{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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 0 1 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 0 0 1 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 0 0 1 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 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 290 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 291 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 292 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 293 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 294 "" 0 1 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 300 "" 0 1 0 0 0 0 1 0 0 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 32 "Special inverse functions .. II \+ " }}{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 procedur es including: " }{TEXT 0 5 "remez" }}{PARA 0 "" 0 "" {TEXT -1 17 "The \+ Maple m-file " }{TEXT 262 10 "fcnapprx.m" }{TEXT -1 37 " contains the \+ code for the procedure " }{TEXT 0 5 "remez" }{TEXT -1 25 " used in thi s worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a M aple session by a command similar to the one that follows, where the f ile 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 12 "Inverse for " }{XPPEDIT 18 0 "g(x)=x*cosh(x)" "6#/-%\"gG6#%\"xG *&F'\"\"\"-%%coshG6#F'F)" }{TEXT -1 2 ": " }{TEXT 0 1 "Y" }{TEXT -1 3 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 13 "The function " }{XPPEDIT 18 0 "g(x) = x*cosh(x);" "6#/- %\"gG6#%\"xG*&F'\"\"\"-%%coshG6#F'F)" }{TEXT -1 47 " is one-to-one, an d so has an inverse function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG " }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 224 "g := x -> x*cosh(x);\np1 := plot([g(x),x],x =-4..4,y=-4..4,color=[red,black],\n linestyle=[1,3]):\np2 \+ := plots[implicitplot](x=y*cosh(y),x=-4..4,\n y=-4..4,grid= [30,30],color=blue):\nplots[display]([p1,p2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\"-%%c oshG6#F-F.F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6'-%'CURVESG6%7W7$$!\"%\"\"!$!3Yf1W8$HB4\"!#:7$$!3cLLL$Q6G\"R!#<$!3 e79-iJl$z*!#;7$$!3ommmmFiDQF1$!3)QV%y[Dew()F47$$!37++]VsF47$$!3nmmmwxE.NF1$!3'e:)=DJ([# eF47$$!3YmmmOk]JLF1$!3)3ph&Hv!om%F47$$!3_LLL[9cgJF1$!3n/y99%3Ot$F47$$! 3smmmhN2-IF1$!3\"3f*Q/)G'GIF47$$!3!******\\`oz$GF1$!3Brp,g(y?V#F47$$!3 !omm;)3DoEF1$!3Bx*RCjhB$>F47$$!3?+++:v2*\\#F1$!3**>lTOD6J:F47$$!3BLLL8 >1DBF1$!3+/vl_PK+7F47$$!3kmmmw))yr@F1$!3m2$H'ROO^'*F17$$!3;+++S(R#**>F 1$!3mLd'4p?g^(F17$$!30++++@)f#=F1$!3'[^8zd1d\"eF17$$!3-+++gi,f;F1$!3Ac W\"GdGi^%F17$$!3qmmm\"G&R2:F1$!3qg]0?Z%)pNF17$$!3XLLLtK5F8F1$!3/N<`6Qm xEF17$$!3eLLL$HsV<\"F1$!3!RdMBiZ;3#F17$$!3+-++]&)4n**!#=$!3X^igK\"eT` \"F17$$!37PLLL\\[%R)F\\r$!3M,iJs')*H:\"F17$$!3G)*****\\&y!pmF\\r$!3RTn %f**ez?)F\\r7$$!3Y******\\O3E]F\\r$!3wjvS,KRucF\\r7$$!3NKLLL3z6LF\\r$! 33)eM\"=[2&\\$F\\r7$$!3sLLL$)[`Pk#HQw\"F\\r7$$!3gSnmmmr[R! #?$!3[&=>;XZ([RF[t7$$\"3yELL$=2Vs\"F\\r$\"3Ivw\"Qh/+v\"F\\r7$$\"3)e*** **\\`pfKF\\r$\"3c(>dbd:WV$F\\r7$$\"36HLLLm&z\"\\F\\r$\"3=uV6]lxCbF\\r7 $$\"3>(******z-6j'F\\r$\"32?!3G;7K9)F\\r7$$\"3q\"******4#32$)F\\r$\"3n Mqdw?F17$$\"35mmm1>qM8F1$\"3xOy%RIY3r#F17$$\"3%)*******HSu ]\"F1$\"3I>\"*o>q4qNF17$$\"3'HLL$ep'Rm\"F1$\"3SizRf%=1b%F17$$\"3')**** **R>4N=F1$\"37\">oNR;b*eF17$$\"3#emm;@2h*>F1$\"3]t77T0i\"[(F17$$\"3]** ***\\c9W;#F1$\"3E\"pG4(>u\\&*F17$$\"3Lmmmmd'*GBF1$\"3'p0F_;`p?\"F47$$ \"3j*****\\iN7]#F1$\"3Ccu,K6qN:F47$$\"3aLLLt>:nEF1$\"3>@dF#ok%H>F47$$ \"35LLL.a#o$GF1$\"3HH#pqiS$GCF47$$\"3ammm^Q40IF1$\"3lB*f!)4,3/$F47$$\" 3y******z]rfJF1$\"3RNed.@YHPF47$$\"3gmmmc%GpL$F1$\"3;N[xD;v*p%F47$$\"3 /LLL8-V&\\$F1$\"3[_t)4X^lw&F47$$\"3=+++XhUkOF1$\"37Yt>m\\]crF47$$\"3=+ ++![,`u$F1$\"3!*zRGZJ\")HzF47$$\"3=+++:oGMq8Fy)F47$$\"34+ +]2%)38RF1$\"3#*\\AnD!eqz*F47$$\"\"%F*$\"3Yf1W8$HB4\"F--%'COLOURG6&%$R GBG$\"*++++\"!\")$F*F*Fg\\l-%*LINESTYLEG6#\"\"\"-F$6%7S7$F(F(7$F6F67$F @F@7$FEFE7$FJFJ7$FOFO7$FTFT7$FYFY7$FhnFhn7$F]oF]o7$FboFbo7$FgoFgo7$F\\ pF\\p7$FapFap7$FfpFfp7$F[qF[q7$F`qF`q7$FeqFeq7$FjqFjq7$F`rF`r7$FerFer7 $FjrFjr7$F_sF_s7$FdsFds7$FisFis7$F_tF_t7$FdtFdt7$FitFit7$F^uF^u7$FcuFc u7$FhuFhu7$F]vF]v7$FbvFbv7$FgvFgv7$F\\wF\\w7$FawFaw7$FfwFfw7$F[xF[x7$F `xF`x7$FexFex7$FjxFjx7$F_yF_y7$FdyFdy7$FiyFiy7$F^zF^z7$FczFcz7$FhzFhz7 $Fb[lFb[l7$F\\\\lF\\\\l-Fa\\l6&Fc\\lF*F*F*-Fi\\l6#\"\"$-F$6_p7$7$F($!3 1:j;IPLr:F17$$!3GBWGFJt_RF1$!3e?f-_#3Xc\"F17$7$$!3l#[M5$z8CPF1$!3Z[/H? 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" }}{PARA 0 "" 0 "" {TEXT -1 57 "The idea is to use Halley's method to solve the equa tion " }{XPPEDIT 18 0 "y*cosh(y)=x" "6#/*&%\"yG\"\"\"-%%coshG6#F%F&%\" xG" }{TEXT -1 17 " numerically for " }{TEXT 263 1 "y" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 11 "We need an " }{TEXT 260 21 "initial a pproximation" }{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 metho d " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Obt ain the first few terms in the Maclaurin series for " }{XPPEDIT 18 0 "f(x) = g^(-1)*``(x);" "6#/-%\"fG6#%\"xG*&)%\"gG,$\"\"\"!\"\"F,-%!G6#F 'F," }{TEXT -1 54 " as the series solution for the differential equati on " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx=1/(cosh( y)+y*sinh(y))" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&,&-%%coshG6#%\"yGF&*&F .F&-%%sinhG6#F.F&F&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "Order := 14:\nde := dif f(y(x),x)=1/(cosh(y(x))+y(x)*sinh(y(x)));\nic := y(0)=0;\ndsolve(\{de, ic\},y(x),type=series):\nconvert(rhs(%),polynom);\nOrder := 6:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xGF,*&\"\" \"F.,&-%%coshG6#F)F.*&F)F.-%%sinhGF2F.F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0%\"xG\"\"\"*&#F%\"\"#F%*$)F$\"\"$F%F%!\"\"*&#\"#<\"#CF%*$)F$\"\"& F%F%F%*&#\"$h*\"$?(F%*$)F$\"\"(F%F%F,*&#\"'Hh6\"&?.%F%*$)F$\"\"*F%F%F% *&#\"(.&)[$\"'+%=&F%*$)F$\"#6F%F%F,*&#\"+@fpNz\"*+;+z%F%*$)F$\"#8F%F%F %" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Alt ernatively, the same Taylor polynomial can be obtained by using the pr ocedures " }{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 74 "RootOf(y*cosh(y)-x,y);\ntaylor(%,x,14);\np := unapply(convert(%,polynom),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%' RootOfG6#,&*&%#_ZG\"\"\"-%%coshG6#F(F)F)%\"xG!\"\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#+3%\"xG\"\"\"F%#!\"\"\"\"#\"\"$#\"#<\"#C\"\"&#!$h*\"$ ?(\"\"(#\"'Hh6\"&?.%\"\"*#!(.&)[$\"'+%=&\"#6#\"+@fpNz\"*+;+z%\"#8-%\"O G6#F%\"#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)ope ratorG%&arrowGF(,09$\"\"\"*&#F.\"\"#F.*$)F-\"\"$F.F.!\"\"*&#\"#<\"#CF. *$)F-\"\"&F.F.F.*&#\"$h*\"$?(F.*$)F-\"\"(F.F.F5*&#\"'Hh6\"&?.%F.*$)F- \"\"*F.F.F.*&#\"(.&)[$\"'+%=&F.*$)F-\"#6F.F.F5*&#\"+@fpNz\"*+;+z%F.*$) F-\"#8F.F.F.F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 53 "The corresponding rational Pade approximation of type" }{XPPEDIT 18 0 "``(7,6);" "6#-%!G6$\"\"(\"\"'" }{TEXT -1 29 " can be o btained 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#*&,**&#\"1rKzOiJ#>#\"1!GLe.9=T$\"\"\"*$)%\"xG\" \"(F)F)F)*&#\"/ZVv.IbF\".?LHs%pnF)*$)F,\"\"&F)F)F)*&#\".$fQ0'Qv#\"-K$H s%pnF)*$)F,\"\"$F)F)F)F,F)F),*F)F)*&#\".fK:MB4$F8F)*$)F,\"\"#F)F)F)*&# \".Py'>^uM\"-?@d1ahF)*$)F,\"\"%F)F)F)*&#\"1p\">i\\ayL&F(F)*$)F,\"\"'F) F)F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiGf*6#%\"xG6\"6$%)op eratorG%&arrowGF(,&*&$\"+s762T!#5\"\"\"9$F1F1*&$\"+b=U>6!\"*F1,&F2F1*& $\"+a.:dBF6F1,&F2F1*&$\"+fT,c:F0F1,&F2F1*&$\"+5S2inF0F1,&F2F1*&$\"+)*f .t=!#6F1,&F2F1*&$\"+D]15SF0F1F2!\"\"F1FLF1FLF1FLF1FLF1FLF1F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "The foll owing picture makes a graphical comparison between the function define d by the truncated Maclaurin series and the graph of " }{XPPEDIT 18 0 "x=y*cosh(y)" 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3v-d-\"F*7$$\"3SLezw5V!f\"F*$\"3-<3j3U(e-\"F*7$$\"3++v$Q#\\\"3f\"F*$\" 3lQt#[WBg-\"F*7$$\"3SL$e\"*[H7f\"F*$\"3%)p0*[g%=E5F*7$$\"35++qvxl\"f\" F*$\"3^vUR^E5F*7$$\"3)**\\i&p@[#f \"F*$\"34JLGd9nE5F*7$$\"33+]2'HKHf\"F*$\"39'Q^TNYo-\"F*7$$\"3!ommZvOLf \"F*$\"3qWC*o[.q-\"F*7$$\"3/++v+'oPf\"F*$\"3k+l))Q71cgY*G5F*7$$\"3ULL`v&Q()f\"F*$\"3NV/=1.5H5F *7$$\"37+DOl5;*f\"F*$\"3_!Q5M;k#H5F*7$$\"31+v.Uac*f\"F*$\"3\"*\\&>n(4U H5F*7$$\"33+++++++;F*$\"3VATPt%*eH5F*-F^fl6&F`fl$FbflFbflF[fm$\"*++++ \"!\")-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$\"$e\" !\"#$\"#;!\"\"Fgfm" 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 13 "The function " }{XPPEDIT 18 0 "psi2(x) = \+ ln(x)-ln(ln(x))+0" "6#/-%%psi2G6#%\"xG,(-%#lnG6#F'\"\"\"-F*6#-F*6#F'! \"\"\"\"!F," }{TEXT -1 60 ".7775630315 provides a suitable starting ap proximation when " }{TEXT 297 1 "x" }{TEXT -1 11 " is large. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "p1 := plots[implicitplot](y*cosh(y)=x,x=10^8..10^9,y=16.2..18.5): \np2 := plot(ln(x)-ln(ln(x))+.7775630315,x=10^8..10^9,color=blue):\npl ots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 477 477 477 {PLOTDATA 2 "6&-%'CURVESG6]q7$7$$\"*++++\"\"\"!$\"3&=G?6qQ?j\"!#;7$$\" 3-R))\\DI'Q0\"!\"*$\"3jt%f/]Bqj\"F-7$7$$\"3E<[\\Hnto5F1$\"3o********** RQ;F-F.7$F57$$\"3aIF#ya*pO6F1$\"3V'>6rc1Tk\"F-7$7$$\"3m]7xm[Iy6F1$\"3c **********fZ;F-F;7$7$$\"3_]7xm[Iy6F1FD7$$\"3menylU%RA\"F1$\"3CQV()zp2^ ;F-7$7$$\"3'RG_;bl!*H\"F1$\"3W**********zc;F-FJ7$FP7$$\"3\\)>)zRE^:8F1 $\"3oc^?**o$zl\"F-7$7$$\"*+++O\"F*$\"3=)3\"Rs?,h;F-FV7$Ffn7$$\"3_'e**> %=o79F1$\"3u4+E'o`Ym\"F-7$7$$\"3DEn_]%e@V\"F1$\"3K***********fm\"F-F\\ o7$Fbo7$$\"3oUTM%e'[::F1$\"3'*[c&RXE7n\"F-7$7$$\"3Kv6GT*Q)y:F1$\"3=*** *******>v;F-Fho7$F^p7$$\"3Q+2Z$R#eA;F1$\"3!yub5c*ox;F-7$7$$\"3q******* *****>(Rp'QB$o\"F-7$$\" 38<(\\]PSTt\"F1$\"3qR)f[jQSo\"F-7$7$$\"3.'z*=J)*[S>/-p\"F-7$7$$\"3'z:r'eUj=>F 1$\"3%*)*********f$p\"F-Far7$Fgr7$$\"3al`*['ydv>F1$\"3E&*[nl&oip\"F-7$ 7$$\"*+++3#F*$\"3W[N?K@;,r\"F-Fet7$7$ $\"3g$p#=+YGJBF1F^u7$$\"3w=\"\\!\\^fuBF1$\"3#o2-tXrOr\"F-7$7$$\"*+++W# F*$\"3w#eyje&>;JA$=j?>@#*HY[KQy HF1$\"3x[P)pJT]t\"F-7$7$$\"3HVg:`\"3=7$F1$\"3J)*********fRz&>^w&*\\u\"F-7$7$$\"3LO]:A:uSMF1$\"3=) *********z[xl$F1 $\"3Q#erg]![an#QA@%F1$\"3SfXsn %4zw\"F-7$7$$\"*+++C%F*$\"3Y$pw2=3&oexuS%F1$\" 3kM-N<+7sFw3UWpr-YF1 $\"3C5f(pcIjx\"F-7$7$$\"3/'\\%p!\\Ofg%F1$\"3!y*********Rw_F1$\"3ekti!*y:)y\"F-7$7$$\"*+++K&F*$\"3Y \\d!)4i$**y\"F-Fbbl7$Fhbl7$$\"3LHv=X$pAV&F1$\"3%=(=^%*3$>z\"F-7$7$$\"3 q_;w8[v$f&F1$\"3c(*********z%z\"F-F^cl7$Fdcl7$$\"3#R%Rv&*=jYcF1$\"3IT= WSFl&z\"F-7$7$$\"*+++o&F*$\"3Y!*oki3>'z\"F-Fjcl7$F`dl7$$\"3Gh6cc&[t'eF 1$\"3mXXb.A@*z\"F-7$7$$\"*+++/'F*$\"3))yXTEl*>!=F-Ffdl7$F\\el7$$\"3ky$ ot@l!)3'F1$\"3-]smm;x-=F-7$7$$\"3!Row'=ABkhF1$\"3V(**********R!=F-Fbel 7$Fhel7$$\"3%=]%[)[MTJ'F1$\"3%z$)\\3M%>1=F-7$7$$\"*+++S'F*$\"3:0+7\"G^ u!=F-F^fl7$Fdfl7$$\"3\"4(e:!>CIa'F1$\"3j!oit$\\a4=F-7$7$$\"*+++w'F*$\" 3h2.lK6s7=F-Fjfl7$F`gl7$$\"3+Ss#=*Q\">x'F1$\"3JBb(Q`&*G\"=F-7$7$$\"3e% *fsAHr#z'F1$\"3I(*********>8=F-Ffgl7$7$F]hl$\"3m(*********>8=F-7$$\"3Y ^_X3&R!3qF1$\"3W'phD@hg\"=F-7$7$$\"*+++7(F*$\"3ff%y!>*[v\"=F-Fehl7$F[i l7$$\"3q\\>fn$))[C(F1$\"3`ePQ3%3#>=F-7$7$$\"*+++[(F*$\"3>BrJ?DLA=F-Fai l7$7$Fhil$\"3aBrJ?DLA=F-7$$\"3wY'GnAP<[(F1$\"3h?e?/cNA=F-7$7$$\"3)Qq^e Ky][(F1$\"3=(*********RA=F-F`jl7$7$$\"330<&eKy][(F1Fijl7$$\"3BpvMm]>Ex F1$\"36[WI[$3`#=F-7$7$$\"*+++%yF*$\"3e=abi5oE=F-F_[m7$Fe[m7$$\"3OWKax< wqzF1$\"3vZ%H,Je#G=F-7$7$$\"*+++?)F*$\"3IV*Q)zL-J=F-F[\\m7$Fa\\m7$$\"3 I=*Q()[G`@)F1$\"30ZW&>F37$=F-7$7$$\"3/+%*4>Z!yC)F1$\"31(*********fJ=F- Fg\\m7$7$$\"3'))R*4>Z!yC)F1F`]m7$$\"3Y7OKu%ejY)F1$\"3zE<@hI*R$=F-7$7$$ \"*+++c)F*$\"3vp,/#R=]$=F-Ff]m7$F\\^m7$$\"3\"pc/]9y$=()F1$\"3pYl^eDvO= F-7$7$$\"*+++#*)F*$\"3!4daLAg*Q=F-Fb^m7$Fh^m7$$\"3)*=bo:yRq*)F1$\"3&pO @e07&R=F-7$7$$\"3/]l.AL-)3*F1$\"3$p*********zS=F-F^_m7$Fd_m7$$\"3tD-J1 -CE#*F1$\"3.\"H%RhQ\"F1$\"3ab9o!4Aak\"F-7$$\"3++]7tY'oO\"F1$\"37bBf'fY!e;F-7$ $\"3+++D^P#)e:F1$\"3Ik%41v)[q;F-7$$\"3+++ve_0_F1$\"3#*[1:*eC9p\"F-7$$\"3++]7VsmA@F1$\"3o*37&p&R(*p\"F-7 $$\"3++]7)R&G2BF1$\"3rbo?o`k2>Vp9A-=F-7$$\"3+++DOnaMk F1$\"3h[J#GN<]!=F-7$$\"3++]7.j(ph'F1$\"3`%eb%>`n2=F-7$$\"3+++vLK`>oF1$ \"37H&[>AU0\"=F-7$$\"3\"))****\\kR:+(F1$\"3Y_6V>n/8=F-7$$\"3+++]P.(e>( F1$\"3\")pqB6,l:=F-7$$\"3++]7GG'>P(F1$\"3vFMY/$\\z\"=F-7$$\"3+++]K%yWc (F1$\"3(\\y%Ro5S?=F-7$$\"3#))*\\781iXxF1$\"3FKU'\\!=lA=F-7$$\"3++]i&Qm \\$zF1$\"3*f0'eG!\\\\#=F-7$$\"3+++](['3?\")F1$\"35yJW?F9F=F-7$$\"3++]7 y+*QJ)F1$\"3$4U?X\\'QH=F-7$$\"*(fa+&)F*$\"3SyXmF()\\J=F-7$$\"3+++vy&G9 p)F1$\"3G$)yYs9hL=F-7$$\"3++]7$eI2)))F1$\"35%pgp\\hc$=F-7$$\"3++++l%zY 0*F1$\"3@2lE%=2v$=F-7$$\"3+++v8X/a#*F1$\"3i=WMS&z&R=F-7$$\"***eBV*F*$ \"3/Y:T#p&RT=F-7$$\"3++]78%zCi*F1$\"3L\"o*pWZHV=F-7$$\"3++](o\"*[W!)*F 1$\"3eu8,Ww2X=F-7$Fiam$\"3uBie%=dp%=F--F^bm6&F`bm$F*F*Fhan$F)!\")-%+AX ESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(FiamFcbn" 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 -1 100 "Mini max rational approximations can be obtained from a provisional version of the inverse function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "remez('Y(x)',x=0.84..10^2,[4,3],inf o=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algorithm:~calcul ating~minimax~error~estimate~by~solving~a~rational~equationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D--~minimising~the~ABSOLUTE~error~--G" }} {PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-ite ration~12G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G---------------------- ----------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprovisional~rat ional~approximation:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,,$\"6g*=&y m!\\e&*yt!#A!\"\"*&$\"69A>O#)e5kQO\"!#?\"\"\"%\"xGF-F-*&$\"6_pz?abe04+ $!#@F-)F.\"\"#F-F-*&$\"6+()Q*=_DdVIk!#BF-)F.\"\"$F-F-*&$\"6O&3*[A%pul9 L!#EF-)F.\"\"%F-F-F-,*$F-\"\"!F-*&$\"6.Rj^#fh[\"Gz*F2F-F.F-F-*&$\"6D9D NJM7XH1\"F2F-F3F-F-*&$\"6cIk&zLMuGd8F8F-F9F-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#$\"39fBv?i!z$G!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1critical~points:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+$\"6yC=Iw7:^XW)!#@$\"6U\"Hv%G`Y`(=9!#?$\"6Bz >D*f&f1Z\"GF($\"6%p>j?tX*p3/'F($\"6'>>pM#QVBHK\"!#>$\"6uUGG!4\")4%**z# F/$\"6q+F]F%Gx)4O&F/$\"6!>agG&QNWQY)F/$\"3++++++++5!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7minimax~a pproximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,,$\"+\\e&*yt!#6! \"\"*&$\"+1T'QO\"!\"*\"\"\"%\"xGF-F-*&$\"+'e04+$!#5F-)F.\"\"#F-F-*&$\" +EdVIk!#7F-)F.\"\"$F-F-*&$\"+pul9L!#:F-)F.\"\"%F-F-F-,*$F-\"\"!F-*&$\" +i[\"Gz*F2F-F.F-F-*&$\"+B^%H1\"F2F-F3F-F-*&$\"+MuGd8F8F-F9F-F-F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%0minimax~error:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&`a*!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*&,,$\"+\\e&*yt!#6!\"\"*&$\"+1T'QO\"!\"*\"\"\"%\"xGF-F-*&$\"+'e04+$!# 5F-)F.\"\"#F-F-*&$\"+EdVIk!#7F-)F.\"\"$F-F-*&$\"+pul9L!#:F-)F.\"\"%F-F -F-,*$F-\"\"!F-*&$\"+i[\"Gz*F2F-F.F-F-*&$\"+B^%H1\"F2F-F3F-F-*&$\"+MuG d8F8F-F9F-F-F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 61 "remez('Y(x)'-(ln(x)-ln(ln(x))),x=10^2..10^8,[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#%-iteration~11G" }}{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#*& ,&$\"5bc=o\\KqPn&)!#?!\"\"*&$\"5q?/,nbWH/9!#D\"\"\"%\"xGF-F(F-,($F-\" \"!F(*&$\"5Y%*=`z%=%f@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"(FjT#!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5relati ve~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2$)z*oCEMrX!#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>goal~for~relative~difference:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2+c\">]_\"p8\"!#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1critical~po ints:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"-+++++5!\"*$\"5]$y`[p75XN '!#<$\"53u*R#4xP6*H#!#:$\"5du&=n')yHNP%!#8$\"2++++++++\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7min imax~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&$\"+KqPn&) !#5!\"\"*&$\"+cWH/9!#:\"\"\"%\"xGF-F(F-,($F-\"\"!F(*&$\"+&=%f@ " 0 "" {MPLTEXT 1 0 62 "remez('Y(x)'-(ln(x)-ln(ln(x))),x=10^8..10^20 ,[1,2],info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algorit hm:~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#%,iteration~8G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G--- -----------------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%Dprovisional~rational~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&$\"2!*[T@(R(41)!#<\"\"\"*&$\"2?s8d[Ph4\"!#GF(%\"xGF(F(F(,($F (\"\"!F(*&$\"2uDRkJ#Q!#]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#$\"/;>]_\"p8\"!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1critical~ points:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"$+\"\"\"'$\"2K'4Y[?[J#Q!#VF()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#$\"&S0)!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% !G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&$\"+sR(41)!#5\"\"\"*&$\"+'[P h4\"!#@F(%\"xGF(F(F(,($F(\"\"!F(*&$\"+w^Z:9F,F(F-F(F(*&$\"+8%>J#Q!#VF( )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 "H alley's formula for solving " }{XPPEDIT 18 0 "x = y*cosh(y);" "6#/%\"x G*&%\"yG\"\"\"-%%coshG6#F&F'" }{TEXT -1 5 " for " }{TEXT 265 1 "y" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi(y) = y*cosh(y)-x;" "6#/-%$phiG6# %\"yG,&*&F'\"\"\"-%%coshG6#F'F*F*%\"xG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Given an approximate zero " }{TEXT 267 1 "a" } {TEXT -1 4 " of " }{XPPEDIT 18 0 "phi(y)" "6#-%$phiG6#%\"yG" }{TEXT -1 36 ", the \"improved\" Halley estimate is " }{XPPEDIT 18 0 "y = y-h ;" "6#/%\"yG,&F$\"\"\"%\"hG!\"\"" }{TEXT -1 24 " where the \"correctio n\" " }{TEXT 266 1 "h" }{TEXT -1 15 " is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h = phi(y)/``(phi*`'`(y)-phi*`''`( y)*phi(y)/(2*phi*`'`(y)));" "6#/%\"hG*&-%$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) = cosh(y)+y*sinh(y);" " 6#/*&%$phiG\"\"\"-%\"'G6#%\"yGF&,&-%%coshG6#F*F&*&F*F&-%%sinhG6#F*F&F& " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi*`\"`(y) = 2*sinh(y)+y*cosh(y );" "6#/*&%$phiG\"\"\"-%\"\"G6#%\"yGF&,&*&\"\"#F&-%%sinhG6#F*F&F&*&F*F &-%%coshG6#F*F&F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "y*cosh(y)-x;\nDiff(%,y)=diff (%,y);\nDiff(%%,y$2)=diff(%%,y$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,&*&%\"yG\"\"\"-%%coshG6#F%F&F&%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,&*&%\"yG\"\"\"-%%coshG6#F)F*F*%\"xG!\"\"F), &F+F**&F)F*-%%sinhGF-F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG 6$,&*&%\"yG\"\"\"-%%coshG6#F)F*F*%\"xG!\"\"-%\"$G6$F)\"\"#,&*&F3F*-%%s inhGF-F*F*F(F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "The following code sets up the starting approximations f rom the previous subsection via the procedure " }{TEXT 0 12 "start_app rox" }{TEXT -1 30 ", together with the procedure " }{TEXT 0 18 "next_h alley_approx" }{TEXT -1 54 " to perform one step of Halley's method fo r a zero of " }{XPPEDIT 18 0 "phi(y) = y*cosh(y)-x;" "6#/-%$phiG6#%\"y G,&*&F'\"\"\"-%%coshG6#F'F*F*%\"xG!\"\"" }{TEXT -1 2 ". " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1800 "start_approx := proc(x)\n local y;\n if x<.84 and x>-.84 then\n y := .4107111272*x+1.119421855 /(x+2.357150354/\n (x+.1556014159/(x+.6762074010/\n \+ (x+.1873035998e-1/(x+.4010065025/x)))));\n else\n if x>0 th en\n if x<100 then\n y := (-.7378955849e-1+(1.36386 4106+(.3000905586+\n (.6430435726e-2+.3314657469e-5*x) *x)*x)*x)/\n (1.+(.9792814862+(.1062945123+.1357287434 e-2*x)*x)*x)\n elif x-100 then\n y := (-.7378955849e-1+(-1.363864106+(.3000905586+\n (-.6430435726e-2+.3314657469e-5*x)*x)*x)*x)/\n \+ (-1.+(.9792814862+(-.1062945123+.1357287434e-2*x)*x)*x)\n \+ elif x>-Float(1,8) then\n y := ln(-x);\n y := \+ ln(y)-y-(-.8567377032+.1404294456e-5*x)/\n (-1.+(.17 21594185e-5+.7399085925e-16*x)*x)\n elif x>-Float(1,20) then\n y := ln(-x);\n y := ln(y)-y+(-.8060973972+.1096 137486e-11*x)/\n (1.+(-.1415475176e-11+.3823119413e- 33*x)*x)\n else\n y := ln(-x);\n y := ln (y)-y-.75;\n end if;\n end if;\n end if;\n y;\nend pr oc: \n\nnext_halley_approx := proc(x,y)\n local snh,csh,t,u,v,h;\n \+ csh := cosh(y);\n snh := sinh(y);\n t := y*csh-x;\n u := csh+y* snh;\n v := 2*snh+y*csh;\n h := t/(u-1/2*v*t/u);\n y-h;\nend pro c:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Te st example: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "xx := evalf(sqrt(14));\ny0 := start_approx(xx);\n y1 := next_halley_approx(xx,y0);\ny2 := next_halley_approx(xx,y1);\nev al(y*cosh(y),y=y2);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+(Qd ;u$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G$\"+7wWP:!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$\"+5ETP:!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#y2G$\"+5ETP:!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+(Qd;u$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "xx := -eval f(sqrt(14)*10^5);\ny0 := start_approx(xx);\ny1 := next_halley_approx(x x,y0);\ny2 := next_halley_approx(xx,y1);\neval(y*cosh(y),y=y2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$!+(Qd;u$!\"%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#y0G$!+QR676!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#y1G$!+Y:r66!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G$!+Y:r66! \")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+-ulTP!\"%" }}}{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*cosh(x);" "6#/-%\"gG6#%\"xG*&F'\"\"\"-%%coshG6#F'F)" } {TEXT -1 2 ": " }{TEXT 0 1 "Y" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7098 "Y := proc( x::algebraic)\n local t,ok,s,terms,ti,i,eq;\n description \"invers e of x -> x*cosh(x)\";\n option `Copyright (c) 2003 Peter Stone.`;\n \n if nargs<>1 then\n error \"expecting 1 argument, got %1\", n args;\n end if;\n if type(x,'float') then evalf('Y'(x))\n elif t ype(x,`*`) and type(op(1,x),'numeric') and \n \+ signum(0,op(1,x),0)=-1 then -Y(-x) \n elif type(x,'realcons') and si gnum(0,x,0)=-1 then -Y(-x)\n elif type(x,And(complexcons,Not(realcon s))) then\n error \"not implemented for complex argument\"\n el if type(x,`*`) and nops(x)=2 then\n ok := false;\n if has(x, 'cosh') then \n if type(op(1,x),'function') and op([1,0],x)='c osh' then\n t := op([1,1],x);\n ok := true;\n \+ elif type(op(2,x),'function') and op([2,0],x)='cosh' then\n \+ t := op([2,1],x);\n ok := true;\n end if;\n \+ eq := t*cosh(t)=x;\n if ok and (evalb(expand(eq)) or t esteq(eq)) then t else 'Y'(x) end if;\n elif has(x,'arccosh') the n \n if type(op(1,x),'function') and op([1,0],x)='arccosh' the n\n t := op([1,1],x);\n ok := true;\n el if type(op(2,x),'function') and op([2,0],x)='arccosh' then\n \+ t := op([2,1],x);\n ok := true;\n end if;\n \+ if ok then\n eq := t*cosh(t)=x; \n if not (ev alb(expand(eq)) or testeq(eq)) then ok := false end if;\n end \+ if;\n if type(t,'realcons') and signum(0,t-1,0)=-1 then ok := \+ false end if;\n if ok then arccosh(t) else 'Y'(x) end if;\n \+ else 'Y'(x) end if;\n elif op(0,x)=`+` then\n ok := false;\n if has(x,'cosh') then\n s := select(has,x,'cosh');\n \+ terms := [op(s)];\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)='cosh' \+ then\n t := op(1,ti);\n eq := t*cosh(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 'Y'(x ) end if;\n elif has(x,'arcsec') then\n s := select(has,x ,'arccosh');\n terms := [op(s)];\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') an d op(0,ti)='arccosh' then\n t := op(1,ti);\n \+ eq := t*arccosh(t)=x;\n if evalb(expand(eq)) or teste q(eq) then\n ok := true;\n break;\n \+ end if;\n end if;\n end do;\n \+ if type(t,'realcons') and \n signum(0,t-1,0)=-1 th en ok := false end if;\n if ok then arccosh(t) else 'Y'(x) end if;\n else 'Y'(x) end if; \n else 'Y'(x) end if;\nend proc:\n \n# construct remember table\nY(0) := 0:\nY('infinity') := 'infinity': \n\n# differentiation\n`diff/Y` := proc(a,x) \n option `Copyright (c ) 2003 Peter Stone.`; \n diff(a,x)/(cosh(Y(a))+Y(a)*sinh(Y(a)))\nend proc:\n\n`D/Y` := 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,linea r(_X)) then return FAIL end if;\n h := subs(inds=u,_X=(u*cosh(u)-coe ff(gx,_X,0))/coeff(gx,_X),f);\n h := h*(cosh(u)+u*sinh(u))/coeff(gx, _X);\n h := int(h,u);\n if has(h,int) then return FAIL end if;\n \+ subs(cosh(u)=gx/u,u=inds,h)\nend proc:\n\n# simplification\n`simplify /Y` := proc(s)\n option remember,system,`Copyright (c) 2003 Peter St one.`;\n if not has(s,'Y') or type(s,'name') then return s\n elif \+ type(s,'function') and op(0,s)='cosh' then\n if type(op(1,s),'fun ction') and op([1,0],s)='Y' then\n return op([1,1],s)/op(1,s) \n else return s\n end if;\n end if;\n map(procname,args )\nend proc:\n\n# numerical evaluation\n`evalf/Y` := proc(x)\n local xx,eps,saveDigits,doY,val,p,q,maxit;\n option `Copyright (c) 2003 P eter Stone.`;\n\n if not type(x,realcons) then return 'Y'(x) end if; \n\n doY := proc(x,eps,maxit)\n local csh,snh,s,t,u,v,h,i; \n \+ # set up a starting approximation\n if x<.84 and x>-.84 then \n s := .4107111272*x+1.119421855/(x+2.357150354/\n \+ (x+.1556014159/(x+.6762074010/\n (x+.1873035998e-1/(x+.401 0065025/x)))));\n else\n if x>0 then\n if x<10 0 then\n s := (-.7378955849e-1+(1.363864106+(.3000905586 +\n (.6430435726e-2+.3314657469e-5*x)*x)*x)*x)/\n \+ (1.+(.9792814862+(.1062945123+.1357287434e-2*x)*x)*x)\n \+ elif x-100 then\n s := (-.7378955849e-1+(-1.363 864106+(.3000905586+\n (-.6430435726e-2+.3314657469e-5*x )*x)*x)*x)/\n (-1.+(.9792814862+(-.1062945123+.135728743 4e-2*x)*x)*x)\n elif x>-Float(1,8) then\n s : = ln(-x);\n s := ln(s)-s-(-.8567377032+.1404294456e-5*x) /\n (-1.+(.1721594185e-5+.7399085925e-16*x)*x)\n \+ elif x>-Float(1,20) then\n s := ln(-x);\n \+ s := ln(s)-s+(-.8060973972+.1096137486e-11*x)/\n \+ (1.+(-.1415475176e-11+.3823119413e-33*x)*x)\n else\n \+ s := ln(-x);\n s := ln(s)-s-.75;\n \+ end if;\n end if;\n end if;\n # solve the equatio n y*cosh(y)=x for y by Halley's method \n for i to maxit do\n \+ csh := cosh(s);\n snh := sinh(s);\n t := s*csh-x; \n u := csh+s*snh;\n v := 2*snh+s*csh;\n h := \+ t/(u-1/2*v*t/u);\n s := s-h;\n if abs(h)<=eps*abs(s) t hen break end if;\n end do;\n s;\n end proc;\n\n p := il og10(Digits);\n q := Float(Digits,-p);\n maxit := trunc((p+(.02331 061386+.1111111111*q))*2.095903274)+2;\n saveDigits := Digits;\n D igits := Digits+min(iquo(Digits,3),5);\n xx := evalf(x);\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(doY(xx,eps, maxit))\n else\n val := doY(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 63 "Comparison of starting approximation with the inverse function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "plot('Y(x)-start_approx(x)',x=0..20 000,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 621 181 181 {PLOTDATA 2 "6&-%'CURVESG6#7jo7$$\"+N@Ki8!\")$!&pZ*!\"*7$$\"+qUkCFF*$\"&oZ*F-7$$ \"+0k'p3%F*$!&#R;F-7$$\"+S&)G\\aF*$!&k]*F-7$$\"+v1h6oF*$!&5D\"F-7$$\"+ 5G$R<)F*$\"&f0*F-7$$\"+X\\DO&*F*$\"%i')F-7$$\"+3x&)*3\"!\"($!(C$oSF-7$ $\"+A**3E7FO$!(=\"pDF-7$$F)FO$!(3*y8F-7$$\"+[Vb)\\\"FO$!'Q]TF-7$$\"+il yM;FO$\"'T\"y$F-7$$\"+w(=5x\"FO$\"()[R5F-7$$\"+*)4D2>FO$\"(lpf\"F-7$$ \"+-K[V?FO$\"(472#F-7$$\"+ ]9QFO$\"(j_t%F-7$$\"+M3VfVFO$\"(c`*\\F-7$$\"+<0dL[FO$\"(Dj8&F-7$$\"+*> 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7$$\"+vjMD;Ffy$\"')eS'F47$$\"+:8#om\"Ffy$\"'l@jF47$$\"+IaB4Ffy$\"'%z&eF47$$\"+\\&[ l&>Ffy$\"'z)y&F47$$\"*++++#F*$\"([hr&F--%'COLOURG6&%$RGBG$F*F*Fbdl$\"* ++++\"F4-%+AXESLABELSG6$Q\"x6\"Q!Fidl-%%VIEWG6$;F(Fjcl%(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 14 "Test exam ples " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "g := x -> x*cosh(x);\nxx := 1.3;\ng(xx);\nY(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&9$\"\"\"-%%coshG6#F-F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#xxG$\"#8!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*\\)=iD!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++8!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "g := x -> x* cosh(x);\nxx := -0.991234567891096;\nevalf(g(xx),18);\nevalf(Y(%),15); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(*&9$\"\"\"-%%coshG6#F-F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$!0'4\"*ycM7**!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!38dfgV ES>:!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0'4\"*ycM7**!#:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "xx := Pi/7;\nyy := xx*cosh(xx);\nY(yy);\nevalf(%,12);\nyy := evalf [13](yy);\nevalf[12](Y(yy));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG ,$*&\"\"(!\"\"%#PiG\"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#yyG, $*&#\"\"\"\"\"(F(*&%#PiGF(-%%coshG6#,$*&F)!\"\"F+F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"(!\"\"%#PiG\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-70&*)z[%!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#yyG$\".f(yPhZ\\!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-80&*)z[% !#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "xx := sqrt(2);\nyy := xx*arccosh(xx);\nY(yy);\nevalf( %,12);\nyy := evalf[14](yy);\nevalf[12](Y(yy));" }}{PARA 11 "" 1 "" {XPPMATH 20 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" }}{PARA 0 "" 0 "" {TEXT -1 57 "The idea is to use Halley's method to solve the equation " }{XPPEDIT 18 0 "sinh(y )-y = x;" "6#/,&-%%sinhG6#%\"yG\"\"\"F(!\"\"%\"xG" }{TEXT -1 17 " nume rically for " }{TEXT 268 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 "starti ng approximations for Halley's root-finding method " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Maclaurin series fo r " }{XPPEDIT 18 0 "g(x) = sinh(x)-x;" "6#/-%\"gG6#%\"xG,&-%%sinhG6#F' \"\"\"F'!\"\"" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^3/3!+x^5/5!+x^7/7!+x^9/9!+` . . . `;" "6#,,*&%\"xG\" \"$-%*factorialG6#F&!\"\"\"\"\"*&F%\"\"&-F(6#F-F*F+*&F%\"\"(-F(6#F1F*F +*&F%\"\"*-F(6#F5F*F+%(~.~.~.~GF+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 276 1 "x" }{TEXT -1 24 " is close to 0 we ha ve: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sinh(x)-x;" " 6#,&-%%sinhG6#%\"xG\"\"\"F'!\"\"" }{TEXT -1 1 " " }{TEXT 274 1 "~" } {TEXT -1 1 " " }{XPPEDIT 18 0 "x^3/6" "6#*&%\"xG\"\"$\"\"'!\"\"" } {TEXT -1 2 ". " }{TEXT 273 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Solving " }{XPPEDIT 18 0 "y=x^3/6" "6#/%\"yG*&%\"xG\"\"$\" \"'!\"\"" }{TEXT -1 5 " for " }{TEXT 275 1 "x" }{TEXT -1 7 " gives " } {XPPEDIT 18 0 "x = 6^(1/3)*y^(1/3);" "6#/%\"xG*&)\"\"'*&\"\"\"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) = 6^(1/3)*y^(1/3);" "6#/- &%$psiG6#\"\"\"6#%\"xG*&)\"\"'*&F(F(\"\"$!\"\"F()%\"yG*&F(F(F/F0F(" } {TEXT -1 77 " can be used to provide an intial approximation for the n umerical inverse of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 7 ", when " }{TEXT 277 1 "x" }{TEXT -1 16 " is close to 0. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "p1 := plots[implicitplot](s inh(y)-y=x,x=0..0.1,y=0..0.85):\np2 := plot(1.817120593*surd(x,3),x=0. .0.1,y=0..0.85,color=blue):\nplots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 307 275 275 {PLOTDATA 2 "6&-%'CURVESG6_q7$7$$\"\"!F)F(7 $$\"3(=uh)*zL.a'!#B$\"3[<,F;2W%R$!#>7$7$$\"3S15i0`/^lF-$\"3C+++++++MF0 F*7$7$$\"3aR3i0`/^lF-F57$$\"3;]Ilj9K#=&!#A$\"3Y\\*evE]fv'F07$7$$\"38#= T!y]uT_F>$\"3[+++++++oF0F;7$7$$\"3\\/7/y]uT_F>FE7$$\"3'R<)f\\j;;%FN$\"35++ ++++g8FQFfn7$F\\o7$$\"3G`D8(*z/auFN$\"3XtV-#fSmj\"FQ7$7$$\"3b*f3*>O<+# )FN$\"37+++++++O<+#)FN$\"3%)*************p\"FQ7$$ \"3:^\"yQ(*=NB\"!#?$\"3Qd.A()3:N>FQ7$7$$\"3Yf!fW8\"*yT\"Ffp$\"3:++++++ S?FQFcp7$Fjp7$$\"3g.M_by(R'=Ffp$\"3$3^!G#)=c@AFQ7$7$$\"3I$=qb'3D`AFfp$ \"3!*************zBFQF`q7$Ffq7$$\"3,)oHT#e^LEFfp$\"3ww*[\\:^h\\#FQ7$7$ $\"37v^j5,PmLFfp$\"3?++++++?FFQF\\r7$Fbr7$$\"3eaPdsUOLNFfp$\"3SJ7$o.k' 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b/(FQ7$$\"3!)****\\i_QQgF0$\"3E(fIqiC*GrFQ7$$\"3j***\\7y%3TiF0$\"3sAO[ Gu\"y?(FQ7$$\"3o****\\P![hY'F0$\"3!))pm6/QMH(FQ7$$\"33KLL$Qx$omF0$\"3I S,ZxJpotFQ7$$\"3k+++v.I%)oF0$\"3#pJ(4oHQZuFQ7$$\"3Amm\"zpe*zqF0$\"3L8g j5)ys^(FQ7$$\"37+++D\\'QH(F0$\"33![TXBNAf(FQ7$$\"3GKLe9S8&\\(F0$\"3%oa /#*pO9m(FQ7$$\"3]++D1#=bq(F0$\"35!3GvKgCt(FQ7$$\"3>LLL3s?6zF0$\"3&Q.5n xf1!yFQ7$$\"3)*)**\\7`Wl7)F0$\"3!>'R;dK!3(yFQ7$$\"3[nmmm*RRL)F0$\"3kT$ z*\\y>PzFQ7$$\"3Smm;a<.Y&)F0$\"3+AI\\!HmR+)FQ7$$\"3-MLe9tOc()F0$\"3eF_ )yb*4p!)FQ7$$\"3u******\\Qk\\*)F0$\"3$3k#RBp.G\")FQ7$$\"3!QLL3dg6<*F0$ \"3I^+Sz8b%>)FQ7$$\"3-mmmmxGp$*F0$\"3WTsN:79`#)FQ7$$\"3!3+]7oK0e*F0$\" 3\"HY[$otq9$)FQ7$$\"3'****\\(=5s#y*F0$\"3')3lvv?zs$)FQ7$$\"3/+++++++5F Q$\"3:Kmz`mKM%)FQ-F\\cm6&F^cmF(F($\"*++++\"!\")-%+AXESLABELSG6%%\"xG% \"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($F_cm!\"\";F($\"#&)!\"#" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 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m;z*ev:J*F*$\"3Ki,BE1;O#)F-7$$\"3)QLL347TL*F*$\"3I[aqU)*zU#)F-7$$\"3/L LLjM?`$*F*$\"3'Q>YRx:%[#)F-7$$\"39+]7o7Tv$*F*$\"3#Q#)Q.$)Q\\D)F-7$$\"3 uKLLQ*o]R*F*$\"3e(R%[^Uqg#)F-7$$\"3I+]7=lj;%*F*$\"3Uq')e(e?qE)F-7$$\"3 &***\\PaRY2a*F*$\"3NjL:&\\\"=.$)F -7$$\"3Ynm\"zXu9c*F*$\"3/))\\a@.>4$)F-7$$\"3C+++&y))Ge*F*$\"3G8F(Q\"*) Q:$)F-7$$\"3z***\\i_QQg*F*$\"3b*Rxu4W9K)F-7$$\"3)***\\7y%3Ti*F*$\"3MU! oGU%HF$)F-7$$\"3[++v.[hY'*F*$\"3c9)*z&f!yL$)F-7$$\"3qLLLQx$om*F*$\"3/f `36,gR$)F-7$$\"3)4++v.I%)o*F*$\"3_<$>Nt/eM)F-7$$\"3Hm;zpe*zq*F*$\"3#e3 '4(3>9N)F-7$$\"35++]#\\'QH(*F*$\"3isw[L%[vN)F-7$$\"3=L$e9S8&\\(*F*$\"3 cCT!GZ2LO)F-7$$\"3%3+D1#=bq(*F*$\"3o+P+i)=$p$)F-7$$\"3oLL$3s?6z*F*$\"3 grt'ov(=v$)F-7$$\"3%)**\\7`Wl7)*F*$\"3!HL%oXJK\"Q)F-7$$\"3Dnmm'*RRL)*F *$\"37&G&euPA(Q)F-7$$\"3gmmTvJga)*F*$\"3=cWaX%\\KR)F-7$$\"3KM$e9tOc()* F*$\"3)ec]$)o;#*R)F-7$$\"3[+++&Qk\\*)*F*$\"39X*y$*\\#p/%)F-7$$\"3LLL3d g6<**F*$\"3N&H1Wjf4T)F-7$$\"3_nmmw(Gp$**F*$\"3Rq$)3Lrb;%)F-7$$\"3/+]7o K0e**F*$\"3U2()fWq^A%)F-7$$\"3k+](=5s#y**F*$\"3P_7:^N@G%)F-7$$\"3/++++ +++5F-$\"3:Kmz`mKM%)F--Fe[m6&Fg[m$Fi[mFi[mFb[n$\"*++++\"!\")-%+AXESLAB ELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$\"\"*!\"#$Fh[m!\"\"F^ \\n" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "psi2(x) = arcsinh(x);" "6#/-%%ps i2G6#%\"xG-%(arcsinhG6#F'" }{TEXT -1 49 " provides a suitable starting approximation when " }{TEXT 296 1 "x" }{TEXT -1 22 " has large magnit ude. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "p1 := plots[implicitplot](sinh(y)-y=x,x=799.9..800,y =7.375..7.39):\np2 := plot([arcsinh(x)],x=799.9..800,y=7.375..7.39,col or=blue):\nplots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 389 269 269 {PLOTDATA 2 "6&-%'CURVESG6U7$7$$\"3w************)*z!#:$\"3Ky\" RbbEoQ(!#<7$$\"3;I&H\"o9,**zF*$\"3[_q0yz#oQ(F-7$7$$\"3'**********R!**z F*$\"39Zy<<:$oQ(F-F.7$F47$$\"3]-l1S60**zF*$\"3UlC+**G$oQ(F-7$7$$\"39++ +++3**zF*$\"3%f^;)yk$oQ(F-F:7$F@7$$\"3'[Z.?\"34**zF*$\"3Czy%*>y$oQ(F-7 $7$$\"3M+++++7**zF*$\"3w%=b/WToQ(F-FF7$7$FM$\"3k&=b/WToQ(F-7$$\"3M[/%R [I\"**zF*$\"31$H$*3uUoQ(F-7$7$$\"3_+++++;**zF*$\"3c`Q4-k%oQ(F-FU7$Fen7 $$\"3q?u(e:q\"**zF*$\"3)oqQ=mZoQ(F-7$7$$\"3s+++++?**zF*$\"3EBDtj8&oQ(F -F[o7$7$Fbo$\"3QADtj8&oQ(F-7$$\"31$R9y#)4#**zF*$\"3q?Ty#e_oQ(F-7$7$$\" 3/-++++C**zF*$\"33#>r`KcoQ(F-Fjo7$7$$\"3#4++++S#**zF*$\"3=\">r`KcoQ(F- 7$$\"3Ul8v*\\\\#**zF*$\"3aM&HP]doQ(F-7$7$$\"35,++++G**zF*$\"3)3')4qGho Q(F-F[q7$Faq7$$\"3wP$)or\"*G**zF*$\"3YZ\\nCC'oQ(F-7$7$$\"3I,++++K**zF* $\"3!)G&['[i'oQ(F-Fgq7$F]r7$$\"3E6`iV)G$**zF*$\"3=i.iXt'oQ(F-7$7$$\"3M +++++O**zF*$\"3i(>(G57(oQ(F-Fcr7$7$$\"3[,++++O**zF*F\\s7$$\"3u%Gib^o$* *zF*$\"3+wdcmA(oQ(F-7$7$$\"3a+++++S**zF*$\"3Ume#>*oQ(F-7$7$$\"3W-++++c**zF*$\"3cU0[=g*oQ(F-Fev7$F[w7$$\"3_YrCvoc**zF *$\"3AWGHro*oQ(F-7$7$$\"3],++++g**zF*$\"3O6#>,)4!pQ(F-Faw7$7$$\"3i-+++ +g**zF*Fjw7$$\"3))=T=Zlg**zF*$\"3%*e#QAz,pQ(F-7$7$$\"3#G++++S'**zF*$\" 3=!)yvTf!pQ(F-F`x7$Ffx7$$\"3A\"4@\">ik**zF*$\"3'=n$=8n!pQ(F-7$7$$\"3-. ++++o**zF*$\"3)*[lR.4\"pQ(F-F\\y7$Fby7$$\"3sk!e5*eo**zF*$\"3o&3HTj6pQ( F-7$7$$\"3M/++++s**zF*$\"3!y@N]'e\"pQ(F-Fhy7$7$$\"3?.++++s**zF*Faz7$$ \"33P]*HcD(**zF*$\"3]*\\u]b;pQ(F-7$7$$\"3S.++++w**zF*$\"3g')QnE3#pQ(F- Fgz7$F][l7$$\"3U4?$\\Bl(**zF*$\"3M8*>gZ@pQ(F-7$7$$\"3e.++++!)**zF*$\"3 IcDJ)yDpQ(F-Fc[l7$Fi[l7$$\"3y\")*oo!\\!)**zF*$\"3;F`'pREpQ(F-7$7$$\"3y .++++%)**zF*$\"3AC7&*\\2$pQ(F-F_\\l7$Fe\\l7$$\"3Gbf!)yX%)**zF*$\"3)4u5 zJJpQ(F-7$7$$\"3#G++++!))**zF*$\"3/$*)*e6d$pQ(F-F[]l7$Fa]l7$$\"3iFHu]U ))**zF*$\"3![:c)Qi$pQ(F-7$7$$\"3;/++++#***zF*$\"3%=cGKnSpQ(F-Fg]l7$F]^ l7$$\"3)***)zE#R#***zF*$\"3un:!)f6%pQ(F-7$7$$\"3M/++++'***zF*$\"3aJs'[ jXpQ(F-Fc^l7$Fi^l7$$\"3[toh%fj***zF*$\"3W#)pu!3YpQ(F-7$7$$\"3a/++++++! 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" }}{PARA 0 " " 0 "" {TEXT -1 26 "Given an approximate zero " }{TEXT 272 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 \"correctio n\" " }{TEXT 271 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) = cosh(y)-1;" "6#/*&%$p hiG\"\"\"-%\"'G6#%\"yGF&,&-%%coshG6#F*F&F&!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "phi*`\"`(y) = sinh(y);" "6#/*&%$phiG\"\"\"-%\"\"G6#%\"y GF&-%%sinhG6#F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "sinh(y)-y-x;\nDiff(%,y)=diff (%,y);\nDiff(%%,y$2)=diff(%%,y$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,(-%%sinhG6#%\"yG\"\"\"F'!\"\"%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(-%%sinhG6#%\"yG\"\"\"F+!\"\"%\"xGF-F+,&-%%coshGF*F,F,F -" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(-%%sinhG6#%\"yG\"\" \"F+!\"\"%\"xGF--%\"$G6$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 procedure " } {TEXT 0 12 "start_approx" }{TEXT -1 30 ", together 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) = sinh(y)-y- x;" "6#/-%$phiG6#%\"yG,(-%%sinhG6#F'\"\"\"F'!\"\"%\"xGF-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 777 "start_approx := proc(x)\n local y;\n if x<.09114 786482 and x>-0.09114786482 then\n y := 1.817120593*surd(x,3);\n \+ elif x>-800 and x<800 then\n if x>0 then \n y := (.5626177561+(4.762139410+(1.325878111+\n (.1640036405e-1+. 2063907279e-5*x)*x)*x)*x)/\n (1.+(2.519357549+(.3152499762+.226 7640910e-2*x)*x)*x);\n else \n y := (.5626177561+(-4.7621 39410+(1.325878111+\n (-.1640036405e-1+.2063907279e-5*x)*x)* x)*x)/\n (-1.+(2.519357549+(-.3152499762+.2267640910e-2*x)*x)*x );\n end if;\n else\n y := arcsinh(x);\n end if;\n y; \nend proc: \n\nnext_halley_approx := proc(x,y)\n local snh,csh,t,u, h;\n csh := cosh(y);\n snh := sinh(y);\n t := snh-y-x;\n u := \+ csh-1;\n h := t/(u-1/2*snh*t/u);\n y-h;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Test examples: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "xx := evalf(sqrt(14/25));\ny0 := start_approx(xx);\ny1 := next_ha lley_approx(xx,y0);\ny2 := next_halley_approx(xx,y1);\ny3 := next_hall ey_approx(xx,y2);\neval(sinh(y)-y,y=y3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+uZJ$[(!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#y0G$\"+Zy)=f\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$\"+\\_ ]#e\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G$\"+.]]#e\"!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G$\"+.]]#e\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*x9L[(!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 169 "xx := -evalf(sqrt(987));\ny 0 := start_approx(xx);\ny1 := next_halley_approx(xx,y0);\ny2 := next_h alley_approx(xx,y1);\ny3 := next_halley_approx(xx,y2);\neval(sinh(y)-y ,y=y3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$!+9clTJ!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G$!+wb(=H%!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#y1G$!+nW1oU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#y2G$!+:K1oU!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G$!+9K1oU! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+8clTJ!\")" }}}{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) = sinh(x)-x;" "6#/-%\"gG6#%\"xG,&-%%sinhG6#F'\"\"\"F'!\"\" " }{TEXT -1 2 ": " }{TEXT 0 1 "Z" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4390 "Z := proc( x::algebraic)\n local t,ok,s,terms,i,ti,eq;\n description \"invers e of x -> sinh(x)-x\";\n option `Copyright (c) 2003 Peter Stone.`;\n \n if nargs<>1 then\n error \"expecting 1 argument, got %1\", n args;\n end if;\n if type(x,'float') then evalf('Z'(x))\n elif t ype(x,`*`) and type(op(1,x),'numeric') and \n \+ signum(0,op(1,x),0)=-1 then -Z(-x) \n elif type(x,'realcons') and si gnum(0,x,0)=-1 then -Z(-x)\n elif type(x,And(complexcons,Not(realcon s))) then\n error \"not implemented for complex argument\"\n el if type(x,`+`) then\n ok := false;\n if has(x,'sinh') then\n s := select(has,x,'sinh');\n if type(s,`+`) then term s := [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)='sinh' then\n t := op(1,ti);\n eq := sinh(t)-t=x;\n if evalb(expand(eq)) or testeq(eq) th en\n ok := true;\n break;\n \+ end if;\n end if;\n end do;\n if ok t hen t else 'Z'(x) end if;\n elif has(x,'arcsinh') then\n \+ s := select(has,x,'arcsinh');\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 if type(ti,'function') and op(0,t i)='arcsinh' then\n t := op(1,ti);\n eq := t-arcsinh(t)=x;\n if evalb(expand(eq)) or testeq(eq) th en\n ok := true;\n break;\n \+ end if;\n end if;\n end do;\n if ok t hen arcsinh(t) else 'Z'(x) end if;\n else 'Z'(x) end if;\n else 'Z'(x) end if;\nend proc:\n\n# construct remember table\nZ(0) := 0:\n Z('infinity') := 'infinity':\n\n# differentiation\n`diff/Z` := proc(a, x) \n option `Copyright (c) 2003 Peter Stone.`; \n diff(a,x)/(cosh (Z(a))-1)\nend proc:\n\n`D/Z` := proc(t)\n option `Copyright (c) 200 3 Peter Stone.`;\n if 1-0.09114786482 then\n s := 1.81712059 3*surd(x,3);\n elif x>-800 and x<800 then\n if x>0 then \+ \n s := (.5626177561+(4.762139410+(1.325878111+\n \+ (.1640036405e-1+.2063907279e-5*x)*x)*x)*x)/\n (1.+ (2.519357549+(.3152499762+.2267640910e-2*x)*x)*x);\n else \n \+ s := (.5626177561+(-4.762139410+(1.325878111+\n \+ (-.1640036405e-1+.2063907279e-5*x)*x)*x)*x)/\n (-1.+(2.5193 57549+(-.3152499762+.2267640910e-2*x)*x)*x);\n end if;\n \+ else\n s := arcsinh(x);\n end if;\n\n # solve the eq uation y*cosh(y)=x for y by Halley's method \n for i to maxit do \n csh := cosh(s);\n snh := sinh(s);\n t := sn h-s-x;\n u := csh-1;\n h := t/(u-1/2*snh*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 := F loat(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 Digits := Digits+max(0,-ilog10( xx)-3);\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(doZ(xx,eps,maxit))\n else\n val := doZ(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 "Comparison of starti ng approximation with the inverse function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "plot('Z(x)-start_a pprox(x)',x=0..900,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 509 170 170 {PLOTDATA 2 "6&-%'CURVESG6#7cr7$$\"\"!F)F(7$$\"+5'\\/8'!#5$!((oVz! \"*7$$\"+A**3E7F0$!((\\YfF07$$\"+$)[8R=F0$\"(Chw\"F07$$\"+W)z@X#F0$\"( []$oF07$$\"+0[AlIF0$\"(c(4\"*F07$$\"+m(p#yOF0$\"(#zS%*F07$$\"+FZJ\"H%F 0$\"(U\\d)F07$$\"+)ofV!\\F0$\"(Xt.(F07$$\"+\\YS*F0$!()z+]F07$$\"+v$>(3)*F 0$!(*32hF07$$\"+I4[.6!\")$!(K#*y(F07$$F3Fhp$!(KS%))F07$$\"+9*)p[8Fhp$! 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\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "secant(Z(x)=2,x=1..2,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\".!GBT<&o\"!#7" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\".Ka\\;>i\"!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\".qn_Gpi\"! #7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\".Nz[ goi\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$ \".ZySgoi\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~6~~- >~~~G$\".ZySgoi\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+3/'oi\"!\" *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The code above contains p rocedures which enable " }{TEXT 0 4 "diff" }{TEXT -1 5 " and " }{TEXT 0 1 "D" }{TEXT -1 18 " to differentiate " }{XPPEDIT 18 0 "Z(x);" "6#-% \"ZG6#%\"xG" }{TEXT -1 8 " and Z. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "diff(Z(x),x);\nD(Z)(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&-%%coshG6#-%\"ZG6#%\"xGF$F $!\"\"F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&-%%coshG6#-%\" ZG6#%\"xGF$F$!\"\"F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "This means that we can make use of Newton's and Halley 's method for root-finding. 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3(eclnAF;%RF^`l7$$\"3wP4rL@[rHFb`l$!3%yfVUA(3mRF^`l7$$\"31v$4Y6cb(HFb` l$!3s,\"*4!\\lR)RF^`l7$$\"3Q7y]&4I'zHFb`l$!3Y$Q0`NK_*RF^`l7$$\"3/]iSwS q$)HFb`l$!3)QerC7p)**RF^`l7$$\"3-DJ?Q?&=*HFb`l$!3gN2@*fK#*)RF^`l7$F]_l $!3IZ9P'\\E@&RF^`l-Fb_l6&Fd_lFg_lFg_lFe_l-%+AXESLABELSG6$Q\"x6\"Q!F\\b t-%%VIEWG6$;F(F]_l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "allroots(Z(x)=4*sin(x),x=-25..25,sensitivity=4);" }} {PARA 12 "" 1 "" {XPPMATH 20 "63$\"\"!F$$!+4,5l?!\")$!+\"e/o,#F'$!+_68 f9F'$!+)H(>l8F'$!+dGD<&)!\"*$!+uh:OrF0$!+YgnODF0$!+?'zQ-$!#5$\"+?'zQ-$ F7$\"+YgnODF0$\"+uh:OrF0$\"+dGD<&)F0$\"+)H(>l8F'$\"+_68f9F'$\"+\"e/o,# F'$\"+4,5l?F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 15 ": The equation " }{XPPEDIT 18 0 "Z(x) =4*sin(x)" "6#/-%\"ZG6#%\"xG*&\"\"%\"\"\"-%$sinG6#F'F*" }{TEXT -1 18 " is equivalent to " }{XPPEDIT 18 0 "x=g(4*sin(x))" "6#/%\"xG-%\"gG6#*& \"\"%\"\"\"-%$sinG6#F$F*" }{TEXT -1 51 ", so the same solutions can be obtained as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "g := x -> sinh(x)-x;\nallroots(g(4*sin(x) )=x,x=-25..25,sensitivity=4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"g Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%%sinhG6#9$\"\"\"F0!\"\"F(F(F( " }}{PARA 12 "" 1 "" {XPPMATH 20 "63$\"\"!F$$!+4,5l?!\")$!+\"e/o,#F'$! +_68f9F'$!+)H(>l8F'$!+dGD<&)!\"*$!+uh:OrF0$!+YgnODF0$!+?'zQ-$!#5$\"+?' zQ-$F7$\"+YgnODF0$\"+uh:OrF0$\"+dGD<&)F0$\"+)H(>l8F'$\"+_68f9F'$\"+\"e /o,#F'$\"+4,5l?F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "I nverse for " }{XPPEDIT 18 0 "g(x) = x+tanh(x);" "6#/-%\"gG6#%\"xG,&F' \"\"\"-%%tanhG6#F'F)" }{TEXT -1 2 ": " }{TEXT 0 1 "U" }{TEXT -1 2 " \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{XPPEDIT 18 0 "g(x) = x+tanh(x);" "6#/-% \"gG6#%\"xG,&F'\"\"\"-%%tanhG6#F'F)" }{TEXT -1 47 " is one-to-one, and so has an inverse function " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 224 "g := x -> x+tanh(x);\np1 := plot([g(x),x],x=- 3..3,y=-3..3,color=[red,black],\n linestyle=[1,2]):\np2 := plots[implicitplot](x=y+tanh(y),x=-3..3,\n y=-3..3,grid=[3 0,30],color=blue):\nplots[display]([p1,p2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"-%%t anhG6#F-F.F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 377 373 373 {PLOTDATA 2 "6'-%'CURVESG6%7S7$$!\"$\"\"!$!3YIno`Z0&*R!#<7$$!3!******\\2<#pGF-$! 3GA#H18)ziQF-7$$!3#)***\\7bBav#F-$!3A_BXV2PZPF-7$$!36++]K3XFEF-$!3TLRK l81iUC\"F-$!3m&flt,H44#F-7$$!3-++DhkaI6F-$!3BYohl EvT>F-7$$!3s******\\XF`**!#=$!3?g\"3#oA&\\v\"F-7$$!3u*******>#z2))Ffp$ !3s<(eFp)e(e\"F-7$$!3S++]7RKvuFfp$!3MlyO&*o?\"Q\"F-7$$!3s,+++P'eH'Ffp$ !3WpIT.PN(=\"F-7$$!3q)***\\7*3=+&Ffp$!3*Ro38)HSC'*Ffp7$$!3[)***\\PFcpP Ffp$!3!ekfZCv,P(Ffp7$$!3;)****\\7VQ[#Ffp$!3:;s^Zg$y\"\\Ffp7$$!32)***\\ i6:.8Ffp$!3GrJSu`(*)f#Ffp7$$!3Wb+++v`hH!#?$!3AhQxTj1BfF_s7$$\"3]****\\ (QIKH\"Ffp$\"3Q-#4#f\"*HzDFfp7$$\"38****\\7:xWCFfp$\"3sr0z[F(>%[Ffp7$$ \"3E,++vuY)o$Ffp$\"3?d&3<5$H=sFfp7$$\"3!z******4FL(\\Ffp$\"3aaxv#*f\\t &*Ffp7$$\"3A)****\\d6.B'Ffp$\"3W(Qmr7mi<\"F-7$$\"3s****\\(o3lW(Ffp$\"3 Cj'*)ov(fw8F-7$$\"35*****\\A))oz)Ffp$\"3/o;\\F-7$$\"3u***\\(=_ (zC\"F-$\"3#G\\MQ5!p&4#F-7$$\"3M+++b*=jP\"F-$\"3w@\\hnKWcAF-7$$\"3g*** \\(3/3(\\\"F-$\"3aw7#F-$\"3#e2U2pO'*4$F-7$$\" 3O++v)Q?QD#F-$\"3E&3Pu$>,KKF-7$$\"3G+++5jypBF-$\"3#os(H!3`CN$F-7$$\"3< ++]Ujp-DF-$\"3%z0\"*>:#Q*[$F-7$$\"3++++gEd@EF-$\"3`#>$[FF-$\"3Qs1^T?:SPF-7$$\"37++D6EjpGF-$\"3!oEr@)*=K'QF-7$$\"\"$F*$ \"3YIno`Z0&*RF--%'COLOURG6&%$RGBG$\"*++++\"!\")$F*F*Fa[l-%*LINESTYLEG6 #\"\"\"-F$6%7S7$F(F(7$F/F/7$F4F47$F9F97$F>F>7$FCFC7$FHFH7$FMFM7$FRFR7$ FWFW7$FfnFfn7$F[oF[o7$F`oF`o7$FeoFeo7$FjoFjo7$F_pF_p7$FdpFdp7$FjpFjp7$ F_qF_q7$FdqFdq7$FiqFiq7$F^rF^r7$FcrFcr7$FhrFhr7$F]sF]s7$FcsFcs7$FhsFhs 7$F]tF]t7$FbtFbt7$FgtFgt7$F\\uF\\u7$FauFau7$FfuFfu7$F[vF[v7$F`vF`v7$Fe vFev7$FjvFjv7$F_wF_w7$FdwFdw7$FiwFiw7$F^xF^x7$FcxFcx7$FhxFhx7$F]yF]y7$ FbyFby7$FgyFgy7$F\\zF\\z7$FazFaz7$FfzFfz-F[[l6&F][lF*F*F*-Fc[l6#\"\"#- F$6_q7$7$F($!3)z&*zQ#f@M?F-7$$!3qz:8[ShkHF-$!3DJxC$>.4+#F-7$7$$!3'>\"* ))y%e-FHF-$!3]5$z8CF-Fe_l7$F[`l7$$!3%Rq%oa85dGF-$!3mo/(\\L>:!>F-7$ 7$$!35ieF[M5$z#F-$!3^?X$fpXH%=F-Fa`l7$Fg`l7$$!3SR*o/4v!\\FF-$!3J&4iu/ \\E!=F-7$7$$!3d*zY[Wp4q#F-$!3gs^l*o?'e!ewl \"F-F\\bl7$Fbbl7$$!31im4w;&)HDF-$!3%o4'Qe$z!3;F-7$7$$!3'Gzzil$zlCF-$!3 qM5$z8C#F-$!3Ge5bqY.D8F-7$7$$!3S[M5$z8C<#F-$!3&Q]q#G+k48F- Fbel7$7$Fiel$!3j.0FG+k48F-7$$!3SMu@-z#e2#F-$!3ns(o`#p^M7F-7$7$F^`l$!3G Kvw&*)=([6F-Fafl7$Fgfl7$$!3c#evc4\\%f>F-$!30(['=!=**R9\"F-7$7$$!3#pIa7 FZ;&>F-$!3!*eF[M5$z8\"F-F[gl7$Fagl7$$!3tNSx&en!R=F-$!3a&*QOQT[d5F-7$7$ Ffal$!2w$\\:lg)*****F-Fggl7$F]hl7$$!3vtsog%)R=&f\"F-$!3Yb#=C4f c())Ffp7$7$Facl$!37O0`LWH$f)FfpF]il7$Fcil7$$!3s*3s!)>p(p9F-$!3#=GM*3(G 41)Ffp7$7$F_el$!3?h6!znG$\\sFfpFgil7$F]jl7$$!3&)[jp%oXVM\"F-$!3G4.XD$) >YsFfp7$7$$!3/adLAQgV8F-$!3&4$[M5$z8C(FfpFajl7$Fgjl7$$!3Lf`eTJB97F-$!3 qF)=$R#e$ykFfp7$7$Fdgl$!3Sj-`gELGgFfpF][m7$Fc[m7$$!31*f,&)p#4%3\"F-$!3 I]]\"H:(z5dFfp7$7$$!3#y*)=dw%4G**Ffp$!3%>X.Jz8C<&FfpFg[m7$F]\\m7$$!3$e :EW\"R[F&*Ffp$!3_0NE1ZFb\\Ffp7$7$Fjhl$!3a\">)3#)4eP[FfpFc\\m7$Fi\\m7$$ !3eZC_@(fd=)Ffp$!3LNe#>QL!GUFfp7$7$Fjjl$!3JpoVF.;;PFfpF]]m7$Fc]m7$$!3J R(='Gb.WoFfp$!3fk\")ed?z+NFfp7$7$$!34l>H[f&46'Ffp$!3&H2ieF[M5$FfpFg]m7 $F]^m7$$!37Y'4&f=)**[&Ffp$!3KzeX4-)ey#Ffp7$7$F`\\m$!3/!)*H(>\\ZBEFfpFc ^m7$Fi^m7$$!3nPU(=+y57%Ffp$!3%z!*\\)\\&=e3#Ffp7$7$F`^m$!3=/szl&*Rl:Ffp F]_m7$Fc_m7$$!3MI)QU9u@v#Ffp$!3cORC!*ov&Q\"Ffp7$7$$!3'yb,s0\"Hl?Ffp$!3 n$p?'eF[M5FfpFg_m7$7$$!39e:?d5Hl?FfpF``m7$$!3'Qw?-$G\\y8Ffp$!3*>B1-(os /p!#>7$7$F``m$!3Ct1t'G;;=&F[amFf`m7$7$F``m$!3%RnInG;;=&F[am7$$!39Qw/Xs L7a!#L$!3yYZ[$oznj#Fgam7$7$$\"3i&o?'eF[M5Ffp$\"3+L1t'G;;=&F[amFdam7$7$ F\\bm$\"3IK1t'G;;=&F[am7$$\"3K`2AIG\\y8Ffp$\"3Ezh?qos/pF[am7$7$$\"3/U: ?d5Hl?FfpF\\bmFdbm7$Fjbm7$$\"3y>)QU9u@v#Ffp$\"3cJRC!*ov&Q\"Ffp7$7$$\"3 =l?'eF[M5$Ffp$\"3I+szl&*Rl:FfpF^cm7$Fdcm7$$\"3oFU(=+y57%Ffp$\"3m-*\\) \\&=e3#Ffp7$7$$\"3=WM5$z8C<&Ffp$\"3:w*H(>\\ZBEFfpFjcm7$F`dm7$$\"3-N'4& f=)**[&Ffp$\"3yteX4-)ey#Ffp7$7$$\"3b\\>H[f&46'FfpFecmFfdm7$F\\em7$$\"3 KH(='Gb.WoFfp$\"3.f\")ed?z+NFfp7$7$$\"3=B[M5$z8C(Ffp$\"3'['oVF.;;PFfpF `em7$Ffem7$$\"3ePC_@(fd=)Ffp$\"3yHe#>QL!GUFfp7$7$$\"3>-ieF[M5$*Ffp$\"3 5(=)3#)4eP[FfpF\\fm7$Fbfm7$$\"3%e9EW\"R[F&*Ffp$\"3U*\\jiqu_&\\Ffp7$7$$ \"3]%))=dw%4G**FfpFadmFhfm7$F^gm7$$\"3=)f,&)p#4%3\"F-$\"3kV]\"H:(z5dFf p7$7$$\"3BeF[M5$z8\"F-$\"32g-`gELGgFfpFbgm7$Fhgm7$$\"3Ae`eTJB97F-$\"3; A)=$R#e$ykFfp7$7$$\"3$HvNB#QgV8F-FgemF^hm7$Fdhm7$$\"3=[jp%oXVM\"F-$\"3 u..XD$)>YsFfp7$7$$\"38'*o?'eF[M\"F-$\"3kb6!znG$\\sFfpFhhm7$7$F_im$\"3w c6!znG$\\sFfp7$$\"3%))3s!)>p(p9F-$\"3EwU$*3(G41)Ffp7$7$$\"3EM5$z8C&f\"F-$\" 3!)[#=C4fc())Ffp7$7$$\"3kn'yRK5@m\"F-$\"3I.ieF[M5$*FfpFfjm7$7$$\"3Un'y RK5@m\"F-Fcfm7$$\"3(GF(og%)R=ls6nDr*Ffp7$7$$\"3%>F-FigmFa\\n7$Fg\\n7$$\"3o\"evc4\\%f>F-$\"3R'['=!=**R9\"F-7 $7$$\"3%)4$z8CF-$\"3%=`nd*)=([6F-F[]n7$Fa]n7$$\"3_Lu@-z#e2#F-$\"3+ s(o`#p^M7F-7$7$$\"3%zW.Jz8C<#F-$\"3>.0FG+k48F-Fg]n7$7$F^^n$\"3T.0FG+k4 8F-7$$\"3Y'Gf(3n?#>#F-$\"3%y0^0nM]K\"F-7$7$$\"3;,AP\"\\aw@#F-F_imFf^n7 $F\\_n7$$\"3/#3^tL3eI#F-$\"3<+Mo$fH$=9F-7$7$$\"3I'eF[M5$zBF-$\"3Ay%f%4 ]hz9F-F`_n7$7$$\"3'eeF[M5$zBF-$\"3+y%f%4]hz9F-7$$\"3C(=G-vK'=CF-$\"3kK /`K!ewl\"F-F]an7$7$$ \"3wB:!>F-7$7$$\"326*))y%e-FHF-Fb]nFecn7$F[dn7$$\"3#)y:8[Shk HF-$\"3OIxC$>.4+#F-7$7$$\"3c**************HF-$\"3`d*zQ#f@M?F-F_dn-F[[l 6&F][lFa[lFa[lF^[l-%+AXESLABELSG6%Q\"x6\"Q\"yF`en-%%FONTG6#%(DEFAULTG- %%VIEWG6$;F(FfzFien" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The idea is to use Halley's method to solve the equati on " }{XPPEDIT 18 0 "y+tanh(y) = x;" "6#/,&%\"yG\"\"\"-%%tanhG6#F%F&% \"xG" }{TEXT -1 17 " numerically for " }{TEXT 284 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 metho d " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We \+ tackle the problem of constructing a numerical 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+tanh(y);" "6#/%\"xG,&% \"yG\"\"\"-%%tanhG6#F&F'" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dy = 2-tanh(y)^2;" "6#/*&%#dxG\"\"\"%#dyG! \"\",&\"\"#F&*$-%%tanhG6#%\"yGF*F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1/(2-tanh(y)^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&,&\"\"#F &*$-%%tanhG6#%\"yGF+F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Obtain the first few terms in the \+ Maclaurin series for " }{XPPEDIT 18 0 "f(x) = g^(-1)*``(x);" "6#/-%\" fG6#%\"xG*&)%\"gG,$\"\"\"!\"\"F,-%!G6#F'F," }{TEXT -1 54 " as the seri es solution for the differential equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1/(2-tanh(y)^2);" "6#/*&%#dyG\"\"\"%# dxG!\"\"*&F&F&,&\"\"#F&*$-%%tanhG6#%\"yGF+F(F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "Order := 10:\nde := diff(y(x),x)=1/(2-tanh(y)^2);\nic := y(0)=0; \ndsolve(\{de,ic\},y(x),type=series):\npx := convert(rhs(%),polynom); \nOrder := 6:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\" yG6#%\"xGF,*&\"\"\"F.,&\"\"#F.*$)-%%tanhG6#F*F0F.!\"\"F6" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#icG/-%\"yG6#\"\"!F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#pxG,,*&#\"\"\"\"\"#F(%\"xGF(F(*&#F(\"#[F(*$)F*\"\"$F (F(F(*&#F(\"%?>F(*$)F*\"\"&F(F(F(*&#F(\"&g,#F(*$)F*\"\"(F(F(!\"\"*&#\" $V\"\")?VABF(*$)F*\"\"*F(F(F=" }}}{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 74 "RootOf(y+tan h(y)-x,y);\ntaylor(%,x,10);\np := unapply(convert(%,polynom),x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RootOfG6#,(%#_ZG\"\"\"-%%tanhG6#F'F (%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"xG#\"\"\"\"\"#F&#F &\"#[\"\"$#F&\"%?>\"\"&#!\"\"\"&g,#\"\"(#!$V\"\")?VAB\"\"*-%\"OG6#F&\" #6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG% &arrowGF(,,*&#\"\"\"\"\"#F/9$F/F/*&#F/\"#[F/*$)F1\"\"$F/F/F/*&#F/\"%?> F/*$)F1\"\"&F/F/F/*&#F/\"&g,#F/*$)F1\"\"(F/F/!\"\"*&#\"$V\"\")?VABF/*$ )F1\"\"*F/F/FDF(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "The Pade approximation of type" }{XPPEDIT 18 0 "``(5, 4);" "6#-%!G6$\"\"&\"\"%" }{TEXT -1 29 " can be obtained as follows. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "numapprox[pade](px,x,[5,4]);\nconvert(%,confrac,x):\nevalf(eva lf(%,15)):\npsi := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*& ,(*&#\"&8H\"\")gp@7\"\"\"*$)%\"xG\"\"&F)F)F)*&#\"#n\"%OOF)*$)F,\"\"$F) F)!\"\"*&#F)\"\"#F)F,F)F)F),(F)F)*&#\"$r&\"%ssF)*$)F,F8F)F)F5*&#\"%p< \"'KsSF)*$)F,\"\"%F)F)F)F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$psiGf *6#%\"xG6\"6$%)operatorG%&arrowGF(,&*&$\"+K9?LC!#5\"\"\"9$F1F1*&$\"+kB \\i:F0F1,&F2F1*&$\"+9^YiR!\"(F1,&F2F1*&$\"+bs^(y$F:F1,&F2F1*&$\"+`=j4e F0F1F2!\"\"FCFCF1FCFCFCF1F(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+tanh(y);" "6#/%\"xG,&%\"yG\"\"\"- %%tanhG6#F&F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "p1 := plots[implicitplot](y +tanh(y)=x,x=0..3,y=0..2.5):\np2 := plot(psi(x),x=0..3,color=blue):\np lots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 307 275 275 {PLOTDATA 2 "6&-%'CURVESG6gp7$7$$\"\"!F)F(7$$\"3YPvRDjK&\\(!#>$\"3y&QN )GZ*Qv$F-7$7$$\"3%**************>\"!#=$\"3_J%zGsw*4gF-F*7$7$F2$\"3AK%z Gsw*4gF-7$$\"3]2&z]El!*\\\"F4$\"3br2nd%*y2vF-7$7$$\"3'zb\\i%*zm*>F4$\" 3/+++++++5F4F;7$7$$\"3Ce&\\i%*zm*>F4FD7$$\"3]VR'*[NmZAF4$\"3NZ+`Uq%p7 \"F47$7$$\"3!**************R#F4$\"3FoQ+Q&)*R?\"F4FJ7$7$FQ$\"3ToQ+Q&)*R ?\"F47$$\"3!HYx'4iT%*HF4$\"3=9@geJl/:F47$7$$\"3')*************f$F4$\"3 ,or&3Kc4\"=F4FY7$Fin7$$\"3*G)4Rq)o6u$F4$\"3-\"=uYFfB)=F47$7$$\"3gR!\\A ?`P(RF4$\"35+++++++?F4F_o7$Feo7$$\"3OOj5?Q<%[%F4$\"35>ZC$[)=jAF47$7$$ \"3#)*************z%F4$\"3g$yY?7QgU#F4F[p7$Fap7$$\"3]%G]1\\$[D_F4$\"3= '4eWUIak#F47$7$$\"3e5f^Ch78fF4$\"3U+++++++IF4Fgp7$F]q7$$\"3wTuc@GAmfF4 $\"3g)z$p[w9GIF47$7$$\"3w**************fF4$\"3;sc6CO0YIF4Fcq7$Fiq7$$\" 3_WIf(F4$\"39&\\x!\\#) >#o$F4F_r7$Fer7$$\"3^\\'=O\"e3LuF4$\"36fW)>#=w0QF47$7$$\"3?\\AbA'*[*z( F4$\"3A+++++++SF4F[s7$Fas7$$\"3K)poOU>:;)F4$\"3m<%4O\"Qt)>%F47$7$$\"3e )************R)F4$\"3+Hq\"\\6Y'HVF4Fgs7$7$$\"3o*************R)F4$\"3cH q\"\\6Y'HVF47$$\"3'o*4@VO'\\)))F4$\"3!=]d1jjef%F47$7$$\"3k************ *f*F4$\"3'3(pC3!y$))\\F4Fht7$F^u7$$\"3]'H`F'yS3'*F4$\"3S&e0xW$*H*\\F47 $7$$\"3#z4+Edr6i*F4$\"3++++++++]F4Fdu7$Fju7$$\"3\\#HaymO?.\"!#<$\"3qhv aMWp*R&F47$7$$\"33++++++!3\"Fcv$\"3;Z7g:l(Qn&F4F`v7$7$Fhv$\"31Y7g:l(Qn &F47$$\"3t$HH=#>@.6Fcv$\"3m>#*3=tc1eF47$7$$\"3RN!)*pc\\q8\"FcvFjqF`w7$ Ffw7$$\"3#p,d@T3P<\"Fcv$\"3A%>e`c'4>iF47$7$$\"3;+++++++7Fcv$\"3sw\"y(4 =BwjF4Fjw7$7$Fax$\"3gv\"y(4=BwjF47$$\"3[9<)\\w*eV7Fcv$\"3w8d[e>vOmF47$ 7$$\"3cjr6xxO/8Fcv$\"3a**************pF4Fix7$F_y7$$\"3^B^dYDH88Fcv$\"3 G0t?Xa*e0(F47$7$$\"3G++++++?8Fcv$\"3#HPwf$R!z4(F4Fey7$7$$\"3^++++++?8F cv$\"3-uj(f$R!z4(F47$$\"3cYE,1J!=Q\"Fcv$\"3a\"Ghl6u\\[(F47$7$$\"3R++++ ++S9Fcv$\"3YTZ;Z(e%\\yF4Ffz7$F\\[l7$$\"3Qp,XlOJ]9Fcv$\"3zd_\"zy_S\"zF4 7$7$$\"3u[yEqn.k9Fcv$\"3U+++++++!)F4Fb[l7$7$Fi[l$\"3K**************zF4 7$$\"30WH:#)Qq<:Fcv$\"3)yYD([wY_$)F47$7$$\"3]++++++g:Fcv$\"3sr4>5WDI') F4Fa\\l7$Fg\\l7$$\"3#HLplZ8[e\"Fcv$\"3'y*))eG5A$z)F47$7$$\"3FC!*>qyH;; 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" }}{PARA 0 "" 0 "" {TEXT -1 48 "The \+ graph of the inverse function has the lines " }{XPPEDIT 18 0 "y=x-1" " 6#/%\"yG,&%\"xG\"\"\"F'!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y=x+ 1" "6#/%\"yG,&%\"xG\"\"\"F'F'" }{TEXT -1 18 " as asympotes, so " } {XPPEDIT 18 0 "x-1" "6#,&%\"xG\"\"\"F%!\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "x+1" "6#,&%\"xG\"\"\"F%F%" }{TEXT -1 54 " can be used t o provide a starting approximation when " }{TEXT 299 1 "x" }{TEXT -1 22 " has large magnitude. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "For intermediate values of " }{TEXT 300 1 "x" } {TEXT -1 48 " a suitable rational approximation can be used. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "remez('U(x)',x=1.9..5,[2,2],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#%Wstandard~Chebyshev~points~fo r~initial~critical~points:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"0+++ +++!>!#9$\"0$)=(eO-'>#F%$\"0%)=(eO-rHF%$\"0;\"GTj(*GRF%$\"0<\"GTj(Rq%F %$\"0+++++++&F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,iteration ~7G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G----------------------------- ---------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprovisional~rational~a pproximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,($\"3K(pmv\"*>z\" **!#=\"\"\"*&$\"3auL$)=oNXzF'F(%\"xGF(!\"\"*&$\"3Sw.E.aqruF'F()F,\"\"# F(F(F(,($F(\"\"!F(*&$\"3]KJ:_!)fTZF'F(F,F(F(*&$\"3zdu%e%***\\@#!#>F(F1 F(F(F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%CYsOi!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%goal~for~relative~difference:G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0+++++++\"!#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1critical~po ints:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6($\"0++++++!>!#9$\"3Iea(Q0987 #!#<$\"3s.f1s*y]s#F($\"3U=Q5\\z$)4OF($\"3_^N'[d>ib%F($\"0+++++++&F%" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%7minimax~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,($\" +=*>z\"**!#5\"\"\"*&$\"+>oNXzF'F(%\"xGF(!\"\"*&$\"+.aqruF'F()F,\"\"#F( F(F(,($F(\"\"!F(*&$\"+_!)fTZF'F(F,F(F(*&$\"+Y***\\@#!#6F(F1F(F(F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%0minimax~error:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&nB'!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*&,($\"+=*>z\"**!#5\"\"\"*&$\"+>oNXzF'F(%\"xGF(!\"\"*&$\"+.aqruF'F()F ,\"\"#F(F(F(,($F(\"\"!F(*&$\"+_!)fTZF'F(F,F(F(*&$\"+Y***\\@#!#6F(F1F(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 f ormula for solving " }{XPPEDIT 18 0 "x = y+tanh(y);" "6#/%\"xG,&%\"yG \"\"\"-%%tanhG6#F&F'" }{TEXT -1 5 " for " }{TEXT 285 1 "y" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Le t " }{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 26 "Given an approximate zero " }{TEXT 287 1 "a" }{TEXT -1 4 " of \+ " }{XPPEDIT 18 0 "phi(y)" "6#-%$phiG6#%\"yG" }{TEXT -1 36 ", the \"imp roved\" Halley estimate is " }{XPPEDIT 18 0 "y = y-h;" "6#/%\"yG,&F$\" \"\"%\"hG!\"\"" }{TEXT -1 24 " where the \"correction\" " }{TEXT 286 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) = 2-tanh(y)^2;" "6#/*&%$phiG\"\"\"-%\"'G6 #%\"yGF&,&\"\"#F&*$-%%tanhG6#F*F,!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi*`\"`(y) = 2*tanh(y)*(tanh(y)^2-1);" "6#/*&%$phiG\"\"\"-%\"\" G6#%\"yGF&*(\"\"#F&-%%tanhG6#F*F&,&*$-F.6#F*F,F&F&!\"\"F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "y+tanh(y)-x;\nDiff(%,y)=diff(%,y);\nDiff(%%,y$2)=diff (%%,y$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"yG\"\"\"-%%tanhG6#F$ F%%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(%\"yG\"\" \"-%%tanhG6#F(F)%\"xG!\"\"F(,&\"\"#F)*$)F*F0F)F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(%\"yG\"\"\"-%%tanhG6#F(F)%\"xG!\"\"-%\"$G6 $F(\"\"#,$*(F2F)F*F),&F)F)*$)F*F2F)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 procedur e " }{TEXT 0 12 "start_approx" }{TEXT -1 30 ", together with the proce dure " }{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+tan h(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 0 752 "start_approx := proc(x)\n local y;\n if x<1.9 an d x>-1.9 then \n y := .2433201432*x+.1562492364/\n (x-39 6.2465114/(x+378.7517255/(x-.5809631853/x)));\n elif x>0 then\n \+ if x<5 then\n y := (.9917919918+(-.7945356819+.7471705403*x)* x)/\n (1.+(.4741598052+.2214999946e-1*x)*x)\n else \n \+ y := x-1;\n end if;\n else\n if x>-5 then\n \+ y := -(.9917919918+(.7945356819+.7471705403*x)*x)/\n (1.+( -.4741598052+.2214999946e-1*x)*x)\n else \n y := x+1;\n \+ end if;\n end if;\n y;\nend proc: \n\nnext_halley_approx := pr oc(x,y)\n local tnh,ts,t,u,v,h;\n tnh := tanh(y);\n ts := tnh^2; \n t := y+tnh-x;\n u := 2-ts;\n v := 2*tnh*(ts-1);\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 0 168 "xx := evalf(sqrt( 21));\ny0 := start_approx(xx);\ny1 := next_halley_approx(xx,y0);\ny2 : = next_halley_approx(xx,y1);\ny3 := next_halley_approx(xx,y2);\neval(y +tanh(y),y=y3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+&pvDe%! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G$\"+.it%e$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$\"+we6%e$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G$\"+xe6%e$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#y3G$\"+xe6%e$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&pvDe%!\" *" }}}{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+tanh(x);" "6#/-%\"gG6#%\"xG,&F'\" \"\"-%%tanhG6#F'F)" }{TEXT -1 2 ": " }{TEXT 0 1 "U" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4463 "U := proc(x::algebraic)\n local t,ok,s,terms,i,ti,eq;\n desc ription \"inverse of x -> x+tanh(x)\";\n option `Copyright (c) 2003 \+ Peter Stone.`;\n\n if nargs<>1 then\n error \"expecting 1 argum ent, got %1\", nargs;\n end if;\n if type(x,'float') then evalf('U '(x))\n elif type(x,`*`) and type(op(1,x),'numeric') and \n \+ signum(0,op(1,x),0)=-1 then -U(-x) \n elif type(x,'r ealcons') and signum(0,x,0)=-1 then -U(-x)\n elif type(x,And(complex cons,Not(realcons))) then\n error \"not implemented for complex a rgument\"\n elif type(x,`+`) then\n ok := false;\n if has( x,'tanh') then\n s := select(has,x,'tanh');\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)='tanh' then\n t := op(1,ti);\n \+ eq := t+tanh(t)=x;\n if evalb(expand(eq)) \+ or testeq(eq) then\n ok := true;\n b reak;\n end if;\n end if;\n end do;\n if ok then t else 'U'(x) end if;\n elif has(x,'arctanh') then\n s := select(has,x,'arctanh');\n if type(s,`+`) then terms := [op(s)] else terms := [s] end if;\n for i to no ps(terms) do\n ti := terms[i];\n if type(ti,`*`) and op(1,ti)=-1 then ti := -ti end if;\n if type(ti,'funct ion') and op(0,ti)='arctanh' then\n t := op(1,ti);\n \+ eq := t+arctanh(t)=x;\n if evalb(expand(eq)) \+ or testeq(eq) then\n ok := true;\n b reak;\n end if;\n end if;\n end do;\n if type(t,'realcons') and (signum(0,t-1,1)=1 or\n \+ signum(0,t+1,-1)=-1) then ok := false end if;\n if ok \+ then arctanh(t) else 'U'(x) end if;\n else 'U'(x) end if;\n els e 'U'(x) end if;\nend proc:\n\n# construct remember table\nU(0) := 0: \nU('infinity') := 'infinity':\n\n# differentiation\n`diff/U` := proc( a,x) \n option `Copyright (c) 2003 Peter Stone.`; \n diff(a,x)/(2- (x-U(a))^2)\nend proc:\n\n`D/U` := proc(t)\n option `Copyright (c) 2 003 Peter Stone.`;\n if 1a)-'U')^2)\n end if\nend proc:\n\n# simplification\n`simplify/U` := proc(s)\n option remembe r,system,`Copyright (c) 2003 Peter Stone.`;\n if not has(s,'U') or t ype(s,'name') then return s\n elif type(s,'function') and op(0,s)='t anh' then\n if type(op(1,s),'function') and op([1,0],s)='U' then \n return op([1,1],s)-op(1,s)\n else return s\n end \+ if;\n end if;\n map(procname,args)\nend proc:\n\n# numerical evalu ation\n`evalf/U` := proc(x)\n local xx,eps,saveDigits,doU,val,p,q,ma xit;\n option `Copyright (c) 2003 Peter Stone.`;\n\n if not type(x ,realcons) then return 'U'(x) end if;\n\n doU := proc(x,eps,maxit)\n local tnh,ts,t,s,u,v,h,i; \n # set up a starting approximat ion\n if x<1.9 and x>-1.9 then \n s := .2433201432*x+.156 2492364/\n (x-396.2465114/(x+378.7517255/(x-.5809631853/x) ));\n elif x>0 then\n if x<5 then\n s := (.991 7919918+(-.7945356819+.7471705403*x)*x)/\n (1.+(.4741598052 +.2214999946e-1*x)*x)\n else \n s := x-1;\n \+ end if;\n else\n if x>-5 then\n s := -(.99179 19918+(.7945356819+.7471705403*x)*x)/\n (1.+(-.4741598052 +.2214999946e-1*x)*x)\n else \n s := x+1;\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 tnh := tanh(s); \n ts := tnh^2;\n t := s+tnh-x;\n u := 2-ts;\n v := 2*tnh*(ts-1);\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 en d do;\n s;\n end proc;\n\n p := ilog10(Digits);\n q := Floa t(Digits,-p);\n maxit := trunc((p+(.02331061386+.1111111111*q))*2.09 5903274)+2;\n saveDigits := Digits;\n Digits := Digits+min(iquo(Di gits,3),5);\n xx := evalf(x);\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(doU(xx,eps,maxit))\n else\n val := doU(xx,eps,maxit)\n end if;\n evalf[saveDigits](val);\nend pro c:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "checking t he procedure " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Comparison of starting approximation with the inverse fun ction. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot('U(x)-start_approx(x)',x=0..8,color=blue);" }} {PARA 13 "" 1 "" {GLPLOT2D 509 170 170 {PLOTDATA 2 "6&-%'CURVESG6#7^u7 $$\"+X&)G\\a!#7$\"\"#F*7$$\"+4x&)*3\"!#6F+7$$\"+klyM;F0$\"\"\"F*7$$\"+ =arz@F0$\"\"!F:7$$\"+FJdpKF0F97$$\"+N3VfVF0$F5F07$$\"+`i9RlF0$F,F07$$ \"+q;')=()F0FA7$$\"+]#HyI\"!#5F97$$\"+MBxV5E$FLF97$$\"+MAKn\\FL$F5FL7$$\"+Nc$\\o'FL$\"#LFL7$$\"+=bQ%R)FL$\"$J%F L7$$\"+&Qk#z**FL$\"%UIFL7$$\"+l9.i6!\"*$\"&\\s\"FL7$$\"+#H!*oC\"Ffo$\" &D(QFL7$$\"+>\"\\W^Ffo7$$\"+r8q.>Ffo$!'VVdFfo7$$\"+KN44>Ffo$!'TZ]Ffo7 $$\"+#p&[9>Ffo$!'wxVFfo7$$\"+_y()>>Ffo$!'#Rt$Ffo7$$\"+8+FD>Ffo$!'Q:JFf o7$$\"+u@mI>Ffo$!'l@DFfo7$$\"+MV0O>Ffo$!'>_>Ffo7$$\"+b'Qo%>Ffo$!&;%))F fo7$$\"+wHid>Ffo$\"%p#*Ffo7$$\"+(H2%o>Ffo$\"&A#)*Ffo7$$\"+=;>z>Ffo$\"' >)y\"Ffo7$$\"+Rf(**)>Ffo$\"'N9DFfo7$$\"+g-w+?Ffo$\"'IkJFfo7$$\"+luTA?F fo$\"'[`UFfo7$$\"+qY2W?Ffo$\"'5x]Ffo7$$\"+v=tl?Ffo$\"'ZhcFfo7$$\"+!3*Q (3#Ffo$\"'cJgFfo7$$\"+#o<#)4#Ffo$\"'rVhFfo7$$\"+&GY!4@Ffo$\"'26iFfo7$$ \"+()[()>@Ffo$\"'POiFfo7$$\"+!\\.28#Ffo$\"'NAiFfo7$$\"+&pgB:#Ffo$\"'c' 3'Ffo7$$\"++z,u@Ffo$\"'dBeFfo7$$\"+g$fd@#Ffo$\"'mD]Ffo7$$\"+?3]dAFfo$ \"'^WRFfo7$$\"+]:PyAFfo$\"':JLFfo7$$\"+!GU#*H#Ffo$\"'=&o#Ffo7$$\"+5I6? 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" }}{PARA 0 "" 0 "" {TEXT -1 28 "The solution is (of course) " }{XPPEDIT 18 0 "x = 2+tanh(2);" "6#/ %\"xG,&\"\"#\"\"\"-%%tanhG6#F&F'" }{TEXT -1 1 " " }{TEXT 283 1 "~" } {TEXT -1 14 " 2.964027580. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalf(2+tanh(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!eFS'H!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "secant(U(x)=2,x=2.9..3,info= true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\" .dTp))Q'H!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~ ~~G$\".%o)GFS'H!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~ 3~~->~~~G$\".y+eFS'H!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxima tion~4~~->~~~G$\".w+eFS'H!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!e FS'H!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The code above con tains procedures which enable " }{TEXT 0 4 "diff" }{TEXT -1 5 " and " }{TEXT 0 1 "D" }{TEXT -1 18 " to differentiate " }{XPPEDIT 18 0 "U(x); " "6#-%\"UG6#%\"xG" }{TEXT -1 8 " and U. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "diff(U(x),x);\nD(U)(x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&\"\"#F$*$),&%\"xGF$- %\"UG6#F*!\"\"F&F$F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,& \"\"#F$*$),&%\"xGF$-%\"UG6#F*!\"\"F&F$F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "This means that we can make use of Newton's and Halley's method for root-finding. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "newton(U(x )=2,x=3,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation ~1~~->~~~G$\".KRk,T'H!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxim ation~2~~->~~~G$\".-/eFS'H!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7app roximation~3~~->~~~G$\".v+eFS'H!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %7approximation~4~~->~~~G$\".w+eFS'H!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!eFS'H!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "halley(U(x)=2,x=3,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\".[F(*GS'H!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\".w+eFS'H!# 7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\".w+eF S'H!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!eFS'H!\"*" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "For another root- finding question consider the problem of finding all the solutions of \+ the equation " }{XPPEDIT 18 0 "U(x) = sin(x);" "6#/-%\"UG6#%\"xG-%$sin G6#F'" }{TEXT -1 2 ". 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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 ap proximations for Halley's root-finding method " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Maclaurin series for \+ " }{XPPEDIT 18 0 "g(x) = x-tanh(x);" "6#/-%\"gG6#%\"xG,&F'\"\"\"-%%tan hG6#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^3-2/15" "6#,&*$%\"xG\"\"$\"\"\"*&\"\"#F'\"#:!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^5+17/315" "6#,&*$%\"xG\"\"&\"\"\"*& \"# " 0 "" {MPLTEXT 1 0 143 "p1 := plots[implicitplot](y-tanh(y)=x,x=0..0.0003,y= 0..0.1):\np2 := plot(1.732050808*surd(x,3),x=0..0.0003,color=blue):\np lots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 336 319 319 {PLOTDATA 2 "6&-%'CURVESG6_q7$7$$\"\"!F)F(7$$\"3B#oaj'Q`H@!#D$\"3/=)yP b,H*R!#?7$7$$\"3N,,6!o>L8#F-$\"35+++++++SF0F*7$7$$\"3Fd,6!o>L8#F-F57$$ \"3'*R!edjYco\"!#C$\"3>l!3)y<\"Q%zF07$7$$\"3*pCRr(Hi1$\"3=+++++++!) 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$$\"3o**\\PfL'zV#F]al$\"379-j]E-#3\"Fcan7$$\"3\")*******)>=+DF]al$\"3S ='z?6]64\"Fcan7$$\"3)****\\i_4Qc#F]al$\"3'*)3![n*G.5\"Fcan7$$\"39+]P%> 5pi#F]al$\"3E_`yIIG46Fcan7$$\"3)******\\:$*[o#F]al$\"3v'e&4D`Q<6Fcan7$ $\"3+++Dr\"[8v#F]al$\"3kSK\\k#Hl7\"Fcan7$$\"3++++Ijy5GF]al$\"3*\\A')Gy $eM6Fcan7$$\"3s**\\P/)fT(GF]al$\"3'>&GSku/V6Fcan7$$\"31+]i0j\"[$HF]al$ \"3_DrFTD.^6Fcan7$Fjcm$\"3UX;4#)=\\f6Fcan-F_dm6&FadmF(F($\"*++++\"!\") -%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"\"$!\"%F dgn" 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 5 "When " }{TEXT 295 1 "x" }{TEXT -1 22 " has large magnitude, " } {XPPEDIT 18 0 "g^(-1)*``(x);" "6#*&)%\"gG,$\"\"\"!\"\"F'-%!G6#%\"xGF' " }{TEXT -1 1 " " }{TEXT 293 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x+1 " "6#,&%\"xG\"\"\"F%F%" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "x>0" "6#2 \"\"!%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g^(-1)*``(x)" "6#*&)% \"gG,$\"\"\"!\"\"F'-%!G6#%\"xGF'" }{TEXT -1 1 " " }{TEXT 294 1 "~" } {TEXT -1 1 " " }{XPPEDIT 18 0 "x-1" "6#,&%\"xG\"\"\"F%!\"\"" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x<0" "6#2%\"xG\"\"!" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "For inter mediate values of " }{TEXT 301 1 "x" }{TEXT -1 48 " a suitable rationa l approximation can be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "remez('P(x)',x=0.0000125..1. 86,[5,2],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~11G" }}{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#*& ,.$\"6,1RiuC)3A*o$!#A\"\"\"*&$\"6$3DNZu_>q)G$!#=F(%\"xGF(F(*&$\"6%zv@^ dn:9_I!#;F()F-\"\"#F(F(*&$\"6#fH<()QTt+blF1F()F-\"\"$F(F(*&$\"6#R\"p*z 1)fbz#>F1F()F-\"\"%F(!\"\"*&$\"6h]=\")\\ww_([[!#!f\"FBF(F-F(F(*&$\"6_2;4:b4G&4SF1F(F2F(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#$\"39fBv?i!z$G!#B" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1critical~poi nts:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+$\"/+++++]7!#=$\"6CBYD([Qd>M7 !#C$\"6#[MbO.'fbJO\"!#B$\"6'4I#ex0\\uP4\"!#A$\"6uEyi5$*4)Q$['F.$\"6%=3 Q#z$4C\\$)G!#@$\"6FgCtaF_n\\T)F3$\"6_*HzS&\\([MO:!#?$\"3+++++++g=!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7minimax~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,.$ \"+#)3A*o$!#6\"\"\"*&$\"+`>q)G$!\"(F(%\"xGF(F(*&$\"+o:9_I!\"&F()F-\"\" #F(F(*&$\"+Tt+blF1F()F-\"\"$F(F(*&$\"+)fbz#>F1F()F-\"\"%F(!\"\"*&$\"+o Fv[[!\"'F()F-\"\"&F(F(F(,($F(\"\"!F(*&$\"+%o)>!f\"FBF(F-F(F(*&$\"+'4G& 4SF1F(F2F(F(F>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0minimax~error:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&q)G$!\"(F(%\"xGF (F(*&$\"+o:9_I!\"&F()F-\"\"#F(F(*&$\"+Tt+blF1F()F-\"\"$F(F(*&$\"+)fbz# >F1F()F-\"\"%F(!\"\"*&$\"+oFv[[!\"'F()F-\"\"&F(F(F(,($F(\"\"!F(*&$\"+% o)>!f\"FBF(F-F(F(*&$\"+'4G&4SF1F(F2F(F(F>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Halley's formula for solving " }{XPPEDIT 18 0 "x = y-tanh(y);" "6#/%\"xG,&%\"yG\"\"\"-%%tanhG6#F&!\"\"" }{TEXT -1 5 " for " }{TEXT 280 1 "y" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi(y) = y -tanh(y)-x;" "6#/-%$phiG6#%\"yG,(F'\"\"\"-%%tanhG6#F'!\"\"%\"xGF-" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Given an approximate ze ro " }{TEXT 282 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 281 1 "h" }{TEXT -1 15 " is given b y: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h = phi(y)/`` (phi*`'`(y)-phi*`''`(y)*phi(y)/(2*phi*`'`(y)));" "6#/%\"hG*&-%$phiG6#% \"yG\"\"\"-%!G6#,&*&F'F*-%\"'G6#F)F*F***F'F*-%#''G6#F)F*-F'6#F)F**(\" \"#F*F'F*-F16#F)F*!\"\"F=F=" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi*`'`(y) = \+ tanh(y)^2;" "6#/*&%$phiG\"\"\"-%\"'G6#%\"yGF&*$-%%tanhG6#F*\"\"#" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "phi*`\"`(y) = 2*tanh(y)*(1-tanh(y)^ 2);" "6#/*&%$phiG\"\"\"-%\"\"G6#%\"yGF&*(\"\"#F&-%%tanhG6#F*F&,&F&F&*$ -F.6#F*F,!\"\"F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "y-tanh(y)-x;\nDiff(%,y)=diff (%,y);\nDiff(%%,y$2)=diff(%%,y$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,(%\"yG\"\"\"-%%tanhG6#F$!\"\"%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(%\"yG\"\"\"-%%tanhG6#F(!\"\"%\"xGF-F(*$)F*\"\"#F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(%\"yG\"\"\"-%%tanhG6#F(! \"\"%\"xGF--%\"$G6$F(\"\"#,$*(F2F)F*F),&F)F)*$)F*F2F)F-F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "The following \+ code sets up the starting approximations from the previous subsection \+ via the procedure " }{TEXT 0 12 "start_approx" }{TEXT -1 30 ", togethe r with the procedure " }{TEXT 0 18 "next_halley_approx" }{TEXT -1 54 " to perform one step of Halley's method for a zero of " }{XPPEDIT 18 0 "phi(y) = y+tanh(y)-x;" "6#/-%$phiG6#%\"yG,(F'\"\"\"-%%tanhG6#F'F)% \"xG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 822 "start_approx := proc(x)\n local \+ y;\n if x<.0000125136 and x>-.0000125136 then \n y := 1.7320508 08*surd(x,3);\n elif x>0 then\n if x<1.859 then\n y := \+ (.3689220882e-1+(328.8701953+(30521.41568+\n (65550.07341 +(-19279.55598+4848.752768*x)*x)*x)*x)*x)/\n (1.+(1590.19 8684+40095.28096*x)*x)\n else \n y := x+1\n end if; \n else\n if x>-1.859 then \n y := (-.3689220882e-1+(32 8.8701953+(-30521.41568+\n (65550.07341+(19279.55598+4848 .752768*x)*x)*x)*x)*x)/\n (1.+(-1590.198684+40095.28096*x )*x)\n else\n y := x-1;\n end if;\n end if;\n y; \nend proc:\n\nnext_halley_approx := proc(x,y)\n local tnh,t,u,v,h; \n tnh := tanh(y);\n t := y-tnh-x;\n u := tnh^2;\n v := 2*tnh* (1-u);\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 0 170 "xx := evalf(sqrt(50)/7);\ny0 := start_approx(xx);\ny1 := next_hal ley_approx(xx,y0);\ny2 := next_halley_approx(xx,y1);\ny3 := next_halle y_approx(xx,y2);\neval(y-tanh(y),y=y3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+WD:55!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#y0G$\"+#[>r(>!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$\"+j%f@ (>!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G$\"+b%f@(>!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G$\"+c%f@(>!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+WD:55!\"*" }}}{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-tanh(x); " "6#/-%\"gG6#%\"xG,&F'\"\"\"-%%tanhG6#F'!\"\"" }{TEXT -1 2 ": " } {TEXT 0 1 "P" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5159 "P := proc(x::algebraic)\n \+ local t,ok,s,terms,i,ti,eq;\n description \"inverse of x -> x-tanh (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,'float') then evalf('P'(x))\n elif type(x,`*`) and t ype(op(1,x),'numeric') and \n signum(0,op(1,x) ,0)=-1 then -P(-x) \n elif type(x,'realcons') and signum(0,x,0)=-1 t hen -P(-x)\n elif type(x,And(complexcons,Not(realcons))) then\n \+ error \"not implemented for complex argument\"\n elif type(x,`+`) t hen\n ok := false;\n if has(x,'tanh') then\n s := se lect(has,x,'tanh');\n if type(s,`+`) then terms := [op(s)] els e 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)='tanh' \+ then\n t := op(1,ti);\n eq := t-tanh(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 'P'(x ) end if;\n elif has(x,'arctanh') then\n s := select(has, x,'arctanh');\n if type(s,`+`) then terms := [op(s)] else term s := [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)='arctanh' the n\n t := op(1,ti);\n eq := arctanh(t)-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') a nd (signum(0,t-1,1)=1 or\n signum(0,t+1,-1)=-1) the n ok := false end if;\n if ok then arctanh(t) else 'P'(x) end \+ if;\n else 'P'(x) end if;\n else 'P'(x) end if;\nend proc:\n\n# construct remember table\nP(0) := 0:\nP('infinity') := 'infinity':\n \n# differentiation\n`diff/P` := proc(a,x) \n option `Copyright (c) \+ 2003 Peter Stone.`; \n diff(a,x)/(x-P(a))^2\nend proc:\n\n`D/P` := p roc(t)\n option `Copyright (c) 2003 Peter Stone.`;\n if 1a)-'P')^2\n end if\nend proc:\n\n#integration\n`int/P` := proc( f)\n local gx,h,inds,u;\n option `Copyright (c) 2003 Peter Stone.` ;\n inds := map(proc(x) if op(0,x) ='P' then x end if end proc,indet s(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,i nds) end if;\n if not type(gx,linear(_X)) then return FAIL end if;\n h := subs(inds=u,_X=(u-tanh(u)-coeff(gx,_X,0))/coeff(gx,_X),f);\n \+ h := h*tanh(u)^2/coeff(gx,_X);\n h := int(h,u);\n if has(h,int) t hen return FAIL end if;\n normal(subs(exp(2*u)=(u-gx+1)/(1-u+gx),u=i nds,h))\nend proc:\n\n# simplification\n`simplify/P` := proc(s)\n op tion remember,system,`Copyright (c) 2003 Peter Stone.`;\n if not has (s,'P') or type(s,'name') then return s\n elif type(s,'function') an d op(0,s)='tanh' then\n if type(op(1,s),'function') and op([1,0], s)='P' 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# num erical evaluation\n`evalf/P` := proc(x)\n local xx,eps,saveDigits,do P,val,p,q,maxit;\n option `Copyright (c) 2003 Peter Stone.`;\n\n i f not type(x,realcons) then return 'P'(x) end if;\n\n doP := proc(x, eps,maxit)\n local tnh,t,s,u,v,h,i; \n # set up a starting a pproximation\n if x<.0000125136 and x>-.0000125136 then \n \+ s := 1.732050808*surd(x,3);\n elif x>0 then\n if x<1.85 9 then\n s := (.3689220882e-1+(328.8701953+(30521.41568+\n \+ (65550.07341+(-19279.55598+4848.752768*x)*x)*x)*x)*x)/ \n (1.+(1590.198684+40095.28096*x)*x)\n else \n \+ s := x+1\n end if;\n else\n if x>-1.85 9 then \n s := (-.3689220882e-1+(328.8701953+(-30521.41568+ \n (65550.07341+(19279.55598+4848.752768*x)*x)*x)*x)*x) /\n (1.+(-1590.198684+40095.28096*x)*x)\n else \n s := x-1;\n end if;\n end if;\n # solv e the equation x=s-tanh(s) for s by Halley's method \n for i to m axit do\n tnh := tanh(s);\n t := s-tnh-x;\n u \+ := tnh^2;\n v := 2*tnh*(1-u);\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+.11111111 11*q))*2.095903274)+2;\n saveDigits := Digits;\n Digits := Digits+ min(iquo(Digits,3),5);\n xx := evalf(x);\n Digits := Digits+max(0, -ilog10(xx)-3);\n eps := Float(3,-saveDigits-1);\n if Digits<=trun c(evalhf(Digits)) and \n ilog10(xx)trunc(evalhf(DBL_MIN_10_EXP)) then\n \+ val := evalhf(doP(xx,eps,maxit))\n else\n val := doP(xx,eps,m axit)\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 "Compariso n of starting approximation with the inverse function. 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" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!1'4\"*ycM7*\\!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "xx := s qrt(3)/3-tanh(sqrt(3)/3);\nP(xx);\nevalf(%,12);\nx1 := evalf[12](xx); \nevalf[12](P(x1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,&*&\"\"$ !\"\"F'#\"\"\"\"\"#F*-%%tanhG6#,$*&F'F(F'F)F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"$!\"\"F%#\"\"\"\"\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-*=p-Nx&!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1 G$\",taQ8m&!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-(=p-Nx&!#7" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "xx := sqrt(3)/3-arctanh(sqrt(3)/3);\nP(xx);\nevalf(%,12);\nx1 := e valf[12](xx);\nevalf[12](P(x1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #xxG,&*&\"\"$!\"\"F'#\"\"\"\"\"#F*-%(arctanhG6#,$*&F'F(F'F)F*F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$-%(arctanhG6#,$*&\"\"$!\"\"F)#\"\"\" \"\"#F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!-i%[*y%e'!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$!,t#z'G6)!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!-j%[*y%e'!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "plot([x-tanh(x),P (x),x],x=-3..3,y=-3..3,color=[red,blue,black],\n linestyle =[1$2,2],thickness=2,scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 486 486 486 {PLOTDATA 2 "6)-%'CURVESG6%7S7$$!\"$\"\"!$!3apKJ Y_%\\+#!#<7$$!3!******\\2<#pGF-$!3]x2P>gjv=F-7$$!3#)***\\7bBav#F-$!3WZ w/fjZjW#\\6k:\"F-7$$!3*****\\7;)=,?F-$!3p DIoKn2P5F-7$$!3/++DO\"3V(=F-$!3&fJ)G[/D.#*!#=7$$!3#******\\V'zVXbE@XHqFen7$$!3!******\\!)H%*\\ \"F-$!3pUXsDk%Q%fFen7$$!3/+++vl[p8F-$!3*)[Q0SW64\\Fen7$$!3\"******\\>i UC\"F-$!3KSSMEP&f(RFen7$$!3-++DhkaI6F-$!3;Q:$)oDS$>$Fen7$$!3s******\\X F`**Fen$!3_(R=zTEqN#Fen7$$!3u*******>#z2))Fen$!3CAGTsupR=(!#>7$$!3q)***\\7 *3=+&Fen$!3?MJ\"pV[@z$Fhq7$$!3[)***\\PFcpPFen$!3k6NS-B]*o\"Fhq7$$!3;)* ***\\7VQ[#Fen$!3%f,y#[--&)\\!#?7$$!32)***\\i6:.8Fen$!30m[#of]pK(!#@7$$ !3Wb+++v`hHFhr$!3oRMn\\iAe')!#E7$$\"3]****\\(QIKH\"Fen$\"3Ygmz!z:;;(F^ s7$$\"38****\\7:xWCFen$\"3UbE%4iFqv%Fhr7$$\"3E,++vuY)o$Fen$\"3)yW9H[=k e\"Fhq7$$\"3!z******4FL(\\Fen$\"3#pSAC2#eJPFhq7$$\"3A)****\\d6.B'Fen$ \"3))=7O$y='zpFhq7$$\"3s****\\(o3lW(Fen$\"32nL51)Rq7\"Fen7$$\"35***** \\A))oz)Fen$\"3'yJ$3vUDM$Fen7$$\"3u***\\(=_(zC\"F-$\"32l]lOLg-SFen7 $$\"3M+++b*=jP\"F-$\"3W!z]QUY>'\\Fen7$$\"3g***\\(3/3(\\\"F-$\"3W;[b59h CfFen7$$\"33++vB4JB;F-$\"3sotdl&y?)pFen7$$\"3u*****\\KCnu\"F-$\"3e2Op( \\Fs0)Fen7$$\"3s***\\(=n#f(=F-$\"3(y]c%\\J)z@*Fen7$$\"3P+++!)RO+?F-$\" 3>Z\"eMp5j.\"F-7$$\"30++]_!>w7#F-$\"3GCzD99gb6F-7$$\"3O++v)Q?QD#F-$\"3 m:H1S)GcF\"F-7$$\"3G+++5jypBF-$\"3utAqR&>rQ\"F-7$$\"3<++]Ujp-DF-$\"3$[FF-$\"3qF$*) f<([c(y'GF_\\ l7$$!+NkzViUC\"F_\\l$!+ $*4*4A#F_\\l7$$!+hkaI6F_\\l$!+a+1,@F_\\l7$$!*buK&**F_\\l$!+?,7c>F_\\l7 $$!*?#z2))F_\\l$!+()3nI=F_\\l7$$!*\"RKvuF_\\l$!+hEX!o\"F_\\l7$$!*qjeH' F_\\l$!+&o]?a\"F_\\l7$$!*\"*3=+&F_\\l$!+'Ge9Q\"F_\\l7$$!*ui&pPF_\\l$!+ #4%=:7F_\\l7$$!*7VQ[#F_\\l$!+'=Wq,\"F_\\l7$$!+Sr\\$*=!#5$!+VBE4\"*Facl 7$$!*;^JI\"F_\\l$!+)*Q\\qyFacl7$$!++F$Qm'!#6$!+)emd7'Facl7$$!(Q:'HF_\\ l$!+,z1$3#Facl7$$\"+D1'4,$F\\dl$\"+guX5YFacl7$$\"+]]2=jF\\dl$\"+'*p(z+ 'Facl7$$\"+v%*=D'*F\\dl$\"+*3Np,(Facl7$$\"*RIKH\"F_\\l$\"+geWZyFacl7$$ \"+]4+p=Facl$\"+.q1i!*Facl7$$\"*^rZW#F_\\l$\"+wxS55F_\\l7$$\"*[n%)o$F_ \\l$\"+cWf.7F_\\l7$$\"*5FL(\\F_\\l$\"++wyx8F_\\l7$$\"*e6.B'F_\\l$\"+jf ;M:F_\\l7$$\"*p3lW(F_\\l$\"+M.9x;F_\\l7$$\"*A))oz)F_\\l$\"+kAYH=F_\\l7 $$\"+Ik-,5F_\\l$\"+p1Hi>F_\\l7$$\"+D-eI6F_\\l$\"++f4,@F_\\l7$$\"+>_(zC \"F_\\l$\"+VB)[A#F_\\l7$$\"+b*=jP\"F_\\l$\"+KpfeBF_\\l7$$\"+4/3(\\\"F_ \\l$\"+:;C$[#F_\\l7$$\"+C4JB;F_\\l$\"+O,h7EF_\\l7$$\"+DVsYn#f(=F_\\l$\"+U:^pGF_\\l7$$\"+!)RO+?F_\\l$\"+rGP&*HF_\\ l7$$\"+_!>w7#F_\\l$\"+-gvBJF_\\l7$$\"+*Q?QD#F_\\l$\"+bH#3D$F_\\l7$$\"+ 5jypBF_\\l$\"+i:TnLF_\\l7$$\"+Ujp-DF_\\l$\"+Ju(3]$F_\\l7$$\"+gEd@EF_\\ l$\"+k%R,i$F_\\l7$$\"+4'>$[FF_\\l$\"+ky?ZPF_\\l7$$\"+6EjpGF_\\l$\"+*og (oQF_\\l7$Fiz$\"+*RG$**RF_\\l-F^[l6&F`[lFd[lFd[lFa[lFe[l-F$6%7S7$F(F(7 $F/F/7$F4F47$F9F97$F>F>7$FCFC7$FHFH7$FMFM7$FRFR7$FWFW7$FgnFgn7$F\\oF\\ o7$FaoFao7$FfoFfo7$F[pF[p7$F`pF`p7$FepFep7$FjpFjp7$F_qF_q7$FdqFdq7$Fjq Fjq7$F_rF_r7$FdrFdr7$FjrFjr7$F`sF`s7$FfsFfs7$F[tF[t7$F`tF`t7$FetFet7$F jtFjt7$F_uF_u7$FduFdu7$FiuFiu7$F^vF^v7$FcvFcv7$FhvFhv7$F]wF]w7$FbwFbw7 $FgwFgw7$F\\xF\\x7$FaxFax7$FfxFfx7$F[yF[y7$F`yF`y7$FeyFey7$FjyFjy7$F_z F_z7$FdzFdz7$FizFiz-F^[l6&F`[lF*F*F*-Ff[l6#\"\"#-%(SCALINGG6#%,CONSTRA INEDG-%*THICKNESSGFg`m-%+AXESLABELSG6$Q\"x6\"Q\"yFcam-%%VIEWG6$;F(FizF ham" 1 2 0 1 10 2 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1 " "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 83 "As an illustration of the secant method for root-fin ding we can solve the equation " }{XPPEDIT 18 0 "P(x) = 2;" "6#/-%\"PG 6#%\"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "The solu tion is (of course) " }{XPPEDIT 18 0 "x = 2-tanh(2);" "6#/%\"xG,&\"\"# \"\"\"-%%tanhG6#F&!\"\"" }{TEXT -1 1 " " }{TEXT 278 1 "~" }{TEXT -1 14 " 1.035972420. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 17 "evalf(2-tanh(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?C(f.\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "secant(P(x)=2,x=1..1.1,info= true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\" .)GR0:O5!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~ ~G$\".h=crf.\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3 ~~->~~~G$\".O*>C(f.\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxim ation~4~~->~~~G$\".C*>C(f.\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7a pproximation~5~~->~~~G$\".C*>C(f.\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?C(f.\"!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The co de above contains procedures which enable " }{TEXT 0 4 "diff" }{TEXT -1 5 " and " }{TEXT 0 1 "D" }{TEXT -1 18 " to differentiate " } {XPPEDIT 18 0 "P(x);" "6#-%\"PG6#%\"xG" }{TEXT -1 8 " and P. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "diff(P(x),x);\nD(P)(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\" \"F$*$),&%\"xGF$-%\"PG6#F(!\"\"\"\"#F$F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$*$),&%\"xGF$-%\"PG6#F(!\"\"\"\"#F$F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "This means that we can make use of Newton's and Halley's method for root-finding. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ne wton(P(x)=2,x=1,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7appro ximation~1~~->~~~G$\".!zFZ'e.\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %7approximation~2~~->~~~G$\".4!>C(f.\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\".C*>C(f.\"!#7" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\".C*>C(f.\"!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?C(f.\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "halley(P(x)= 2,x=1,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1 ~~->~~~G$\".%)zMvf.\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxim ation~2~~->~~~G$\".C*>C(f.\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7a pproximation~3~~->~~~G$\".C*>C(f.\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?C(f.\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "For another root-finding question consider the probl em of finding the solution of the equation " }{XPPEDIT 18 0 "P(x) = 3* exp(-x);" "6#/-%\"PG6#%\"xG*&\"\"$\"\"\"-%$expG6#,$F'!\"\"F*" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "g := x -> x-tanh(x); \nevalf(secant(g(x)=-ln(x/3),x=1..2),13);\nevalf(evalf(g(%),13));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&9$\"\"\"-%%tanhG6#F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\".Zc@@jc\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Of#))\\'!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }