+'*G'Rw$ F1$!+f/,%\\'F17$$!+\\ymPKF1$!+rk&>+&F17$$!+CSzXEF1$!+jP1%o$F17$$!+,-#R 0#F1$!+:gu:EF17$$!*/fKi)F1$!+^\")Q^%*!#67$$\"*_\\gB$F1$\"+b77NJFjo7$$ \"+mY6B9F1$\"+3?%*[7F17$$\"+)G$fhDF1$\"+4mg_?F17$$\"+3&4!RPF1$\"+aX]_F F17$$\"*u\\E\"\\F-$\"+'zvzM$F17$$\"*)H))>hF-$\"+DjNwQF17$$\"*n9K=(F-$ \"+&\\F]N&F17$$\"+BXCz6F-$\"+BR*[g&F17$$\"+/@K/8F-$\"+^#**4(eF1 7$$\"+A#z-T\"F-$\"+&H'=tgF17$$\"+QD`L:F-$\"+')\\u%G'F17$$\"+3EjU;F-$\" +,57`kF17$$\"+wELimF17$$\"+ZaJw=F-$\"+FCYinF17$$\"+6XC&*>F- $\"+f%zm*oF17$$\"+<&eW5#F-$\"+Q$yz+(F17$$\"+y0EAAF-$\"+^ZY;rF17$$\"+6Q iWBF-$\"+GEp4xF17$$\"+xqPzLF-$\"+(z+Tu(F17$$\"+L]4)\\$F-$\"+:EWyxF17$$\"+y$ *z4OF-$\"+1cY2yF17$$\"+/EcEPF-$\"+l(GZ$yF17$$\"+,+sSQF-$\"+yBkeyF17$$ \"+:ABgRF-$\"+-`1\")yF17$$\"+#oO`2%F-$\"+H))Q+zF17$$\"+Cw/$>%F-$\"+$>z !=zF17$$\"+gQy4VF-$\"+1cwLzF17$$\"+PB01(zF17$$\"+Qc>nZF-$\"+3pw!)zF17$$\"+l, Tz[F-$\"+RlX*)zF17$$\"\"&\"\"!$\"+/oz(*zF1-%'COLOURG6&%$RGBG$\"#5!\"\" $Fe]lFe]lF_^l-%*THICKNESSG6#\"\"\"-F$6%F&-Fi]l6&F[^lF_^lF\\^lF_^l-Fa^l 6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!F__l-%%VIEWG6$;F(Fc]l%(DEFAULTG" 1 2 0 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 38 "The local Taylor series approximation " }{TEXT 271 2 "fn" }{TEXT -1 50 " \+ is faster to evaluate than the original function " }{XPPEDIT 18 0 "f(x )" "6#-%\"fG6#%\"xG" }{TEXT -1 57 ", which is evaluated by performing \+ numerical integration." }}{PARA 0 "" 0 "" {TEXT -1 295 "In Maple 8, wh ich uses fast numerical integration using hardware floating point arit hmetic, the difference in the evaluation times is more pronounced when the software floating point numerical integration is performed. This \+ happens when Digits>trunc(evalhf(Digits)), typically when Digits>=15. \+ \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 279 "Digits := trunc(evalh f(Digits))+1;\nrandomize():\nst := time():\nfor i from 1 to 50 do\n \+ xx := -.55+rand()*Float(1,-12)*4.45;\n fn(xx);\nend do:\ntime()-st; \nst := time():\nfor i from 1 to 50 do\n xx := -.55+rand()*Float(1,- 12)*4.45;\n evalf(f(xx));\nend do:\ntime()-st;\nDigits := 10:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#z!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&o?\"! \"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "A nother test is to evaluate a definite integral such as " }{XPPEDIT 18 0 "Int(f(x),x=0..2)" "6#-%$IntG6$-%\"fG6#%\"xG/F);\"\"!\"\"#" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "st := time():\ne valf[15](Int(fn,0..2));\ntime()-st;\nst := time():\nevalf[15](Int(f(x) ,x=0..2));\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0n!*[#G?( H*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$a$!\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"0n!*[#G?(H*!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"&Hv\"!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Examp le 5 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 " We construct a local Taylor series approximation for the inverse funct ion " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " of " } {XPPEDIT 18 0 "g(x) = x+exp(x);" "6#/-%\"gG6#%\"xG,&F'\"\"\"-%$expG6#F 'F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "g := x -> x+exp(x);\nf := x -> RootOf(y+exp (y)=x,y);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)ope ratorG%&arrowGF(,&9$\"\"\"-%$expG6#F-F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(-%'RootOfG6$/, &%\"yG\"\"\"-%$expG6#F1F29$F1F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "xx := 0.5;\nevalf(g(xx)); \nevalf(f(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"&!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+r7s[@!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,+++]!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "plot([g(x),x,f(x)],x=-2..5,y =-2..5,\n color=[red,black,blue],linestyle=[1,3,1]);" }}{PARA 13 "" 1 "" {GLPLOT2D 477 477 477 {PLOTDATA 2 "6'-%'CURVESG6%7W7$$!\"# \"\"!$!3=(QjnrkY'=!#<7$$!3GLL$3#*>u%=F-$!3)oV$QYlx*o\"F-7$$!3om;z43m9< F-$!30]IzOgjM:F-7$$!3QLLe/$f`c\"F-$!3U6l>/kMc8F-7$$!3PLL3K\"o]T\"F-$!3 )\\\"*)4p!f@<\"F-7$$!3]m;Hn7\\l7F-$!3%[>I.#f!R$)*!#=7$$!3WL$ekO9o7\"F- $!3HV^/OS]F!)FG7$$!3:***\\7oCA$)*FG$!3+[UY2&)=\"4'FG7$$!3qJLe9_>Z$)FG$ !3\\H'eZILs+%FG7$$!3-++Dc#Gp'oFG$!3K0DS5@bM=FG7$$!3lkmmT7$$!3/KL$3n_J+%FG$\"3UTm:C[$zp#FG7$$!3!p****\\sZL\\#FG$\"3yV(z iiU)*H&FG7$$!3]$*****\\PVt(*F[o$\"3[+Bl3Yb\"4)FG7$$\"3([,++Dxg$[F[o$\" 3Yw7.3+\"z4\"F-7$$\"3^nmT&G\"H5=FG$\"3Wm(Hk@z%z8F-7$$\"39NLLej%yQ$FG$ 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"Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "fn := loctaylor(f(x),x=-2..5,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%Xfinding~degree~9~local~Taylor~polynomial~approximationsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Owhile~attempting~to~control~the~absolute~e rrorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~1G%.~~~centre~..~G$ !\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~2G%.~~~centre~. .~G$!0(z>AP2-:!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~3G%.~~~ centre~..~G$!0=gH$=b(3\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~ ..~4G%.~~~centre~..~G$!0w1.'oyjn!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %%*step~..~5G%.~~~centre~..~G$!05noA9Bl\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~6G%.~~~centre~..~G$\"0;ad%)=g]#!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~7G%.~~~centre~..~G$\"0qbWX*3]q!#: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~8G%.~~~centre~..~G$\"0is 1+b)=7!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%*step~..~9G%.~~~centre~. .~G$\"0/3WPCmq\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~10G%. ~~~centre~..~G$\"0Ul,_*))QA!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+st ep~..~11G%.~~~centre~..~G$\"0chV6;[!H!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~12G%.~~~centre~..~G$\"0E#)eeOMd$!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%%+step~..~13G%.~~~centre~..~G$\"0()=8e]TE%!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%+step~..~14G%.~~~centre~..~G$\"\"&\" \"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(['fn'(x),f(x)],x=-2..5,thickness=[1,2]);" }} {PARA 13 "" 1 "" {GLPLOT2D 477 477 477 {PLOTDATA 2 "6&-%'CURVESG6%7S7$ $!\"#\"\"!$!+R#G+7#!\"*7$$!+@*>u%=F-$!+IY#[)>F-7$$!+53m9Z$)F- $!+Y-.^6F-7$$!*EGp'oF-$!+))*3,/\"F-7$$!*v\"HW`F-$!+vsq#H*!#57$$!*n_J+% F-$!+*p\\VM)Fjn7$$!*tZL\\#F-$!+CTR3tFjn7$$!)QVt(*F-$!+,\"G@I'Fjn7$$\") s2O[F-$!+]Dck`Fjn7$$\"*G\"H5=F-$!+YyKSXFjn7$$\"*NYyQ$F-$!+)[sMf$Fjn7$$ 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F(Ffz%(DEFAULTG" 1 2 0 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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "xx := 2.3;\nevalf(f(xx));\nfn(xx); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#B!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+MC:hb!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ MC:hb!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 6 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "We co nstruct a local Taylor series approximation for the inverse function \+ " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " of " } {XPPEDIT 18 0 "g(x) = x*cosh(x);" "6#/-%\"gG6#%\"xG*&F'\"\"\"-%%coshG6 #F'F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "We need to give " }{TEXT 0 6 "RootOf" }{TEXT -1 71 " a starting approximation to obtain the \"correct\" real number values. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 145 "g := x -> x*cosh(x);\ntaylor(g(y),y,4):\nconvert(%,p olynom)=x;\n[solve(%,y)]:\ncx := op(remove(has,%,Complex(1)));\ndx := \+ ln(x)-ln(ln(x))+.7775630315;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG f*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\"-%%coshG6#F-F.F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"yG\"\"\"*&#F&\"\"#F&*$)F%\"\"$F& F&F&%\"xG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#cxG,&*&\"\"$!\"\",&*& \"#F\"\"\"%\"xGF,F,*&F'F,,&\"#CF,*&\"#\")F,)F-\"\"#F,F,#F,F4F,#F,F'F,* &F4F,F)#F(F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dxG,(-%#lnG6#%\"x G\"\"\"-F'6#F&!\"\"$\"+:.jvx!#5F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 511 "eta := proc(x)\n if type( x,'float') then\n evalf('eta'(x))\n elif x=0 then 0\n elif x= infinity then infinity\n elif x=-infinity then -infinity\n else\n \+ 'eta'(x)\n end if;\nend proc;\n# numerical evaluation\n`evalf/e ta` := subs(\{cx_=eval(cx),dx_=eval(dx)\},proc(x)\n if not type(x,r ealcons) then \n 'eta'(x)\n else\n evalf(RootOf(y*cosh(y )-x,y,`if`(x<5.8,cx_,dx_)))\n end if;\nend proc);\n# differentiatio n\n`diff/eta` := proc(a,x) \n diff(a,x)/(cosh(eta(a))+eta(a)*sinh(et a(a)))\nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$etaGf*6#%\"xG6 \"F(F(@+-%%typeG6$9$.%&floatG-%&evalfG6#-.F$6#F-/F-\"\"!F7/F-%)infinit yGF9/F-,$F9!\"\"F;F3F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%*evalf /etaGf*6#%\"xG6\"F(F(@%4-%%typeG6$9$%)realconsG-.%$etaG6#F.-%&evalfG6# -%'RootOfG6%,&*&%\"yG\"\"\"-%%coshG6#F%)diff/etaGf*6$%\"aG%\"xG6\"F)F )*&-%%diffG6$9$9%\"\"\",&-%%coshG6#-%$etaG6#F.F0*&F5F0-%%sinhGF4F0F0! \"\"F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "Diff(eta(x),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$-%$etaG6#%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&-%%coshG6#-%$etaG6#%\"xGF$*&F)F$-%%sinhGF(F $F$!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "xx := 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Fddl$\"0B8r`D&GAF`il>81%'DigitsG>F\\[m-%$maxG6$-%&truncG6#-%'evalhfG6# F\\[m-Fb[m6#,$*(FEFK\"\"$!\"\"F\\[mFKFK>80-%&evalfG6#9$@$52F^\\mF?2FCF ^\\mY6$Q\\oindependent~variable~is~outside~of~the~interpolation~interv al:~%1F<-&F`\\m6#F[[m6#;F?FC>8+-%%nopsG6#FG>8(FA>8*,&F`]mFKFKFK?(F8)-Fb[m6#,&*&#FK\"\"#FKFe]mFKFK*&Fc^mFKFg]m FKFK@%1&FG6$F^^mFKF^\\m>Fe]mF^^m>Fg]mF^^m>8&&FG6$Fe]mFd^m>8,,&F^\\mFKF ]_mF\\\\m>83-Fb]m6#&FG6#Fe]m>82&FG6$Fe]mFd_m?(8-,&Fd_mFKFKF\\\\mF\\\\m F[\\m%%trueG>Fj_m,&*&Fa_mFKFj_mFKFK&FG6$Fe]mF^`mFK>F\\[mF[[mO-F`\\m6#F j_mF " 0 "" {MPLTEXT 1 0 42 "xx := 2.635865514;\nevalf(et a(xx));\nfn(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+9b'ej#! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\"=8uJ\"!\"*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+\"=8uJ\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "fn_ := x -> fn(abs(x))*signum(0,x,0):\nplot(['fn_(x)' ,eta(x)],x=-4..4,thickness=[1,2]);" }}{PARA 13 "" 1 "" 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@F-$\"+1Zo(>\"F-7$$\"+od'*GBF-$\"+B,%=C\"F-7$$\"+EcB,DF-$\"+Y*f_G\"F-7 $$\"+v>:nEF-$\"+r)yYK\"F-7$$\"+0a#o$GF-$\"+(HDGO\"F-7$$\"+`Q40IF-$\"+% =>()R\"F-7$$\"+\"3:(fJF-$\"+zp;I9F-7$$\"+e%GpL$F-$\"+O:dk9F-7$$\"+:-V& \\$F-$\"+H;*R\\\"F-7$$\"+ZhUkOF-$\"+Tb2C:F-7$$\"+ " 0 "" {MPLTEXT 1 0 22 "GKint('fn(x) ',x=0..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+s4xxT!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "In t(eta(x),x=0..4);\nevalf(evalf[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-%$etaG6#%\"xG/F);\"\"!\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+s4xxT!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "Int(eta(x),x \+ = 0 .. 4)=4*eta(4)-Int(g(x),x=0..eta(4))" "6#/-%$IntG6$-%$etaG6#%\"xG/ F*;\"\"!\"\"%,&*&F.\"\"\"-F(6#F.F1F1-F%6$-%\"gG6#F*/F*;F--F(6#F.!\"\" " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "4*eta(4)-Int(g(x),x = 0 .. eta(4));\nvalue(%) ;\nevalf(evalf[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"%\" \"\"-%$etaG6#F%F&F&-%$IntG6$*&%\"xGF&-%%coshG6#F.F&/F.;\"\"!F'!\"\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"%\"\"\"-%$etaG6#F%F&F&*&#F&\" \"#F&*&,,F'!\"\"*&F'F&)-%$expG6#F'F,F&F&F&F/*$F1F&F/*&F,F&F2F&F&F&F2F/ F&F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+s4xxT!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "This last integral m ay also be evaluated numerically. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "a := eta(4.);\n4*a-GKint(g(x ),x=0..a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"+B0I!e\"!\"*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+s4xxT!\"*" }}}{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 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 84 "(a) Construct a procedure which uses \+ local Taylor series to approximate the function" }}{PARA 257 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "f(x) = ln(1+x)/3-ln(x^2-x+1)/6+arcta n((2*x-1)/sqrt(3))/sqrt(3);" "6#/-%\"fG6#%\"xG,(*&-%#lnG6#,&\"\"\"F.F' F.F.\"\"$!\"\"F.*&-F+6#,(*$F'\"\"#F.F'F0F.F.F.\"\"'F0F0*&-%'arctanG6#* &,&*&F6F.F'F.F.F.F0F.-%%sqrtG6#F/F0F.-F@6#F/F0F." }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 23 "over the interval from " }{XPPEDIT 18 0 " x=0" "6#/%\"xG\"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=3" "6#/%\"xG \"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 76 "(b) Compare the values given by your procedure and the original formula for " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x=pi/4" "6#/%\"xG*&%#piG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 48 "(c) Use the approximating procedure to ev aluate " }{XPPEDIT 18 0 "Int(f(x),x=0..3)" "6#-%$IntG6$-%\"fG6#%\"xG/F );\"\"!\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "________ __________________________" }}{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 34 "__________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 84 "(a) Construct a proce dure which uses local Taylor series to approximate the function" }} {PARA 257 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "g(x) = Int(ln(1+t) /3-ln(t^2-t+1)/6+arctan((2*t-1)/sqrt(3))/sqrt(3),t = 0 .. x);" "6#/-% \"gG6#%\"xG-%$IntG6$,(*&-%#lnG6#,&\"\"\"F1%\"tGF1F1\"\"$!\"\"F1*&-F.6# ,(*$F2\"\"#F1F2F4F1F1F1\"\"'F4F4*&-%'arctanG6#*&,&*&F:F1F2F1F1F1F4F1-% %sqrtG6#F3F4F1-FD6#F3F4F1/F2;\"\"!F'" }{TEXT -1 1 "," }}{PARA 0 "" 0 " " {TEXT -1 23 "over the interval from " }{XPPEDIT 18 0 "x=0" "6#/%\"xG \"\"!" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "x=3" "6#/%\"xG\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 86 "(b) Compare the values given \+ by your procedure and the original formula for g(x) when " }{XPPEDIT 18 0 "x = sqrt(2);" "6#/%\"xG-%%sqrtG6#\"\"#" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 81 "(c) Plot the graph of your approximating \+ procedure over the interval from 0 to 3." }}{PARA 0 "" 0 "" {TEXT -1 8 "(d) Use " }{TEXT 0 6 "fsolve" }{TEXT -1 138 " in conjunction with b oth the the original formula, and also the approximating procedure, to find the first positive root of the equation " }{XPPEDIT 18 0 "g(x)=0 " "6#/-%\"gG6#%\"xG\"\"!" }{TEXT -1 60 ". Compare the times required b y each of the two schemes. " }}{PARA 0 "" 0 "" {TEXT -1 34 "_______ ___________________________" }}{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 34 "_______________________________ ___" }}{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 }