{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 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 0 1 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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 286 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 287 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 288 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 289 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1 " -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 31 "Special inverse functions .. IV" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 25.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 68 "load interpolation and function approximation procedure s including: " }{TEXT 0 5 "remez" }}{PARA 0 "" 0 "" {TEXT -1 17 "The M aple m-file " }{TEXT 262 10 "fcnapprx.m" }{TEXT -1 37 " contains the c ode for the procedure " }{TEXT 0 5 "remez" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Ma ple session by a command similar to the one that follows, where the fi le path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procdrs/fcnapprx.m\";" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "load inverse functions" }} {PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 9 "invfcns. m" }{TEXT -1 52 " contains the code for the special inverse function. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple session \+ by a command similar to the one that follows, where the file path give s its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "read \" K:\\\\Maple/procdrs/invfcns.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "load root-finding procedures" }} {PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 7 "roots.m " }{TEXT -1 38 " contains the code for the procedures " }{TEXT 0 6 "se cant" }{TEXT -1 2 ", " }{TEXT 0 6 "newton" }{TEXT -1 5 " and " }{TEXT 0 6 "halley" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 " " {TEXT -1 121 "It can be read into a Maple session by a command simil ar to the one that follows, where the file path gives its location." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/ roots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Inverse for " }{XPPEDIT 18 0 "g(x) = exp(x)-1-x-x^2/2;" "6#/-% \"gG6#%\"xG,*-%$expG6#F'\"\"\"F,!\"\"F'F-*&F'\"\"#F/F-F-" }{TEXT -1 2 ": " }{TEXT 0 1 "V" }{TEXT -1 3 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "The function " } {XPPEDIT 18 0 "g(x) = exp(x)-1-x-x^2/2;" "6#/-%\"gG6#%\"xG,*-%$expG6#F '\"\"\"F,!\"\"F'F-*&F'\"\"#F/F-F-" }{TEXT -1 47 " is one-to-one, and s o 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 0 238 "g := x -> exp(x)-1-x-x^2/2;\np1 := plot([g(x) ,x],x=-4..8,y=-4..8,color=[red,black],\n linestyle=[1,3]): \np2 := plots[implicitplot](x=exp(y)-1-y-y^2/2,x=-4..8,\n y =-4..8,grid=[50,50],color=blue):\nplots[display]([p1,p2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,*- %$expG6#9$\"\"\"F1!\"\"F0F2*&#F1\"\"#F1*$)F0F5F1F1F2F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6'-%'CURVESG6%7Z7$$!\"% \"\"!$!3yl76hVo\")\\!#<7$$!3z******\\TVQPF-$!3mKIJp%>dA%F-7$$!34++]-r% 3^$F-$!3.Lml')\\IAOF-7$$!3A+++l;!\\D$F-$!3ms\\rWhq.IF-7$$!3o*****\\lfs *HF-$!32*)\\_v$*fWCF-7$$!3%)****\\s@%3u#F-$!3'=eiVu\\2&>F-7$$!3J++]U.6 .DF-$!334z0o-%ya\"F-7$$!3')****\\-G&pD#F-$!3U\\TgFiH&=\"F-7$$!3(***** \\AjP-?F-$!3WU!e=\"3lt')!#=7$$!33++]sih[1'FV7$$!3%)***** *pGf([\"F-$!38=)y:$fdHRFV7$$!3)******\\J$od7F-$!3G9k3,U)))[#FV7$$!3'y* 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$\"3#G^xj\"3V5FF-Faio7$Fgio7$$\"3#QQEX+q-*yF-$\"33+GV$*[(>s#F-7$7$$\"3 y,++++++!)F-$\"3R_@1IkMJFF-F]jo-Fc]l6&Fe]lFi]lFi]lFf]l-%+AXESLABELSG6% Q\"x6\"Q\"yF^[p-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F^]lFg[p" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We tackle the problem o f constructing a numerical inverse for " }{XPPEDIT 18 0 "g(x)" "6#-%\" gG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "The idea is to use Halley's method to solve the equation " }{XPPEDIT 18 0 "exp(y) -1-y-y^2/2 = x;" "6#/,*-%$expG6#%\"yG\"\"\"F)!\"\"F(F**&F(\"\"#F,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 \+ 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 25 "The Maclaurin series for " }{XPPEDIT 18 0 "g(x) = exp(x)-1-x-x^2/2;" "6#/ -%\"gG6#%\"xG,*-%$expG6#F'\"\"\"F,!\"\"F'F-*&F'\"\"#F/F-F-" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^3/3!+x^ 4/4!+x^5/5!+x^6/6!+` . . . `;" "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 268 1 "x" }{TEXT -1 24 " is close to 0 we have: " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "exp(x)-1-x-x^2/2;" "6#,*-%$expG6 #%\"xG\"\"\"F(!\"\"F'F)*&F'\"\"#F+F)F)" }{TEXT -1 1 " " }{TEXT 266 1 " ~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^3/6" "6#*&%\"xG\"\"$\"\"'!\"\"" } {TEXT -1 2 ". " }{TEXT 265 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 267 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 269 1 "x" }{TEXT -1 16 " is close to 0. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "The follo wing picture makes a graphical comparison between the function " } {XPPEDIT 18 0 "psi[1](x)" "6#-&%$psiG6#\"\"\"6#%\"xG" }{TEXT -1 18 " a nd the graph of " }{XPPEDIT 18 0 "x = exp(y)-1-y-y^2/2;" "6#/%\"xG,*-% $expG6#%\"yG\"\"\"F*!\"\"F)F+*&F)\"\"#F-F+F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 151 "p 1 := plots[implicitplot](x=exp(y)-1-y-y^2/2,x=-.2..0.2,y=-1.3..1.1):\n p2 := plot(1.817120593*surd(x,3),x=-.2..0.2,color=blue):\nplots[displa y]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 307 275 275 {PLOTDATA 2 "6& -%'CURVESG6[q7$7$$!35+++++++?!#=$!3My4JB:zf6!#<7$$!32mF*$!3IS4t '*=DY6F-7$7$$!3(*************R=F*$!3T@7U+9\"e7\"F-F.7$F47$$!3gV>/C\\2= =F*$!3\"Q$[dX]:@6F-7$7$$!3Gl5X9E8cZ4\"F-7$7$$!3#)************z;F*$!3UuX&HGv#*3\"F- FF7$FL7$$!3cq#*4=Dl)e\"F*$!3YP/9*[3o1\"F-7$7$$!3o************>:F*$!3;b )=?\\B*\\5F-FR7$FX7$$!3!>6V&e/,\"F-Fhn7$F^o7$$!3U/CTV(*)=O\"F*$!3(pb_R:m3,\"F-7$7$$ !3a************f8F*$!3<-U37#>.,\"F-Fdo7$7$$!3#)************f8F*F]p7$$! 3aXGnE]&RD\"F*$!3MBH'*R)piz*F*7$7$$!3_*************>\"F*$!3eoMQ\"o4,k* F*Fcp7$7$$!3Q*************>\"F*F\\q7$$!3!eGL*4.-Y6F*$!3?z-SS\"yQ[*F*7$ 7$$!3`************R5F*$!3g;\\#>CFq<*F*Fbq7$Fhq7$$!3iEP>$f&3Q5F*$!3?Ow$ 3W'[r\"*F*7$7$$!3X]pV.p6M5F*$!39)***********f\"*F*F^r7$Fdr7$$!3iNmk>\\ +l$*!#>$!3p8?@[q*4#))F*7$7$$!3K&************z)F]s$!3e,VFZ)F*Fgs7$F]t7$$!3C%f#yk$\\mN(F]s$!3fR/8\"Q5g5)F*7$7$$!3I&******** ****>(F]s$!3%z!e\"z0U\"R!)F*Fct7$Fit7$$!3iy(G*[hw@kF]s$!36GFk5.%pq(F*7 $7$$!3E&************f&F]s$!3sOhl&4chN(F*F_u7$7$Ffu$!3gNhl&4chN(F*7$$!3 Ag\\2LH)o[&F]s$!3'*>]:S-(yI(F*7$7$$!39hcEBk)yK&F]s$!3m)***********RsF* F^v7$Fdv7$$!3&\\hmHR6'HYF]s$!3EE+AkJBioF*7$7$$!3C&*************RF]s$!3 8Io>./fF*7$7$$!3a&************R#F]s$!37pkd+K\"oX&F*Fex7$7$$!3?&**** ********R#F]sF^y7$$!3s^nH%>deH#F]s$!3)e%>U$o&[#Q&F*7$7$$!3_`wa$Q1$3AF] s$!3<************>`F*Fdy7$Fjy7$$!3)*4.d\"=!Qy;F]s$!335y0\"*=(Hz%F*7$7$ $!3*y-ckSFIC\"F]s$!3)))***********fVF*F`z7$7$$!3sFgX1u-V7F]s$!3K)***** ******fVF*7$$!3%*4mA;(Rk6\"F]s$!3%*HSEqh8qTF*7$7$$!3g^************z!#? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "x>100" "6#2\"$+\"%\"xG" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "psi4(x)=ln(x)" "6#/-%%psi4G6#%\"xG-%#l nG6#F'" }{TEXT -1 39 " is a suitable starting approximation. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "For " } {XPPEDIT 18 0 "-9.377<=x" "6#1,$-%&FloatG6$\"%x$*!\"$!\"\"%\"xG" } {XPPEDIT 18 0 "``<-1/5" "6#2%!G,$*&\"\"\"F'\"\"&!\"\"F)" }{TEXT -1 14 " the function " }{XPPEDIT 18 0 "psi5(x)=3.03*(x-.33)^(1/3)+1.25" "6#/ -%%psi5G6#%\"xG,&*&-%&FloatG6$\"$.$!\"#\"\"\"),&F'F/-F+6$\"#LF.!\"\"*& F/F/\"\"$F5F/F/-F+6$\"$D\"F.F/" }{TEXT -1 45 " provides a suitable sta rting approximation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "When " }{XPPEDIT 18 0 "-9.377 #)\\B!#7F**$F0\"\"$F*F*F*,*-F&6$F*\"\"!F**&-F&6$\"+r)G4C#F/F*F0F*F**&- F&6$\"+i-F+9F)F**$F0F6F*F**&-F&6$\"+@<_xG!#8F**$F0F=F*F*!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-4<=x" "6#1,$\"\"%!\"\"%\"xG" }{XPPEDIT 18 0 "``<= -0" "6#1%!G,$\"\"!!\"\"" }{TEXT -1 9 ".02 --- " }{XPPEDIT 18 0 "(-.37 37507846+11.33661192*x-9.066891785*x^2)/(1.-7.556746459*x+1.319797681* x^2)" "6#*&,(-%&FloatG6$\"+Yy]PP!#5!\"\"*&-F&6$\"+#>hO8\"!\")\"\"\"%\" xGF0F0*&-F&6$\"+&y\"*o1*!\"*F0*$F1\"\"#F0F*F0,(-F&6$F0\"\"!F0*&-F&6$\" +fkucvF6F0F1F0F**&-F&6$\"+\"o(z>8F6F0*$F1F8F0F0F*" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Halley's formula for so lving " }{XPPEDIT 18 0 "x = exp(y)-1-y-y^2/2;" "6#/%\"xG,*-%$expG6#%\" yG\"\"\"F*!\"\"F)F+*&F)\"\"#F-F+F+" }{TEXT -1 5 " for " }{TEXT 270 1 " y" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi(y) = exp(y)-1-y-y^2/2-x;" "6#/-% $phiG6#%\"yG,,-%$expG6#F'\"\"\"F,!\"\"F'F-*&F'\"\"#F/F-F-%\"xGF-" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Given an approximate ze ro " }{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 \"correction\" " }{TEXT 271 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*&-%$phiG6#% \"yG\"\"\"-%!G6#,&*&F'F*-%\"'G6#F)F*F***F'F*-%#''G6#F)F*-F'6#F)F**(\" \"#F*F'F*-F16#F)F*!\"\"F=F=" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi*`'`(y) = \+ exp(y)-1-y;" "6#/*&%$phiG\"\"\"-%\"'G6#%\"yGF&,(-%$expG6#F*F&F&!\"\"F* F/" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi*`\"`(y) = exp(y)-1;" "6#/* &%$phiG\"\"\"-%\"\"G6#%\"yGF&,&-%$expG6#F*F&F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "exp(y)-1-y-y^2/2-x;\nDiff(%,y)=diff(%,y);\nDiff(%%,y$2)=diff(%%, y$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,-%$expG6#%\"yG\"\"\"F(!\"\" F'F)*&\"\"#F)F'F+F)%\"xGF)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%Diff G6$,,-%$expG6#%\"yG\"\"\"F,!\"\"F+F-*&\"\"#F-F+F/F-%\"xGF-F+,(F(F,F,F- F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,,-%$expG6#%\"yG\" \"\"F,!\"\"F+F-*&\"\"#F-F+F/F-%\"xGF--%\"$G6$F+F/,&F(F,F,F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "The following \+ code sets up the starting approximations from the previous subsection \+ via the 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) = exp(y)-1-y-y^2/2-x;" "6#/-%$phiG6#%\"yG,,-%$expG6#F'\"\"\" F,!\"\"F'F-*&F'\"\"#F/F-F-%\"xGF-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 762 "start_appro x := proc(x)\n local y;\n if x<.04 and x>-.017 then \n y := 1 .817120593*surd(x,3);\n else\n if x>0 then\n if x<1200 \+ then \n y := (.5027799463+(4.262325156+(.6304891014+.234982 1998e-2*x)*x)*x)/\n (1.+(2.240928871+(.1400270262+.287752 1721e-3*x)*x)*x); \n else\n y := ln(x)\n \+ end if;\n else\n if x>-4 then\n y := (-. 3737507846+(11.33661192-9.066891785*x)*x)/\n (1.+(-7.5567 46459+1.319797681*x)*x)\n else\n y := -1-sqrt(-1-2* x)\n end if;\n end if;\n end if;\n y;\nend proc:\n\nn ext_halley_approx := proc(x,y)\n local t,u,v,h;\n v := exp(y)-1;\n t := v-y*(1+y/2)-x;\n u := v-y;\n h := t/(u-1/2*v*t/u);\n y-h ;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Test example: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "xx := -evalf(sqrt(14)/100);\ny0 := start_approx(xx);\ny1 := next_halley_approx(xx,y0);\ny2 := next_halle y_approx(xx,y1);\ny3 := next_halley_approx(xx,y2);\neval(exp(y)-1-y-y^ 2/2,y=y3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$!+(Qd;u$!#6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G$!+qoK5j!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$!+Fra%R'!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#y2G$!+6hb%R'!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G$!+4hb%R'!# 5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!*Rd;u$!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "code for inverse of " }{XPPEDIT 18 0 "g (x) = exp(x)-1-x-x^2/2;" "6#/-%\"gG6#%\"xG,*-%$expG6#F'\"\"\"F,!\"\"F' F-*&F'\"\"#F/F-F-" }{TEXT -1 2 ": " }{TEXT 0 1 "V" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4580 "V := proc(x::algebraic)\n local st,ok,t,expterms,i,ti;\n des cription \"inverse of x -> exp(x)-1-x-x^2/2\";\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('V'(x))\n elif type(x,And(complexcons,Not(realcons))) then\n \+ error \"not implemented for complex argument\"\n elif type(x,`+` ) and has(x,'exp') then\n st := select(has,x,'exp');\n ok := false;\n if type(st,'function') and op(0,st)='exp' then \n \+ t := op(1,st);\n if testeq(x-st+1+t+1/2*t^2=0) then ok := t rue end if;\n elif op(0,st)=`+` then\n expterms := [op(st )];\n for i to nops(expterms) do\n ti := expterms[i ];\n if type(ti,`*`) and op(1,ti)=-1 then ti := -ti end if; \n if type(ti,'function') and op(0,ti)='exp' then\n \+ t := op(1,ti);\n if testeq(x-ti+1+t+1/2*t^2=0) th en\n ok := true;\n break;\n \+ end if;\n end if;\n end do;\n end if;\n \+ if ok then t else 'V'(x) end if;\n else 'V'(x)\n end if;\nend proc:\n\n# construct remember table\nV(0) := 0:\nV('infinity') := 'in finity':\nV(-'infinity') := -'infinity':\n\n# differentiation\n`diff/V ` := proc(a,x) \n option `Copyright (c) 2003 Peter Stone.`; \n dif f(a,x)*2/(2*a+V(a)^2)\nend proc:\n\n`D/V` := proc(t)\n option `Copyr ight (c) 2003 Peter Stone.`;\n if 1a)+'V'^2)\n end if\nend proc:\n\n#integration\n`int/V` := proc(f)\n local gx,h,inds ,u;\n option `Copyright (c) 2003 Peter Stone.`;\n inds := map(proc (x) if op(0,x) ='V' then x end if end proc,indets(f,function));\n if nops(inds)<>1 then return FAIL end if;\n inds := inds[1];\n if no ps(inds)=1 then gx := op(inds) else gx := op(2,inds) end if;\n if no t type(gx,linear(_X)) then return FAIL end if;\n h := subs(inds=u,_X =(exp(u)-1-u-u^2/2-coeff(gx,_X,0))/coeff(gx,_X),f);\n h := h*(exp(u) -1-u)/coeff(gx,_X);\n h := int(h,u);\n if has(h,int) then return F AIL end if;\n subs(exp(u)=gx+1+u+u^2/2,u=inds,h)\nend proc:\n\n# sim plification\n`simplify/V` := proc(s)\n local t,sl,rm;\n option rem ember,system,`Copyright (c) 2003 Peter Stone.`;\n if not has(s,'V') \+ or type(s,'name') then return s\n elif type(s,'function') and op(0,s )='exp' then\n if type(op(1,s),'function') and op([1,0],s)='V' th en\n t := op(1,s);\n return op([1,1],s)+1+t+1/2*t^2;\n else return s\n end if;\n elif type(s,'function') and op( 0,s)='ln' and type(op(1,s),`+`) then\n sl,rm := selectremove(has, op(1,s),'V');\n t := rm-1;\n if testeq(V(t)+1/2*V(t)^2=sl) t hen\n return V(t)\n else return s\n end if;\n else \n map(procname,args)\n end if;\nend proc:\n\n# numerical evalu ation\n`evalf/V` := proc(x)\n local xx,eps,saveDigits,doV,val,p,q,ma xit;\n option `Copyright (c) 2003 Peter Stone.`;\n\n if not type(x ,realcons) then return 'V'(x) end if;\n \n doV := proc(x,eps,maxit) \n local s,t,u,v,h,i; \n # set up a starting approximation\n if x<.04 and x>-.017 then \n s := 1.817120593*surd(x,3); \n else\n if x>0 then\n if x<1200 then \n \+ s := (.5027799463+(4.262325156+(.6304891014+.2349821998e-2*x )*x)*x)/\n (1.+(2.240928871+(.1400270262+.2877521721e-3* x)*x)*x); \n else\n s := ln(x)\n \+ end if;\n else\n if x>-4 then\n \+ s := (-.3737507846+(11.33661192-9.066891785*x)*x)/\n \+ (1.+(-7.556746459+1.319797681*x)*x)\n else\n \+ s := -1-sqrt(-1-2*x)\n end if;\n end if;\n en d if;\n # solve the equation exp(y)-1-y-y^2/2=x for y by Halley's method \n for i to maxit do\n v := exp(s)-1;\n t := v-s*(1+s/2)-x;\n u := v-s;\n h := t/(u-1/2*v*t/u); \n s := s-h;\n if abs(h)<=eps*abs(s) then break end if ;\n end do;\n s;\n end proc;\n\n p := ilog10(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(doV(xx,eps,maxit))\n else\n val := doV(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 63 "Compariso n of starting approximation with the inverse function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot('V (x)-start_approx(x)',x=-10..10,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 663 156 156 {PLOTDATA 2 "6&-%'CURVESG6#7\\s7$$!#5\"\"!$!(_$y 5!\"*7$$!+s%HaF-$!(9@/&F-7$$!+]$*4)*\\F-$!(ws1'F -7$$!+]_&\\c%F-$!(J!*Q(F-7$$!+]zCcVF-$!(?E;)F-7$$!+]1aZTF-$!(Vv/*F-7$$ !+&4f,5%F-$!(\"3m#*F-7$$!+Rvx_SF-$!(O<\\*F-7$$!+in3HSF-$!(Ftg*F-7$$!+% )fR0SF-$!(5[s*F-7$$!+&f]N*RF-$!)CXM5F-7$$!+1_q\")RF-$!(S!o**F-7$$!+<)f )pRF-$!(gZf*F-7$$!+FW,eRF-$!(*pC#*F-7$$!+;8DjQF-$!(c3Q'F-7$$!+/#)[oPF- $!(9qu$F-7$$!+%>BJa$F-$\"(.El\"F-7$$!+$=exJ$F-$\"(4:!eF-7$$!+eW%o7$F-$ \"()e7$)F-7$$!+L2$f$HF-$\"(H,*)*F-7$$!+?7T!)GF-$\")9K<5F-7$$!+3<*[#GF- $\"))Qx.\"F-7$$!+'>s$pFF-$\")eH]5F-7$$!+%o_Qr#F-$\")%Q]0\"F-7$$!+sJLeE F-$\")c-_5F-7$$!+gO\"Gg#F-$\")OLT5F-7$$!+[THZDF-$\")\\0B5F-7$$!+PYx\" \\#F-$\"(DI(**F-7$$!+Mz>&H#F-$\"(ZD[)F-7$$!+L7i)4#F-$\"(=2;'F-7$$!+Ma% H)=F-$\"(+G#GF-7$$!+P'psm\"F-$!(ZL1\"F-7$$!+u_*=Y\"F-$!(z`([F-7$$!+74_ c7F-$!(#=G#)F-7$$!+6wP\\6F-$!(5q`*F-7$$!+5VBU5F-$!)U!e.\"F-7$$!+&fEm)) *F)$!)oC`5F-7$$!+!454N*F)$!)C!=0\"F-7$$!+&e$>:))F)$!)H%*G5F-7$$!*3x%z# )F-$!(I0#)*F-7$$!+5rc&H(F)$!(xlC)F-7$$!+Srl6jF)$!(*RHcF-7$$!+qruF`F)$! 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-0.991234567891096;\nevalf(g(xx),18);\nevalf(V(%),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arr owGF(,*-%$expG6#9$\"\"\"F1!\"\"F0F2*&#F1\"\"#F1*$)F0F5F1F1F2F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$!0'4\"*ycM7**!#:" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$!3^D_v*y,#*G\"!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0'4\"*ycM7**!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "g := x -> exp(x)-1-x-x^2/2;\nDigit s := 55:\nxx := evalf(sqrt(14)*10^(-8));\nevalf(g(xx),Digits+24);\neva lf(V(%));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6# %\"xG6\"6$%)operatorG%&arrowGF(,*-%$expG6#9$\"\"\"F1!\"\"F0F2*&#F1\"\" #F1*$)F0F5F1F1F2F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"XYp syx!)>gv,$\\lJK([Pe&QTRx'Qd;u$!#i" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"bovz@cB6(\\@6L\"zz)zJL^;1NfftX*\\xr>RT)R`I()!#%*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"XYpsyx!)>gv,$\\lJK([Pe&QTRx'Qd;u$!#i" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "xx := exp(sqrt(3))-5/2-sqrt(3);\nV(xx);\nevalf[12](%);\nx1 := evalf[14](xx) ;\nevalf[12](V(x1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,(-%$exp G6#*$\"\"$#\"\"\"\"\"#F,#\"\"&F-!\"\"F)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$\"\"$#\"\"\"\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"-d230K%#x1G$\"/`YmG=?9!#8 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-d230K " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "p lot([exp(x)-1-x-x^2/2,V(x),x],x=-5..5,y=-5..5,color=[red,blue,black], \n linestyle=[1$2,2],thickness=2,scaling=constrained);" }} {PARA 13 "" 1 "" {GLPLOT2D 466 402 402 {PLOTDATA 2 "6)-%'CURVESG6%7Y7$ $!\"&\"\"!$!3I:4+`?E$\\)!#<7$$!3YLLLe%G?y%F-$!3L*eEXk!\\VwF-7$$!3OmmT& esBf%F-$!3X,N#yfTC%pF-7$$!3ALL$3s%3zVF-$!3_%p:s%*pl>'F-7$$!3_LL$e/$QkT F-$!3MP,SJ27\"\\&F-7$$!3ommT5=q]RF-$!38gAj$RzS$[F-7$$!3ILL3_>f_PF-$!3+ \"*3UfV#\\E%F-7$$!3K++vo1YZNF-$!3nW^pb-)fr$F-7$$!3;LL3-OJNLF-$!3')zO)3 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" }}{PARA 0 "" 0 "" {TEXT -1 28 "The solution is (of course) " }{XPPEDIT 18 0 "x = exp(2)- 5;" "6#/%\"xG,&-%$expG6#\"\"#\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }{TEXT 264 1 "~" }{TEXT -1 14 " 2.389056099. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf(exp(2)-5);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*4c!*Q#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "secant(V(x)= 2,x=2..2.5,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximat ion~1~~->~~~G$\".\"*y=*Q'R#!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7ap proximation~2~~->~~~G$\".GMSA*)Q#!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%7approximation~3~~->~~~G$\".>9Ec!*Q#!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\".N*)4c!*Q#!#7" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$\".J*)4c!*Q#!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*4c!*Q#!\"*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 48 "The code above contains procedures which enable " }{TEXT 0 4 "diff" }{TEXT -1 5 " and " }{TEXT 0 1 "D" }{TEXT -1 18 " to differentiate " }{XPPEDIT 18 0 "V(x);" "6#-%\"VG6#%\"xG" }{TEXT -1 8 " and V. 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'eFFhpFacm7$$\"3]=nOs+P)R%Fhp$\"3Qqu[W&fqw\"F-7$7$$\"3]I?5Oot'=&Fhp$\" 3xA'eF[M5$>F-Fgcm7$F]dm7$$\"3#HNbc(*R!GjFhp$\"3`T'*3#p;e7#F-7$7$$\"3MR b'*o?'eF)Fhp$\"3!)[vOmYCeCF-Fcdm7$7$FhamF\\em7$$\"3of%)Fhp$\"3[d'*o?'eF[#F-F`em7$Ffem7$$\"3%\\$ )yG%3Gb5F-$\"3/6%QE7)y1GF-7$7$$\"3!=y!*3$z=07F-$\"3<#p?'eF[MIF-F\\fm7$ 7$$\"3-#y!*3$z=07F-Fefm7$$\"33$3-WM,vF\"F-$\"3R(>Y!fmrizKF-F[gm7$Fagm7$$\"3IM,7nNj1:F-$\"33\"=fUn$))eMF-7$7$$\"35Pfi^$ pqf\"F-$\"3'os^l*o?'e$F-Fegm7$7$$\"3)o$fi^$pqf\"F-$\"3UEF-$\"3yR2'Qh6E4%F-7$7$$\"3qleZBqH5?F-$\"3chF[M5$z8%F-F`im7$F fim7$$\"3)R>'fL#\\r@#F-$\"3%fAw:USNS%F-7$7$$\"39go\"Goq*RCF-$\"3E'z8C< b'*o%F-F\\jm7$Fbjm7$$\"3W!yJ`2>!fCF-$\"3Ht;x]F-7$7$$\"3I\"R1)[!pE) GF-$\"3'4$[M5$z8C&F-Fb[n7$7$$\"3'3R1)[!pE)GF-F[\\n7$$\"35z/$F -$\"3qH>,+jhFcF-7$7$$\"3P(3%zOFY%F-$\"3gOXY-,ZBrF-7$7$Fejm$\"3Sl$HQ,jnQ(F- Fc`n7$Fi`n7$$\"3'3p)>;%H\"=ZF-$\"39vSG=;!)>uF-7$7$$\"3&G>Uk()pEu%F-$\" 3up*o?'eF[uF-F]an7$Fcan7$$\"3?:%zBL6\\(\\F-$\"3^&QM+%Qu9xF-7$7$$\"3X&G w#>x_B_F-$\"3c.++++++!)F-Fian-F][l6&F_[lFb[lFb[lF`[l-%+AXESLABELSG6%Q \"x6\"Q\"yFjbn-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(FhzFccn" 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 62 "We tackle the problem of \+ constructing a numerical inverse for " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG 6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 57 "The idea is t o use Halley's method to solve the equation " }{XPPEDIT 18 0 "y-arcsin h(y) = x;" "6#/,&%\"yG\"\"\"-%(arcsinhG6#F%!\"\"%\"xG" }{TEXT -1 17 " \+ numerically for " }{TEXT 273 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) = x-arcsinh(x);" "6#/-%\"gG6#%\"xG,&F'\"\"\"- %(arcsinhG6#F'!\"\"" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x-1/6;" "6#,&%\"xG\"\"\"*&F%F%\"\"'!\"\"F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "x^3+3/40;" "6#,&*$%\"xG\"\"$\"\"\"*&F&F '\"#S!\"\"F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^5-5/112;" "6#,&*$%\"xG \"\"&\"\"\"*&F&F'\"$7\"!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^7+35 /1152" "6#,&*$%\"xG\"\"(\"\"\"*&\"#NF'\"%_6!\"\"F'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x^9-63/2816;" "6#,&*$%\"xG\"\"*\"\"\"*&\"#jF'\"%;G!\"\" F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^11+` . . . `" "6#,&*$%\"xG\"#6\" \"\"%(~.~.~.~GF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 281 1 "x" }{TEXT -1 24 " is \+ close to 0 we have: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x-sinh(x);" "6#,&%\"xG\"\"\"-%%sinhG6#F$!\"\"" }{TEXT -1 1 " " } {TEXT 279 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^3/6" "6#*&%\"xG\"\"$ \"\"'!\"\"" }{TEXT -1 2 ". " }{TEXT 278 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "Solving " }{XPPEDIT 18 0 "y=x^3/6" "6#/%\"yG*&%\"x G\"\"$\"\"'!\"\"" }{TEXT -1 5 " for " }{TEXT 280 1 "x" }{TEXT -1 7 " g ives " }{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 f or the numerical inverse of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG " }{TEXT -1 7 ", when " }{TEXT 282 1 "x" }{TEXT -1 16 " is close to 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "p1 := plots[implicitplot](y-arcsinh(y)=x,x=0..0.02,y =0..0.52):\np2 := plot([1.817120593*surd(x,3)],x=0..0.02,y=0..0.52,col or=blue):\nplots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 316 293 293 {PLOTDATA 2 "6&-%'CURVESG6_q7$7$$\"\"!F)F(7$$\"33%pX#y7s'\\\"! #B$\"3+hlnC&3h2#!#>7$7$$\"3?2$H_uE&*\\\"F-$\"3#*************z?F0F*7$7$ $\"3sB$H_uE&*\\\"F-F57$$\"3[Pp'Qd/M=\"!#A$\"3[f%z5[J#HTF07$7$$\"3s!zo_ ]@*)>\"F>$\"3#)************fTF0F;7$FB7$$\"3PnEV+%zO!RF>$\"3s]()eNV]QhF 07$7$$\"3.(p'>#GJC/%F>$\"3s************RiF0FH7$7$$\"3o&p'>#GJC/%F>FQ7$ $\"3!>X&R6,s]*)F>$\"3q;P5x7G(3)F07$7$$\"3q@\"*yG;1p&*F>$\"3i********** **>$)F0FW7$Fgn7$$\"3!3$)z$4JQv;!#@$\"3?W7c\"R+W'**F07$7$$\"3)e`0Xv1d'= F`o$\"3\")************R5!#=F]o7$7$$\"39Ob]anql=F`o$\"3&*************R5 Fio7$$\"3wd$)>s\\>_FF`o$\"3uUGsIHWw6Fio7$7$$\"3%)>S(=&46S(=&46(=FioFcs7$Fis7$$\"3c'*[A^p[\\7F\\t$\"3UE:o#RLJ'> 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M(=%F\\t$\"3GrS?9p%)GHFio7$$\"3)QLL$3y_qXF\\t$\"3>Z[xs;f:IFio7$$\"3(4+ ++l+>+&F\\t$\"3YfVzthi2JFio7$$\"3Y+++]Z/NaF\\t$\"3'oQI3Dd[>$Fio7$$\"3I +++]$fC&eF\\t$\"33\"yP'3ujuKFio7$$\"3hLL$ez6:B'F\\t$\"3tyV'31hQM$Fio7$ $\"3#ommm\"=C#o'F\\t$\"3g^)oa88EU$Fio7$$\"39nmmm#pS1(F\\t$\"3ii?#R_*f' [$Fio7$$\"3&*****\\i`A3vF\\t$\"3=F!p\"HJ>eNFio7$$\"3[mmmm(y8!zF\\t$\"3 S)*\\71bC>OFio7$$\"3^++]i.tK$)F\\t$\"3!\\$eHUE%Ro$Fio7$$\"3\\++](3zMu) F\\t$\"3`'=(*fy0Nu$Fio7$$\"3Comm\"H_?<*F\\t$\"3>\\.(*)3(p.QFio7$$\"3)z mm\"zihl&*F\\t$\"3-Y%3CJTt&QFio7$$\"37MLL3#G,***F\\t$\"3+oh8#**yN\"RFi o7$$\"3ULLezw5V5F0$\"3_VmUf>PH? d^@%Fio7$$\"3/++]2'HKH\"F0$\"3i&e&)4WG_E%Fio7$$\"3qmmmwanL8F0$\"3:kTn6 zB4VFio7$$\"3?+++v+'oP\"F0$\"3m(='R!)fDbVFio7$$\"3ULLeR<*fT\"F0$\"3_%G yqJJhR%Fio7$$\"3=+++&)Hxe9F0$\"37-op*4m*RWFio7$$\"3mmm\"H!o-*\\\"F0$\" 3]DQth`V![%Fio7$$\"32++DTO5T:F0$\"3zuQaJ-(>_%Fio7$$\"3kmmmT9C#e\"F0$\" 3)4vs8^`=c%Fio7$$\"3,++D1*3`i\"F0$\"3U1ienO(Gg%Fio7$$\"3WLLL$*zym;F0$ \"3cB*35Y,Emk#[Fio7$$\"3-+ +DOl5;>F0$\"3ID+im.Zi[Fio7$$\"3?++v.Uac>F0$\"35Kgd4&Qk*[Fio7$$\"3/++++ +++?F0$\"3y$e;\"\\TUK\\Fio-Fbcm6&FdcmF(F($\"*++++\"!\")-%+AXESLABELSG6 %%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F($\"\"#!\"#;F($\"#_F_gn" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "T he function " }{XPPEDIT 18 0 "psi2(x) = x+arcsinh(x);" "6#/-%%psi2G6#% \"xG,&F'\"\"\"-%(arcsinhG6#F'F)" }{TEXT -1 49 " provides a suitable st arting approximation when " }{TEXT 283 1 "x" }{TEXT -1 22 " has large \+ magnitude. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 162 "p1 := plots[implicitplot](y-arcsinh(y)=x,x=499.9.. 500,y=506.8..506.9):\np2 := plot([x+arcsinh(x)],x=499.9..500,y=506.8.. 506.9,color=blue):\nplots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 472 308 308 {PLOTDATA 2 "6&-%'CURVESG6[p7$7$$\"3w*********** *)*\\!#:$\"3u\"**Gk18#o]F*7$$\"3!H*o[`M,**\\F*$\"393J^YlAo]F*7$7$$\"3; T!H/)o-**\\F*$\"3#=++++S#o]F*F-7$F37$$\"330ntLM.**\\F*$\"39'HjicY#o]F* 7$7$$\"3'**********R!**\\F*$\"3YMy]XJDo]F*F97$7$F@$\"3-Ny]XJDo]F*7$$\" 3WAm)RT`!**\\F*$\"3azL,'em#o]F*7$7$$\"3cXm],o1**\\F*$\"3Y,++++Go]F*FH7 $FN7$$\"3Q!fOURt!**\\F*$\"335Mw0mGo]F*7$7$$\"39+++++3**\\F*$\"3?]geCKH o]F*FT7$FZ7$$\"31km[uL4**\\F*$\"3uPL^DmIo]F*7$7$$\"3%H([eAn5**\\F*$\"3 k,++++Ko]F*Fjn7$F`o7$$\"3Y*yOZN8\"**\\F*$\"358KEXmKo]F*7$7$$\"3M+++++7 **\\F*$\"35SOm.LLo]F*Ffo7$7$$\"3w*********>\"**\\F*$\"3_ROm.LLo]F*7$$ \"3I=q)\\LL\"**\\F*$\"3W%)H,lmMo]F*7$7$$\"3MBPmVm9**\\F*$\"3E,++++Oo]F *Fgp7$7$F^q$\"3%=++++g$o]F*7$$\"3Y*HP_J`\"**\\F*$\"3!HqiZom$o]F*7$7$$ \"3_+++++;**\\F*$\"3W-1u#Qt$o]F*Ffq7$F\\r7$$\"31%o([&Ht\"**\\F*$\"3#zJ 7Xq'Qo]F*7$7$$\"3u'>VZc'=**\\F*$\"3g-++++So]F*Fbr7$7$$\"3='>VZc'=**\\F *$\"3--++++So]F*7$$\"35A\"QdF$>**\\F*$\"3k!)=ECnSo]F*7$7$$\"3s+++++?** \\F*$\"3%*Qp\"=Y8%o]F*Fcs7$Fis7$$\"3gj'))fD8#**\\F*$\"3!*R8,WnUo]F*7$7 $$\"3-#HBe[E#**\\F*$\"3k,++++Wo]F*F_t7$Fet7$$\"3Qd#RiBL#**\\F*$\"3IY2w jnWo]F*7$7$$\"3#4++++S#**\\F*$\"3-\\E*3a`%o]F*F[u7$Fau7$$\"3ka**[;KD** \\F*$\"3o[+^$ym%o]F*7$7$$\"3U6S!pSm#**\\F*$\"3S-++++[o]F*Fgu7$7$$\"3'3 ,/pSm#**\\F*$\"3%=++++![o]F*7$$\"3u/2u'>t#**\\F*$\"3w)HfK!o[o]F*7$7$$ \"35,++++G**\\F*$\"37Kx'*>O\\o]F*Fhv7$F^w7$$\"3ud:*p<$H**\\F*$\"3'fW3I #o]o]F*7$7$$\"3q_`)zK1$**\\F*$\"3;.++++_o]F*Fdw7$Fjw7$$\"3ukCCdJJ**\\F *$\"3sRvvUo_o]F*7$7$$\"3I,++++K**\\F*$\"3%**=U!*pL&o]F*F`x7$Ffx7$$\"3[ tM\\PJL**\\F*$\"3/Il]ioao]F*7$7$$\"3U;t1\\iM**\\F*$\"3y-++++co]F*F\\y7 $Fby7$$\"3AOXug6yP do]F*Fhy7$F^z7$$\"3'=q&*z4t$**\\F*$\"3g-V+-peo]F*7$7$$\"3E/*\\,<'Q**\\ F*$\"3)H+++++'o]F*Fdz7$Fjz7$$\"3#4#pCyIR**\\F*$\"3s$3`<#pgo]F*7$7$$\"3 o,++++S**\\F*$\"3]B#*=dQho]F*F`[l7$7$Fg[l$\"31C#*=dQho]F*7$$\"3KU#)\\e IT**\\F*$\"3ai<]Tpio]F*7$7$$\"3U8JB\"4E%**\\F*$\"3g-++++ko]F*F_\\l7$7$ $\"3)R6L74E%**\\F*$\"3;.++++ko]F*7$$\"3q<'\\(QIV**\\F*$\"3!zQ]7'pko]F* 7$7$$\"3'=++++S%**\\F*$\"3%4!=EORlo]F*F`]l7$Ff]l7$$\"3%[4,!>IX**\\F*$ \"3Q5*)*4)pmo]F*7$7$$\"3qYpJ7gY**\\F*$\"3#R++++!oo]F*F\\^l7$7$Fc^l$\"3 O.++++oo]F*7$$\"3aEED**HZ**\\F*$\"3Iytu+qoo]F*7$7$$\"31-++++[**\\F*$\" 3S^PL:Spo]F*F[_l7$Fa_l7$$\"3+gU]zH\\**\\F*$\"3gXd\\?qqo]F*7$7$$\"3'=S, M$f]**\\F*$\"3)H++++?(o]F*Fg_l7$7$$\"3W-9SLf]**\\F*F``l7$$\"3-[fvfH^** \\F*$\"3ydSCSqso]F*7$7$$\"3C-++++_**\\F*$\"3+w]S%4M(o]F*Ff`l7$F\\al7$$ \"3CPx+SH`**\\F*$\"3woA**fquo]F*7$7$$\"3g!['[aea**\\F*$\"3u.++++wo]F*F bal7$7$$\"3/!['[aea**\\F*$\"3=.++++wo]F*7$$\"3e\"ef-#Hb**\\F*$\"3gC/uz qwo]F*7$7$$\"3W-++++c**\\F*$\"3?udZtTxo]F*Fcbl7$Fibl7$$\"35F:^+Hd**\\F *$\"3Gz%)[*4(yo]F*7$7$$\"3w\"=sbx&e**\\F*$\"3]/++++!)o]F*F_cl7$Fecl7$$ \"3wFNw!)Gf**\\F*$\"3AykB>r!)o]F*7$7$$\"3i-++++g**\\F*$\"3UXea_U\")o]F *F[dl7$Fadl7$$\"3/Hc,hGh**\\F*$\"3GyV)*Qr#)o]F*7$7$$\"3O0&empD'**\\F*$ \"3q/++++%)o]F*Fgdl7$F]el7$$\"3-'yn7%Gj**\\F*$\"3#4AK(er%)o]F*7$7$$\"3 #G++++S'**\\F*$\"3A!H:;La)o]F*Fcel7$7$$\"3Q.++++k**\\F*F\\fl7$$\"31V+_ @Gl**\\F*$\"3_j*z%yr')o]F*7$7$$\"3%3XXxhl'**\\F*$\"3K/++++))o]F*Fbfl7$ Fhfl7$$\"3#pNs())o]F*7$7$$\"3-.++++o**\\F*$\"3g3To 5W*)o]F*F^gl7$Fdgl7$$\"3qpZ-#y#p**\\F*$\"3#yBvz@2*o]F*7$7$$\"3W?I$)Qbq **\\F*$\"3%R++++?*o]F*Fjgl7$7$Fahl$\"3]/++++#*o]F*7$$\"3KRsFiFr**\\F*$ \"3#ywAxBF*o]F*7$7$$\"3?.++++s**\\F*$\"3s,Bv*[M*o]F*Fihl7$7$$\"3y.++++ s**\\F*Fbil7$$\"3'y!)HDuK(**\\F*$\"3[*>quDZ*o]F*7$7$$\"3]77#*fau**\\F* $\"3E0++++'*o]F*Fhil7$7$$\"3#>@@*fau**\\F*$\"3q/++++'*o]F*7$$\"3%\\V#y AFv**\\F*$\"3stv@xs'*o]F*7$7$$\"3S.++++w**\\F*$\"3Gn)>)oX(*o]F*Fijl7$F _[m7$$\"3yf^..Fx**\\F*$\"31\\['pH()*o]F*7$7$$\"3)p-55Q&y**\\F*$\"3K/++ +++p]F*Fe[m-%'COLOURG6&%$RGBG\"\"\"\"\"!Fe\\m-F$6$7S7$F($\"3)3y$fib2o] F*7$$\"3]mT:(z@!**\\F*$\"3E%z]LS(4o]F*7$$\"3Oe9ui2/**\\F*$\"3%\\`woS; \"o]F*7$$\"3K;z_\"4i!**\\F*$\"3au&G$yx8o]F*7$$\"3i;aphN3**\\F*$\"3A)GW 9Hf\"o]F*7$$\"3+e*=)H\\5**\\F*$\"33MY2a**\\F*$\"3+thf'o]F*7$$\"3+]i_QQg**\\F*$\"3 [.L'*31oo]F*7$$\"3kC\"y%3Ti**\\F*$\"3WI\"e%>4qo]F*7$$\"39]P![hY'**\\F* $\"3O+'*zqMso]F*7$$\"3CL$Qx$om**\\F*$\"3U[F=MPuo]F*7$$\"33+v.I%)o**\\F *$\"337#p'p`wo]F*7$$\"3#=zpe*zq**\\F*$\"3;_aju\\yo]F*7$$\"35+D\\'QH(** \\F*$\"3UFWdEo)o]F*7$$\"3)[7`W l7)**\\F*$\"3/nbaU)*))o]F*7$$\"3Ymm*RRL)**\\F*$\"3;\"epNi5*o]F*7$$\"3] ;a<.Y&)**\\F*$\"3%p#z;v=$*o]F*7$$\"39e9tOc()**\\F*$\"3cJ@z]H&*o]F*7$$ \"3)***\\Qk\\*)**\\F*$\"3s*y,rJs*o]F*7$$\"3-$3dg6<***\\F*$\"3Uhz28X**o ]F*7$$\"3gmmxGp$***\\F*$\"3y#[BaO9!p]F*7$$\"3+D\"oK0e***\\F*$\"3]$Gk@` N!p]F*7$$\"3%\\(=5s#y***\\F*$\"3sxeV\"zb!p]F*7$$\"$+&Fe\\m$\"3G1)*yiv2 p]F*-Fa\\m6&Fc\\m$Fe\\mFe\\mF^\\n$\"*++++\"!\")-%+AXESLABELSG6%%\"xG% \"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;$\"%**\\!\"\"Fh[n;$\"%o]Fa]n$\"%p]F a]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 -1 101 "A minimax rational approximation can be obtained from a provis ional version of the inverse function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "remez('Q(x)',x=.02..500 ,[4,3],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#%Wstandard~Chebyshev~points~for~initial~critical~points :G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+$\"/++++++?!#:$\"2CM]ncN\\!>F%$ \"2%)\\_Wp(Q(y1\"!#>F(%\"xGF(F(*&$\"63D`\"G$oI'[A8F,F()F-\"\"#F(F(*&$\"6N \\8.9gm\"G]>!#?F()F-\"\"$F(F(*&$\"6Q*[pt;$y9&=C!#AF()F-\"\"%F(F(F(,*$F (\"\"!F(*&$\"6*fv1S>\"=7\"RoF6F(F-F(F(*&$\"6Q8df" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %,difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\")3m%z#!#>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%5relative~difference:G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"31O!o(o6WQ@!#F" }}{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+$\"/++++++?!#:$\"6I3!oMj*\\\\f\\&!#A$\"6Sy![HP4froG!# @$\"6KI!QvH?Y5([\"!#?$\"63#4=#H6Z$>8pF.$\"6\"QFyh&Rt60'H!#>$\"6?dcpS\\ `g[8\"!#=$\"6W7ZG7\\2))RG$F6$\"3++++++++]F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7minimax~approx imation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,,$\"+g3YvP!#5\"\"\"*&$ \"+xQ(y1\"!\")F(%\"xGF(F(*&$\"+2j[A8F,F()F-\"\"#F(F(*&$\"+m;G]>!\"*F() F-\"\"$F(F(*&$\"+$y9&=C!#6F()F-\"\"%F(F(F(,*$F(\"\"!F(*&$\"+\"=7\"RoF6 F(F-F(F(*&$\"+PQy*y\"F6F(F1F(F(*&$\"+xUo9CF!\"*F()F-\"\"$F(F(*&$\"+$y9&=C!#6F()F-\"\"%F(F(F(,*$F(\" \"!F(*&$\"+\"=7\"RoF6F(F-F(F(*&$\"+PQy*y\"F6F(F1F(F(*&$\"+xUo9CF " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Halley's f ormula for solving " }{XPPEDIT 18 0 "x = y-arcsinh(y);" "6#/%\"xG,&%\" yG\"\"\"-%(arcsinhG6#F&!\"\"" }{TEXT -1 5 " for " }{TEXT 275 1 "y" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi(y) = y-arcsinh(y)-x;" "6#/-%$phi G6#%\"yG,(F'\"\"\"-%(arcsinhG6#F'!\"\"%\"xGF-" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 26 "Given an approximate zero " }{TEXT 277 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 \"cor rection\" " }{TEXT 276 1 "h" }{TEXT -1 15 " is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h = phi(y)/``(phi*`'`(y)-ph i*`''`(y)*phi(y)/(2*phi*`'`(y)));" "6#/%\"hG*&-%$phiG6#%\"yG\"\"\"-%!G 6#,&*&F'F*-%\"'G6#F)F*F***F'F*-%#''G6#F)F*-F'6#F)F**(\"\"#F*F'F*-F16#F )F*!\"\"F=F=" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi*`'`(y) = 1-1/sqrt(1 +y^2);" "6#/*&%$phiG\"\"\"-%\"'G6#%\"yGF&,&F&F&*&F&F&-%%sqrtG6#,&F&F&* $F*\"\"#F&!\"\"F3" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi*`\"`(y) = y /((1+y^2)^(3/2));" "6#/*&%$phiG\"\"\"-%\"\"G6#%\"yGF&*&F*F&),&F&F&*$F* \"\"#F&*&\"\"$F&F/!\"\"F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "y-arcsinh(y)-x;\nD iff(%,y)=diff(%,y);\nDiff(%%,y$2)=diff(%%,y$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"yG\"\"\"-%(arcsinhG6#F$!\"\"%\"xGF)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(%\"yG\"\"\"-%(arcsinhG6#F(!\"\"%\"x GF-F(,&F)F)*&F)F)*$,&F)F)*$)F(\"\"#F)F)#F)F5F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(%\"yG\"\"\"-%(arcsinhG6#F(!\"\"%\"xGF--%\" $G6$F(\"\"#*&,&F)F)*$)F(F2F)F)#!\"$F2F(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-arc sinh(y)-x;" "6#/-%$phiG6#%\"yG,(F'\"\"\"-%(arcsinhG6#F'!\"\"%\"xGF-" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 770 "start_approx := proc(x)\n local y;\n if x<0 .019 and x>-0.019 then\n y := 1.817120593*surd(x,3);\n elif x>- 500 and x<500 then\n if x>0 then \n y := (.3775460 860+(10.67873877+(13.22486307+\n (1.950281666+.2418514783e-1 *x)*x)*x)*x)/\n (1.+(6.839112181+(1.789783837+.2414684277e-1*x)* x)*x);\n else \n y := (.3775460860+(-10.67873877+(13.2248 6307+\n (-1.950281666+.2418514783e-1*x)*x)*x)*x)/\n (- 1.+(6.839112181+(-1.789783837+.2414684277e-1*x)*x)*x);\n end if; \n else\n y := x+arcsinh(x);\n end if;\n y;\nend proc: \n\n next_halley_approx := proc(x,y)\n local p,q,t,u,v,h;\n p := 1+y^2; \n q := sqrt(p);\n t := y-arcsinh(y)-x;\n u := 1-1/q;\n v := y /(p*q);\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 15 "Test examples: " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 174 "xx := evalf(sqrt(37/25));\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(y-arcsinh(y),y=y3);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+1Db;7!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#y0G$\"+Nd+RI!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$\"+Mj+^I !\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G$\"+jj+^I!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G$\"+jj+^I!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+1Db;7!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "xx := -evalf(sqrt(789));\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(y-arcsinh (y),y=y3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$!+\"Q9*3G!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G$!+6jKCK!\")" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#y1G$!+yUiDK!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#y2G$!+yUiDK!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y3G$!+yUiDK! \")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+\"Q9*3G!\")" }}}{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-arcsinh(x);" "6#/-%\"gG6#%\"xG,&F'\"\"\"-%(arcsinhG6#F' !\"\"" }{TEXT -1 2 ": " }{TEXT 0 1 "Q" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4499 "Q := \+ proc(x::algebraic)\n local t,ok,s,terms,i,ti,eq;\n description \"i nverse of x -> x-arcsinh(x)\";\n option `Copyright (c) 2003 Peter St one.`;\n\n if nargs<>1 then\n error \"expecting 1 argument, got %1\", nargs;\n end if;\n if type(x,'float') then evalf('Q'(x))\n \+ elif type(x,`*`) and type(op(1,x),'numeric') and \n \+ signum(0,op(1,x),0)=-1 then -Q(-x) \n elif type(x,'realcons' ) and signum(0,x,0)=-1 then -Q(-x)\n elif type(x,And(complexcons,Not (realcons))) then\n error \"not implemented for complex argument \"\n elif type(x,`+`) then\n ok := false;\n if has(x,'arcs inh') 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,ti)='arcsinh' then\n t := op(1,ti); \n eq := t-arcsinh(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 'Q'(x) end if;\n elif has(x,'sin h') then\n s := select(has,x,'sinh');\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)='sinh' then\n t := op(1,ti);\n \+ eq := sinh(t)-t=x;\n if evalb(expand(eq)) or tes teq(eq) then\n ok := true;\n break; \n end if;\n end if;\n end do;\n \+ if ok then sinh(t) else 'Q'(x) end if;\n else 'Q'(x) end if; \n else 'Q'(x) end if;\nend proc:\n\n# construct remember table\nQ(0 ) := 0:\nQ('infinity') := 'infinity':\nQ(1-ln(1+sqrt(2))) := 1:\n\n# d ifferentiation\n`diff/Q` := proc(a,x) \n option `Copyright (c) 2003 \+ Peter Stone.`; \n diff(a,x)*sqrt(1+Q(a)^2)/(sqrt(1+Q(a)^2)-1)\nend p roc:\n\n`D/Q` := proc(t)\n option `Copyright (c) 2003 Peter Stone.`; \n if 1-0. 019 then\n s := 1.817120593*surd(x,3);\n elif x>-500 and \+ x<500 then\n if x>0 then \n s := (.377546086 0+(10.67873877+(13.22486307+\n (1.950281666+.2418514783e- 1*x)*x)*x)*x)/\n (1.+(6.839112181+(1.789783837+.2414684277e-1 *x)*x)*x);\n else \n s := (.3775460860+(-10.6787387 7+(13.22486307+\n (-1.950281666+.2418514783e-1*x)*x)*x)*x )/\n (-1.+(6.839112181+(-1.789783837+.2414684277e-1*x)*x)*x); \n end if;\n else\n s := x+arcsinh(x);\n end if;\n\n # solve the equation y*cosh(y)=x for y by Halley's metho d \n for i to maxit do\n p := 1+s^2;\n q := sqrt( p);\n t := s-arcsinh(s)-x;\n u := 1-1/q;\n v : = s/(p*q);\n h := t/(u-1/2*v*t/u);\n s := s-h;\n \+ if abs(h)<=eps*abs(s) then break end if;\n end do;\n s;\n end proc;\n\n p := ilog10(Digits);\n q := Float(Digits,-p);\n \+ maxit := trunc((p+(.02331061386+.1111111111*q))*2.095903274)+2;\n s aveDigits := Digits;\n Digits := Digits+min(iquo(Digits,3),5);\n x x := evalf(x);\n Digits := Digits+max(0,-ilog10(xx)-3);\n eps := F loat(3,-saveDigits-1);\n if Digits<=trunc(evalhf(Digits)) and \n \+ ilog10(xx)trunc(evalhf(DBL_MIN_10_EXP)) then\n val := evalhf(doQ(xx,eps,m axit))\n else\n val := doQ(xx,eps,maxit)\n end if;\n evalf[ saveDigits](val);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 23 "checking the procedure " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 63 "Comparison of starting approximation \+ with the inverse function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot('Q(x)-start_approx(x)',x=0..1, color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 432 145 145 {PLOTDATA 2 "6& 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45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Test examples " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "g := x -> x-arcsinh(x);\nxx := 1.3;\ng(xx);\nQ(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"-%(a rcsinhG6#F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#8! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*T*[:A!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++8!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "g := x -> x-arcsinh(x);\nxx \+ := -0.991234567891096;\nevalf(g(xx),18);\nevalf(Q(%),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\" \"\"-%(arcsinhG6#F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x xG$!0'4\"*ycM7**!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!3KOo!QzE2;\"! #=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0'4\"*ycM7**!#:" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "g := x -> x-arcsinh(x);\nDigits := 60:\nxx := evalf(sqrt(5)*10^(-8));\neva lf(g(xx),Digits+16);\nevalf(Q(%));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"-%(a rcsinhG6#F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"gn! 4FCd_6'f$=1WNiFJ(oO<4kp*y*\\xz1OA!#n" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"gnuZ7yDQGjRbf+!pGKF%*)*[uFV#)\\7)**Qj=!#$)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"gn!4FCd_6'f$=1WNiFJ(oO<4kp*y*\\xz1OA!#n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "xx : = Pi/4-arcsinh(Pi/4);\nQ(xx);\nevalf[12](%);\nx1 := evalf[12](xx);\nev alf[12](Q(x1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,&*&\"\"%!\" \"%#PiG\"\"\"F*-%(arcsinhG6#,$*&F'F(F)F*F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%!\"\"%#PiG\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-)Rj\")R&y!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# x1G$\",rYnsT'!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 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" }}{PARA 0 "" 0 "" {TEXT -1 28 "The solution is (of course) " }{XPPEDIT 18 0 "x = 2-arcsinh(2);" "6#/%\"xG,&\"\"#\"\"\"-% (arcsinhG6#F&!\"\"" }{TEXT -1 1 " " }{TEXT 274 1 "~" }{TEXT -1 14 " 1. 626860408. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 30 "evalf(evalf(2-arcsinh(2),13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+[_kjb!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "secant(Q(x)=2,x=0.5..0.6,inf o=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$ \".%*yPF4d&!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~- >~~~G$\".$en]bjb!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation ~3~~->~~~G$\".CuEXOc&!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxim ation~4~~->~~~G$\".4#[_kjb!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7app roximation~5~~->~~~G$\".:#[_kjb!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+[_kjb!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "The code above \+ contains procedures which enable " }{TEXT 0 4 "diff" }{TEXT -1 5 " and " }{TEXT 0 1 "D" }{TEXT -1 18 " to differentiate " }{XPPEDIT 18 0 "Q( x);" "6#-%\"QG6#%\"xG" }{TEXT -1 8 " and Q. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "diff(Q(x),x);\nD(Q )(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&F$F$*&F$F$*$,&F$F $*$)-%\"QG6#%\"xG\"\"#F$F$#F$F/!\"\"F1F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"\"F$,&F$F$*&F$F$*$,&F$F$*$)-%\"QG6#%\"xG\"\"#F$F$#F$F/!\" \"F1F1" }}}{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 fo r root-finding. 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>M#))**zF\\]l7$$\"3d+++?EdRZF\\]l$\"3c5ebIDTtzF\\]l7$$\"3^++DOw-1[F\\] l$\"3gT\"G*=_T'o(F\\]l7$$\"3Y++]_E[s[F\\]l$\"3'e,\"4i4#\\4(F\\]l7$$\"3 S++vow$*Q\\F\\]l$\"3JcMtu2jWF\\]l7$ $\"3G++]<#Rm\\&F\\]l$!3K(ez*)yRtj&F\\]l7$$\"3%)***\\(=dHdbF\\]l$!3AxE9 va[rlF\\]l7$$\"3G+++?A&zh&F\\]l$!3u$4wa+I')G(F\\]l7$$\"3q++D@(3'ycF\\] l$!3m[iI*4#4lxF\\]l7$$\"3F++]A_ERdF\\]l$!3mVD5-#Q^)zF\\]l7$$\"37++D6Ej peF\\]l$!3?[J#GJ!)zc(F\\]l7$F[\\l$!3V3M " 0 "" {MPLTEXT 1 0 50 "allroots(Q(x)=8*sin(3*x),x=-5..5,miniterations=9);" } }{PARA 12 "" 1 "" {XPPMATH 20 "6/$\"\"!F$$!+2&4b#[!\"*$!+%Gbjc%F'$!+NI 8-HF'$!+p&GPH#F'$!+F)y`N*!#5$!+[4%*3A!#6$\"+[4%*3AF3$\"+F)y`N*F0$\"+p& GPH#F'$\"+NI8-HF'$\"+%Gbjc%F'$\"+2&4b#[F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 15 ": The equat ion " }{XPPEDIT 18 0 "Q(x) = 8*sin(3*x);" "6#/-%\"QG6#%\"xG*&\"\")\"\" \"-%$sinG6#*&\"\"$F*F'F*F*" }{TEXT -1 18 " is equivalent to " } {XPPEDIT 18 0 "x = g(8*sin(3*x));" "6#/%\"xG-%\"gG6#*&\"\")\"\"\"-%$si nG6#*&\"\"$F*F$F*F*" }{TEXT -1 51 ", so the same solutions can be obta ined as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 74 "g := x -> x-arcsinh(x);\nallroots(g(8*sin(3*x) )=x,x=-5..5,miniterations=9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"g Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"-%(arcsinhG6#F-!\"\"F(F( F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6/$\"\"!F$$!+2&4b#[!\"*$!+%Gbjc%F' $!+NI8-HF'$!+p&GPH#F'$!+F)y`N*!#5$!+[4%*3A!#6$\"+[4%*3AF3$\"+F)y`N*F0$ \"+p&GPH#F'$\"+NI8-HF'$\"+%Gbjc%F'$\"+2&4b#[F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Inverse for " }{XPPEDIT 18 0 "g(x) = x+ exp(x);" "6#/-%\"gG6#%\"xG,&F'\"\"\"-%$expG6#F'F)" }{TEXT -1 2 ": " } {TEXT 0 1 "K" }{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+exp(x);" "6#/-%\"gG6#%\"xG,&F'\"\"\"-%$expG6#F'F)" }{TEXT -1 47 " is one-to-one, and so has an inverse function " }{XPPEDIT 18 0 "K(x);" "6#-%\"KG6#%\"xG" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "g := x -> x+exp(x );\np1 := plot([g(x),x],x=-4..6,y=-4..6,color=[red,black],\n \+ linestyle=[1,3]):\np2 := plots[implicitplot](x=y+exp(y),x=-4..6,\n \+ y=-4..6,grid=[30,30],color=blue):\nplots[display]([p1,p2]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&a rrowGF(,&9$\"\"\"-%$expG6#F-F.F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 375 375 375 {PLOTDATA 2 "6'-%'CURVESG6%7Y7$$!\"%\"\"!$!3yl76hVo\")R!#< 7$$!3YLLLe%G?y$F-$!3=m\\'[,_#fPF-7$$!3#om;aesBf$F-$!3cpLxc'R[c$F-7$$!3 ALL$3s%3zLF-$!3d'*p!*)31]M$F-7$$!31LL$e/$QkJF-$!3y`Gz^G9AJF-7$$!3ommT5 =q]HF-$!3kXC8C))R)*GF-7$$!3ILL3_>f_FF-$!3I)\\uUdH))o#F-7$$!3K++vo1YZDF -$!3Osydv/=pCF-7$$!3;LL3-OJNBF-$!3q/fwuM`QAF-7$$!3p***\\P*o%Q7#F-$!3#Q K[r7wU+#F-7$$!3Kmmm\"RFj!>F-$!3%>tFR;-xv\"F-7$$!33LL$e4OZr\"F-$!3.,rnv [sM:F-7$$!3u*****\\n\\!*\\\"F-$!3+9Q*yO2dF\"F-7$$!3%)*****\\ixCG\"F-$! 3+UU.W$G^+\"F-7$$!3#******\\KqP2\"F-$!3n\\oPii_?t!#=7$$!3]JL$3-TC%))F^ p$!3*p`@3Ty@r%F^p7$$!3skmm;4z)e'F^p$!3!o30R*3[99F^p7$$!3GjmmmOlzYF^p$ \"3qd^8m%)4$e\"F^p7$$!3=****\\(=t)eCF^p$\"37%HM6>I7O&F^p7$$!3n!ommmh5$ \\!#>$\"34?-LczuD!*F^p7$$\"3g1+]7=lj;F^p$\"3k6H'[[ptM\"F-7$$\"3,/+]PaR #=F-7$$\"3MULLe9EgeF^p$\"3/9zN***fGQ#F-7$$\"31NL$eR \"3GyF^p$\"3[#3=H*[TqHF-7$$\"3ulmmT5k]**F^p$\"3s>N5N='**p$F-7$$\"3=nm \"zRQb@\"F-$\"3'>@1,^Zwe%F-7$$\"3:++v=>Y29F-$\"3YHK$ojMI\\&F-7$$\"3Znm ;zXu9;F-$\"3)\\M2$R%[8k'F-7$$\"34+++]y))G=F-$\"3wwgKX`%e0)F-7$$\"3H++] i_QQ?F-$\"3D\")G2nhe;(*F-7$$\"3b++D\"y%3TAF-$\"3EV]rI6Yk6!