{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 "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 "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{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 "Grey Emphasis" -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 1 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 0 1 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 "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 0 0 1 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 0 0 1 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 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 291 "" 0 1 0 0 0 0 0 0 1 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 1 0 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 }{CSTYLE "" -1 295 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 296 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 297 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 298 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 299 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 301 "" 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 O utput" -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 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 30 "Special inverse functions .. I" } }{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" } }{PARA 0 "" 0 "" {TEXT -1 19 "Version: 25.3.2007" }}{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 68 "load interpolation and function approximation procedure s including: " }{TEXT 0 5 "remez" }}{PARA 0 "" 0 "" {TEXT -1 17 "The M aple m-file " }{TEXT 262 10 "fcnapprx.m" }{TEXT -1 37 " contains the c ode for the procedure " }{TEXT 0 5 "remez" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Ma ple session by a command similar to the one that follows, where the fi le path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procdrs/fcnapprx.m\";" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "load inverse functions" }} {PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 9 "invfcns. m" }{TEXT -1 52 " contains the code for the special inverse function. " }}{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 give s its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \" K:\\\\Maple/procdrs/invfcns.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "load root-finding procedures" }} {PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 7 "roots.m " }{TEXT -1 38 " contains the code for the procedures " }{TEXT 0 6 "se cant" }{TEXT -1 2 ", " }{TEXT 0 6 "newton" }{TEXT -1 5 " and " }{TEXT 0 6 "halley" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 " " {TEXT -1 121 "It can be read into a Maple session by a command simil ar to the one that follows, where the file path gives its location." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/ roots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Introduction " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " } {XPPEDIT 18 0 "g(x)=x*exp(x)" "6#/-%\"gG6#%\"xG*&F'\"\"\"-%$expG6#F'F) " }{TEXT -1 31 " is one-to-one on the interval " }{XPPEDIT 18 0 "-1 <= x;" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "g(x)=x*exp(x)" "6#/-%\"gG6#%\"xG *&F'\"\"\"-%$expG6#F'F)" }{TEXT -1 31 " is one-to-one on the interval \+ " }{XPPEDIT 18 0 "-1 <= x;" "6#1,$\"\"\"!\"\"%\"xG" }{XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 238 "g := x -> x*exp(x);\np1 := plot([g(x),x],x =-1.2..2.72,y=-1.2..2.72,color=[red,black],\n linestyle=[1 ,2]):\np2 := plots[implicitplot](x=y*exp(y),x=-2..2.72,\n y =-1..2.72,grid=[30,30],color=blue):\nplots[display]([p1,p2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&9$\"\"\"-%$expG6#F-F.F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 335 282 282 {PLOTDATA 2 "6'-%'CURVESG6%7U7$$!3%**************>\"!#<$!3kDk% HaIVh$!#=7$$!3jmmmb^b96F*$!3/MJWCGUcOF-7$$!3SLL[`+@S5F*$!3O`(p(>*)*en$ F-7$$!3#emmc57gc*F-$!3[!\\&>0\"G_n$F-7$$!3ulmmR:QC()F-$!3+c&yvL#>YOF-7 $$!3=LL$o4^n)yF-$!3CCf]Sj5%e$F-7$$!3Dnm;_/;5rF-$!3CL')=v\">@\\$F-7$$!3 P++]@e/1jF-$!31HF-7$$!3KKLLvL!Gz$F-$!3%)oX\")e\">cf#F-7$$!3@mm m&\\l+N6N+WAF-7$$!3I*****fsui>#F-$!3AbDCu%3Kw\"F-7$$!33+++!H 7tM\"F-$!3Uk\"yph%[x6F-7$$!3y)*****Rnz\"H&!#>$!3Zs\"Qo'f/>]F\\p7$$\"39 XLL)>Jw8#F\\p$\"3-[B>gx\"Q=#F\\p7$$\"3'pmm1'R>(4\"F-$\"3XJ%o#H*HWA\"F- 7$$\"3!pmmEwvb%=F-$\"36$QGya['>AF-7$$\"31****\\5<7;FF-$\"3mAgT_OwjNF-7 $$\"3ammm#Q-n[$F-$\"3-24M#4\"HT\\F-7$$\"3\">++0^^@L%F-$\"3I&4%)\\oy5o' F-7$$\"3\\++]6!>s8&F-$\"3Q\"4J-lloe)F-7$$\"3?nmm\"\\As(fF-$\"3;(=*eCIk '3\"F*7$$\"3Eom;2zg[nF-$\"3Z,]@XVED8F*7$$\"3CLLL)G^1e(F-$\"3#Q!\\HO$>y h\"F*7$$\"3=NL$)>0\"\\W)F-$\"354$o&=h&\\'>F*7$$\"3+,+]@2D(>*F-$\"3i(3+ z(GA2BF*7$$\"3jLL.v)z4+\"F*$\"3]Qs6)*GhBFF*7$$\"3u****>PS#\\3\"F*$\"3l ?\"zj]B0@$F*7$$\"31++!H-Zq;\"F*$\"39mKrg%H\"\\PF*7$$\"3/++DM_]Y7F*$\"3 ^/Mf=DbNVF*7$$\"32++qI+tM8F*$\"3U9u>:'*fq]F*7$$\"3gmmEMR+99F*$\"3%36Iq 6L\\\"eF*7$$\"33+++Zdk)\\\"F*$\"3!Q7\")*GYP2nF*7$$\"3?LLe4QMv:F*$\"3*e **p;x&p7wF*7$$\"3Q++g]]>f;F*$\"3Oj1Q\"=,#>()F*7$$\"3hmmrLD4Q,>F*$ \"3oV0nd`js7Fbw7$$\"3%)***\\iX0c)>F*$\"3MP^^qp?Y9Fbw7$$\"3'QLLpY/p1#F* $\"3isI^KE#Hj\"Fbw7$$\"3OLLjZW/]@F*$\"3o*f`bvke%=Fbw7$$\"3xmmJ()f\\KAF *$\"3G8=]!**z83#Fbw7$$\"3******>*Qg#3BF*$\"39>oT%p(R@BFbw7$$\"3'ommP% \\4&R#F*$\"3-E\"*G**pBFEFbw7$$\"3OLL`/3wsCF*$\"3\\nB@H/\\JHFbw7$$\"3m* **\\5\")obb#F*$\"3[dJkY2A\"H$Fbw7$$\"3!)****>Dx>&f#F*$\"3k8Il*zmtZ$Fbw 7$$\"3&****\\$Rm#[j#F*$\"3qi%f5F%=tOFbw7$$\"3%)**\\n>LTxEF*$\"3eb!Q&)G W\\*QFbw7$$\"3>++++++?FF*$\"34gui]w/HTFbw-%'COLOURG6&%$RGBG$\"*++++\"! 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" }}{PARA 0 "" 0 "" {TEXT -1 58 "The idea is to use \+ Halley's method to solve the equation " }{XPPEDIT 18 0 "x = y*exp(y); " "6#/%\"xG*&%\"yG\"\"\"-%$expG6#F&F'" }{TEXT -1 17 " numerically for \+ " }{TEXT 264 1 "y" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "We \+ need an " }{TEXT 260 21 "initial approximation" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 57 "starting approximations \+ for Halley's root-finding method " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "x=y*exp(y)" "6#/% \"xG*&%\"yG\"\"\"-%$expG6#F&F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "dx/d y=exp(y)+y*exp(y)" "6#/*&%#dxG\"\"\"%#dyG!\"\",&-%$expG6#%\"yGF&*&F-F& -F+6#F-F&F&" }{XPPEDIT 18 0 "``=exp(y)*(1+y)" "6#/%!G*&-%$expG6#%\"yG \"\"\",&F*F*F)F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hen ce " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=1/(exp(y )*(1+y))" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&*&-%$expG6#%\"yGF&,&F&F&F.F &F&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "The first few terms in the Maclaurin series for " } {XPPEDIT 18 0 "f(x) = g^(-1)*``(x);" "6#/-%\"fG6#%\"xG*&)%\"gG,$\"\"\" !\"\"F,-%!G6#F'F," }{TEXT -1 72 " can be obtained as the series soluti on for the differential equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx = 1/(exp(y)*(1+y));" "6#/*&%#dyG\"\"\"%#dxG!\" \"*&F&F&*&-%$expG6#%\"yGF&,&F&F&F.F&F&F(" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "Orde r := 9:\nde := diff(y(x),x)=1/(exp(y(x))*(1+y(x)));\nic := y(0)=0;\nds olve(\{de,ic\},y(x),type=series);\nconvert(rhs(%),polynom);\nOrder := \+ 6:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF, *&\"\"\"F.*&-%$expG6#F)F.,&F.F.F)F.F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG+5F'\"\"\"F)!\"\"\"\"##\"\"$F+F-#!\")F-\"\"%#\"$D\"\" #C\"\"&#!#aF4\"\"'#\"&2o\"\"$?(\"\"(#!&%Q;\"$:$\"\")-%\"OG6#F)\"\"*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,2%\"xG\"\"\"*$)F$\"\"#F%!\"\"*&#\"\" $F(F%*$)F$F,F%F%F%*&#\"\")F,F%*$)F$\"\"%F%F%F)*&#\"$D\"\"#CF%*$)F$\"\" &F%F%F%*&#\"#aF;F%*$)F$\"\"'F%F%F)*&#\"&2o\"\"$?(F%*$)F$\"\"(F%F%F%*&# \"&%Q;\"$:$F%*$)F$F1F%F%F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Alternatively, the same Taylor polynomial can b e obtained by using the procedures " }{TEXT 0 6 "RootOf" }{TEXT -1 5 " and " }{TEXT 0 6 "taylor" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "RootOf(y*exp(y)-x, y);\ntaylor(%,x,9);\np := unapply(convert(%,polynom),x);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'RootOfG6#,&*&%#_ZG\"\"\"-%$expG6#F(F)F)%\"xG! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+5%\"xG\"\"\"F%!\"\"\"\"##\"\" $F'F)#!\")F)\"\"%#\"$D\"\"#C\"\"&#!#aF0\"\"'#\"&2o\"\"$?(\"\"(#!&%Q;\" $:$\"\")-%\"OG6#F%\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#% \"xG6\"6$%)operatorG%&arrowGF(,29$\"\"\"*$)F-\"\"#F.!\"\"*&#\"\"$F1F.* $)F-F5F.F.F.*&#\"\")F5F.*$)F-\"\"%F.F.F2*&#\"$D\"\"#CF.*$)F-\"\"&F.F.F .*&#\"#aFDF.*$)F-\"\"'F.F.F2*&#\"&2o\"\"$?(F.*$)F-\"\"(F.F.F.*&#\"&%Q; \"$:$F.*$)F-F:F.F.F2F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 51 "The corresponding rational Pade approximation \+ for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 8 " of type" } {XPPEDIT 18 0 " ``(m,n)" "6#-%!G6$%\"mG%\"nG" }{TEXT -1 40 " is define d to be the rational function " }{XPPEDIT 18 0 "r(x) = p(x)/q(x);" "6# /-%\"rG6#%\"xG*&-%\"pG6#F'\"\"\"-%\"qG6#F'!\"\"" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "degree(p(x))<=m" "6#1-%'degreeG6#-%\"pG6#%\"xG%\"mG" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "degree(q(x))<=n" "6#1-%'degreeG6#-% \"qG6#%\"xG%\"nG" }{TEXT -1 45 " such that the Maclaurin series expans ion of " }{XPPEDIT 18 0 "r(x)" "6#-%\"rG6#%\"xG" }{TEXT -1 60 " has ma ximal initial agreement with the series expansion of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 34 ". The Pade approximation of t ype" }{XPPEDIT 18 0 "``(4,3)" "6#-%!G6$\"\"%\"\"$" }{TEXT -1 29 " can \+ be obtained as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "numapprox[pade](p(x),x,[4,3]);\ncon vert(%,confrac,x):\nevalf(evalf(%,15)):\npsi := unapply(%,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,**&#\"%F>\"&+9\"\"\"\"*$)%\"xG\"\"% F)F)F)*&#\"$B\"\"#]F)*$)F,\"\"$F)F)F)*&#\"$B'\"$!>F)*$)F,\"\"#F)F)F)F, F)F),*F)F)*&#\"$8)F8F)F,F)F)*&#\"%x\\\"$]*F)F9F)F)*&#\"&\"))=F(F)F2F)F )!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiGf*6#%\"xG6\"6$%)opera torG%&arrowGF(,(*&$\"+AFg?5!#5\"\"\"9$F1F1$\"+#onC;\"!\"*F1*&$\"+zT+h> F5F1,(F2F1$\"+(eU\"3>F5F1*&$\"+X\\d\"p\"F0F1,(F2F1$\"+mx#)pyF0F1*&$\"+ )G&RV5!#6F1,&$\"+Dba!o%F0F1F2F1!\"\"FIFIFIFIFIF(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The following picture makes a graphical comparison between the function defined by the Pade approximation and the graph of " }{XPPEDIT 18 0 "x = y*exp(y);" "6#/% \"xG*&%\"yG\"\"\"-%$expG6#F&F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "p1 := plots [implicitplot](y*exp(y)=x,x=-exp(-1)..6,y=-1..1.6):\np2 := plot(psi(x) ,x=-exp(-1)..6,color=blue):\nplots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 324 273 273 {PLOTDATA 2 "6&-%'CURVESG6]q7$7$$!3+?Wr6Wzy O!#=$!\"\"\"\"!7$$!3G:8y=>kdOF*$!3#4yrcaO'o*)F*7$7$$!3I'=!p*ykul$F*$!3 =++++++g*)F*F.7$7$F5$!3I,+++++g*)F*7$$!3+ZQ`h!pse$F*$!3P++++++?zF*F=7$FC7$$!39ZeiBwWoMF*$!3_/hgAW)e'pF*7 $7$$!3!p%\\<[>vdMF*$!3c++++++!)oF*FI7$FO7$$!3/x=*yE@wG$F*$!3/:hnwcr**f F*7$7$$!3U\"e2&eLvcKF*$!3v++++++SeF*FU7$Fen7$$!3mOAnm&G=/$F*$!3s&4.ojs +1&F*7$7$$!3ew%p1Gg,(HF*$!3%4++++++![F*F[o7$Fao7$$!3V(3YN;Mos#F*$!35n6 d&f%o[TF*7$7$$!3G_!RAxC;e#F*$!37,+++++gPF*Fgo7$F]p7$$!3(fSb?_!)*RBF*$! 3g&3cHiOmE$F*7$7$$!31fV;!fVA2#F*$!3J,+++++?FF*Fcp7$Fip7$$!3[uI6FQ`!)=F *$!3Wv91N$GUT#F*7$7$$!3cF-FUW>?9F*$!3A,+++++!o\"F*F_q7$Feq7$$!3O[Byz>$ *\\8F*$!3iQ>0QE(3f\"F*7$7$$!32VeCNEkJ6F*$!3y.Mq!QuRJ\"F*F[r7$Far7$$!3K BTV$!36cm!Q*)H6/)Fjr7$7$$!3Uxm*p*>B.gFjr$!3C6++++++kFjrFgr7$F ^s7$$!3txYg*4Il_#!#?$!3#3te2j'Ht^Fgs7$7$$\"3]T&pn4VK;%Fjr$\"3o*)****** ******RFjrFds7$7$$\"3!3apn4VK;%FjrF^t7$$\"3!)p;qzhoYuFjr$\"3N,'HQSO!Rn Fjr7$7$$\"3'QtA794bT\"F*$\"3@i?w!R;NB\"F*Fdt7$Fjt7$$\"3)*[KfAPp\"e\"F* $\"3C7<]qp9s8F*7$7$$\"3kueF;J.j;F*$\"31************R9F*F`u7$7$$\"3OueF ;J.j;F*Fiu7$$\"3/!)\\`vko?DF*$\"33^:p#*pvG?F*7$7$$\"3Mr6%=j?!yJF*$\"3; ************zCF*F_v7$Fev7$$\"3D([\">E,w0NF*$\"3aI\"*z@AbmEF*7$7$$\"3y5 8p<4miRF*$\"3>!zau>Cm#HF*F[w7$Faw7$$\"352fb$***opXF*$\"3+B%><_]@F$F*7$ 7$$\"3i\"R*[.!=^+&F*$\"3C************>NF*Fgw7$F]x7$$\"3#e*\\(odW1q&F*$ \"3Y,Lf%o#Q]QF*7$7$$\"3s())fTp7)4lF*$\"31hk7QjuMUF*Fcx7$Fix7$$\"3yV8\\ )3M!yoF*$\"3#)\\e)R]b'4WF*7$7$$\"3!p8N@dTX>(F*$\"3i************fXF*F_y 7$Fey7$$\"3NogD!)H'p8)F*$\"3)>DR2'ejN\\F*7$7$$\"3mk%G1Zkp0*F*$\"3d]^'* eaL-`F*F[z7$7$$\"3yl%G1Zkp0*F*Fdz7$$\"3$p(Q^B-(eU*F*$\"3)z^W43w$\\aF*7 $7$$\"3Dm\"e;+mP!)*F*$\"3U*************f&F*Fjz7$F`[l7$$\"3k&>]*z58z5!# <$\"3'=_-O!y'=$fF*7$7$$\"3G/(4Zi6/;\"Fi[l$\"3@hn\\^Q,0iF*Ff[l7$F]\\l7$ $\"3+(\\m#4)\\)=7Fi[l$\"3?@=WntR,kF*7$7$$\"3e/s\")47&)*G\"Fi[l$\"3C*** *********RmF*Fc\\l7$Fi\\l7$$\"3%H\"3s)[!oj8Fi[l$\"3uzwT!)[0]oF*7$7$$\" 3'>ccB!o7:9Fi[l$\"3O??];qU'*pF*F_]l7$Fe]l7$$\"3'p;vE.$z8:Fi[l$\"3%p-rN eYrF(F*7$7$$\"3_r98(R'Qb;Fi[l$\"3/************zwF*F[^l7$Fa^l7$$\"39*Q$ **)ylWm\"Fi[l$\"3.Ah>W4&>q(F*7$7$$\"3m>M+!)>%)p;Fi[l$\"38G&oEy[\\r(F*F g^l7$F]_l7$$\"3%H8ptd\\W#=Fi[l$\"3Q(o5mVQ()3)F*7$7$$\"3Nx-ldrbC>Fi[l$ \"3K#*R34_wI$)F*Fc_l7$Fi_l7$$\"3sw[ulLV%)>Fi[l$\"3[]_-Hf_v%)F*7$7$$\"3 U83N-Kb&3#Fi[l$\"3'))***********>()F*F_`l7$Fe`l7$$\"3E*GY=&G$y9#Fi[l$ \"3q(z)QbrO[))F*7$7$$\"3/NrHNBFz@Fi[l$\"3+P3+s$oJ\"*)F*F[al7$Faal7$$\" 3$4$eK<+5*F*7$7$$\"3H#*R%H^()RV#Fi[l$\"3+,B$[Ls\"Q% *F*Fgal7$7$$\"3u#*R%H^()RV#Fi[l$\"36-B$[Ls\"Q%*F*7$$\"3is`!G=nj[#Fi[l$ \"3i7%*pLU8Y&*F*7$7$$\"3e0zJJ!G,f#Fi[l$\"3y************f(*F*Fhbl7$7$F_ cl$\"3o)***********f(*F*7$$\"35lN[WH)*eEFi[l$\"3Y8Jz\"[X8))*F*7$7$$\"3 U]3f!p-()o#Fi[l$\"301fvb;sL**F*Fgcl7$7$F^dl$\"3;2fvb;sL**F*7$$\"3C@tOB G!p$GFi[l$\"3(3&[C1+\\>5Fi[l7$7$$\"373xBoyTVHFi[l$\"3T#4RW`h#Q5Fi[lFfd l7$F\\el7$$\"3Hy5D-F#[,$Fi[l$\"3Q!R5VYX30\"Fi[l7$7$$\"3]k2::RD!=$Fi[l$ \"3%)************z5Fi[lFbel7$Fhel7$$\"3Ug\"zg6/L>$Fi[l$\"3ezX6[;(>3\"F i[l7$7$$\"3!ec%)e/L\")>$Fi[l$\"3![!3i67q#3\"Fi[lF^fl7$Fdfl7$$\"3g7g*p! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "psi2 := x -> ln(x)-ln(ln(x) )+0.3:\np1 := plots[implicitplot](y*exp(y)=x,x=6..200,y=1..4,grid=[30, 30]):\np2 := plot(psi2(x),x=6..200,color=blue):\nplots[display]([p1,p2 ]);" }}{PARA 13 "" 1 "" {GLPLOT2D 343 292 292 {PLOTDATA 2 "6&-%'CURVES G6gq7$7$$\"\"'\"\"!$\"3_<5P87JJ9!#<7$$\"3y\"[lk2*)yy'F-$\"3&\\U7w_d]] \"F-7$7$$\"3+qZd9'e!=pF-$\"3vW.Jz8C<:F-F.7$F47$$\"3H$p9%))fDVyF-$\"3O \"o)e'p&=#f\"F-7$7$$\"31mCov59&>)F-$\"3q8CUo'*F-$\"3l#[M5$z8C;-TSh\"eExbx*=F-F\\o7$Fbo7$$\"3X]h?Z*yvJ\"FU$\"3u=C/Up ^B>F-7$7$$\"3]skKg^wJ8FU$\"3b?'eF[M5$>F-Fho7$7$F_p$\"3x?'eF[M5$>F-7$$ \"3cT!yZ[dR[\"FU$\"33$yt!ylB,?F-7$7$$\"37'4N)[C.c:FU$\"3]*o?'eF[M?F-Fg p7$F]q7$$\"3]?$[9)o7i;FU$\"3U#fy`)G8x?F-7$7$$\"3\\g%oPZlL\"=FU$\"3XeF[ M5$z8#F-Fcq7$Fiq7$$\"3m'R$H)z![^=FU$\"3]F U$\"3)zEn'Q7i\"=#F-F_r7$Fer7$$\"3u*\\:z\"\\b#F-7$7$$\"3W< b'*o?'eF$FU$\"3Imv;hj<_DF-Fjv7$F`w7$$\"3iH%\\$\\u.iNFU$\"3hn9**)[=4h#F -7$7$$\"3iP(flJIwx$FU$\"3?.Jz8CF:TD!zFF-7$7$$\"3;w#[M5$z8YFU$\"3!z s&yyF&)*z#F-F]z7$7$Fdz$\"3OGdyyF&)*z#F-7$$\"3xF\"*QXn*G\"[FU$\"3*opODu z7$GF-7$7$$\"3C([p*=+'z+&FU$\"35Ts^l*o?'GF-F\\[l7$Fb[l7$$\"3mlT:L+*G:& FU$\"3)fh&yO<:#)GF-7$7$$\"3_b'*o?'eFG&FU$\"3oI?&\\YZ,!HF-Fh[l7$7$F_\\l $\"38J?&\\YZ,!HF-7$$\"3$ooE_W0d]&FU$\"3EtlIG4/JHF-7$7$$\"3W!\\7cQ3Xv&F U$\"315$z8C22$F-F[_l7$Fa_l7$$ \"3Ng.P$>\\$4qFU$\"3fm/m-=J7JF-7$7$$\"3g$z8Cr/uFU$\"3kViN0:iaJF-7$7$$\"3!ov%)o*=(4d(FU$\"3%z W.Jz8C<$F-Fc`l7$Fi`l7$$\"3jTKcQh$>\"yFU$\"3Mbt5rp4&>$F-7$7$$\"3'Hqn!*p')FU$\"3!p^l*o?'eF$F-Fgbl7$7$F^cl$\"3O\"Fbfl$ \"3D#4PF.nO_$F-Fihl7$F_il7$$\"3vO;/%z1eA\"Fbfl$\"3_.B![dL?a$F-7$7$$\"3 %G[M5$z8k7Fbfl$\"3UjjqF\"zmc$F-Feil7$F[jl7$$\"3Qo$*4Vt0t7Fbfl$\"33YnH \"*RTsNF-7$7$$\"3#)3'ouR4XH\"Fbfl$\"3wB9Fbfl$\"3?Q\")4\"\\aol$F-7$7$$\"3%f*o?'eF[Y\"Fbfl$\"3_Sqh56v# o$F-F[]m7$7$Fb]m$\"3'4/<16^Fo$F-7$$\"37Hx!*RY4o9Fbfl$\"3'yf:v-.Yo$F-7$ 7$$\"3g6rLmt+x9Fbfl$\"3q#z8C.92*o:Fbfl$\"3m*>Y+'RgNPF-7$7$$\"3!=;Fbfl$\"31p9Gy#z3w$F-7$7$$\"3g4$z8Cep[&y!Ry$F-F^`m7$Fd`m7$$\"3SUN=K3,q;Fbfl$\"3#*Qn^'fahy$F-7$7$$\" 3/\\Y@o)3Ro\"Fbfl$\"3mheF[M5$z$F-Fj`m7$7$Faam$\"35ieF[M5$z$F-7$$\"3\\] ek?jk@P=F$QF-7$7$$\"3=&eF[M5$*z\"Fbfl$ \"3@jVgN:-WQF-Febm7$7$$\"3!\\eF[M5$*z\"FbflF^cm7$$\"3GOKpn!Rd#=Fbfl$\" 3'eLn]c#obQF-7$7$$\"3+BFbfl$\"3gIz8CFbfl$\"3s&RKEiP6!RF-7$7$$\"3zgeF[M5L>Fbfl$\"3&\\l`E%G H-RF-F\\em7$7$$\"3]geF[M5L>FbflFeem7$$\"3_Dvf>Od$)>Fbfl$\"3S7qJVL&>#RF -7$7$$\"3e)*************>Fbfl$\"3n>P\\*=_$GRF-F[fm-%'COLOURG6&%$RGBG\" \"\"F*F*-F$6$7U7$F($\"3*eRX%)Qh&3:F-7$$\"3Imm;aRK9\")F-$\"3y]5.j5ta;F- 7$$\"3ELL$3zkG-\"FU$\"3U[0\"Gh*Q\"y\"F-7$$\"36+Dc()4$o?\"FU$\"3eCMWzh1 y=F-7$$\"3!om\"H%=(z!R\"FU$\"311Th(eUX'>F-7$$\"3[LLeTcd/=FU$\"3fr\"evW Z18#F-7$$\"3qLL3\"*o4@AFU$\"3-$pr:0&**oAF-7$$\"3xm;z([Qcj#FU$\"3[hs/-w R'Q#F-7$$\"3IL$eHhr*>IFU$\"3!**G@(egv\"[#F-7$$\"3%)**\\i-j#zT$FU$\"3oB y0xE')pDF-7$$\"3TL$e>h\"\\HQFU$\"3W8])zxt=l#F-7$$\"3M+]7EqtRUFU$\"3;#Q \"owx5EFF-7$$\"3Qnm;g[shYFU$\"3OMZ`)>&)fz#F-7$$\"3sLL3u>TL]FU$\"3/!\\a ,]FH&GF-7$$\"3)3++0jV=X&FU$\"35$3v&**)4E\"HF-7$$\"3[++]2M*>(eFU$\"3CKu $*HOXoHF-7$$\"3k++]pb)oF'FU$\"3IXQu8R*)=IF-7$$\"3xL$e>WmXk'FU$\"3/(omr ?h@1$F-7$$\"3Unm;iXx\"3(FU$\"3!zM4Gs936$F-7$$\"3Wnmm)=Z@X(FU$\"3-.Qp^r *)\\JF-7$$\"3u**\\i,'yH)yFU$\"3gU3MI![J>$F-7$$\"37nmm.uLk#)FU$\"3XP#o^ AK'HKF-7$$\"3+,]i^%[Fo)FU$\"3-*z(z*=&*yE$F-7$$\"3a,]([ru63*FU$\"3!)=_X D,w-LF-7$$\"3[nm\"Hs!*o\\*FU$\"3y4<%p\\UwL$F-7$$\"3mn;z!zZ'y)*FU$\"3i& [3j(eVoLF-7$$\"3TL$3iVU!H5Fbfl$\"3\\\"pIw&=U+MF-7$$\"3ULe>\\W\"=2\"Fbf l$\"3jYIq))fRKMF-7$$\"3++vB7w/46Fbfl$\"3c9!))z%fEfMF-7$$\"3_L$e$[/E\\6 Fbfl$\"3k'y!R*GZt[$F-7$$\"3'******GC/3>\"Fbfl$\"3:\\B/$e+a^$F-7$$\"3(* **\\#4uY9B\"Fbfl$\"3B+11a$F-7$$\"3/+DcZ/xq7Fbfl$\"3UAGz>*))oc$F-7$ $\"32+]F>FV98Fbfl$\"3q\"zMA7APf$F-7$$\"3qmmO7_m`8Fbfl$\"3**z\"=FQArh$F -7$$\"3)****\\FFabR\"Fbfl$\"3Pls1MzSTOF-7$$\"3RLeR()>^L9Fbfl$\"3wNF[GV #Gm$F-7$$\"3O++X&z4]Z\"Fbfl$\"3;kPx,2j&o$F-7$$\"3vm\"H))*f09:Fbfl$\"3: 'GK%)*)Rlq$F-7$$\"31+D,K0([b\"Fbfl$\"3=C#*pc^&ys$F-7$$\"3emmT)>uZf\"Fb fl$\"3#e*H%3Q'=[PF-7$$\"3.+D1R'\\lj\"Fbfl$\"3S6,2S!f*oPF-7$$\"3cLL``Vy w;Fbfl$\"31&e9sp$\\)y$F-7$$\"3VL$3.;Izr\"Fbfl$\"3SqP*3A9!3QF-7$$\"3'p; H!f_tehR%QF-7$$ \"3tm;u]^?R=Fbfl$\"3C)f#\\o(fI'QF-7$$\"3_LLtE=kx=Fbfl$\"3+R%egH%yzQF-7 $$\"3J+D;SLi=>Fbfl$\"3k815$zfs*QF-7$$\"3=+vjxy%y&>Fbfl$\"3EGcRT_l8RF-7 $$\"$+#F*$\"3lwlSu!G4$RF--Fgfm6&Fifm$F*F*F]gn$\"*++++\"!\")-%+AXESLABE LSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(FgfnFign" 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 68 "remez('W(x)',x=2.844968958..41.19355573,[2,1],info=true,maxgraph=9 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algorithm:~calculating~ minimax~error~estimate~by~solving~a~rational~equationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D--~minimising~the~ABSOLUTE~error~--G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,iteration~6G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G--------------------------------------G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%Dprovisional~rational~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,($\"2$3)f>!4%e)R!#<\"\"\"*&$\"2#))Q1#)*\\ Yc$F'F(%\"xGF(F(*&$\"2]ipxAv$RF()F,\"\"#F(F(F(,&$F(\"\"!F(*&$\"2qF bhI[\\O\"F'F(F,F(F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%go al~for~relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/;> ]_\"p8\"!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%1critical~points:G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6'$\".+!e*o\\%G!#7$\"2@,7PqzVj%!#;$\"2KQ'GU`TF%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7minimax~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,($\"+-4%e)R!#5\"\"\"*&$\"+#)*\\Yc$F'F(%\"xGF(F(*&$\"+G_PR " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Halley's formula for solving " } {XPPEDIT 18 0 "x=y*exp(y)" "6#/%\"xG*&%\"yG\"\"\"-%$expG6#F&F'" } {TEXT -1 5 " for " }{TEXT 271 1 "y" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi( y) = y*exp(y)-x;" "6#/-%$phiG6#%\"yG,&*&F'\"\"\"-%$expG6#F'F*F*%\"xG! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Given an approxi mate zero " }{TEXT 273 1 "a" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "phi(y) " "6#-%$phiG6#%\"yG" }{TEXT -1 36 ", the \"improved\" Halley estimate \+ is " }{XPPEDIT 18 0 "y = y-h;" "6#/%\"yG,&F$\"\"\"%\"hG!\"\"" }{TEXT -1 24 " where the \"correction\" " }{TEXT 272 1 "h" }{TEXT -1 15 " is \+ given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h = ph i(y)/``(phi*`'`(y)-phi*`''`(y)*phi(y)/(2*phi*`'`(y)));" "6#/%\"hG*&-%$ phiG6#%\"yG\"\"\"-%!G6#,&*&F'F*-%\"'G6#F)F*F***F'F*-%#''G6#F)F*-F'6#F) F**(\"\"#F*F'F*-F16#F)F*!\"\"F=F=" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi*`'` (y) = exp(y)*(1+y)" "6#/*&%$phiG\"\"\"-%\"'G6#%\"yGF&*&-%$expG6#F*F&,& F&F&F*F&F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi*`\"`(y) = exp(y)*( 2+y)" "6#/*&%$phiG\"\"\"-%\"\"G6#%\"yGF&*&-%$expG6#F*F&,&\"\"#F&F*F&F& " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "y*exp(y)-x;\nDiff(%,y)=factor(diff(%,y));\nDi ff(%%,y$2)=factor(diff(%%,y$2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& *&%\"yG\"\"\"-%$expG6#F%F&F&%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,&*&%\"yG\"\"\"-%$expG6#F)F*F*%\"xG!\"\"F)*&F+F*,&F*F *F)F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,&*&%\"yG\"\"\"- %$expG6#F)F*F*%\"xG!\"\"-%\"$G6$F)\"\"#*&F+F*,&F3F*F)F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "The following co de sets up the starting approximations from the previous subsection vi a the procedure " }{TEXT 0 12 "start_approx" }{TEXT -1 30 ", together \+ with the procedure " }{TEXT 0 18 "next_halley_approx" }{TEXT -1 54 " t o perform one step of Halley's method for a zero of " }{XPPEDIT 18 0 " phi(y) = y*exp(y)-x;" "6#/-%$phiG6#%\"yG,&*&F'\"\"\"-%$expG6#F'F*F*%\" xG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 711 "start_approx := proc(x)\n local \+ y;\n if x>945 then\n y := ln(x);\n y := y-ln(y)+(.25079063 53+.1169032816e-6*x)/\n (1.+(.8463520229e-6+.7588315325e-22 *x)*x);\n elif x>45 then\n y := ln(x);\n y := y-ln(y)+.3; \n elif x>2.567437424 then\n y := (.3985840902+(.3564649982+.17 39375228e-2*x)*x)/\n (1.+.1364948306*x) \n elif x>-. 367879441171442321595523770161 then \n y := 0.1020602722*x+1.1624 67682-1.961004179/\n (x+1.908142587-0.1691574945/\n (x +0.7869827766-0.01043395288/(0.4680545525+x)));\n end if;\n y;\nen d proc:\n\nnext_halley_approx := proc(x,y)\n local t,h;\n t := y*e xp(y)-x;\n h := t/(exp(y)*(y+1)-1/2*(y+2)*t/(y+1));\n y-h;\nend pr oc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "T est example: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 163 "xx := 3.567891234;\ny0 := start_approx(xx);\ny1 := next_halley_approx(xx,y0);\ny2 := next_halley_approx(xx,y1);\ny3 : = next_halley_approx(xx,y2);\neval(y*exp(y),y=y3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+M7*yc$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G$\"+%HN#Q6!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$ \"+VU]S6!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G$\"+XU]S6!\"*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G$\"+XU]S6!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+J7*yc$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 20 "code for inverse of " }{XPPEDIT 18 0 "g(x)=x*exp(x)" "6 #/-%\"gG6#%\"xG*&F'\"\"\"-%$expG6#F'F)" }{TEXT -1 2 ": " }{TEXT 0 1 "W " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 6722 "W := proc(x::algebraic)\n local t,ok,s,ter ms,ti,i,eq;\n description \"inverse of x -> x*exp(x)\";\n option ` Copyright (c) 2003 Peter Stone.`;\n\n if nargs<>1 then\n error \+ \"expecting 1 argument, got %1\", nargs;\n end if;\n if type(x,'na me') then 'W'(x)\n elif type(x,'float') then evalf('W'(x))\n elif \+ type(x,And(complexcons,Not(realcons))) then\n error \"not impleme nted for complex argument\" \n elif type(x,`*`) and nops(x)= 2 then\n ok := false;\n if has(x,'exp') then \n if t ype(op(1,x),'function') and op([1,0],x)='exp' then\n t := o p([1,1],x);\n ok := true;\n elif type(op(2,x),'func tion') and op([2,0],x)='exp' then\n t := op([2,1],x);\n \+ ok := true;\n end if;\n if type(t,'realcons') \+ and signum(0,t+1,0)=-1 then\n ok := false\n end if; \n eq := t*exp(t)=x;\n if ok and (evalb(expand(eq)) or testeq(eq)) then t else 'W'(x) end if;\n elif has(x,'ln') then \+ \n if type(op(1,x),'function') and op([1,0],x)='ln' then\n \+ t := op([1,1],x);\n ok := true;\n elif type (op(2,x),'function') and op([2,0],x)='ln' then\n t := op([2 ,1],x);\n ok := true;\n end if;\n if ok the n\n eq := t*ln(t)=x; \n if not (evalb(expand(eq) ) or testeq(eq)) then ok := false end if;\n end if;\n \+ if type(t,'realcons') and signum(0,t,0)=1 and\n signum (0,t-exp(-1),0)=-1 then ok := false end if;\n if ok then ln(t) else 'W'(x) end if;\n else 'W'(x) end if;\n elif op(0,x)=`+` t hen\n ok := false;\n if has(x,'exp') then\n s := sel ect(has,x,'exp');\n terms := [op(s)];\n for i to nops( terms) do\n ti := terms[i];\n if type(ti,`*`) an d op(1,ti)=-1 then ti := -ti end if;\n if type(ti,'function ') and op(0,ti)='exp' then\n t := op(1,ti);\n \+ eq := t*exp(t)=x;\n if evalb(expand(eq)) or testeq(e q) then\n ok := true;\n break;\n \+ end if;\n end if;\n end do;\n if type(t,'realcons') and signum(0,t+1,0)=-1 then\n ok := fal se\n end if;\n if ok then t else 'W'(x) end if;\n \+ elif has(x,'ln') then\n s := select(has,x,'ln');\n te rms := [op(s)];\n for i to nops(terms) do\n ti := t erms[i];\n if type(ti,`*`) and op(1,ti)=-1 then ti := -ti e nd if;\n if type(ti,'function') and op(0,ti)='ln' then\n \+ t := op(1,ti);\n eq := t*ln(t)=x;\n \+ if evalb(expand(eq)) or testeq(eq) then\n ok := true;\n break;\n end if;\n \+ end if;\n end do;\n if type(t,'realcons') and signum(0 ,t,0)=1 and\n signum(0,t-exp(-1),0)=-1 then ok := fals e end if;\n if ok then ln(t) else 'W'(x) end if;\n else ' W'(x) end if;\n else 'W'(x) end if;\nend proc:\n\n# construct rememb er table\nW(0) := 0:\nW('infinity') := 'infinity':\nW(-1/exp(1)) := -1 :\nW(-exp(-1)) := -1:\nW(-1/2*ln(2)) := -ln(2):\n\n# differentiation\n `diff/W` := proc(a,x) \n option `Copyright (c) 2003 Peter Stone.`; \+ \n diff(a,x)*W(a)/(1+W(a))/a\nend proc:\n\n`D/W` := proc(t)\n opti on `Copyright (c) 2003 Peter Stone.`;\n if 1 a)\n end if\nend proc:\n\n# integration\n`int/W` := proc(f)\n loca l gx,h,inds,u;\n option `Copyright (c) 2003 Peter Stone.`;\n inds \+ := map(proc(x) if op(0,x)='W' then x end if end proc,indets(f,function ));\n if nops(inds)<>1 then return FAIL end if;\n inds := inds[1]; \n if nops(inds)=1 then gx := op(inds) else gx := op(2,inds) end if; \n if not type(gx,linear(_X)) then return FAIL end if;\n h := subs (inds=u,_X=(u*exp(u)-coeff(gx,_X,0))/coeff(gx,_X),f);\n h := h*(u+1) *exp(u)/coeff(gx,_X);\n h := int(h,u);\n if has(h,int) then return FAIL end if;\n subs(exp(u)=gx/u,u=inds,h)\nend proc:\n\n# simplific ation\n`simplify/W` := proc(s)\n local Wterm,a;\n option remember, system,`Copyright (c) 2003 Peter Stone.`;\n if not has(s,'W') or typ e(s,'name') then return s\n elif type(s,'function') and op(0,s)='exp ' then\n if type(op(1,s),'function') and op([1,0],s)='W' then\n \+ return op([1,1],s)/op(1,s)\n elif type(op(1,s),`*`) then\n Wterm := map(proc(f) if type(f,'function') and \n \+ op(0,f)='W' then f else 1 end if end proc,op(1,s));\n a := \+ op(1,s)/Wterm;\n if type(a,'rational') and type(Wterm,'functio n') then\n return (op(nops(Wterm),Wterm)/Wterm)^a end i f\n elif type(op(1,s),`+`) then\n Wterm := map(proc(f) if has(f,'W') then f else 0 end if end proc,op(1,s));\n if type( Wterm,`+`) then Wterm := op(1,Wterm) end if;\n return `simplif y/W`(exp(op(1,s)-Wterm)*exp(Wterm))\n else return s\n end if ;\n end if;\n map(procname,args)\nend proc:\n\n# numerical evaluat ion\n`evalf/W` := proc(x)\n local xx,eps,saveDigits,doW,val,p,q,maxi t;\n option `Copyright (c) 2003 Peter Stone.`;\n\n if not type(x,r ealcons) then return 'W'(x) end if;\n\n saveDigits := Digits;\n Di gits := Digits+min(iquo(Digits,3),5);\n xx := evalf(x);\n if xx<-. 3678794411714423215955237701615 or xx945 then\n s := ln(x);\n s := s-ln(s)+(.2507906353+.1169032816e- 6*x)/\n (1.+(.8463520229e-6+.7588315325e-22*x)*x);\n e lif x>45 then\n s := ln(x);\n s := s-ln(s)+.3;\n \+ elif x>2.567437424 then\n s := (.3985840902+(.3564649982+.1739 375228e-2*x)*x)/\n (1.+.1364948306*x) \n elif x>- .367879441171442321595523770161 then \n s := 0.1020602722*x+1. 162467682-1.961004179/\n (x+1.908142587-0.1691574945/\n \+ (x+0.7869827766-0.01043395288/(0.4680545525+x)));\n end i f;\n\n # solve x=s*exp(s) for s by Halley's method \n for i \+ to maxit do\n p := exp(s);\n t := s*p-x;\n h : = t/(p*(s+1)-1/2*(s+2)*t/(s+1));\n s := s-h;\n if abs( h)<=eps*abs(s) then break end if;\n end do;\n s;\n end pro c;\n\n p := ilog10(saveDigits);\n q := Float(saveDigits,-p);\n m axit := trunc((p+(.02331061386+.1111111111*q))*2.095903274)+2;\n eps := Float(3,-saveDigits-1);\n if Digits<=trunc(evalhf(Digits)) and \+ \n ilog10(xx)trunc(evalhf(DBL_MIN_10_EXP)) then\n val := evalhf(doW(xx ,eps,maxit))\n else\n val := doW(xx,eps,maxit)\n end if;\n \+ evalf[saveDigits](val);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "checking the procedure " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Comparison of starting approximati on with the inverse function. 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" }}{PARA 0 "" 0 "" {TEXT -1 28 "The solution is (of co urse) " }{XPPEDIT 18 0 "x = 2*exp(2);" "6#/%\"xG*&\"\"#\"\"\"-%$expG6# F&F'" }{TEXT -1 1 " " }{TEXT 265 1 "~" }{TEXT -1 14 " 14.77811220. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf(2*exp(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?7\"yZ\" !\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "secant(W(x)=2,x=14..15,info=true);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%7approximation~1~~->~~~G$\".*R]QLy9!#6" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\".b5u2yZ\"!#6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\".M.A6yZ\"! #6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\".'y> 7\"yZ\"!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~ G$\".'y>7\"yZ\"!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?7\"yZ\"!\") " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The code above contains p rocedures which enable " }{TEXT 0 4 "diff" }{TEXT -1 5 " and " }{TEXT 0 1 "D" }{TEXT -1 18 " to differentiate " }{XPPEDIT 18 0 "W(x);" "6#-% \"WG6#%\"xG" }{TEXT -1 8 " and W. 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" }}{PARA 0 "" 0 "" {TEXT -1 11 "We need an " }{TEXT 260 21 "i nitial approximation" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 57 "starting approximations for Halley's root-finding me thod " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 " The Maclaurin series for " }{XPPEDIT 18 0 "g(x) = x-arctan(x);" "6#/-% \"gG6#%\"xG,&F'\"\"\"-%'arctanG6#F'!\"\"" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/3" "6#*&\"\"\"F$\"\"$!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^3-1/5;" "6#,&*$%\"xG\"\"$\"\"\"* &F'F'\"\"&!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^5+1/7;" "6#,&*$% \"xG\"\"&\"\"\"*&F'F'\"\"(!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^7 -1/9;" "6#,&*$%\"xG\"\"(\"\"\"*&F'F'\"\"*!\"\"F*" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x^9+` . . . `" "6#,&*$%\"xG\"\"*\"\"\"%(~.~.~.~GF'" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 288 1 "x" }{TEXT -1 24 " is close to 0 we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x-arctan(x);" "6#,&%\"xG\"\"\"-%'arctanG6#F$!\" \"" }{TEXT -1 1 " " }{TEXT 286 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x ^3/3;" "6#*&%\"xG\"\"$F%!\"\"" }{TEXT -1 2 ". " }{TEXT 285 0 "" } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "Solving " }{XPPEDIT 18 0 "y = x^3/3;" "6#/%\"yG*&%\"xG\"\"$F'!\"\"" }{TEXT -1 5 " for " }{TEXT 287 1 "x" }{TEXT -1 7 " gives " }{XPPEDIT 18 0 "x = 3^(1/3)*y^(1/3);" "6#/%\"xG*&)\"\"$*&\"\"\"F)F'!\"\"F))%\"yG*&F)F)F'F*F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "psi[1 ](x) = 3^(1/3)*y^(1/3);" "6#/-&%$psiG6#\"\"\"6#%\"xG*&)\"\"$*&F(F(F-! \"\"F()%\"yG*&F(F(F-F/F(" }{TEXT -1 77 " can be used to provide an int ial approximation for the numerical inverse of " }{XPPEDIT 18 0 "g(x); " "6#-%\"gG6#%\"xG" }{TEXT -1 7 ", when " }{TEXT 289 1 "x" }{TEXT -1 16 " is close to 0. 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" }}{PARA 0 "" 0 "" {TEXT -1 26 "For intermediate vaues of " }{TEXT 297 1 "x" }{TEXT -1 111 ", a suitable rational approximation for the d eviation from the previous asymptotic approximations can be used. " }} {PARA 0 "" 0 "" {TEXT -1 59 "The rational approximation constructed be low tends to 0 as " }{TEXT 298 1 "x" }{TEXT -1 20 " tends to infinity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "remez('R(x)-x-1.570796327',x=0.1..500,[1,2],info=true );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algorithm:~calculating~ minimax~error~estimate~by~solving~a~rational~equationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D--~minimising~the~ABSOLUTE~error~--G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-ite ration~13G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G---------------------- ----------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprovisional~rat ional~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&$\"2s2vW> dm>\"!#;!\"\"*&$\"2MhtT=+ne$F'\"\"\"%\"xGF,F(F,,($F,\"\"!F,*&$\"2%)o'Q FY.5iF'F,F-F,F,*&$\"2j@h5)y_%o$F'F,)F-\"\"#F,F,F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%pI/5!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>goal~for~relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/;>]_\"p8\"!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1critical~points:G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6'$\",+++++\"!#6$\"2+\\I,'R2[:!#<$\"2&[p!faCeD%F($ \"2*\\3ZP9-f:!#;$\"23y$[YF=j5!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%! G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7minimax~approximation:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&$\"+%>dm>\"!\"*!\"\"*&$\"+%=+ne$F' \"\"\"%\"xGF,F(F,,($F,\"\"!F,*&$\"+FY.5iF'F,F-F,F,*&$\"+\")y_%o$F'F,)F -\"\"#F,F,F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%0minimax~error:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"&&p(*!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&$\"+%>dm>\"!\"*!\"\"*&$\"+%=+ne$F'\"\"\"%\"xGF,F (F,,($F,\"\"!F,*&$\"+FY.5iF'F,F-F,F,*&$\"+\")y_%o$F'F,)F-\"\"#F,F,F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "The n ext rational approximation fills in a range of positive values for the starting approximation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "remez('R(x)',x=3*10^(-8)..0.1,[5,4] ,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algorithm:~ca lculating~minimax~error~estimate~by~solving~a~rational~equationG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%D--~minimising~the~ABSOLUTE~error~--G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-iteration~12G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G----------- ---------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprov isional~rational~approximation:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*& ,.$\"8-*ffEad<\"zop%!#D\"\"\"*&$\"8%30`%>;')p!ph@!#=F(%\"xGF(F(*&$\"8' R:([vw%*)**en6!#8F()F-\"\"#F(F(*&$\"8e@0$>[JPP_h?!#5F()F-\"\"$F(F(*&$ \"8*Q@V#z(R[p'3'=!\")F()F-\"\"%F(F(*&$\"8u9/&=()*\\J&=@lF=F()F-\"\"&F( F(F(,,$F(\"\"!F(*&$\"8'p%='HOAMZ'ym)!#i?b_dBGF=F(F>F(F(!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%$Q6!*>;!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5%fgpecp@2A#!#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>goal~for ~relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5Ry(zx&)Q <=P$!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1critical~points:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6- $\"/++++++I!#@$\"87:L_1VeS[dM#!#H$\"8UU'y !#F$\"8]%>FzJ(pr%3A6!#E$\"8VHQ&Ry5@Vc]dF1$\"80nj*4)f]yzI`#!#D$\"8Y_q!p GM3V#\\x*F6$\"8)\\mhi#RtWnQ8$!#C$\"8go_qVt._o[G(F;$\"5+++++++++5!#?" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%7minimax~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,.$\" +=\"zop%!#7\"\"\"*&$\"+*p!ph@!\"&F(%\"xGF(F(*&$\"+*)**en6\"\"!F()F-\" \"#F(F(*&$\"+PP_h?\"\"$F()F-F7F(F(*&$\"+[p'3'=\"\"&F()F-\"\"%F(F(*&$\" +:`=@lF " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Halley's formula for solving " } {XPPEDIT 18 0 "x = y-arctan(y);" "6#/%\"xG,&%\"yG\"\"\"-%'arctanG6#F&! \"\"" }{TEXT -1 5 " for " }{TEXT 282 1 "y" }{TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi(y) = y-tanh(y)-x;" "6#/-%$phiG6#%\"yG,(F'\"\"\"-%%tanhG6#F'!\" \"%\"xGF-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Given an ap proximate zero " }{TEXT 284 1 "a" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "p hi(y)" "6#-%$phiG6#%\"yG" }{TEXT -1 36 ", the \"improved\" Halley esti mate is " }{XPPEDIT 18 0 "y = y-h;" "6#/%\"yG,&F$\"\"\"%\"hG!\"\"" } {TEXT -1 24 " where the \"correction\" " }{TEXT 283 1 "h" }{TEXT -1 15 " is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h = phi(y)/``(phi*`'`(y)-phi*`''`(y)*phi(y)/(2*phi*`'`(y)));" "6#/% \"hG*&-%$phiG6#%\"yG\"\"\"-%!G6#,&*&F'F*-%\"'G6#F)F*F***F'F*-%#''G6#F) F*-F'6#F)F**(\"\"#F*F'F*-F16#F)F*!\"\"F=F=" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi*`'`(y) = 1-1/(1+y^2);" "6#/*&%$phiG\"\"\"-%\"'G6#%\"yGF&,&F&F&* &F&F&,&F&F&*$F*\"\"#F&!\"\"F0" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi *`\"`(y) = 2*y/((1+y^2)^2);" "6#/*&%$phiG\"\"\"-%\"\"G6#%\"yGF&*(\"\"# 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 62 "y-arctan(y)- x;\nDiff(%,y)=diff(%,y);\nDiff(%%,y$2)=diff(%%,y$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"yG\"\"\"-%'arctanG6#F$!\"\"%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(%\"yG\"\"\"-%'arctanG6#F(!\"\"%\"x GF-F(,&F)F)*&F)F),&F)F)*$)F(\"\"#F)F)F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(%\"yG\"\"\"-%'arctanG6#F(!\"\"%\"xGF--%\"$ G6$F(\"\"#,$*(F2F),&F)F)*$)F(F2F)F)!\"#F(F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "The following code sets \+ up the starting approximations from the previous subsection via the pr ocedure " }{TEXT 0 12 "start_approx" }{TEXT -1 30 ", together with the procedure " }{TEXT 0 18 "next_halley_approx" }{TEXT -1 54 " to perfor m one step of Halley's method for a zero of " }{XPPEDIT 18 0 "phi(y) = y+tanh(y)-x;" "6#/-%$phiG6#%\"yG,(F'\"\"\"-%%tanhG6#F'F)%\"xG!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1112 "start_approx := proc(x)\n local y;\n if x< 3e-8 and x>-3e-8 then \n y := 1.732050808*surd(x,3);\n elif x>0 then\n if x<.1 then \n y := (.4696879118e-2+(21616.9069 9+(1167589989.+(.2061523737e13+\n (.1860866948e15+.652118 5315e15*x)*x)*x)*x)*x)/\n (1.+(866786.4734+(.1386705651e1 1+\n (.8768401337e13+.2823575255e15*x)*x)*x)*x);\n e lse\n y := x+1.570796327+(-1.196657194-3.586700184*x)/\n \+ (1.+(6.210034627+3.684527881*x)*x);\n end if;\n else\n \+ if x>-.1 then \n y := (-.4696879118e-2+(21616.90699+(-11 67589989.+(.2061523737e13+\n (-.1860866948e15+.6521185315 e15*x)*x)*x)*x)*x)/\n (1.+(-866786.4734+(.1386705651e11+ \n (-.8768401337e13+.2823575255e15*x)*x)*x)*x)\n els e\n y := x-1.570796327-(-1.196657194+3.586700184*x)/\n \+ (1.+(-6.210034627+3.684527881*x)*x);\n end if;\n end if; \n y;\nend proc:\n\nnext_halley_approx := proc(x,y)\n local t,p,u, v,h;\n t := y-arctan(y)-x;\n p := 1+y^2;\n u := 1-1/p;\n v := \+ 2*y/p^2;\n h := t/(u-1/2*v*t/u);\n y-h;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Test example: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "xx := evalf(sqrt(5));\ny0 := start_approx(xx);\ny1 := next_halley _approx(xx,y0);\ny2 := next_halley_approx(xx,y1);\neval(y-arctan(y),y= y2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+xz1OA!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G$\"+Ip:IN!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$\"+dg(3`$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#y2G$\"+dg(3`$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+xz1OA!\"* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "code for i nverse of " }{XPPEDIT 18 0 "g(x) = x-arctan(x);" "6#/-%\"gG6#%\"xG,&F' \"\"\"-%'arctanG6#F'!\"\"" }{TEXT -1 2 ": " }{TEXT 0 1 "R" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5709 "R := proc(x::algebraic)\n local t,ok,s,terms,i,ti ,eq;\n description \"inverse of x -> x-arctan(x)\";\n option `Copy right (c) 2003 Peter Stone.`;\n\n if nargs<>1 then\n error \"ex pecting 1 argument, got %1\", nargs;\n end if;\n if type(x,'name') then 'R'(x)\n elif type(x,'float') then evalf('R'(x))\n elif type (x,`*`) and type(op(1,x),'numeric') and \n sig num(0,op(1,x),0)=-1 then -R(-x) \n elif type(x,'realcons') and signu m(0,x,0)=-1 then -R(-x)\n elif type(x,And(complexcons,Not(realcons)) ) then\n error \"not implemented for complex argument\"\n elif \+ type(x,`+`) then\n ok := false;\n if has(x,'arctan') then\n \+ s := select(has,x,'arctan');\n if type(s,`+`) then ter ms := [op(s)] else terms := [s] end if;\n for i to nops(terms) do\n ti := terms[i];\n if type(ti,`*`) and op(1 ,ti)=-1 then ti := -ti end if;\n if type(ti,'function') and op(0,ti)='arctan' then\n t := op(1,ti);\n \+ eq := t-arctan(t)=x;\n if evalb(expand(eq)) or testeq(e q) then\n ok := true;\n break;\n \+ end if;\n end if;\n end do;\n if ok then t else 'R'(x) end if;\n elif has(x,'tan') then\n \+ s := select(has,x,'tan');\n if type(s,`+`) then terms := [op( s)] else terms := [s] end if;\n for i to nops(terms) do\n \+ ti := terms[i];\n if type(ti,`*`) and op(1,ti)=-1 th en ti := -ti end if;\n if type(ti,'function') and op(0,ti)= 'tan' then\n t := op(1,ti);\n eq := tan(t) -t=x;\n if evalb(expand(eq)) or testeq(eq) then\n \+ ok := true;\n break;\n end i f;\n end if;\n end do;\n if type(t,'realcon s') and (signum(0,t-Pi/2,1)=1 or\n signum(0,t+Pi/2, -1)=-1) then ok := false end if;\n if ok then tan(t) else 'R'( x) end if;\n else 'R'(x) end if;\n else 'R'(x) end if;\nend pro c:\n\n# construct remember table\nR(0) := 0:\nR('infinity') := 'infini ty':\nR(1-Pi/4) := 1:\nR(sqrt(3)-Pi/3) := sqrt(3):\nR(sqrt(3)/3-Pi/6) \+ := sqrt(3)/3:\nR(1+sqrt(2)-3/8*Pi) := 1+sqrt(2):\nR(sqrt(2)-1-1/8*Pi) \+ := sqrt(2)-1:\nR(2+sqrt(3)-5/12*Pi) := 2+sqrt(3):\nR(2-sqrt(3)-1/12*Pi ) := 2-sqrt(3):\n\n# differentiation\n`diff/R` := proc(a,x) \n optio n `Copyright (c) 2003 Peter Stone.`; \n diff(a,x)*(1+1/R(a)^2)\nend \+ proc:\n\n`D/R` := proc(t)\n option `Copyright (c) 2003 Peter Stone.` ;\n if 11 then return FAIL end \+ if;\n inds := inds[1];\n if nops(inds)=1 then gx := op(inds) else \+ gx := op(2,inds) end if;\n if not type(gx,linear(_X)) then return FA IL end if;\n h := subs(inds=u,_X=(u-arctan(u)-coeff(gx,_X,0))/coeff( gx,_X),f);\n h := h*u^2/(1+u^2)/coeff(gx,_X);\n h := int(h,u);\n \+ if has(h,int) then return FAIL end if;\n subs(arctan(u)=u-gx,u=inds ,h)\nend proc:\n\n# simplification\n`simplify/R` := proc(s)\n option remember,system,`Copyright (c) 2003 Peter Stone.`;\n if not has(s,' R') or type(s,'name') then return s\n elif type(s,'function') and op (0,s)='arctan' and nops(s)=1 then\n if type(op(1,s),'function') a nd op([1,0],s)='R' then\n return op(1,s)-op([1,1],s)\n el se return s\n end if;\n end if;\n map(procname,args)\nend pro c:\n\n# numerical evaluation\n`evalf/R` := proc(x)\n local xx,eps,sa veDigits,doR,val,p,q,maxit;\n option `Copyright (c) 2003 Peter Stone .`;\n \n if not type(x,realcons) then return 'R'(x) end if;\n\n \+ doR := proc(x,eps,maxit)\n local t,p,s,u,v,h,i; \n # set up \+ a starting approximation\n if x<3e-8 and x>-3e-8 then \n \+ s := 1.732050808*surd(x,3);\n elif x>0 then\n if x<.1 the n \n s := (.4696879118e-2+(21616.90699+(1167589989.+(.2061 523737e13+\n (.1860866948e15+.6521185315e15*x)*x)*x)*x )*x)/\n (1.+(866786.4734+(.1386705651e11+\n \+ (.8768401337e13+.2823575255e15*x)*x)*x)*x);\n else\n \+ s := x+1.570796327+(-1.196657194-3.586700184*x)/\n \+ (1.+(6.210034627+3.684527881*x)*x);\n end if;\n else \n if x>-.1 then \n s := (-.4696879118e-2+(21616.9 0699+(-1167589989.+(.2061523737e13+\n (-.1860866948e15 +.6521185315e15*x)*x)*x)*x)*x)/\n (1.+(-866786.4734+(. 1386705651e11+\n (-.8768401337e13+.2823575255e15*x)*x) *x)*x)\n else\n s := x-1.570796327-(-1.196657194+3. 586700184*x)/\n (1.+(-6.210034627+3.684527881*x)*x);\n end if;\n end if;\n # solve the equation x=s-tanh(s ) for s by Halley's method \n for i to maxit do\n t := s- arctan(s)-x;\n p := 1+s^2;\n u := 1-1/p;\n v : = 2*s/p^2;\n h := t/(u-1/2*v*t/u);\n s := s-h;\n \+ if abs(h)<=eps*abs(s) then break end if;\n end do;\n s;\n end proc;\n\n p := ilog10(Digits);\n q := Float(Digits,-p);\n \+ maxit := trunc((p+(.02331061386+.1111111111*q))*2.095903274)+2;\n s aveDigits := Digits;\n Digits := Digits+min(iquo(Digits,3),5);\n x x := evalf(x);\n Digits := Digits+max(0,-ilog10(xx)-3);\n eps := F loat(3,-saveDigits-1);\n if Digits<=trunc(evalhf(Digits)) and \n \+ ilog10(xx)trunc(evalhf(DBL_MIN_10_EXP)) then\n val := evalhf(doR(xx,eps,m axit))\n else\n val := doR(xx,eps,maxit)\n end if;\n evalf[ saveDigits](val);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 23 "checking the procedure " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 63 "Comparison of starting approximation \+ with the inverse function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot('R(x)-start_approx(x)',x=0..0. 5,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 561 178 178 {PLOTDATA 2 " 6&-%'CURVESG6#7bs7$$\"\"!F)F(7$$\"+S`!eS$!#8$\"(@4m$!#57$$\"+!o5;\"oF- $\"(t;(oF07$$\"+-;u@5!#7$\"('znFF07$$\"+O@Ki8F9$!(`!*e\"F07$$\"+qE!Hq \"F9$!(^cp%F07$$\"+/K[V?F9$!(X$)['F07$$\"+QP1%Q#F9$!(KvA(F07$$\"+sUkCF F9$!(#e+sF07$$\"+1[AlIF9$!(v'[mF07$$F,F9$!(Wsv&F07$$\"+ueQYPF9$!(*RjYF 07$$\"+3k'p3%F9$!(uiY$F07$$\"+UpaFWF9$!(SjB#F07$$\"+wu7oZF9$!(HG-\"F07 $$\"+5!3(3^F9$\"'Q19F07$$\"+X&)G\\aF9$\"(W=B\"F07$$\"+7'\\/8'F9$\"(#yZ JF07$$F3F9$\"(-hn%F07$$\"+[>Fjr$!(8\\B\"F07$$\"+uq8Q?Fjr$!(\")))Q#F07$ $\"+*)>Wr@Fjr$!(4?b$F07$$\"+0pu/BFjr$!(^!fXF07$$\"+?=0QCFjr$!(KOS&F07$ $\"+NnNrDFjr$!(%=&3'F07$$\"+ml'z$GFjr$!(0l(pF07$$\"+(RwX5$Fjr$!(@;H(F0 7$$\"+q1nrJFjr$!(4&*G(F07$$\"+V\\wQKFjr$!(8uD(F07$$\"+<#feI$Fjr$!([n>( F07$$\"+!\\`HP$Fjr$!(S*3rF07$$\"+P?92NFjr$!(6w&oF07$$\"+%eI8k$Fjr$!(VZ 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examples " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "g := x -> x-arctan (x);\nxx := evalf(sqrt(2));\ng(xx);\nR(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"-%'a rctanG6#F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+iN@ 99!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+S%p*)e%!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "g := x -> x-arctan(x);\nxx := -0.4991234567891096;\nevalf(g(xx),20);\nevalf(R(%),16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9 $\"\"\"-%'arctanG6#F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #xxG$!1'4\"*ycM7*\\!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!4&GV]J>Gt< O!#?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!1'4\"*ycM7*\\!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "xx := 1;\nyy := xx-tan(xx);\nR(yy);\nevalf(%,12);\nyy := 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jrFjr7$F`sF`s7$FfsFfs7$F[tF[t7$F`tF`t7$FetFet7$FjtFjt7$F_uF_u7$FduFdu7 $FiuFiu7$F^vF^v7$FcvFcv7$FhvFhv7$F]wF]w7$FbwFbw7$FgwFgw7$F\\xF\\x7$Fax Fax7$FfxFfx7$F[yF[y7$F`yF`y7$FeyFey7$FjyFjy7$F_zF_z7$FdzFdz7$FizFiz-F^ [l6&F`[lF*F*F*-Ff[l6#\"\"#-%(SCALINGG6#%,CONSTRAINEDG-%*THICKNESSGFb`m -%+AXESLABELSG6$Q\"x6\"Q\"yF^am-%%VIEWG6$;F(FizFcam" 1 2 0 1 10 2 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "As an \+ illustration of the secant method for root-finding we can solve the eq uation " }{XPPEDIT 18 0 "R(x) = 2;" "6#/-%\"RG6#%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "The solution is (of course) " } {XPPEDIT 18 0 "x = 2-arctan(2);" "6#/%\"xG,&\"\"#\"\"\"-%'arctanG6#F&! \"\"" }{TEXT -1 1 " " }{TEXT 280 1 "~" }{TEXT -1 15 " 0.8928512822. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalf(2-arctan(2),11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+AG ^G*)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "secant(R(x)=2,x=0.8..0.9,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\".Ml%oQH*)!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\".>V/7&G*)!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\".r?#G^G*)! #8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\".j?# G^G*)!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+AG^G*)!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The code above contains procedures whic h enable " }{TEXT 0 4 "diff" }{TEXT -1 5 " and " }{TEXT 0 1 "D" } {TEXT -1 18 " to differentiate " }{XPPEDIT 18 0 "R(x);" "6#-%\"RG6#%\" xG" }{TEXT -1 8 " and R. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "diff(R(x),x);\nD(R)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&F$F$*$)-%\"RG6#%\"xG\"\"#F$!\"\" F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$*&F$F$*$)-%\"RG6#%\"xG \"\"#F$!\"\"F$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "This means that we can make use of Newton's and Halley's \+ method for root-finding. 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" }}{PARA 0 "" 0 "" {TEXT -1 11 "We \+ need an " }{TEXT 260 21 "initial approximation" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 57 "starting approximations \+ for Halley's root-finding method " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 62 "We tackle the problem of constructing a n umerical inverse for " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "x = y+ arctan(y);" "6#/%\"xG,&%\"yG\"\"\"-%'arctanG6#F&F'" }{TEXT -1 1 "," }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dy = 1+1/(1+y^2); " "6#/*&%#dxG\"\"\"%#dyG!\"\",&F&F&*&F&F&,&F&F&*$%\"yG\"\"#F&F(F&" } {XPPEDIT 18 0 "`` = (2+y^2)/(1+y^2);" "6#/%!G*&,&\"\"#\"\"\"*$%\"yGF'F (F(,&F(F(*$F*F'F(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = (1 +y^2)/(2+y^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&F&F&*$%\"yG\"\"#F&F&,&F -F&*$F,F-F&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 56 "Obtain the first few terms in the Maclaur in series for " }{XPPEDIT 18 0 "f(x) = g^(-1)*``(x);" "6#/-%\"fG6#%\" xG*&)%\"gG,$\"\"\"!\"\"F,-%!G6#F'F," }{TEXT -1 54 " as the series solu tion for the differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx = (1+y^2)/(2+y^2);" "6#/*&%#dyG\"\"\"%#dxG!\" \"*&,&F&F&*$%\"yG\"\"#F&F&,&F-F&*$F,F-F&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "O rder := 14:\nde := diff(y(x),x)=(1+y(x)^2)/(2+y(x)^2);\nic := y(0)=0; \ndsolve(\{de,ic\},y(x),type=series):\nconvert(rhs(%),polynom);\nOrder := 6:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\" xGF,*&,&\"\"\"F/*$)F)\"\"#F/F/F/,&F2F/F0F/!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&#\"\"\"\"\"#F&%\"xGF&F&*&#F&\"#[F&*$)F(\"\"$F&F&F&*&#F&\"%?>F&* $)F(\"\"&F&F&!\"\"*&#F&\"&g,#F&*$)F(\"\"(F&F&F5*&#\"$V\"\")?VABF&*$)F( \"\"*F&F&F&*&#\"#(*\",+3q=-\"F&*$)F(\"#6F&F&F5*&#\"&()\\%\"-+CmeqzF&*$ )F(\"#8F&F&F5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Alternatively, the same Taylor polynomial can be obtained by using the procedures " }{TEXT 0 6 "RootOf" }{TEXT -1 5 " and " } {TEXT 0 6 "taylor" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "RootOf(y+arctan(y)-x,y);\nta ylor(%,x,14);\np := unapply(convert(%,polynom),x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'RootOfG6#,(%#_ZG\"\"\"-%'arctanG6#F'F(%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG#\"\"\"\"\"#F&#F&\"#[\"\"$#! \"\"\"%?>\"\"&#F,\"&g,#\"\"(#\"$V\"\")?VAB\"\"*#!#(*\",+3q=-\"\"#6#!&( )\\%\"-+Cmeqz\"#8-%\"OG6#F&\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,0*&#\"\"\"\"\"#F/9$F/F/*&#F/\" #[F/*$)F1\"\"$F/F/F/*&#F/\"%?>F/*$)F1\"\"&F/F/!\"\"*&#F/\"&g,#F/*$)F1 \"\"(F/F/F>*&#\"$V\"\")?VABF/*$)F1\"\"*F/F/F/*&#\"#(*\",+3q=-\"F/*$)F1 \"#6F/F/F>*&#\"&()\\%\"-+CmeqzF/*$)F1\"#8F/F/F>F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "The Pade approximati on of type" }{XPPEDIT 18 0 "``(7, 6);" "6#-%!G6$\"\"(\"\"'" }{TEXT -1 29 " can be obtained as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "numapprox[pade](p(x),x,[7,6] );\nconvert(%,confrac,x):\nevalf(evalf(%,15)):\npsi := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,**&#\"+&47!pZ\"/s)**>+#*y\"\"\"\" *$)%\"xG\"\"(F)F)F)*&#\"+.C.Hf\"-SEvm\"p(F)*$)F,\"\"&F)F)F)*&#\"*@W,=% \"+o(=VC&F)*$)F,\"\"$F)F)F)*&#F)\"\"#F)F,F)F)F),*F)F)*&#\"):)Hd#\"*#G8 &=#F)*$)F,F>F)F)F)*&#\")G6lT\"+`\"pag$F)*$)F,\"\"%F)F)F)*&#\",\\*z'GF# \"/?^G4+2$)F)*$)F,\"\"'F)F)F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%$psiGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&$\"+Ks!=u*!#5\"\"\"9$F1F 1*&$\"+l$[eH\"!\")F1,&F2F1*&$\"+)Qy\\B$F6F1,&F2F1*&$\"+6$zx-(F0F1,&F2F 1*&$\"+cM7P?F6F1,&F2F1*&$\"+[Cj3NF6F1,&F2F1*&$\"+&[=ga&!\"*F1F2!\"\"FL FLF1FLFLFLF1FLF1FLFLF(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The following picture makes a graphical compar ison between the function defined by the Pade approximation and the gr aph of " }{XPPEDIT 18 0 "x = y+arctan(y);" "6#/%\"xG,&%\"yG\"\"\"-%'ar ctanG6#F&F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "p1 := plots[implicitplot](y+arctan (y)=x,x=2.5995..2.6,y=1.59..1.5906):\np2 := plot(psi(x),x=2.5995..2.6, color=blue):\nplots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 307 275 275 {PLOTDATA 2 "6&-%'CURVESG6_p7$7$$\"3#***********\\*f#!#<$ \"3#QF**R7(4!f\"F*7$$\"3K]+Oe:^*f#F*$\"3)*Rz'*Hh5!f\"F*7$7$$\"3/+++++_ *f#F*$\"3+y1Z2F6!f\"F*F-7$F37$$\"3KUNj%oB&*f#F*$\"3#)\\(R%yb6!f\"F*7$7 $$\"3m'*HJf$H&*f#F*$\"3?+++++7!f\"F*F97$F?7$$\"3#=!\\*3\"e`*f#F*$\"3o= h#p-D,f\"F*7$7$$\"3=+++++a*f#F*$\"3i&o\")4HG,f\"F*FE7$FK7$$\"3%)*eXr$z a*f#F*$\"31$HDaZM,f\"F*7$7$$\"3I+++++c*f#F*$\"3!y_FX(Q9!f\"F*FQ7$FW7$$ \"3UxiRj+c*f#F*$\"3YnW#R#R9!f\"F*7$7$$\"3Q$*>.h,c*f#F*$\"3A++++S9!f\"F *Fgn7$F]o7$$\"3wXVi*=s&*f#F*$\"3o&y]CP`,f\"F*7$7$$\"3W+++++e*f#F*$\"3Q 2s9e%f,f\"F*Fco7$Fio7$$\"3qIA&eJ%e*f#F*$\"3'RKx4#G;!f\"F*7$7$$\"3!H'Qg i4f*f#F*$\"3C++++!o,f\"F*F_p7$Fep7$$\"3)=\")p?W'f*f#F*$\"3CFi^pA!f\"F*Fgq7$F]r7$$\"37v*zW p?'*f#F*$\"3#4.Cm;\">!f\"F*7$7$$\"3M3'GSw@'*f#F*$\"3E++++?>!f\"F*Fcr7$ 7$$\"3y3'GSw@'*f#F*F\\s7$$\"3/mUm?Gj*f#F*$\"3C#)G?:1?!f\"F*7$7$$\"3%3+ +++S'*f#F*$\"3/>%e#4i?!f\"F*Fbs7$7$Fis$\"3E>%e#4i?!f\"F*7$$\"3AXl%o%\\ k*f#F*$\"3?ZTyj+@!f\"F*7$7$$\"3cMiIlDl*f#F*$\"3G++++g@!f\"F*Fat7$Fgt7$ $\"3a[.-tql*f#F*$\"3M$evB^>-f\"F*7$7$$\"3'4++++g'*f#F*$\"3qO_0$z@-f\"F *F]u7$Fcu7$$\"3'f#)z\"*>p'*f#F*$\"3=5U)4'*G-f\"F*7$7$$\"3a,++++o*f#F*$ \"3#y\"))*oPP-f\"F*Fiu7$7$$\"35,++++o*f#F*$\"3g<))*oPP-f\"F*7$$\"3%QIR `K\"o*f#F*$\"3-PGf4%Q-f\"F*7$7$$\"3eXnVmLo*f#F*$\"3K+++++C!f\"F*Fjv7$F `w7$$\"3A=)z9X$p*f#F*$\"3))>UAeyC!f\"F*7$7$$\"3A,++++q*f#F*$\"3)=I/3'H D!f\"F*Ffw7$F\\x7$$\"3M%\\;wd0(*f#F*$\"3g3-'oId-f\"F*7$7$$\"3%\\9?u;9( *f#F*$\"3M++++SE!f\"F*Fbx7$Fhx7$$\"35@lu.xr*f#F*$\"3GwT]bnE!f\"F*7$7$$ \"3O,++++s*f#F*$\"3M?SuW&o-f\"F*F^y7$Fdy7$$\"3!oRg)H)H(*f#F*$\"3!e_nT? 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" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 294 1 "x" }{TEXT -1 49 " has large magnitude the starting approximatio ns " }{XPPEDIT 18 0 "x-Pi/2" "6#,&%\"xG\"\"\"*&%#PiGF%\"\"#!\"\"F)" } {TEXT -1 8 " ( when " }{XPPEDIT 18 0 "x>0" "6#2\"\"!%\"xG" }{TEXT -1 7 " ) and " }{XPPEDIT 18 0 "x+Pi/2" "6#,&%\"xG\"\"\"*&%#PiGF%\"\"#!\" \"F%" }{TEXT -1 8 " ( when " }{XPPEDIT 18 0 "x<0" "6#2%\"xG\"\"!" } {TEXT -1 15 ") can be used. " }}{PARA 0 "" 0 "" {TEXT -1 26 "For inter mediate vaues of " }{TEXT 295 1 "x" }{TEXT -1 111 ", a suitable ration al approximation for the deviation from the previous asymptotic approx imations can be used. " }}{PARA 0 "" 0 "" {TEXT -1 59 "The rational ap proximation constructed below tends to 0 as " }{TEXT 296 1 "x" }{TEXT -1 20 " tends to infinity. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "remez('S(x)-x+1.570796327',x=2.5..5 00,[1,2],info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algor ithm:~calculating~minimax~error~estimate~by~solving~a~rational~equatio nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D--~minimising~the~ABSOLUTE~err or~--G" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-iteration~12G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G----------- ---------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprov isional~rational~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*& ,&$\"2R#HczT+YB!#<\"\"\"*&$\"2mrI9Aekp%F'F(%\"xGF(F(F(,($F(\"\"!F(*&$ \"2Ob\"=:NRJiF'F(F,F(!\"\"*&$\"2yDF>'e\\`ZF'F()F,\"\"#F(F(F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!#<" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%,difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$!$E#!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5relative~difference:G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/ioVqX!=\"!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>goal~for~relative~difference:G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"/;>]_\"p8\"!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1critical~points:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"-+++++D!#6$\"2B^m)3Q=\"y#!#;$\"2O&zsX!p=# QF($\"27lXq)REvpF($\"23%HOT7*e$G!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7minimax~approximation:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&$\"+!=/gM#!#5\"\"\"*&$\"+@#ekp%F'F (%\"xGF(F(F(,($F(\"\"!F(*&$\"+:NRJiF'F(F,F(!\"\"*&$\"+ie\\`ZF'F()F,\" \"#F(F(F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0minimax~error:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"&X\">!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#*&,&$\"+!=/gM#!#5\"\"\"*&$\"+@#ekp%F'F(%\"xGF(F(F(,($ F(\"\"!F(*&$\"+:NRJiF'F(F,F(!\"\"*&$\"+ie\\`ZF'F()F,\"\"#F(F(F3" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Halley's formula for so lving " }{XPPEDIT 18 0 "x = y+arctan(y);" "6#/%\"xG,&%\"yG\"\"\"-%'arc tanG6#F&F'" }{TEXT -1 5 " for " }{TEXT 277 1 "y" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "phi(y) = y+arctan(y)-x;" "6#/-%$phiG6#%\"yG,(F'\"\"\"-% 'arctanG6#F'F)%\"xG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Given an approximate zero " }{TEXT 279 1 "a" }{TEXT -1 4 " of " } {XPPEDIT 18 0 "phi(y)" "6#-%$phiG6#%\"yG" }{TEXT -1 36 ", the \"improv ed\" Halley estimate is " }{XPPEDIT 18 0 "y = y-h;" "6#/%\"yG,&F$\"\" \"%\"hG!\"\"" }{TEXT -1 24 " where the \"correction\" " }{TEXT 278 1 " h" }{TEXT -1 15 " is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "h = phi(y)/``(phi*`'`(y)-phi*`''`(y)*phi(y)/(2*phi*`'`( y)));" "6#/%\"hG*&-%$phiG6#%\"yG\"\"\"-%!G6#,&*&F'F*-%\"'G6#F)F*F***F' F*-%#''G6#F)F*-F'6#F)F**(\"\"#F*F'F*-F16#F)F*!\"\"F=F=" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "phi*`'`(y) = 1+1/(1+y^2);" "6#/*&%$phiG\"\"\"-%\"'G6#% \"yGF&,&F&F&*&F&F&,&F&F&*$F*\"\"#F&!\"\"F&" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "phi*`\"`(y) = -2*y/((1+y^2)^2);" "6#/*&%$phiG\"\"\"-%\" \"G6#%\"yGF&,$*(\"\"#F&F*F&*$,&F&F&*$F*F-F&F-!\"\"F1" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "y+arctan(y)-x;\nDiff(%,y)=diff(%,y);\nDiff(%%,y$2)=diff(%%,y$2 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"yG\"\"\"-%'arctanG6#F$F%%\" xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(%\"yG\"\"\"-%' arctanG6#F(F)%\"xG!\"\"F(,&F)F)*&F)F),&F)F)*$)F(\"\"#F)F)F.F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(%\"yG\"\"\"-%'arctanG6#F( F)%\"xG!\"\"-%\"$G6$F(\"\"#,$*(F2F),&F)F)*$)F(F2F)F)!\"#F(F)F." }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "The foll owing code sets up the starting approximations from the previous subse ction via the procedure " }{TEXT 0 12 "start_approx" }{TEXT -1 30 ", t ogether with the procedure " }{TEXT 0 18 "next_halley_approx" }{TEXT -1 54 " to perform one step of Halley's method for a zero of " } {XPPEDIT 18 0 "phi(y) = y+arctan(y)-x;" "6#/-%$phiG6#%\"yG,(F'\"\"\"-% 'arctanG6#F'F)%\"xG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 618 "start_approx := proc (x)\n local y;\n if x<2.6 and x>-2.6 then \n y := .9741807232 *x-12.95848365/(x+32.34978388/\n (x+.7027779311/(x-20.37123456/ (x+35.08632448/(x-5.546018485/x)))));\n elif x>0 then \n y := x -1.570796327+(.2346004180+.4696458221*x)/\n (1.+(-.623139351 5+.4753495862*x)*x);\n else\n y := x+1.570796327+(-.2346004180+ .4696458221*x)/\n (1.+(.6231393515+.4753495862*x)*x);\n en d if;\n y;\nend proc: \n\nnext_halley_approx := proc(x,y)\n local \+ t,p,u,v,h;\n t := y+arctan(y)-x;\n p := 1+y^2;\n u := 1+1/p;\n \+ v := -2*y/p^2;\n h := t/(u-1/2*v*t/u);\n y-h;\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Test exam ple: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "xx := evalf(sqrt(23)/10);\ny0 := start_approx(xx);\n y1 := next_halley_approx(xx,y0);\ny2 := next_halley_approx(xx,y1);\ny3 := next_halley_approx(xx,y2);\neval(y+arctan(y),y=y3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+B:$ez%!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G$\"+R3w?C!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#y1G$\"+T3w?C!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G$\"+S3w?C!# 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G$\"+T3w?C!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+C:$ez%!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "code for inverse of " }{XPPEDIT 18 0 "g(x) = x+arcta n(x);" "6#/-%\"gG6#%\"xG,&F'\"\"\"-%'arctanG6#F'F)" }{TEXT -1 2 ": " } {TEXT 0 1 "S" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4557 "S := proc(x::algebraic)\n \+ local t,ok,s,terms,i,ti,eq;\n description \"inverse of x -> x+arct an(x)\";\n option `Copyright (c) 2003 Peter Stone.`;\n\n if nargs< >1 then\n error \"expecting 1 argument, got %1\", nargs;\n end \+ if;\n if type(x,'name') then 'S'(x)\n elif type(x,'float') then ev alf('S'(x))\n elif type(x,`*`) and type(op(1,x),'numeric') and \n \+ signum(0,op(1,x),0)=-1 then -S(-x) \n elif typ e(x,'realcons') and signum(0,x,0)=-1 then -S(-x)\n elif type(x,And(c omplexcons,Not(realcons))) then\n error \"not implemented for com plex argument\"\n elif type(x,`+`) then\n ok := false;\n i f has(x,'arctan') then\n s := select(has,x,'arctan');\n \+ if type(s,`+`) then terms := [op(s)] else terms := [s] end if;\n \+ for i to nops(terms) do\n ti := terms[i];\n \+ if type(ti,`*`) and op(1,ti)=-1 then ti := -ti end if;\n i f type(ti,'function') and op(0,ti)='arctan' then\n t := \+ op(1,ti);\n eq := t+arctan(t)=x;\n if eval b(expand(eq)) or testeq(eq) then\n ok := true;\n \+ break;\n end if;\n end if;\n \+ end do;\n if ok then t else 'S'(x) end if;\n elif ha s(x,'tan') then\n s := select(has,x,'tan');\n if type( s,`+`) then terms := [op(s)] else terms := [s] end if;\n for i to nops(terms) do\n ti := terms[i];\n if type(t i,`*`) and op(1,ti)=-1 then ti := -ti end if;\n if type(ti, 'function') and op(0,ti)='tan' then\n t := op(1,ti);\n \+ eq := t+tan(t)=x;\n if evalb(expand(eq)) or testeq(eq) then\n ok := true;\n bre ak;\n end if;\n end if;\n end do;\n \+ if type(t,'realcons') and (signum(0,t-Pi/2,1)=1 or\n \+ signum(0,t+Pi/2,-1)=-1) then ok := false end if;\n if ok then tan(t) else 'S'(x) end if;\n else 'S'(x) end if;\n els e 'S'(x) end if;\nend proc:\n\n# construct remember table\nS(0) := 0: \nS('infinity') := 'infinity':\nS(1+Pi/4) := 1:\nS(sqrt(3)+Pi/3) := sq rt(3):\nS(sqrt(3)/3+Pi/6) := sqrt(3)/3:\nS(1+sqrt(2)+3/8*Pi) := 1+sqrt (2):\nS(sqrt(2)-1+1/8*Pi) := sqrt(2)-1:\nS(2+sqrt(3)+5/12*Pi) := 2+sqr t(3):\nS(2-sqrt(3)+1/12*Pi) := 2-sqrt(3):\n\n# differentiation\n`diff/ S` := proc(a,x) \n option `Copyright (c) 2003 Peter Stone.`; \n di ff(a,x)*(1+S(a)^2)/(2+S(a)^2)\nend proc:\n\n`D/S` := proc(t)\n optio n `Copyright (c) 2003 Peter Stone.`;\n if 1-2.6 then \n s := .97 41807232*x-12.95848365/(x+32.34978388/\n (x+.7027779311/(x-2 0.37123456/(x+35.08632448/(x-5.546018485/x)))));\n elif x>0 then \+ \n s := x-1.570796327+(.2346004180+.4696458221*x)/\n \+ (1.+(-.6231393515+.4753495862*x)*x);\n else\n s := x+ 1.570796327+(-.2346004180+.4696458221*x)/\n (1.+(.6231393 515+.4753495862*x)*x);\n end if;\n # solve the equation x=s+ arctan(s) for s by Halley's method \n for i to maxit do\n \+ t := s+arctan(s)-x;\n p := 1+s^2;\n u := 1+1/p;\n \+ v := -2*s/p^2;\n h := t/(u-1/2*v*t/u);\n s := s-h ;\n if abs(h)<=eps*abs(s) then break end if;\n end do;\n \+ s;\n end proc;\n\n p := ilog10(Digits);\n q := Float(Digits ,-p);\n maxit := trunc((p+(.02331061386+.1111111111*q))*2.095903274) +2;\n saveDigits := Digits;\n Digits := Digits+min(iquo(Digits,3), 5);\n xx := evalf(x);\n eps := Float(3,-saveDigits-1);\n if Digi ts<=trunc(evalhf(Digits)) and \n ilog10(xx)trunc(evalhf(DBL_MIN_10_EXP)) th en\n val := evalhf(doS(xx,eps,maxit))\n else\n val := doS( xx,eps,maxit)\n end if;\n evalf[saveDigits](val);\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "checking the procedure \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Compa rison of starting approximation with the inverse function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "pl ot('S(x)-start_approx(x)',x=0..40,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 489 222 222 {PLOTDATA 2 "6&-%'CURVESG6#7^r7$$\"+qUkCF!#6$\" \"#F*7$$\"+S&)G\\aF*$\"\"!F17$$\"+5G$R<)F*$\"\"\"F*7$$\"+3x&)*3\"!#5F+ 7$$\"+ilyM;F:$\"\"'F*7$$\"+;arz@F:$F6F:7$$\"+CJdpKF:FC7$$\"+L3VfVF:F07 $$\"+\\i9RlF:F07$$\"+m;')=()F:FC7$$\"+7z>^7!\"*$!$U\"F:7$$\"+e'40j\"FS $!%S_F:7$$\"+!f`rt\"FS$!&')>\"F:7$$\"+BvzV=FS$!%oDFS7$$\"+c9W]>FS$!%$> &FS7$$\"+(Q&3d?FS$!%d**FS7$$\"+?$HP;#FS$!&%>=FS7$$\"+_KPqAFS$!&@=$FS7$ $\"+%=,\"f#FS$!'ag8FS7$$\"+j'\\yh#FS$!&%o*)FS7$$\"+ttoWEFS$!'1T7FS7$$\" +#3D:n#FS$!']-:FS7$$\"+\"zi$)p#FS$!'d\"p\"FS7$$\"+,0?DFFS$!'t<=FS7$$\" +5#Q?v#FS$!'X*)=FS7$$\"+>f()yFFS$!'M9>FS7$$\"+HOr0GFS$!'A**=FS7$$\"+Z! *QfGFS$!'ysFS7$$\"+B:^$)RFS$\"'(>%FS$\"'\"f`\"FS7$$\"+(o ZiH%FS$\"'#3N\"FS7$$\"+K'*)\\FS$!&NV\"FS7$$\"+My>#4&FS$!&mW$FS7 $$\"+vMw%>&FS$!&:M&FS7$$\"+<\"HtH&FS$!&%4rFS7$$\"+eZ*)*R&FS$!&^u)FS7$$ \"+Tg-0cFS$!'Kh6FS7$$\"+Dt:5eFS$!'H&R\"FS7$$\"+e9XMiFS$!'*es\"FS7$$\"+ \"fX(emFS$!'x$)=FS7$$\"+Y*yWw'FS$!'F,>FS7$$\"++B@qoFS$!'K6>FS7$$\"+ac% f(pFS$!'^9>FS7$$\"+3!z;3(FS$!'U6>FS7$$\"+;d9$H(FS$!'[))=FS7$$\"+DCh/vF S$!'TY=FS7$$\"+G9lRzFS$!'&Qr\"FS7$$\"+L/pu$)FS$!'sQ:FS7$$\"+CI(yv)FS$! 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" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(2+arctan(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+=([r5$! \"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "secant(S(x)=2,x=3..3.2,info=true);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%7approximation~1~~->~~~G$\".mk)Qf1J!#7" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\".U3#f92J!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\".!)yr[r5$! #7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\".%z< ([r5$!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$ \".%z<([r5$!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+=([r5$!\"*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The code above contains procedu res which enable " }{TEXT 0 4 "diff" }{TEXT -1 5 " and " }{TEXT 0 1 "D " }{TEXT -1 18 " to differentiate " }{XPPEDIT 18 0 "S(x);" "6#-%\"SG6# %\"xG" }{TEXT -1 8 " and S. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "diff(S(x),x);\nD(S)(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%*$)-%\"SG6#%\"xG\"\"#F%F%F %,&F,F%F&F%!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&\"\"\"F%*$)-% \"SG6#%\"xG\"\"#F%F%F%,&F,F%F&F%!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 82 "This means that we can make use of Ne wton's and Halley's method for root-finding. 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0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We t ackle the problem of constructing a numerical inverse for " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 58 "The idea is to use Halley's method to solve the equation \+ " }{XPPEDIT 18 0 "x = y*sec(y);" "6#/%\"xG*&%\"yG\"\"\"-%$secG6#F&F' " }{TEXT -1 17 " numerically for " }{TEXT 266 1 "y" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 11 "We need an " }{TEXT 260 21 "initial appr oximation" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 57 "starting approximations for Halley's root-finding method " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " } {XPPEDIT 18 0 "x = y*sec(y);" "6#/%\"xG*&%\"yG\"\"\"-%$secG6#F&F'" } {TEXT -1 4 ", " }{XPPEDIT 18 0 "dx/dy = sec(y)+y*sec(y)*tan(y);" "6# /*&%#dxG\"\"\"%#dyG!\"\",&-%$secG6#%\"yGF&*(F-F&-F+6#F-F&-%$tanG6#F-F& F&" }{XPPEDIT 18 0 "`` = sec(y)*(1+y*tan(y));" "6#/%!G*&-%$secG6#%\"yG \"\"\",&F*F**&F)F*-%$tanG6#F)F*F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1/(sec(y)*(1+y*tan(y)));" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F& *&-%$secG6#%\"yGF&,&F&F&*&F.F&-%$tanG6#F.F&F&F&F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Obtain th e first few terms in the Maclaurin series for " }{XPPEDIT 18 0 "f(x) \+ = g^(-1)*``(x);" "6#/-%\"fG6#%\"xG*&)%\"gG,$\"\"\"!\"\"F,-%!G6#F'F," } {TEXT -1 54 " as the series solution for the differential equation " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1/(sec(y)*(1 +y*tan(y)));" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&*&-%$secG6#%\"yGF&,&F&F &*&F.F&-%$tanG6#F.F&F&F&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "Order := 14:\nde \+ := diff(y(x),x)=1/(sec(y)*(1+y*tan(y)));\nic := y(0)=0;\ndsolve(\{de,i c\},y(x),type=series);\npx := convert(rhs(%),polynom);\nOrder := 6:" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&\" \"\"F.*&-%$secG6#F*F.,&F.F.*&F*F.-%$tanGF2F.F.F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG+3F'\"\"\"F)#!\"\"\"\"#\"\"$#\"#8\"#C\"\" &#!$T&\"$?(\"\"(#\"%4&*\"%k!)\"\"*#!(,=B(\"(+)GO\"#6#\"+%#pxG,0%\"xG\"\" \"*&#F'\"\"#F'*$)F&\"\"$F'F'!\"\"*&#\"#8\"#CF'*$)F&\"\"&F'F'F'*&#\"$T& \"$?(F'*$)F&\"\"(F'F'F.*&#\"%4&*\"%k!)F'*$)F&\"\"*F'F'F'*&#\"(,=B(\"(+ )GOF'*$)F&\"#6F'F'F.*&#\"+ " 0 "" {MPLTEXT 1 0 73 "RootOf(y*sec(y)-x,y);\ntaylor(%,x,14);\np := unapply( convert(%,polynom),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RootOfG6# ,&*&%#_ZG\"\"\"-%$secG6#F(F)F)%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG\"\"\"F%#!\"\"\"\"#\"\"$#\"#8\"#C\"\"&#!$T&\"$?(\"\"(#\"% 4&*\"%k!)\"\"*#!(,=B(\"(+)GO\"#6#\"+%\"pGf*6#%\"xG6\"6$%)operatorG%&arro wGF(,09$\"\"\"*&#F.\"\"#F.*$)F-\"\"$F.F.!\"\"*&#\"#8\"#CF.*$)F-\"\"&F. F.F.*&#\"$T&\"$?(F.*$)F-\"\"(F.F.F5*&#\"%4&*\"%k!)F.*$)F-\"\"*F.F.F.*& #\"(,=B(\"(+)GOF.*$)F-\"#6F.F.F5*&#\"+ " 0 "" {MPLTEXT 1 0 93 "numapprox[pade](p(x),x,[7,6]);\nconvert(%,confrac,x): \nevalf(evalf(%,15)):\npsi := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,**&#\"/>^$*)fG7)\"0!3yu*o#zU\"\"\"*$)%\"xG\"\"(F)F)F )*&#\",$Hssy()\",!3PuoWF)*$)F,\"\"&F)F)F)*&#\"-p#oSbU#\",_/81\\)F)*$)F ,\"\"$F)F)F)F,F)F),*F)F)*&#\"-&\\Lr+&GF8F)*$)F,\"\"#F)F)F)*&#\".FWj&3L E\"-?XIh!\\)F)*$)F,\"\"%F)F)F)*&#\"/FGw#H.i#\"/!GD_W-*QF)*$)F,\"\"'F)F )F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiGf*6#%\"xG6\"6$%)ope ratorG%&arrowGF(,&*&$\"+4g7=G!#5\"\"\"9$F1F1*&$\"+7Y.>;!\"*F1,&F2F1*&$ \"+F?(>&GF6F1,&F2F1*&$\"+!4)ocBF0F1,&F2F1*&$\"+/I$*3$*F0F1,&F2F1*&$\"+ ^s\"yj#!#6F1,&F2F1*&$\"+Y?6#f&F0F1F2!\"\"F1FLF1FLF1FLF1FLF1FLF1F(F(F( " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "The following picture makes a graphical comparison between the function d efined by the Pade approximation and the graph of " }{XPPEDIT 18 0 "x \+ = y*sec(y);" "6#/%\"xG*&%\"yG\"\"\"-%$secG6#F&F'" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "p1 := plots[implicitplot](x=y*sec(y),x=0..1.65,y=0..1):\np2 := pl ot(psi(x),x=0..1.65,color=blue):\nplots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 324 273 273 {PLOTDATA 2 "6&-%'CURVESG6]q7$7$$\"\" !F)F(7$$\"33CaEuq!=\\#!#>$\"3]\"f%Q%*Q\")*[#F-7$7$$\"3.BBsM@?.SF-$\"35 +++++++SF-F*7$F17$$\"3s9$**[^Xl)\\F-$\"3]D84.7&y(\\F-7$7$$\"3I+++++++m F-$\"3)GOo'4UH#e'F-F77$F=7$$\"3+#G3d#H!e[(F-$\"3A!)\\*>u\\JY(F-7$7$$\" 3Q$G/[Woc-)F-$\"3!))*************zF-FC7$FI7$$\"3/9Tt1mv'***F-$\"3EbBGK 'f8%**F-7$7$$\"3vRyNX@p37!#=$\"3%**************>\"FXFO7$FU7$$\"31()*Q0 Yr9D\"FX$\"3y*=e\"[C`T7FX7$7$$\"31++++++?8FX$\"3#=].5ij!38FXFfn7$F\\o7 $$\"3(=Bw!R980:FX$\"3m'eKy7*z([\"FX7$7$$\"3tN4V^2q?;FX$\"3-+++++++;FXF bo7$7$Fio$\"3I+++++++;FX7$$\"3)p'oI[NUgFX$\"3W%p#o]&3A%>FXFap7$Fgp7$$\"3#ys(H-;3FX7$7 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_*)p%)p-7#)FX7$$\"35+++7rpO7Fdfl$\"3m$y\"3=/1K$)FX7$$\"3))*****R]59F\" Fdfl$\"3#>7cg\\aRX)FX7$$\"3(*******)=\\`I\"Fdfl$\"3oITvv:yp&)FX7$$\"3$ ******z%)z3M\"Fdfl$\"3')yCq&e!p(o)FX7$$\"3++++%4+^P\"Fdfl$\"3c/IZP&*=) z)FX7$$\"3*)******Q_459Fdfl$\"3N+3O%4g#3*)FX7$$\"3!******pg+[W\"Fdfl$ \"3*z'pC7'fY,*FX7$$\"35+++N7pw9Fdfl$\"3o;0EAU95\"*FX7$$\"33+++%\\TK^\" Fdfl$\"3H\"f:+VXq@*FX7$$\"3++++#[Kfa\"Fdfl$\"3geH!4h)\\5$*FX7$$\"3-+++ #*yy!e\"Fdfl$\"3w^!\\7IG!3%*FX7$$\"32+++o*[Th\"Fdfl$\"3[Kou7\"e%*\\*FX 7$$\"3#*************\\;Fdfl$\"3y:8(z\"3v&f*FX-F^bm6&F`bmF(F($\"*++++\" !\")-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"$l\" !\"#Feen" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur ve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "For larger values of " }{TEXT 270 1 "x" }{TEXT -1 19 " th e approximation " }{XPPEDIT 18 0 "psi[2](x) = r(x)+arctan(x);" "6#/-&% $psiG6#\"\"#6#%\"xG,&-%\"rG6#F*\"\"\"-%'arctanG6#F*F/" }{TEXT -1 20 " \+ can be used, where " }{XPPEDIT 18 0 "r(x)" "6#-%\"rG6#%\"xG" }{TEXT -1 102 " is the rational function constructed as follows, using a prel iminary version of the inverse function " }{XPPEDIT 18 0 "T(x)" "6#-% \"TG6#%\"xG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "remez('T(x)-arctan(x)',x=1.0 3379..100,[1,2],info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pReme z~algorithm:~calculating~minimax~error~estimate~by~solving~a~rational~ equationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D--~minimising~the~ABSOL UTE~error~--G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Wstandard~Chebyshev~points~for~initial~critical~poi nts:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$\".+++!zL5!#7$\"/M.!f0Fb\"F% $\"/+++&*o^]F%$\"/m'*4Mn]&)F%$\"/++++++5!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-iteration~11G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G--------------------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprovisional~rational~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&$\"2\"35oPQ')4\"*!#=\"\"\"*&$\"2#39d:x)ey\"!#< F(%\"xGF(!\"\"F(,($F(\"\"!F(*&$\"2gz:z55v%eF,F(F-F(F(*&$\"2Hs$enL/0KF, F()F-\"\"#F(F(F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%goal~for~ relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/;>]_\"p8 \"!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1critical~points:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6' $\".+++!zL5!#7$\"2OE2@*o_!G\"!#;$\"22_5]\"oJKAF($\"2!f>&))Rd%R_F($\"2A WFtX:5k#!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%7minimax~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&$\"+QQ')4\"*!#6\"\"\"*&$\"+;x)ey\"!#5F(%\"xGF(!\"\" F(,($F(\"\"!F(*&$\"+3,^ZeF,F(F-F(F(*&$\"+oL/0KF,F()F-\"\"#F(F(F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%0minimax~error:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&*z@!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*&,&$\"+QQ')4\"*!#6\"\"\"*&$\"+;x)ey\"!#5F(%\"xGF(!\"\"F(,($F(\"\"!F( *&$\"+3,^ZeF,F(F-F(F(*&$\"+oL/0KF,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 29 "Halley's formula for solving " } {XPPEDIT 18 0 "x = y*sec(y);" "6#/%\"xG*&%\"yG\"\"\"-%$secG6#F&F'" } {TEXT -1 5 " for " }{TEXT 274 1 "y" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi( y) = y*sec(y)-x;" "6#/-%$phiG6#%\"yG,&*&F'\"\"\"-%$secG6#F'F*F*%\"xG! \"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Given an approxi mate zero " }{TEXT 276 1 "a" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "phi(y) " "6#-%$phiG6#%\"yG" }{TEXT -1 36 ", the \"improved\" Halley estimate \+ is " }{XPPEDIT 18 0 "y = y-h;" "6#/%\"yG,&F$\"\"\"%\"hG!\"\"" }{TEXT -1 24 " where the \"correction\" " }{TEXT 275 1 "h" }{TEXT -1 15 " is \+ given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h = ph i(y)/``(phi*`'`(y)-phi*`''`(y)*phi(y)/(2*phi*`'`(y)));" "6#/%\"hG*&-%$ phiG6#%\"yG\"\"\"-%!G6#,&*&F'F*-%\"'G6#F)F*F***F'F*-%#''G6#F)F*-F'6#F) F**(\"\"#F*F'F*-F16#F)F*!\"\"F=F=" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi*`'` (y)=sec(y)*(1+y*tan(y))" "6#/*&%$phiG\"\"\"-%\"'G6#%\"yGF&*&-%$secG6#F *F&,&F&F&*&F*F&-%$tanG6#F*F&F&F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " phi*`\"`(y) =sec(y)*(2*tan(y)+2*y*tan(y)^2+y)" "6#/*&%$phiG\"\"\"-%\" \"G6#%\"yGF&*&-%$secG6#F*F&,(*&\"\"#F&-%$tanG6#F*F&F&*(F1F&F*F&-F36#F* F1F&F*F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "y*sec(y)-x;\nDiff(%,y)=factor(diff( %,y));\nDiff(%%,y$2)=factor(diff(%%,y$2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"yG\"\"\"-%$secG6#F%F&F&%\"xG!\"\"" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%%DiffG6$,&*&%\"yG\"\"\"-%$secG6#F)F*F*%\"xG! \"\"F)*&F+F*,&F*F**&F)F*-%$tanGF-F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,&*&%\"yG\"\"\"-%$secG6#F)F*F*%\"xG!\"\"-%\"$G6$F)\" \"#*&F+F*,(*&F3F*-%$tanGF-F*F**(F3F*F)F*)F7F3F*F*F)F*F*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "The following cod e sets up the starting approximations from the previous subsection via the procedure " }{TEXT 0 12 "start_approx" }{TEXT -1 30 ", together w ith the procedure " }{TEXT 0 18 "next_halley_approx" }{TEXT -1 54 " to perform one step of Halley's method for a zero of " }{XPPEDIT 18 0 "p hi(y) = y*sec(y)-x;" "6#/-%$phiG6#%\"yG,&*&F'\"\"\"-%$secG6#F'F*F*%\"x G!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 652 "start_approx := proc(x)\n local \+ y;\n if x<1.038 and x>-1.038 then \n y := .2818126009*x+1.61903 4612/(x+2.851972027/(x+.2356688090/\n (x+.9308933004/(x+.2637 817251e-1/(x+.5592112046/x)))));\n elif x>0 then\n y := (.91098 63838e-1-.1785887716*x)/\n (1.+(.5847510108+.3205043368*x)*x)+a rctan(x);\n else\n y := -(.9109863838e-1+.1785887716*x)/\n \+ (1.+(-.5847510108+.3205043368*x)*x)+arctan(x);\n end if;\n y;\n end proc: \n\nnext_halley_approx := proc(x,y)\n local sc,tn,t,u,v,h; \n sc := sec(y);\n tn := tan(y);\n t := y*sc-x;\n u := sc*(1+y *tn);\n v := sc*(2*tn*(1+y*tn)+y);\n h := t/(u-1/2*v*t/u);\n y-h ;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Test example: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "xx := evalf(sqrt(14));\ny0 := star t_approx(xx);\ny1 := next_halley_approx(xx,y0);\ny2 := next_halley_app rox(xx,y1);\ny3 := next_halley_approx(xx,y2);\neval(y*sec(y),y=y3);\n " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+(Qd;u$!\"*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#y0G$\"+;*Q'*G\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$\"+\"ydXB\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G$\"+Q)[XB\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G$\"+ P)[XB\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!Qd;u$!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "code for inverse of " } {XPPEDIT 18 0 "g(x) = x*sec(x);" "6#/-%\"gG6#%\"xG*&F'\"\"\"-%$secG6#F 'F)" }{TEXT -1 2 ": " }{TEXT 0 1 "T" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6199 "T := pro c(x::algebraic)\n local t,ok,s,terms,ti,i,eq;\n description \"inve rse of x -> x*sec(x)\";\n option `Copyright (c) 2003 Peter Stone.`; \n\n if nargs<>1 then\n error \"expecting 1 argument, got %1\", nargs;\n end if;\n if type(x,'name') then 'T'(x)\n elif type(x, 'float') then evalf('T'(x))\n elif type(x,`*`) and type(op(1,x),'num eric') and \n signum(0,op(1,x),0)=-1 then -T(- x) \n elif type(x,'realcons') and signum(0,x,0)=-1 then -T(-x)\n e lif type(x,And(complexcons,Not(realcons))) then\n error \"not imp lemented for complex argument\"\n elif type(x,`*`) and nops(x)=2 the n\n ok := false;\n if has(x,'sec') then \n if type(o p(1,x),'function') and op([1,0],x)='sec' then\n t := op([1, 1],x);\n ok := true;\n elif type(op(2,x),'function' ) and op([2,0],x)='sec' then\n t := op([2,1],x);\n \+ ok := true;\n end if;\n eq := t*sec(t)=x;\n \+ if ok and (evalb(expand(eq)) or testeq(eq)) then t else 'T'(x) end if ;\n elif has(x,'arcsec') then \n if type(op(1,x),'functio n') and op([1,0],x)='arcsec' then\n t := op([1,1],x);\n \+ ok := true;\n elif type(op(2,x),'function') and op([2, 0],x)='arcsec' then\n t := op([2,1],x);\n ok := \+ true;\n end if;\n if ok then\n eq := t*arcs ec(t)=x; \n if not (evalb(expand(eq)) or testeq(eq)) then o k := false end if;\n end if;\n if type(t,'realcons') a nd signum(0,t-1,0)=-1 and\n signum(0,t+1,0)=1 the n ok := false end if;\n if ok then arcsec(t) else 'T'(x) end i f;\n else 'T'(x) end if;\n elif op(0,x)=`+` then\n ok := f alse;\n if has(x,'sec') then\n s := select(has,x,'sec'); \n terms := [op(s)];\n for i to nops(terms) do\n \+ ti := terms[i];\n if type(ti,`*`) and op(1,ti)=-1 the n ti := -ti end if;\n if type(ti,'function') and op(0,ti)=' sec' then\n t := op(1,ti);\n eq := t*sec(t )=x;\n if evalb(expand(eq)) or testeq(eq) then\n \+ ok := true;\n break;\n end if ;\n end if;\n end do;\n if ok then t else ' T'(x) end if;\n elif has(x,'arcsec') then\n s := select(h as,x,'arcsec');\n terms := [op(s)];\n for i to nops(te rms) do\n ti := terms[i];\n if type(ti,`*`) and \+ op(1,ti)=-1 then ti := -ti end if;\n if type(ti,'function') and op(0,ti)='arcsec' then\n t := op(1,ti);\n \+ eq := t*arcsec(t)=x;\n if evalb(expand(eq)) or test eq(eq) then\n ok := true;\n break;\n end if;\n end if;\n end do;\n \+ if type(t,'realcons') and signum(0,t-1,0)=-1 and\n \+ signum(0,t+1,0)=1 then ok := false end if;\n if ok then ar csec(t) else 'T'(x) end if;\n else 'T'(x) end if;\n else 'T'(x) end if;\nend proc:\n\n# construct remember table\nT(0) := 0:\nT('infi nity') := Pi/2:\nT(sqrt(2)/4*Pi) := Pi/4:\nT(2/3*Pi) := Pi/3:\nT(sqrt( 3)/9*Pi) := Pi/6:\n\n# differentiation\n`diff/T` := proc(a,x) \n opt ion `Copyright (c) 2003 Peter Stone.`; \n diff(a,x)*'T'(a)/x/(1+'T'( a)*tan('T'(a)))\nend proc:\n\n`D/T` := proc(t)\noption `Copyright (c) \+ 2003 Peter Stone.`;\n if 1a)/(1+'T'*tan@'T')\n end \+ if\nend proc:\n\n# integration\n`int/T` := proc(f)\n local gx,h,inds ,u;\n option `Copyright (c) 2003 Peter Stone.`;\n inds := map(proc (x) if op(0,x) ='T' then x end if end proc,indets(f,function));\n if nops(inds)<>1 then return FAIL end if;\n inds := inds[1];\n if no ps(inds)=1 then gx := op(inds) else gx := op(2,inds) end if;\n if no t type(gx,linear(_X)) then return FAIL end if;\n h := subs(inds=u,_X =(u*sec(u)-coeff(gx,_X,0))/coeff(gx,_X),f);\n h := h*(sec(u)*(1+u*ta n(u)))/coeff(gx,_X);\n h := int(h,u);\n if has(h,int) then return \+ FAIL end if;\n subs(sec(u)=gx/u,u=inds,h)\nend proc:\n\n# simplifica tion\n`simplify/T` := proc(s)\n option remember,system,`Copyright (c ) 2003 Peter Stone.`;\n if not has(s,'T') or type(s,'name') then ret urn s\n elif type(s,'function') and op(0,s)='sec' then\n if typ e(op(1,s),'function') and op([1,0],s)='T' then\n return op([1, 1],s)/op(1,s)\n else return s\n end if;\n elif type(s,'fun ction') and op(0,s)='cos' then\n if type(op(1,s),'function') and \+ op([1,0],s)='T' then\n return op(1,s)/op([1,1],s)\n else \+ return s\n end if;\n end if;\n map(procname,args)\nend proc: \n\n# numerical evaluation\n`evalf/T` := proc(x)\n local xx,eps,save Digits,doT,val,p,q,maxit;\n option `Copyright (c) 2003 Peter Stone.` ;\n\n if not type(x,realcons) then return 'T'(x) end if;\n\n doT : = proc(x,eps,maxit)\n local s,sc,tn,t,u,v,h,i;\n # set up a \+ starting approximation\n if x<1.038 and x>-1.038 then \n \+ s := .2818126009*x+1.619034612/(x+2.851972027/(x+.2356688090/\n \+ (x+.9308933004/(x+.2637817251e-1/(x+.5592112046/x)))));\n e lif x>0 then\n s := (.9109863838e-1-.1785887716*x)/\n \+ (1.+(.5847510108+.3205043368*x)*x)+arctan(x);\n else\n \+ s := -(.9109863838e-1+.1785887716*x)/\n (1.+(-.5847510108+.3 205043368*x)*x)+arctan(x);\n end if;\n # solve x=s*sec(s) fo r s by Halley's method \n for i to maxit do\n sc := sec(s );\n tn := tan(s);\n t := s*sc-x;\n u := sc*(1 +s*tn);\n v := sc*(2*tn*(1+s*tn)+s);\n h := t/(u-1/2*v *t/u);\n s := s-h;\n if abs(h)<=eps*abs(s) then break end if;\n end do;\n s;\n end proc;\n\n p := ilog10(Digi ts);\n q := Float(Digits,-p);\n maxit := trunc((p+(.02331061386+.1 111111111*q))*2.095903274)+2;\n saveDigits := Digits;\n Digits := \+ Digits+min(iquo(Digits,3),5);\n xx := evalf(x);\n eps := Float(3,- saveDigits-1);\n if Digits<=trunc(evalhf(Digits)) and \n ilo g10(xx)trunc( evalhf(DBL_MIN_10_EXP)) then\n val := evalhf(doT(xx,eps,maxit))\n else\n val := doT(xx,eps,maxit)\n end if;\n evalf[saveDigi ts](val);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "checking the procedure " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Comparison of starting approximation with the i nverse function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 48 "plot('T(x)-start_approx(x)',x=0..30,color=blue );" }}{PARA 13 "" 1 "" {GLPLOT2D 446 212 212 {PLOTDATA 2 "6&-%'CURVESG 6#7`r7$$\"+-K[V?!#6$\"\"!F,7$$\"+0k'p3%F*$!\"\"F*7$$\"+2'\\/8'F*$\"\" \"F*7$$\"+5G$R<)F*$!\"$F*7$$\"+A**3E7!#5F+7$$\"+ilyM;F?F+7$$\"+V)z@X#F ?F+7$$\"+DJdpKF?$!\"'F?7$$\"+1k'p3%F?$!#\")F?7$$\"+(ofV!\\F?$!$(zF?7$$ \"+pHv@dF?$!%:]F?7$$\"+]i9RlF?$!&sD#F?7$$\"+tdN]sF?$!&?z'F?7$$\"+(Hl:' zF?$!'>b&45!\"*$!(?$p:F?7$$\"+D'*HF5Fep$!(p0#=F?7$$\"+j)z ]/\"Fep$!(2_o\"F?7$$\"+,,'G1\"Fep$!'!H()*F?7$$\"+R.k!3\"Fep$!'.5QF?7$$ 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" }}{PARA 0 "" 0 "" {TEXT -1 28 "The solution is (of course) " } {XPPEDIT 18 0 "x = sec(1);" "6#/%\"xG-%$secG6#\"\"\"" }{TEXT -1 1 " " }{TEXT 267 1 "~" }{TEXT -1 14 " 1.850815718. 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