{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 "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis " -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{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 "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 } 1 1 0 0 0 0 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 "Map le 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 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Norma l" -1 258 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 66 "A procedure for evaluating the in verse hyperbolic tangent function" }}{PARA 0 "" 0 "" {TEXT -1 37 "by P eter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Vers ion: 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 a nd function approximation procedures including: " }{TEXT 0 5 "remez" } }{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 10 "fcnapp rx.m" }{TEXT -1 37 " contains the code 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 Maple session by a command similar to the one that follows, where the file 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 24 "load numerical functions" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 262 8 "numfcn.m" }{TEXT -1 72 " contains the code for the alternative mathematical functions including " } {TEXT 0 8 "arctanh_" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 123 "I t can be read into a Maple session by a command similar to the one tha t follows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "read \"K:\\\\Maple/procdrs/numfcn.m\";" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 46 "load n umerical integration procedures and data" }}{PARA 0 "" 0 "" {TEXT -1 18 "The Maple m-files " }{TEXT 262 6 "intg.m" }{TEXT -1 5 " and " } {TEXT 262 8 "gkdata.m" }{TEXT -1 67 " contain the code and data for th e numerical integration procedure " }{TEXT 0 8 "quad/Int" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "They ca n be read into a Maple session by commands similar to those that follo w, where the file paths give their location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "read \"K:\\\\Maple/procdrs/intg.m\";\nread \"K: \\\\Maple/procdrs/gkdata.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Graphing partial sums of the Maclaurin se ries for " }{XPPEDIT 18 0 "arctanh(x);" "6#-%(arctanhG6#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 25 "The Maclaurin series for " }{XPPEDIT 18 0 "arctanh(x );" "6#-%(arctanhG6#%\"xG" }{TEXT -1 5 " is " }{XPPEDIT 18 0 "Sum(x^( 2*n+1)/(2*n+1),n = 0 .. infinity) = x+x^3/3+x^5/5+x^7/7+` . . . `;" "6 #/-%$SumG6$*&)%\"xG,&*&\"\"#\"\"\"%\"nGF-F-F-F-F-,&*&F,F-F.F-F-F-F-!\" \"/F.;\"\"!%)infinityG,,F)F-*&F)\"\"$F8F1F-*&F)\"\"&F:F1F-*&F)\"\"(F " 0 "" {MPLTEXT 1 0 106 "x := 'x': i := 'i': n := 'n':\nATH := (n,x) -> sum(x^(2*i+1)/(2*i+1),i=0..n);\nAT H(1,x);\nATH(2,x);\nATH(3,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$AT HGf*6$%\"nG%\"xG6\"6$%)operatorG%&arrowGF)-%$sumG6$*&)9%,&%\"iG\"\"#\" \"\"F6F6F3!\"\"/F4;\"\"!9$F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,& %\"xG\"\"\"*&#F%\"\"$F%)F$F(F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,( %\"xG\"\"\"*&#F%\"\"$F%)F$F(F%F%*&#F%\"\"&F%)F$F,F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"xG\"\"\"*&#F%\"\"$F%)F$F(F%F%*&#F%\"\"&F%)F$F ,F%F%*&#F%\"\"(F%)F$F0F%F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "evalf(ATH(12,0.5));\nevalf(arctanh( 0.5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+S91$\\&!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+V91$\\&!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "The degree 1, 3 and 5 approximations \+ " }{XPPEDIT 18 0 "y = x;" "6#/%\"yG%\"xG" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "y = x+x^3/3;" "6#/%\"yG,&%\"xG\"\"\"*&F&\"\"$F)!\"\"F' " }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = x+x^3/3+x^5/5;" "6#/%\"yG,( %\"xG\"\"\"*&F&\"\"$F)!\"\"F'*&F&\"\"&F,F*F'" }{TEXT -1 28 " can be p lotted as follows." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = arctanh(x);" "6#/%\"yG-%(arctanhG6#%\"xG" }{TEXT -1 33 " is also pl otted for comparison. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 182 "ATH := (n,x) -> sum(x^(2*i+1)/(2*i+1),i=0..n):\nplot([arctanh(x),ATH(0,x),AT H(1,x),ATH(2,x)],x=-1.2..1.2,-1.2..1.2,\n color=[black,red,green,b lue],linestyle=[2,1$3],numpoints=100);" }}{PARA 13 "" 1 "" {GLPLOT2D 426 302 302 {PLOTDATA 2 "6(-%'CURVESG6%7`q7$$!3Y*****>3.P***!#=$!3TEp& [v)R**F*$!33yp)z:F?!HF-7$$!3E+++4wNy)*F*$!3QO')G r1:[DF-7$$!3h*****>-So\")*F*$!3wYn%opj>M#F-7$$!31+++@KG!p*F*$!3S!f\"*e 96h2#F-7$$!3]+++?ksj&*F*$!3+y3/d9e,>F-7$$!3[+++@QyS$*F*$!3Q-\")\\())f% *o\"F-7$$!3))*****zD.)*3*F*$!3#Gx\\sTC;_\"F-7$$!3u*****f9#zP))F*$!3'oe MqxrFR\"F-7$$!3!******>uK\\f)F*$!3'R/P\"p>S\"H\"F-7$$!3?+++PJRu$)F*$!3 q$R)))3%QD@\"F-7$$!3I+++;-:7\")F*$!3\"z#yNZAdI6F-7$$!3))*****H1'***)yF *$!3od_/Kfxo5F-7$$!3z*****4(zdJwF*$!3-Lr@(yLP+\"F-7$$!3C+++tT$GS(F*$!3 \")y!REYh5^*F*7$$!3))*****4(f'=:(F*$!3)=`,ue9s(*)F*7$$!3#******\\*[)G 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" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "A rational approximation for " }{XPPEDIT 18 0 "arctanh(x) " "6#-%(arctanhG6#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 31 "We choose a rationa l function " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "s(x) = (a[0]*x+a[1]*x^2+a[2]*x^3+` . . . `+a[m]*x^m)/(b[0]+b[1]*x+a[2]*x^2 +` . . . `+b[n]*x^n);" "6#/-%\"sG6#%\"xG*&,,*&&%\"aG6#\"\"!\"\"\"F'F/F /*&&F,6#F/F/*$F'\"\"#F/F/*&&F,6#F4F/*$F'\"\"$F/F/%(~.~.~.~GF/*&&F,6#% \"mGF/)F'F>F/F/F/,,&%\"bG6#F.F/*&&FB6#F/F/F'F/F/*&&F,6#F4F/*$F'F4F/F/F :F/*&&FB6#%\"nGF/)F'FNF/F/!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "to approximate " }{XPPEDIT 18 0 "arctanh(x)" "6#-%(arctan hG6#%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-3/5, 3/5 ];" "6#7$,$*&\"\"$\"\"\"\"\"&!\"\"F)*&F&F'F(F)" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 56 "First we find a rational approximation for the function " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "f(x) = PIECEWISE([(arctanh(x)-x)/(x^3) , x <> 0],[1/3, x = 0]);" "6#/-%\"fG6#%\"xG-%*PIECEWISEG6$7$*&,&-%(arc tanhG6#F'\"\"\"F'!\"\"F1*$F'\"\"$F20F'\"\"!7$*&F1F1F4F2/F'F6" }{TEXT -1 2 ". 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rational function " }{XPPEDIT 18 0 "s(x) = x+r(x)*` .`*x^3;" "6#/-%\"sG6#%\"xG,&F'\"\"\"*(-%\"rG6#F'F)%\".GF)F'\"\"$F)" } {TEXT -1 14 " approximates " }{XPPEDIT 18 0 "arctanh(x)" "6#-%(arctanh G6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "The relative error graph is drawn for " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" } {TEXT -1 11 ", which is " }{XPPEDIT 18 0 "s(x)" "6#-%\"sG6#%\"xG" } {TEXT -1 73 " with the coefficients rounded to 20 digits and arranged \+ in nested form. 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++++++](=dHdbF*$!'`J^F77$$\":++++++v$pRi(e&F*$!'(Q\"oF77$$\":++++++++A _zh&F*$!'\"zl(F77$$\":++++++D1Z!G[cF*$!'.WuF77$$\":++++++]7s3'ycF*$!'p ngF77$$\":+++++]il%Gx$p&F*$!'\\c\\F77$$\":++++++v=(p$*3dF*$!'['f$F77$$ \":+++++](=(4,Ts&F*$!'fJ?F77$$\":+++++++DAl#RdF*$!&1A$F77$$\":++++](oH %=8uu&F*$\"'k+jF*7$$\":+++++v$4Y6cbdF*$\"((=)e\"F*7$$\":++++]i!*y54Pw& F*$\"(#=PDF*7$$\":+++++](opq&=x&F*$\"(Y2Y$F*7$$\":+++++D\"G$*H:)y&F*$ \"(68;&F*7$$\":++++++vo\"*[W!eF*$\"('4UlF*7$$\":+++++vo/%[u?eF*$\"(Q* \\uF*7$$\":+++++]iSwSq$eF*$\"(+%QxF*7$$\":+++++DcwoOL&eF*$\"(lZG(F*7$$ \":++++++]7hK'peF*$\"(kK,'F*7$$\":++++](o/t0yxeF*$\"(&Qn]F*7$$\":+++++ vV[`Gf)eF*$\"(Rf#RF*7$$\":++++]iSm\\wS*eF*$\"(;zg#F*7$$\":+++++]P%eWA- fF*$\"(CG9\"F*7$$\":++++]PM-Us.\"fF*$!&WF%F77$$\":+++++DJ?Q?&=fF*$!'SY ?F77$$\":++++]7GQMom#fF*$!'jSOF77$$\":++++++DcI;[$fF*$!'0<^F77$$\":+++ +](=UnU'H%fF*$!'!*fjF77$$\":+++++v=#HA6^fF*$!'YFsF77$$\":++++]i:5>g#ff F*$!'F[vF77$$\":+++++]7G:3u'fF*$!'8 " 0 "" {MPLTEXT 1 0 69 "evalf(evalf( numapprox[infnorm](1-h(x)/arctanh(x),x=-3/5..3/5),25),5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&'o()!#B" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 36 "The maximum relative error in using " } {XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 16 " to approximate \+ " }{XPPEDIT 18 0 "arctanh(x);" "6#-%(arctanhG6#%\"xG" }{TEXT -1 17 " i n the interval " }{XPPEDIT 18 0 "[-3/5, 3/5];" "6#7$,$*&\"\"$\"\"\"\" \"&!\"\"F)*&F&F'F(F)" }{TEXT -1 12 " is about " }{XPPEDIT 18 0 "9*`. `*10^(-19);" "6#*(\"\"*\"\"\"%#.~GF%)\"#5,$\"#>!\"\"F%" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "xx := 0.6;\nevalf(evalf(h(xx),20),17);\nevalf(evalf(a rctanh(xx),20),17);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"'!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2JX*f0=ZJp!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"2JX*f0=ZJp!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "We can test the accuracy of the appro ximation " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 5 " for \+ " }{XPPEDIT 18 0 "arctanh(x);" "6#-%(arctanhG6#%\"xG" }{TEXT -1 35 " w ith random numbers between 0 and " }{XPPEDIT 18 0 "3/5;" "6#*&\"\"$\" \"\"\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 92 "The pr intout occurs only when the relative error is greater than or equal to the specified \"" }{TEXT 262 3 "eps" }{TEXT -1 3 "\".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 372 "randomize():\neps := Float(3,-16); \nfor i from 1 to 100 do\n xx := evalf(rand()/Float(1,12)*.6,20);\n \+ axx := evalf(arctanh(xx),20);\n hxx := evalf(h(xx),16);\n e := \+ evalf(abs((axx-hxx)/axx),20);\n if e>=eps then \n printf(\" tr ial no. %d, x = %.16f,\\n\",i,xx);\n printf(\" arctanh(x) = %.1 6f, h(x) = %.16f, rel error = %.2e\\n\\n\",axx,hxx,e); \n end if;\ne nd do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"$!#;" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 22, x = .1104654200658000," }}{PARA 6 "" 1 "" {TEXT -1 80 " arctanh(x) = .1109180608666014, h(x) = .11091 80608666014, rel error = 4.40e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 39 " trial no. 29, x = .1014543483318000," }}{PARA 6 "" 1 "" {TEXT -1 80 " arctanh(x) = .1018046033464812, h(x) \+ = .1018046033464812, rel error = 4.06e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 50, x = .594448887898 8000," }}{PARA 6 "" 1 "" {TEXT -1 80 " arctanh(x) = .6845182589828240 , h(x) = .6845182589828243, rel error = 3.80e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 60, x = .001 0528711134000," }}{PARA 6 "" 1 "" {TEXT -1 80 " arctanh(x) = .0010528 715024493, h(x) = .0010528715024493, rel error = 3.47e-16" }}{PARA 6 " " 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 80, x = .1498203019218000," }}{PARA 6 "" 1 "" {TEXT -1 80 " arctanh(x) = . 1509566066524168, h(x) = .1509566066524168, rel error = 3.25e-16" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial n o. 88, x = .1214358270084000," }}{PARA 6 "" 1 "" {TEXT -1 80 " arcta nh(x) = .1220380891582010, h(x) = .1220380891582010, rel error = 3.97e -16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "arctanh16" }}{PARA 0 "" 0 "" {TEXT -1 78 "Here is the code for the fixed precision versions of the function to evaluate " } {TEXT 262 10 "arctanh(x)" }{TEXT -1 65 ", which can be evaluated with \+ hardware floating point arithmetic." }}{PARA 0 "" 0 "" {TEXT 259 4 "No te" }{TEXT -1 45 ": These procedures make use of the procedure " } {TEXT 0 4 "ln16" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 909 "arctanh16 := proc(xx)\n l ocal x,z,a1,a2,a3,a4,a5,a6,\n b1,b2,b3,b4,b5,num,den; \n \n \+ x := evalf(xx); \n\n if abs(x)<0.6 then\n # coefficients of nu merator in rational approximation\n a1 := .33333333333333369102; \n a2 := -.79597883432769854579;\n a3 := .660861183050422127 84;\n a4 := -.2199978121091297804; \n a5 := .238105519169652 21691e-1;\n a6 := -.99082402325989929717e-4;\n # coefficient s of denominator in rational approximation\n b1 := -2.98793650298 29547281;\n b2 := 3.3467740223631470994;\n b3 := -1.72084696 73672415413;\n b4 := .39285967198094191770;\n b5 := -.299759 89951303021460e-1;\n # evaluate the rational approximation\n \+ z := x*x;\n num := (a1+(a2+(a3+(a4+(a5+a6*z)*z)*z)*z)*z)*z;\n \+ den := 1+(b1+(b2+(b3+(b4+b5*z)*z)*z)*z)*z;\n (1+num/den)*x;\n \+ else\n ln16((1+x)/(1-x))*0.5;\n end if;\nend proc: # arctanh1 6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Testing th e procedure arctanh16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "We can use Maple's function " }{TEXT 0 7 "arctanh" } {TEXT -1 41 " to check the accuracy of the procedures " }{TEXT 0 9 "ar ctanh16" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "xx := 0.59;\nevalf(evalhf(arctanh16(xx)),16);\nevalf(evalhf(arctanh(x x)),16);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#f!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1='zvngmx'!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1='zvngmx'!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "evalf(plot(1-'arctanh16'(x)/ arctanh(x),x=0..0.99,color=blue),20);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6#7bfm7$$\"5+++DJqX\\Vn!#B$\"\"!F ,7$$\"5+++D19*)p[8!#AF+7$$\"5++]P4r$[I-#F0$\"\"&!#?7$$\"5+++]7GyR(p#F0 F+7$$\"5+++v=Un4YSF0F+7$$\"5++++Dccz%R&F0F+7$$\"5+++]P%[$>#4)F0F+7$$\" 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6FJ7$$\"5](oaN%ftEO')F6FJ7$$\"5+]Pf$ehuFk)F6FJ7$$\"5+v=njG\"*yb')F6FJ7 $$\"5+++vVTO!)o')F6FJ7$$\"5++](ok_vWw)F6FJ7$$\"5++++]6u9g))F6FJ7$$\"5+ D\"y]]e+q'))F6F+7$$\"5+]i:geP&Q())F6FJ7$$\"5+vVB:Kpq!)))F6F+7$$\"5++DJ q0,c()))F6F+7$$\"5+D1RDzKT%*))F6FJ7$$\"5+](o/GXm7!*)F6F+7$$\"5+voaNE'> \"3*)F6F+7$$\"5++]i!**zs\\\"*)F6FJ7$$\"5+]7y+Z\"z'G*)F6F+7$$\"5++v$4T \\&QU*)F6F_`p7$$\"5+Dc,mn'Q#\\*)F6F+7$$\"5+]P4@T=4c*)F6F_`p7$$\"5+v=_x\"*F6F+7$$\"5+++++*)[fv#*F6F+7$$\"5++v$f`+yyK*F6F jbo7$$\"5++](=<7h,Q*F6Fhdp7$$\"5++D\"y!QUWK%*F6F+7$$\"5+++vVats%[*F6FJ 7$$\"5++++]\"e5[e*F6F+7$$\"5+++Dc3Q*[o*F6F\\cm7$$\"5+vVt_XzL)p*F6F47$$ \"5+](=#\\#3#y6(*F6F+7$$\"5+DJqX>iAD(*F6F+7$$\"5++v=Uc.nQ(*F6F+7$$\"5+ ]i:NI'ebw*F6F+7$$\"5++]7G/pW#z*F6F+7$$\"5](=njF(*o\"*z*F6Fg_p7$$\"5+v$ 4Y7/\"*e!)*F6Fg_p7$$\"5]i:&G(4Jh7)*F6F+7$$\"5+]P4@y^L>)*F6F+7$$\"5]PfL pYs0E)*F6Fg_p7$$\"5+D\"yv^JzF$)*F6F+7$$\"5]7.#eOQ,&R)*F6F][l7$$\"5++D1 9_MAY)*F6Fg_p7$$\"5](o/B1_XH&)*F6F+7$$\"5+voa5*en'f)*F6F+7$$\"5]i!*yed '*Qm)*F6F+7$$\"5+]7.2E<6t)*F6F+7$$\"5]PMFb%zL)z)*F6Fg_p7$$\"5+Dc^.jeb' ))*F6F][l7$$\"5]7yv^JzF$*)*F6F+7$$\"#**FQFg_p-%+AXESLABELSG6$Q\"x6\"Q! Fbis-%'COLOURG6&%$RGBGF+F+$\"*++++\"F[t-%%VIEWG6$;F+F\\is%(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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "A procedure for evaluating the inverse hy perbolic tangent function: " }{TEXT 0 8 "arctanh_" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "arc tanh_: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 20 "Calling Sequence: " }{TEXT -1 14 "arctanh_( x ) " }{TEXT 264 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 15 "Parameters: " }{TEXT 265 21 "x - a real constant" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 268 12 "Description:" }{TEXT -1 1 " " }{TEXT 267 14 "The procedure " }{TEXT 0 8 "arctanh_" } {TEXT 266 44 " calculates arctanh(x) for a real number x." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure active open the subsection, place the cursor anywh ere after the prompt [ > and press [Enter].\nYou can then close up th e subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 24 "arctanh_: implem entation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 16 ": The procedure " }{TEXT 0 16 "`evalf/arctanh_` " }{TEXT -1 22 " calls the procedures " }{TEXT 0 4 "ln25" }{TEXT -1 5 " and " }{TEXT 0 4 "ln55" }{TEXT -1 40 ", which give 25 and 55 digit v alues for " }{XPPEDIT 18 0 "f(x)=2*arctanh(x)" "6#/-%\"fG6#%\"xG*&\"\" #\"\"\"-%(arctanhG6#F'F*" }{XPPEDIT 18 0 "``=ln((1+x)/(1-x))" "6#/%!G- %#lnG6#*&,&\"\"\"F*%\"xGF*F*,&F*F*F+!\"\"F-" }{TEXT -1 29 ", when x ha s small magnitude." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 5701 "arctanh_ := proc(x::algebraic)\n local n ;\n\n if nargs <> 1 then\n error \"expecting 1 argument, got %1 \", nargs;\n end if;\n if type(x,'float') then evalf('arctanh_'(x) )\n elif (type(x,'numeric') or type(x,'radnum')) \n \+ and signum(0,x,0)<0 then -arctanh_(-x)\n elif type(x,`*`) a nd type(op(1,x),'numeric') \n and signum(0,op(1,x), 0)<0 then -arctanh_(-x)\n elif type(x,'function') and nops(x)=1 then \n n := op(0,x);\n if n='tanh_' or n='tanh' then op(1,x) \n \+ else 'arctanh_'(x)\n end if\n elif type(x,'realcons') then \n if x=0 then 0\n elif x='infinity' then 'infinity'\n \+ elif type(x,'rational') then ln((1+x)/(1-x))/2 \n else 'arctanh_' (x)\n end if\n else 'arctanh_'(x)\n end if;\nend proc: # of a rctanh_\n\n`evalf/arctanh_` := proc(xx)\n local hfDigits,x,z,saveDig its,eps,maxit,\n k,sum,term,pow,fact,val;\n\n if not type(xx,re alcons) then return 'arctanh_'(xx) end if;\n\n if abs(evalf(xx))>1 t hen\n error \"argument must in the interval [-1,1]\"\n end if; \n\n # Use the fixed precision log function for low precision values \n hfDigits := trunc(evalhf(Digits));\n if Digits<=hfDigits then\n x := evalf(xx,hfDigits+2);\n if abs(x)evalhf(DBL_MIN*10) then\n return evalf(evalhf(arctan h16(x)));\n end if;\n end if;\n \n\011\011 # Increase precisio n for the computation by a few digits\n\011\011 saveDigits := Digits; \n Digits := Digits+length(Digits)+1;\n x := evalf(xx);\n\n if a bs(x)<.142926235 then\n if saveDigits<27 then \n val := 0 .5*ln25(x);\n return evalf[saveDigits](val);\n elif save Digits<57 then \n val := 0.5*ln55(x);\n return evalf[s aveDigits](val);\n else\n # use Maclaurin series\n \+ eps := Float(1,-saveDigits);\n maxit := Digits*3;\n \+ pow := x;\n z := x*x;\n sum := x; \n for k fro m 3 to maxit by 2 do\n pow := pow*z;\n term := p ow/k;\n sum := sum + term;\n if abs(term)<=eps*a bs(sum) then break end if;\n end do;\n end if;\n ret urn evalf[saveDigits](sum)\n else \n # use the formula arctanh( x)=ln((1+x)/(1-x))/2 \n Digits := Digits-1;\n x := (1+x)/(1- x);\n sum := `evalf/ln_`(x)*0.5;\n Digits := saveDigits;\n \+ return evalf(sum)\n end if;\nend proc: # of `evalf/arctanh_`\n\n ln55 := proc(x)\n local a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,\n b1,b2 ,b3,b4,b5,b6,b7,b8,b9,b10,num,den,z;\n\n a1:=.6666666666666666666666 66666666666666666666666666666666629320;\n a2:=-3.0236719106603269537 4131357837970900929046094017020955983410;\n a3:=5.759265864316915949 76838022357015201757299310460552013015450;\n a4:=-5.9817477224574319 6802350394240008949133082311184652539030090;\n a5:=3.680203026552955 04297682303530772108130921487817409523520081;\n a6:=-1.3634589329129 9499809977611952490346537230565968815304818524;\n a7:=.2954893913414 70810823863848579141174585905442723820950292210;\n a8:=\n -.34521 8307298122912747982961243678449589476265469236438942557e-1;\n a9:=\n .182311944616087243686809701698491402559491532501548316428824e-2; \n a10:=\n -.2805953797335146834107676898806613181994378048324284 65245728e-4; \n b1:=-5.135507865990490430611970367569563513935691410 25531482367664;\n b2:=11.2916320874982396115911811273255375632923330 744915142229073;\n b3:=-13.88000222695107201063245348008469145947538 58151261832560495;\n b4:=10.4481436639608638297607013886211764621819 878400996024825408;\n b5:=-4.959555753674581061688602810843142219091 09173633790418206902;\n b6:=1.47344026819822124028258428849727917681 078450178610613406050;\n b7:=-.2627126804287995740189799979037847653 30071519158327209078978;\n b8:=\n .257017881054974116836861028950 396828946662746815094303063510e-1;\n b9:=\n -.1150722174405968665 06002711755418289182221522714374675952815e-2;\n b10:=\n .15141274 8080791346839127036171599604505552523448918494509310e-4;\n\n z := x* x;\n num := (a1+(a2+(a3+(a4+(a5+(a6+(a7+(a8+\n (a9+a10*z)* z)*z)*z)*z)*z)*z)*z)*z)*z;\n den := 1+(b1+(b2+(b3+(b4+(b5+(b6+(b7+(b 8+\n (b9+b10*z)*z)*z)*z)*z)*z)*z)*z)*z)*z;\n (2+num/den)*x; \nend proc: # of ln55\n\nln25 := proc(x)\n local a1,a2,a3,a4,a5,b1,b 2,b3,b4,num,den,z;\n\n a1 := .666666666666666666666667294110;\n a2 := -1.14797224113924323579879814202;\n a3 := .590350305326987361465 764591737;\n a4 := -.865664194907786855930604193839e-1;\n a5 := +. 464652936012806969997236389703e-3;\n b1 := -2.3219583617088648536963 6882863;\n b2 := 1.85012904644437138176470533310;\n b3 := -.578135 378275181709679450917850;\n b4 := .559246057352437269125330586694e-1 ;\n\n z := x*x;\n num := (a1+(a2+(a3+(a4+a5*z)*z)*z)*z)*z;\n den := 1+(b1+(b2+(b3+b4*z)*z)*z)*z;\n (2+num/den)*x;\nend proc: # of ln 25 \n\narctanh16 := proc(xx)\n local x,z,a1,a2,a3,a4,a5,a6,\n \+ b1,b2,b3,b4,b5,num,den; \n \n x := evalf(xx); \n\n if abs(x)<0 .6 then\n # coefficients of numerator in rational approximation\n a1 := .33333333333333369102;\n a2 := -.79597883432769854579 ;\n a3 := .66086118305042212784;\n a4 := -.21999781210912978 04; \n a5 := .23810551916965221691e-1;\n a6 := -.99082402325 989929717e-4;\n # coefficients of denominator in rational approxi mation\n b1 := -2.9879365029829547281;\n b2 := 3.34677402236 31470994;\n b3 := -1.7208469673672415413;\n b4 := .392859671 98094191770;\n b5 := -.29975989951303021460e-1;\n # evaluate the rational approximation\n z := x*x;\n num := (a1+(a2+(a3 +(a4+(a5+a6*z)*z)*z)*z)*z)*z;\n den := 1+(b1+(b2+(b3+(b4+b5*z)*z) *z)*z)*z;\n (1+num/den)*x;\n else\n ln16((1+x)/(1-x))*0.5; \n end if;\nend proc: # arctanh16 " }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 8 "arctanh _" }{TEXT -1 11 ": examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Here is a graph and \+ some numerical examples." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 178 "p1 := plot(arctanh_(x),x=-1..1,y=- 3..3,color=red,thickness=2):\np2 := plots[implicitplot](\{x=-1,x=1\},x =-1..1,y=-3..3,\n linestyle=4,color=black):\nplots[display]( [p1,p2]);" }}{PARA 13 "" 1 "" {GLPLOT2D 254 327 327 {PLOTDATA 2 "6'-%' CURVESG6%7`o7$$!+ynP')**!#5$!+h^^XO!\"*7$$!+dNvs**F*$!+h1g)H$F-7$$!+N. 8f**F*$!+Rp_&4$F-7$$!+9r]X**F*$!+9XM^HF-7$$!+r1E=**F*$!+r%Gzu#F-7$$!+H U,\"*)*F*$!+:IS.EF-7$$!+V8_O)*F*$!+$z)H*R#F-7$$!+e%G?y*F*$!+>B3aAF-7$$ !+(oUIn*F*$!+$[(e[?F-7$$!+!>F-7$$!+W5Su$*F*$!+5#*\\;\"F-7$$!+@OS,zF*$!+(>0=2\"F-7$$!+/R=0vF*$!+8-TT(*F*7$$!+P8#\\4(F*$! +mZgh))F*7$$!+/siqmF*$!+?>Ka!)F*7$$!+(y$pZiF*$!+%o,zK(F*7$$!+$yaE\"eF* $!+4\"=Pk'F*7$$!+\">s%HaF*$!+LQD$3'F*7$$!+]$*4)*\\F*$!+kv_!\\&F*7$$!+] _&\\c%F*$!+>GvG\\F*7$$!+]1aZTF*$!+WTR8WF*7$$!+/#)[oPF*$!+_x\"Q'RF*7$$! +$=exJ$F*$!+pSA[MF*7$$!+L2$f$HF*$!+%\\P\\-$F*7$$!+PYx\"\\#F*$!+?jNXDF* 7$$!+L7i)4#F*$!+M6FI@F*7$$!+P'psm\"F*$!+)Q\")Ho\"F*7$$!+74_c7F*$!+4r>j 7F*7$$!*3x%z#)F*$!+3QZ)H)!#67$$!*?PQM%F*$!+:CdYVFcu7$$!(\"zr)*F*$!+2Uz r)*!#87$$\")!o2J%F-$\"+u6W8VFcu7$$\")%Q#\\\")F-$\"+X.Nn\")Fcu7$$\"*;*[ H7F-$\"+G6uN7F*7$$\"*qvxl\"F-$\"+!eF-7$$\"+I:3u'*F*$ \"+m]?]?F-7$$\"+?5s#y*F*$\"+05pbAF-7$$\"+l2/P)*F*$\"+pI!4S#F-7$$\"+50O \"*)*F*$\"+,H+0EF-7$$\"+$Q?&=**F*$\"+!=E&\\FF-7$$\"+b-oX**F*$\"+T+%H&H F-7$$\"+#>g#f**F*$\"+$R@r4$F-7$$\"+G,%G(**F*$\"+BS>+LF-7$$\"+k+U')**F* $\"+eu5ZOF-7$%%FAILGFe`l-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F^alF]al -%*THICKNESSG6#\"\"#-F$6=7$7$$!\"\"F^al$!\"$F^al7$Fgal$!3!)*********** *fF!#<7$F[bl7$Fgal$!3d************>DF^bl7$F`bl7$Fgal$!3O************zA F^bl7$Fdbl7$Fgal$!3:************R?F^bl7$Fhbl7$Fgal$!3;*************z\" F^bl7$F\\cl7$Fgal$!3;************f:F^bl7$F`cl7$Fgal$!3<************>8F ^bl7$Fdcl7$Fgal$!3=************z5F^bl7$Fhcl7$Fgal$!3#>************R)!# =7$F\\dl7$Fgal$!3+#*************fF_dl7$Fadl7$Fgal$!33#************f$F_ dl7$Fedl7$Fgal$!3=#************>\"F_dl7$Fidl7$Fgal$\"3s2++++++7F_dl7$F ]el7$Fgal$\"3i2++++++OF_dl7$Fael7$Fgal$\"3a2++++++gF_dl7$Feel7$Fgal$\" 3Y2++++++%)F_dl7$Fiel7$Fgal$\"3u++++++!3\"F^bl7$F]fl7$Fgal$\"3t++++++? 8F^bl7$Fafl7$Fgal$\"3s++++++g:F^bl7$Fefl7$Fgal$\"3q+++++++=F^bl7$Fifl7 $Fgal$\"3#4++++++/#F^bl7$F]gl7$Fgal$\"39,+++++!G#F^bl7$Fagl7$Fgal$\"3N ,+++++?DF^bl7$Fegl7$Fgal$\"3c,+++++gFF^bl7$Figl7$Fgal$\"3y,++++++IF^bl -Fg`l6&Fi`lF^alF^alF^al-%*LINESTYLEG6#\"\"%-F$6=7$7$$\"2m************* **F^blF\\bl7$FjhlFial7$7$FjhlFablFihl7$7$FjhlFeblF^il7$7$FjhlFiblF`il7 $7$FjhlF]clFbil7$7$FjhlFaclFdil7$7$FjhlFeclFfil7$7$FjhlFiclFhil7$7$Fjh lF]dlFjil7$7$FjhlFbdlF\\jl7$7$FjhlFfdlF^jl7$7$FjhlFjdlF`jl7$7$FjhlF^el Fbjl7$7$FjhlFbelFdjl7$7$FjhlFfelFfjl7$7$FjhlFjelFhjl7$7$FjhlF^flFjjl7$ 7$FjhlFbflF\\[m7$7$FjhlFfflF^[m7$7$FjhlFjflF`[m7$7$FjhlF^glFb[m7$7$Fjh lFbglFd[m7$7$FjhlFfglFf[m7$7$FjhlFjglFh[m7$7$FjhlF^hlFj[mF`hlFbhl-%+AX ESLABELSG6%Q\"x6\"Q\"yFa\\m-%%FONTG6#%(DEFAULTG-%%VIEWG6$;Fgal$\"\"\"F ^al;Fial$\"\"$F^al" 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" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "Digits := 100:\nx x := sqrt(2)/10;\nevalf(arctanh_(xx));\nevalf(evalf(arctanh(xx),Digits +2));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG,$*&\"#5 !\"\"\"\"##\"\"\"F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q%3O$=(zf d*y)\\y&H\"\\X(G_(*R'e!Rpi1`nVcH&z-f1IqSEtnR/y;VcPU\"!$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q%3O$=(zfd*y)\\y&H\"\\X(G_(*R'e!Rpi1`nVc H&z-f1IqSEtnR/y;VcPU\"!$+\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "D igits := 100:\nxx := sqrt(2)/2;\nevalf(arctanh_(xx));\nevalf(evalf(arc tanh(xx),Digits+2));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#xxG,$*&\"\"#!\"\"F'#\"\"\"F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"_qH/c-\">*oqLy3E?A%=xLl3cH`2TN;EG.;G!4Bzz\\K4EBDIa>qet8))!$+\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qH/c-\">*oqLy3E?A%=xLl3cH`2TN;EG.; G!4Bzz\\K4EBDIa>qet8))!$+\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "Digits := 23:\nxx := 0.691; \nevalf(arctanh_(xx));\nevalf(evalf(arctanh(xx),Digits+2));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"$\"p!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"8]\"3=uT+g.n)\\)!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"8]\"3=uT+g.n)\\)!#B" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 23 "We solve \+ the equation " }{XPPEDIT 18 0 "arctanh(x) = 2*x-x^3;" "6#/-%(arctanhG 6#%\"xG,&*&\"\"#\"\"\"F'F+F+*$F'\"\"$!\"\"" }{TEXT -1 21 " using the \+ function " }{TEXT 0 8 "arctanh_" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot([arctanh_(x),2*x-x^3],x=0..1.2,-0.2..2.5 );" }}{PARA 13 "" 1 "" {GLPLOT2D 334 280 280 {PLOTDATA 2 "6&-%'CURVESG 6$7V7$$\"\"!F)F(7$$\"++&ech#!#6$\"+h_D;EF-7$$\"+v*G:*[F-$\"+:fV&*[F-7$ $\"+]L)4X(F-$\"+U!=[Y(F-7$$\"+X.u-5!#5$\"+z:715F=7$$\"+Gy:f7F=$\"+;k(e E\"F=7$$\"+e'*)o\\\"F=$\"+wDA3:F=7$$\"+)>ZIu\"F=$\"+U'G5w\"F=7$$\"+yOi (*>F=$\"+G.&[-#F=7$$\"+GPQ^AF=$\"+JIi!H#F=7$$\"+IrS7DF=$\"+znOnDF=7$$ \"+&o;Bu#F=$\"+4!RV\"GF=7$$\"+!RS6+$F=$\"+l#\\k4$F=7$$\"+]o-hKF=$\"+*H 4YQ$F=7$$\"+5cZ6NF=$\"+d6_nOF=7$$\"+yq!*QPF=$\"+7\\QHRF=7$$\"+!4X$4SF= $\"+n$>wC%F=7$$\"+g:WQUF=$\"+#H(oBXF=7$$\"+=_$\\]%F=$\"+sG>`[F=7$$\"+g s#3u%F=$\"+SjB`^F=7$$\"+=#Q'**\\F=$\"+z!zD\\&F=7$$\"+_u3Y_F=$\"+\"oun# eF=7$$\"+v8B.bF=$\"+8rW)='F=7$$\"+o(p$RdF=$\"+b0uLlF=7$$\"+Dp2%*fF=$\" +N@AApF=7$$\"+ygkeiF=$\"+#Q')eM(F=7$$\"+-V&*)['F=$\"+RS)Qt(F=7$$\"+&\\ $pPnF=$\"+ZS9w\")F=7$$\"+?am%*pF=$\"+mHbi')F=7$$\"+:B1YsF=$\"+k.ws\"*F =7$$\"+Q<:b8Fau7$$\"+=3;%**)F=$\"+.Z:p9Fau7$$\"+[=iY#*F=$\"+ qFau7$$\"+QM&=v*F=$\"+K:\\)=#Fau7$$\"+p?29)*F=$\"+sURM BFau7$$\"+*p!Hw)*F=$\"+1UnRDFau7$$\"+:+S2**F=$\"+6*y_o#Fau7$$\"+I$4&Q* *F=$\"+89w!*GFau7$$\"+()R1a**F=$\"+_$pp.$Fau7$$\"+X'='p**F=$\"+;\"pSC$ Fau7$$\"+tfRx**F=$\"+=V6#R$Fau7$$\"+-L<&)**F=$\"+M#eJg$Fau7$$\"+J1&H** *F=$\"+c%)4vRFau7$%%FAILGFb[l-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6$7SF' 7$$\"3h*******\\ech#!#>$\"3KX4$HXF&H_F`\\l7$$\"3-+++v*G:*[F`\\l$\"3q7# p8'RNr(*F`\\l7$$\"3u******\\L)4X(F`\\l$\"3upB='4gg[\"!#=7$$\"3)****** \\MSF+\"F]]l$\"3]#\\fTD)R&*>F]]l7$$\"3#)****\\Fy:f7F]]l$\"3%=)z=z>N)\\ #F]]l7$$\"3')****\\d'*)o\\\"F]]l$\"3=!Gz!G)Q-'HF]]l7$$\"3w****\\(>ZIu \"F]]l$\"3c')R;**o8LMF]]l7$$\"3u****\\xOi(*>F]]l$\"3)3n>]t&F]]l7$$\"3[******\\o-hKF]]l$\"3;1]L[kEvhF]]l7$$\"3(***** **4cZ6NF]]l$\"3w+A!34q**e'F]]l7$$\"3S****\\xq!*QPF]]l$\"3st_(=SO^&pF]] l7$$\"3&********3X$4SF]]l$\"3m8S&*[S>utF]]l7$$\"3s******f:WQUF]]l$\"3$ z/J93tar(F]]l7$$\"3f****\\<_$\\]%F]]l$\"3))>l0,!>c4)F]]l7$$\"3**)***** fs#3u%F]]l$\"3M#>vieKhT)F]]l7$$\"3!)****\\<#Q'**\\F]]l$\"3%p[UAvZ&\\() F]]l7$$\"33++]_u3Y_F]]l$\"3-'f)HKkP[!*F]]l7$$\"3[*****\\PJK]&F]]l$\"39 ic&Rby(R$*F]]l7$$\"3%*****\\n(p$RdF]]l$\"3_x`0m,<)e*F]]l7$$\"3A*****\\ #p2%*fF]]l$\"3bQ3am%*pF]]l$\"3w!>*zpmrc5F\\dl7$$\"3?*****\\Jig C(F]]l$\"3/WG_C\\vo5F\\dl7$$\"3G****\\PEdR2\"F\\dl$\"3'4To)e= H#4*F]]l7$$\"3&*****\\o#R05\"F\\dl$\"3[_Z8D.?\"o)F]]l7$$\"3()*****>`9V 7\"F\\dl$\"3c$\\b$)*)>SF)F]]l7$$\"3)****\\<#Rm\\6F\\dl$\"3s()>AV%eyz(F ]]l7$$\"3%****\\A_ER<\"F\\dl$\"3krEV$>31I(F]]l7$$\"3%**************>\" F\\dl$\"3`,+++++?nF]]l-Fd[l6&Ff[lF(Fg[lF(-%+AXESLABELSG6$Q\"x6\"Q!Ff[m -%%VIEWG6$;F($\"#7Fi[l;$!\"#Fi[l$\"#DFi[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "fsolve(ar ctanh_(x)=2*x-x^3,x=0.8);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+9O;gz !#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "fsolve(arctanh(x)=2*x-x^3,x=0.8);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+9O;gz!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 22 "We find the va lue of " }{XPPEDIT 18 0 "Int(arctanh(1/(2+2*x^2)),x=0..3)" "6#-%$IntG 6$-%(arctanhG6#*&\"\"\"F*,&\"\"#F**&F,F**$%\"xGF,F*F*!\"\"/F/;\"\"!\" \"$" }{TEXT -1 20 " using the function " }{TEXT 0 8 "arctanh_" }{TEXT -1 2 ". 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