#;7$$\"3+++] P![hY#F-$\"3zMlJ'G9VU\"F`u7$$\"3iKLL$Qx$oEF-$\"3!o$o0/N\\3()RVqQ &F`u7$$\"3a***\\7`Wl7%F-$\"3dU'ezV)**3mF`u7$$\"3enmmm*RRL%F-$\"3oicFsM !y0)F`u7$$\"3%zmmTvJga%F-$\"3')e%f6&fO!))*F`u7$$\"3]MLe9tOcZF-$\"3IfMs 8'*y57!#:7$$\"31,++]Qk\\\\F-$\"3y*G0EPV2Y\"F]y7$$\"3![LL3dg6<&F-$\"3'y %H-#y-H\"=F]y7$$\"3K,+voTAq_F-$\"3!))[E[t(H(*>F]y7$$\"3%ymmmw(Gp`F-$\" 3KU,fuF>7$FCFC7$FHFH7$FMFM7$FR FR7$FWFW7$FfnFfn7$F[oF[o7$F`oF`o7$FeoFeo7$FjoFjo7$F`pF`p7$FepFep7$FjpF jp7$F_qF_q7$FdqFdq7$FjqFjq7$F_rF_r7$FdrFdr7$FirFir7$F^sF^s7$FcsFcs7$Fh sFhs7$F]tF]t7$FbtFbt7$FgtFgt7$F\\uF\\u7$FbuFbu7$FguFgu7$F\\vF\\v7$FavF av7$FfvFfv7$F[wF[w7$F`wF`w7$FewFew7$FjwFjw7$F_xF_x7$FdxFdx7$FixFix7$F_ yF_y7$FdyFdy7$F^zF^z7$FhzFhz7$Fb[lFb[l7$Ff\\lFf\\l-F[]l6&F]]lF*F*F*-Fc ]l6#\"\"$-F$6ep7$7$F+F(7$$!3'z*eh)[im\"QF-$!3C0s^vXLF-Fibl7$F_cl7$$!3;gN\"Q^qBH$F-$!3@_)e89>$GLF-7$7$$ !35rdWV7%QF$F-F`clFecl7$7$F\\dl$!3'f?'eF[M5LF-7$$!3L?\"=P7\"R;JF-$!3C& RZ_%4ZfJF-7$7$$!3i4$z8C**HSHF-$!3;w0+jXt!*HF-7$7$$!3^0kWMM)R\"HF-FidlFael7$Fg el7$$!3cR;&[GfHw#F-$!3(G3+V>!)fl#F-7$7$$!3Gok!R)p$za#F-Fbf lFgfl7$F]gl7$$!3:D@I;mB1CF-$!37/e$y5:.\\#F-7$7$$!3-;b'*o?'eF#F-$!3E=79 J6OqBF-Fagl7$Fggl7$$!3'*4eCackEAF-$!3_A_o$[y]K#F-7$7$$!37#Ho.u`J<#F-Fh glF]hl7$Fchl7$$!3a=rn9'*4X?F-$!3q;q/Ppzh@F-7$7$$!3A>'eF[M5$>F-$!3uIex, E>g?F-Fghl7$F]il7$$!3Sc7T?Usi=F-$!3[#)f5XZM**>F-7$7$$!3]T]%*fj.'y\"F-$ !3m>'eF[M5$>F-Fcil7$Fiil7$$!3sDl%)o7;y;F-$!3e;QY5,3R=F-7$7$$!3'Gs^l*o? 'e\"F-$!3#oW,;!opgeJgt9F-Fe[m7$7$$!32E[M5$z8C\"F-F^\\m7$$!3KLQ %QRAE6\"F-$!3/>e%oAO,P\"F-7$7$$!3!R@9!z'RR_*F^p$!3_E[M5$z8C\"F-Fd\\m7$ Fj\\m7$$!3Lcag\\Ok.#*F^p$!3E5A_pmc<7F-7$7$$!3=(Hz8C\"F-F`]m7$7$$!31'Hz8CFCCmT)\\V*F^pF_^m7$7$Ff^m$!3IGCCmT)\\V*F^p7$$!3 )=(\\oHy'zC&F^p$!3scY+\"z!zM#*F^p7$7$$!3s\"HF4!ox&)[F^pFg]mF^_m7$7$Fe_ m$!3G)Hz8C-p.2`(F^p$!3:d[\\Qr!QK\"F^p7$7$$\" 3cJb'*o?'eF)F^p$!3C$)3+QhEd%*FfqFicm7$F_dm7$$\"3fR)Qf3&H=)*F^p$!3!34d9 sE7j\"Ffq7$7$$\"3s\\M5$z8C<\"F-$\"3u_$f\"f:pQ!)FfqFedm7$F[em7$$\"3(e'e ;[')e57F-$\"3!**QM?%zhv**Ffq7$7$$\"3M(HfnTFeG\"F-F[bmFaem7$Fgem7$$\"3U ,!*e[ge[9F-$\"3%*[5/_O'e1#F^p7$7$$\"3JY.Jz8C<:F-$\"37.^U7wYbBF^pF[fm7$ Fafm7$$\"3+DiYE56\"p\"F-$\"37yxLN(*)))3$F^p7$7$$\"3)GC6 7U/'H+\"QF^pFgfm7$F]gm7$$\"3%)[MM/gjL>F-$\"3G2Xj=e\">6%F^p7$7$$\"3k^i( RJ(H.@F-FgbmFcgm7$Figm7$$\"3EjhM2\"p0=#F-$\"3wGjn]1'34&F^p7$7$$\"3oRTs ^l*o?#F-$\"3AJ%z7Zh0=&F^pF]hm7$Fchm7$$\"3LxF^p7$7$$\"34u@sDgP:JF-F`dmFajm7$Fgjm7$$\"3$>:.%*4] Z@$F-$\"3]3BQyT:U&)F^p7$7$$\"32I[M5$z8C$F-$\"3'yaz_J7Nh)F^pF[[n7$Fa[n7 $$\"3/]:63Fq'[$F-$\"3W&HnL&R!4F*F^p7$7$$\"3'os^l*o?'e$F-$\"3`aT_IZaP&* F^pFg[n7$F]\\n7$$\"39[*>oJb'ePF-$\"3]$G_$GPl****F^p7$7$$\"3mB'eF[M5$RF -$\"37wod9x:Y5F-Fc\\n7$Fi\\n7$$\"3CY$Gb#zgISF-$\"3;FPL].%G2\"F-7$7$$\" 3Y?b'*o?'eF%F-$\"3zO85c4cQ6F-F_]n7$Fe]n7$$\"3MWnBM0c-VF-$\"3'eAKyK:d9 \"F-7$7$$\"3A6L5\\I>-WF-F\\emF[^n7$Fa^n7$$\"3+([(4\"o\"Q$e%F-$\"3U!Qyr w@(47F-7$7$$\"3E " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We tackle the problem of constructing a numerical inve rse for " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 57 "The idea is to use Halley's method to sol ve the equation " }{XPPEDIT 18 0 "y+exp(y) = x;" "6#/,&%\"yG\"\"\"-%$e xpG6#F%F&%\"xG" }{TEXT -1 17 " numerically for " }{TEXT 284 1 "y" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "We need an " }{TEXT 260 21 "initial approximation" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 57 "starting approximations for Halley's ro ot-finding method " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "x = y+exp(y);" "6#/%\"xG,&%\"yG\" \"\"-%$expG6#F&F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "dx/dy = 1+exp(y); " "6#/*&%#dxG\"\"\"%#dyG!\"\",&F&F&-%$expG6#%\"yGF&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1/(1+exp(y));" "6#/*&%#dyG\"\"\"%#dxG!\"\"* &F&F&,&F&F&-%$expG6#%\"yGF&F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "The first few terms in th e Taylor series for " }{XPPEDIT 18 0 "f(x) = g^(-1)*``(x);" "6#/-%\"f G6#%\"xG*&)%\"gG,$\"\"\"!\"\"F,-%!G6#F'F," }{TEXT -1 7 " about " } {XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 72 " can be obtained as the series solution for the differential equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 1/(1+exp(y));" "6#/*&%#dyG \"\"\"%#dxG!\"\"*&F&F&,&F&F&-%$expG6#%\"yGF&F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "Order := 7:\nde := diff(y(x),x)=1/(1+exp(y(x)));\nic := y(1)=0;\n dsolve(\{de,ic\},y(x),type=series);\nconvert(rhs(%),polynom);\nOrder : = 6:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/-%%diffG6$-%\"yG6#%\"xG F,*&\"\"\"F.,&F.F.-%$expG6#F)F.!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#icG/-%\"yG6#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"y G6#%\"xG+1,&F'\"\"\"F*!\"\"#F*\"\"#F*#F+\"#;F-#F*\"$#>\"\"$#F*\"%sI\" \"%#!#8\"&S9'\"\"&#\"#Z\"(gXZ\"\"\"'-%\"OG6#F*\"\"(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,0*&\"\"#!\"\"%\"xG\"\"\"F(#F(F%F&*&\"#;F&,&F'F(F(F&F %F&*&\"$#>F&F,\"\"$F(*&\"%sIF&F,\"\"%F(*(\"#8F(\"&S9'F&F,\"\"&F&*(\"#Z F(\"(gXZ\"F&F,\"\"'F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Alternatively, the same Taylor polynomial can be obt ained by using the procedure " }{TEXT 0 6 "taylor" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "RootOf(y+exp(y)-x,y);\ntaylor(%,x=1,7);\nconvert(%,polynom);\nexp and(%);\nconvert(%,horner);\nevalf(evalf(%,15)):\npsi := unapply(%,x); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RootOfG6#,(%#_ZG!\"\"-%$expG6#F 'F(%\"xG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+1,&%\"xG\"\"\"F&!\" \"#F&\"\"#F&#F'\"#;F)#F&\"$#>\"\"$#F&\"%sI\"\"%#!#8\"&S9'\"\"&#\"#Z\"( gXZ\"\"\"'-%\"OG6#F&\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&\"\"# !\"\"%\"xG\"\"\"F(#F(F%F&*&\"#;F&,&F'F(F(F&F%F&*&\"$#>F&F,\"\"$F(*&\"% sIF&F,\"\"%F(*(\"#8F(\"&S9'F&F,\"\"&F&*(\"#ZF(\"(gXZ\"F&F,\"\"'F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,0*&#\"&rA&\"&?>)\"\"\"%\"xGF(F(*&#\"# LF'F(*$)F)\"\"&F(F(!\"\"*&#\"#h\"&oF$F(*$)F)\"\"%F(F(F(*&#\"#&)\"&GP(F (*$)F)\"\"$F(F(F(#\"'\"GO)\"(gXZ\"F0*&#\"#ZFAF(*$)F)\"\"'F(F(F(*&#\"%6 CF4F(*$)F)\"\"#F(F(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"'\"GO)\" (gXZ\"!\"\"*&,&#\"&rA&\"&?>)\"\"\"*&,&#\"%6C\"&oF$F'*&,&#\"#&)\"&GP(F- *&,&#\"#hF2F-*&,&#\"#LF,F'*(\"#ZF-F&F'%\"xGF-F-F-FBF-F-F-FBF-F-F-FBF-F -F-FBF-F-F-FBF-F-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$psiGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(,&$\"++NRrc!#5!\"\"*&,&$\"+0tt!Q'F/\"\"\"*&,& $\"+'3)ydt!#6F0*&,&$\"+&G')G:\"!#7F5*&,&$\"+mAdh=F?F5*&,&$\"+7.KGS!#8F 0*&$\"+$\\\"R(=$!#9F59$F5F5F5FMF5F5F5FMF5F5F5FMF5F5F5FMF5F5F5FMF5F5F(F (F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 98 "T he following picture makes a graphical comparison between the Taylor p olynomial and the graph of " }{XPPEDIT 18 0 "x = y+exp(y);" "6#/%\"xG, &%\"yG\"\"\"-%$expG6#F&F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "p1 := plots[implicit plot](y+exp(y)=x,x=-3..5,y=-4..1.4):\np2 := plot(psi(x),x=-3..5,color= blue):\nplots[display]([p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 324 273 273 {PLOTDATA 2 "6&-%'CURVESG6gp7$7$$!\"$\"\"!$!3e.N&*zvtZI!#<7$$! 3')R_OTEd@HF-$!3Ai%yqYQH(HF-7$7$$!31V'*p7j1mGF-$!3/************>HF-F.7 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C'\\V.;qF&F-7$$\"$+#F*$\"3IO![lO<$)H&F--F`fm6&Fbfm$F*F*Fffn$\"*++++\"! \")-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F`fnFbgn " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "C urve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "The following rational approximations were constructed using a pr eliminary version of the numerical procedure to evaluate " }{TEXT 262 4 "K(x)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "remez('K(x)'-ln(x),x=8..1000,[1,2], info=true,maxgraph=9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~alg orithm:~calculating~minimax~error~estimate~by~solving~a~rational~equat ionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D--~minimising~the~ABSOLUTE~e rror~--G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-iteration~11G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G-- ------------------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprovisional~rational~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&$\"2U;*4t)3?'\\!#\"\"\"%\" xGF-F(F-,($F-\"\"!F-*&$\"2//VS2#zH7F'F-F.F-F-*&$\"23RA2L$R\"[$!#?F-)F. \"\"#F-F-F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%=$!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% =$!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,difference:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$\"y!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/Bb*))pWX#!#C" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>goal~for~r elative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/;>]_\"p8\" !#A" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1critical~points:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6' $\"-+++++!)!#6$\"2R&4E+\\rw6!#:$\"25&)y@v#pCAF($\"2g$oGI`'*QpF($\"2+)4 EpFJMQ!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7minimax~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&$\"+t)3?'\\!#5!\"\"*&$\"+(oK)GG!#7\"\"\"%\"xGF-F(F-,($F-\"\" !F-*&$\"+u?zH7F'F-F.F-F-*&$\"+JLR\"[$!#8F-)F.\"\"#F-F-F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%0minimax ~error:~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&>=$!\")" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&$\"+t )3?'\\!#5!\"\"*&$\"+(oK)GG!#7\"\"\"%\"xGF-F(F-,($F-\"\"!F-*&$\"+u?zH7F 'F-F.F-F-*&$\"+JLR\"[$!#8F-)F.\"\"#F-F-F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "remez('K(x)',x=2.7 ..8,[2,1],info=true,maxgraph=9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%] pRemez~algorithm:~calculating~minimax~error~estimate~by~solving~a~rati onal~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~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G--------------------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Dprovisional~rational~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,($\"23$4/IEoUh!#mfGpY*fF'\"\"\"% \"xGF,F,*&$\"2C]TAcd[!\\!#>F,)F-\"\"#F,F,F,,&$F,\"\"!F,*&$\"2TY.DP9c$= F'F,F-F,F,F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%goal~for~relative~differen ce:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/;>]_\"p8\"!#A" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1critical~poi nts:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"/++++++F!#8$\"2k.(zf!pP9$! #;$\"2'HlK=<^*[%F($\"2vh%=hWzolF($\"/++++++!)F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7minimax~approx imation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,($\"+IEoUh!#5!\"\"*&$ \"+'GpY*fF'\"\"\"%\"xGF,F,*&$\"+iv&[!\\!#7F,)F-\"\"#F,F,F,,&$F,\"\"!F, *&$\"+tVhN=F'F,F-F,F,F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%0minimax~error:~G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"&R8$!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,($\"+IEoUh!#5!\"\"*&$\"+'GpY*fF'\" \"\"%\"xGF,F,*&$\"+iv&[!\\!#7F,)F-\"\"#F,F,F,,&$F,\"\"!F,*&$\"+tVhN=F' F,F-F,F,F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "remez('K(x)',x=-8. 5..-0.6,[3,2],info=true,maxgraph=9);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%]pRemez~algorithm:~calculating~minimax~error~estimate~by~solving~a~ rational~equationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D--~minimising~ the~ABSOLUTE~error~--G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%,iteration~9G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G--------------------------------------G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%Dprovisional~rational~approximation:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,*$\"426=@4(*R$\\b!#>!\"\"*&$\"4+!G%e-Y%39 yF'\"\"\"%\"xGF,F,*&$\"4.kk,c#[#\\1#F'F,)F-\"\"#F,F(*&$\"4+M&3%\\c:#48 F'F,)F-\"\"$F,F,F,,($F,\"\"!F,*&$\"4uUGQU5Usw\"F'F,F-F,F(*&$\"4=.6Nk@B IK\"F'F,F1F,F,F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,difference:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&Nc\"!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%5relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1sJ-#3[5&o!#F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>goal~for~r elative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1'z_z&*y/\\ )!#B" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%1critical~points:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6)$!1+++++++&) !#:$!41'>#f=vUuA(!#=$!4h!p&y>;e=%[F($!4kQz*36g/+HF($!4ZtA#)\\%)e[h\"F( $!4!ojZU(RT,e)!#>$!0+++++++'F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7minimax~approximation:G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,*$\"+r*R$\\b!#5!\"\"*&$\"+gW39yF'\" \"\"%\"xGF,F,*&$\"+E[#\\1#F'F,)F-\"\"#F,F(*&$\"+lb@48F'F,)F-\"\"$F,F,F ,,($F,\"\"!F,*&$\"+/@Cn " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "H alley's formula for solving " }{XPPEDIT 18 0 "x = y+exp(y);" "6#/%\"xG ,&%\"yG\"\"\"-%$expG6#F&F'" }{TEXT -1 5 " for " }{TEXT 286 1 "y" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "phi(y) = y+exp(y)-x;" "6#/-%$phiG6#% \"yG,(F'\"\"\"-%$expG6#F'F)%\"xG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "Given an approximate zero " }{TEXT 288 1 "a" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "phi(y)" "6#-%$phiG6#%\"yG" }{TEXT -1 36 ", the \"improved\" Halley estimate is " }{XPPEDIT 18 0 "y = y-h;" "6#/% \"yG,&F$\"\"\"%\"hG!\"\"" }{TEXT -1 24 " where the \"correction\" " } {TEXT 287 1 "h" }{TEXT -1 15 " is given by: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h = phi(y)/``(phi*`'`(y)-phi*`''`(y)*ph i(y)/(2*phi*`'`(y)));" "6#/%\"hG*&-%$phiG6#%\"yG\"\"\"-%!G6#,&*&F'F*-% \"'G6#F)F*F***F'F*-%#''G6#F)F*-F'6#F)F**(\"\"#F*F'F*-F16#F)F*!\"\"F=F= " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "phi*`'`(y) = 1+exp(y);" "6#/*&%$phiG\" \"\"-%\"'G6#%\"yGF&,&F&F&-%$expG6#F*F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi*`\"`(y) = exp(y);" "6#/*&%$phiG\"\"\"-%\"\"G6#%\"yGF&-%$expG 6#F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "y+exp(y)-x;\nDiff(%,y)=diff(%,y);\nDiff(% %,y$2)=diff(%%,y$2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"yG\"\"\"- %$expG6#F$F%%\"xG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$, (%\"yG\"\"\"-%$expG6#F(F)%\"xG!\"\"F(,&F)F)F*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$,(%\"yG\"\"\"-%$expG6#F(F)%\"xG!\"\"-%\"$G6$ F(\"\"#F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "The following code sets up the starting approximations from th e 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 ze ro of " }{XPPEDIT 18 0 "phi(y) = y+exp(y)-x;" "6#/-%$phiG6#%\"yG,(F'\" \"\"-%$expG6#F'F)%\"xG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 862 "start_approx := p roc(x)\n local y;\n if x<2.7 and x>-.61712 then\n y := -.5671 393500+(.6380737305+(-.7357788086e-1+\n (.1152886285e-2+(.186 1572266e-2+\n (-.4028320312e-3+.3187391493e-4*x)*x)*x)*x)*x)* x\n elif x>0 then\n if x<8.34856 then\n y := (-.6142682 630+(.5994669286+.4904857562e-2*x)*x)/\n (1.+.1835614373*x) \+ \n else\n y := ln(x)+(-.4962008873-.2828832687e-2*x)/\n \+ (1.+(.1229792074+.3481393331e-3*x)*x)\n end if;\n els e\n if x>-8.5 then\n y := (-.5549339971+(.7814084460+\n \+ (-.2064924826+.1309215565*x)*x)*x)/\n (1.+(-.17 67242104+.1323023216*x)*x)\n else\n y := x\n end if; \n end if;\n y;\nend proc: \n\nnext_halley_approx := proc(x,y)\n \+ local p,t,u,h;\n p := exp(y);\n t := y+p-x;\n u := 1+p;\n h : = t/(u-1/2*p*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 0 137 "xx := evalf (sqrt(37/25));\ny0 := start_approx(xx);\ny1 := next_halley_approx(xx,y 0);\ny2 := next_halley_approx(xx,y1);\neval(y+exp(y),y=y2);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+1Db;7!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#y0G$\"+m#))R0\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$\"+h#))R0\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G$ \"+m#))R0\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+2Db;7!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "xx := evalf(sqrt(78));\ny0 := start_approx(xx);\ny1 := next_halle y_approx(xx,y0);\ny2 := next_halley_approx(xx,y1);\ny3 := next_halley_ approx(xx,y2);\neval(y+exp(y),y=y3);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#xxG$\"+m3wJ))!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G$\"+#o I<$>!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y1G$\"+A!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y2G$\"+A!\"*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#y3G$\"+A!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"+n3wJ))!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "co de for inverse of " }{XPPEDIT 18 0 "g(x) = x+exp(x);" "6#/-%\"gG6#%\"x G,&F'\"\"\"-%$expG6#F'F)" }{TEXT -1 2 ": " }{TEXT 0 1 "K" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4944 "K := proc(x::algebraic)\n local t,ok,s,terms,i,ti ,eq;\n description \"inverse of x -> x+exp(x)\";\n option `Copyrig ht (c) 2003 Peter Stone.`;\n\n if nargs<>1 then\n error \"expec ting 1 argument, got %1\", nargs;\n end if;\n if type(x,'float') t hen evalf('K'(x))\n elif type(x,And(complexcons,Not(realcons))) then \n error \"not implemented for complex argument\"\n elif type(x ,`+`) then\n ok := false;\n if has(x,'exp') then\n s := select(has,x,'exp');\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,ti)='e xp' then\n t := op(1,ti);\n eq := t+exp(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 'K '(x) end if;\n elif has(x,'ln') then\n s := select(has,x, 'ln');\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,ti)='ln' then\n \+ t := op(1,ti);\n eq := t+ln(t)=x;\n \+ if evalb(expand(eq)) or testeq(eq) then\n ok := true; \n break;\n end if;\n end if ;\n end do;\n if type(t,'realcons') and \n \+ signum(0,t,-1)=-1 then ok := false end if;\n if ok t hen ln(t) else 'K'(x) end if;\n else 'K'(x) end if;\n else 'K'( x) end if;\nend proc:\n\n# construct remember table\nK(1) := 0:\nK('in finity') := 'infinity':\nK(-'infinity') := -'infinity':\n\n# different iation\n`diff/K` := proc(a,x) \n option `Copyright (c) 2003 Peter St one.`; \n diff(a,x)/(1+a-K(a))\nend proc:\n\n`D/K` := proc(t)\n op tion `Copyright (c) 2003 Peter Stone.`;\n if 1a)-'K' )\n end if\nend proc:\n\n#integration\n`int/K` := proc(f)\n local \+ gx,h,inds,u;\n option `Copyright (c) 2003 Peter Stone.`;\n inds := map(proc(x) if op(0,x) ='K' then x end if end proc,indets(f,function) );\n if nops(inds)<>1 then return FAIL end if;\n inds := inds[1]; \n if nops(inds)=1 then gx := op(inds) else gx := op(2,inds) end if; \n if not type(gx,linear(_X)) then return FAIL end if;\n h := subs (inds=u,_X=(u+exp(u)-coeff(gx,_X,0))/coeff(gx,_X),f);\n h := h*(1+ex p(u))/coeff(gx,_X);\n h := int(h,u);\n if has(h,int) then return F AIL end if;\n subs(exp(u)=gx-u,u=inds,h)\nend proc:\n\n# simplificat ion\n`simplify/K` := proc(s)\n option remember,system,`Copyright (c) 2003 Peter Stone.`;\n if not has(s,'K') or type(s,'name') then retu rn s\n elif type(s,'function') and op(0,s)='exp' then\n if type (op(1,s),'function') and op([1,0],s)='K' then\n return op([1,1 ],s)-op(1,s)\n else return s\n end if;\n end if;\n map(p rocname,args)\nend proc:\n\n# numerical evaluation\n`evalf/K` := proc( x)\n local xx,eps,saveDigits,doK,val,p,q,maxit;\n option `Copyrigh t (c) 2003 Peter Stone.`;\n\n if not type(x,realcons) then return 'K '(x) end if;\n \n doK := proc(x,eps,maxit)\n local p,s,t,u,h, i; \n # set up a starting approximation\n if x<2.7 and x>-.6 1712 then\n s := -.5671393500+(.6380737305+(-.7357788086e-1+\n (.1152886285e-2+(.1861572266e-2+\n (-.4028320 312e-3+.3187391493e-4*x)*x)*x)*x)*x)*x\n elif x>0 then\n \+ if x<8.34856 then\n s := (-.6142682630+(.5994669286+.490485 7562e-2*x)*x)/\n (1.+.1835614373*x) \n else\n \+ s := ln(x)+(-.4962008873-.2828832687e-2*x)/\n (1. +(.1229792074+.3481393331e-3*x)*x)\n end if;\n else\n \+ if x>-8.5 then\n s := (-.5549339971+(.7814084460+\n \+ (-.2064924826+.1309215565*x)*x)*x)/\n (1.+ (-.1767242104+.1323023216*x)*x)\n else\n s := x\n \+ end if;\n end if;\n\n # solve the equation y+exp(y)=x for y by Halley's method \n for i to maxit do\n p := exp (s);\n t := s+p-x;\n u := 1+p;\n h := t/(u-1/2 *p*t/u);\n s := s-h;\n if abs(h)<=eps*abs(s) then brea k end if;\n end do;\n s;\n end proc;\n\n p := ilog10(Dig its);\n q := Float(Digits,-p);\n maxit := trunc((p+(.02331061386+. 1111111111*q))*2.095903274)+2;\n saveDigits := Digits;\n Digits := Digits+min(iquo(Digits,3),5);\n xx := evalf(x);\n eps := Float(3, -saveDigits-1);\n if Digits<=trunc(evalhf(Digits)) and \n il og10(xx)trunc (evalhf(DBL_MIN_10_EXP)) then\n val := evalhf(doK(xx,eps,maxit)) \n else\n val := doK(xx,eps,maxit)\n end if;\n evalf[saveDi gits](val);\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "checking the procedure " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "Comparison of starting approximation wi th the inverse function." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "plot('K(x)-start_approx(x)',x=-10.. 16,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 541 200 200 {PLOTDATA 2 "6&-%'CURVESG6#7ex7$$!#5\"\"!$!%SX!\")7$$!+#*RFL%*!\"*$!&5+)F17$$!+d8s '=*F1$!'!Q-\"F17$$!+A(o,%*)F1$!'-58F17$$!+5;`,))F1$!'![]\"F17$$!+)\\%* Gm)F1$!'`GF17$$!+Zx#p])F1$ !'B??F17$$!+3\")f*[)F1$!'@-AF17$$!+p%oAZ)F1$!'4r?F17$$!+I)Q\\X)F1$!'$= %>F17$$!+_&z-U)F1$!'-!p\"F17$$!+u-i&Q)F1$!'aY9F17$$!+HA%eJ)F1$!&&>)*F1 7$$!+&=kgC)F1$!&$4bF17$$!+ShGw\")F1$!&,`\"F17$$!+(43l5)F1$\"&B7#F17$$! +_+tO!)F1$\"&NX&F17$$!+3?&p'zF1$\"&!p%)F17$$!+kR<(*yF1$\"'[<6F17$$!+>f RFyF1$\"'(zN\"F17$$!+F17$$!+B>+A^F1$! 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#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"-%$expG6#F-F.F(F(F (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$!0'4\"*ycM7**!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!3\\1fm/L;,i!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0'4\"*ycM7**!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "xx := Pi/4+exp(Pi/4);\nK(xx) ;\nevalf[12](%);\nx1 := evalf[12](xx);\nevalf[12](K(x1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,&*&\"\"%!\"\"%#PiG\"\"\"F*-%$expG6#,$*&F 'F(F)F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%!\"\"%#PiG\"\" \"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-)Rj\")R&y!#7" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#x1G$\"-99#y'yH!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-*Rj\")R&y!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "xx := ln(3)+ln(ln(3));\nK(xx );\nevalf[12](%);\nx1 := evalf[14](xx);\nevalf[12](K(x1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,&-%#lnG6#\"\"$\"\"\"-F'6#F&F*" }} {PARA 11 "" 1 "" {XPPMATH 20 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" 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-findin g we can solve the equation " }{XPPEDIT 18 0 "K(x) = 2;" "6#/-%\"KG6#% \"xG\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 28 "The solutio n is (of course) " }{XPPEDIT 18 0 "x = 2+exp(2);" "6#/%\"xG,&\"\"#\"\" \"-%$expG6#F&F'" }{TEXT -1 1 " " }{TEXT 285 1 "~" }{TEXT -1 14 " 9.389 056099. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "evalf(evalf(2+exp(2),13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*4c!*Q*!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "secant(K(x)=2,x=9..9.5,info= true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\" .c$p-L\"R*!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~-> ~~~G$\".\"ysG/*Q*!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximatio n~3~~->~~~G$\".605c!*Q*!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approx imation~4~~->~~~G$\".M*)4c!*Q*!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% 7approximation~5~~->~~~G$\".M*)4c!*Q*!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*4c!*Q*!\"*" }}}{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 "K(x)" "6#-%\"KG6#%\"xG" }{TEXT -1 8 " and K. 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'3**)Ffal$\"3LXW!)[50G>Ffal7$$\"35++]#Re`2*Ffal$\"3ghgz&)46LrUwcFhel7$$\"3E,+]d**Gp$*Ffa l$\"35P](>hDRJ$Fhel7$$\"3P+++Dws4%*Ffal$\"3E'\\XEfLr-*F]gl7$$\"3[**** \\#Hl,X*Ffal$!3&Rsa2qn<_\"Fhel7$$\"3O+++gHg!\\*Ffal$!3\"QpTYf!)Q#RFhel 7$$\"3D,+]F1/J&*Ffal$!3+2#fU8<$oiFhel7$$\"3O+++&Hy9d*Ffal$!3<^N$\\!ph? &)Fhel7$$\"3Z****\\if\">h*Ffal$!31VdBztwk5Ffal7$$\"3O+++ION_'*Ffal$!3= Z'*GgA#=E\"Ffal7$$\"3F++]A_ER(*Ffal$!3K(RDQ&**=>;Ffal7$$\"3=+++:oFfal7$$\"34++]2%) 38**Ffal$!3YCY%y1.)))>Ffal7$$\"3t)*\\i0j\"[$**Ffal$!31&>$pg_M)*>Ffal7$ $\"3;***\\P?Wl&**Ffal$!3%*4_Ki'*R**>Ffal7$$\"3e**\\(=5s#y**Ffal$!3kc[E %zh>*>Ffal7$$\"#5F]al$!3mBd=[K1w>Ffal-Fg`l6&Fi`lF\\alF\\alFj`l-%+AXESL ABELSG6$Q\"x6\"Q\"yF`[s-%%VIEWG6$;FbalFfjr;$!\"$F]al$\"#B!\"\"" 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 35 "allroots(K(x)=2*sin(3*x),x=- 2..10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-$!+]2.<%)!#5$!+N@R!3\"F%$\" +0^eV5!\"*$\"+MNp!=#F*$\"+\\!\\b+$F*$\"+tRM(R%F*$\"+#4W()*\\F*$\"+wo?% f'F*$\"+#4U,+(F*$\"+le87))F*$\"+%G6h(*)F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 15 ": The equat ion " }{XPPEDIT 18 0 "K(x) = 2*sin(3*x);" "6#/-%\"KG6#%\"xG*&\"\"#\"\" \"-%$sinG6#*&\"\"$F*F'F*F*" }{TEXT -1 18 " is equivalent to " } {XPPEDIT 18 0 "x = g(2*sin(3*x));" "6#/%\"xG-%\"gG6#*&\"\"#\"\"\"-%$si nG6#*&\"\"$F*F$F*F*" }{TEXT -1 51 ", so the same solutions can be obta ined as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 71 "g := x -> x+exp(x);\nallroots(g(2*sin(3*x))=x, x=-2..10,miniterations=9);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf* 6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\"\"-%$expG6#F-F.F(F(F(" }} {PARA 12 "" 1 "" {XPPMATH 20 "6-$!+]2.<%)!#5$!+N@R!3\"F%$\"+0^eV5!\"*$ \"+MNp!=#F*$\"+\\!\\b+$F*$\"+tRM(R%F*$\"+#4W()*\\F*$\"+wo?%f'F*$\"+#4U ,+(F*$\"+le87))F*$\"+%G6h(*)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 102 ": The inverse function K can be expressed \+ in terms of the Lambert W function by means of the formula: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "K(x)=x-LambertW(exp(x))" " 6#/-%\"KG6#%\"xG,&F'\"\"\"-%)LambertWG6#-%$expG6#F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "g := x -> x-LambertW(exp(x));\nxx := 12.34567891;\ng( xx);\nK(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%) operatorG%&arrowGF(,&9$\"\"\"-%)LambertWG6#-%$expG6#F-!\"\"F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+\"*ycM7!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*v&\\1B!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"+_d\\1B!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "There are, however, subtraction error and overflow proble ms in evaluating " }{XPPEDIT 18 0 "K(x)" "6#-%\"KG6#%\"xG" }{TEXT -1 18 " by this formula. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "xx := 10.^6;\ng(xx);\nK(xx);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"(+++\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"'b\"Q\"!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+u'\\:Q\"!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 27 "xx := 10.^10;\ng(xx);\nK(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$\"\" \"-%)LambertWG6#-%$expG6#F-!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"+++++5\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$!\" \"%)infinityG$!\"!\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+$4&e-B! \")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }