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" }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple session by a command similar to the o ne 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 Ma ple m-file " }{TEXT 267 8 "numfcn.m" }{TEXT -1 72 " contains the code \+ for the alternative mathematical functions including " }{TEXT 0 3 "ln_ " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into \+ a Maple session by a command similar to the one that follows, where th e 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 numeri cal integration procedures and data" }}{PARA 0 "" 0 "" {TEXT -1 18 "Th e Maple m-files " }{TEXT 267 6 "intg.m" }{TEXT -1 5 " and " }{TEXT 267 8 "gkdata.m" }{TEXT -1 67 " contain the code and data for the nume rical integration procedure " }{TEXT 0 8 "quad/Int" }{TEXT -1 25 " use d in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "They can be rea d into a Maple session by commands similar to those that follow, where the file paths give their location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "read \"K:\\\\Maple/procdrs/intg.m\";\nread \"K:\\\\Ma ple/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 19 "Using the function " }{XPPEDIT 18 0 "2*arctanh(x) \+ = ln((1+x)/(1-x));" "6#/*&\"\"#\"\"\"-%(arctanhG6#%\"xGF&-%#lnG6#*&,&F &F&F*F&F&,&F&F&F*!\"\"F1" }{TEXT -1 43 " to evaluate the natural logar ithm function" }}{PARA 0 "" 0 "" {TEXT -1 72 "We can evaluate the natu ral logarithm function by means of the function " }{XPPEDIT 18 0 "f(x) = ln((1+x)/(1-x))" "6#/-%\"fG6#%\"xG-%#lnG6#*&,&\"\"\"F-F'F-F-,&F-F-F '!\"\"F/" }{XPPEDIT 18 0 "``= 2*arctanh(x)" "6#/%!G*&\"\"#\"\"\"-%(arc tanhG6#%\"xGF'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "x=(1+s)/(1-s)" "6#/%\"xG*&,&\"\"\"F'%\"sGF'F',&F'F'F(! \"\"F*" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "x-s*x=1+s" "6#/,&%\" xG\"\"\"*&%\"sGF&F%F&!\"\",&F&F&F(F&" }{TEXT -1 10 ", so that " } {XPPEDIT 18 0 "x-1=s*x+s" "6#/,&%\"xG\"\"\"F&!\"\",&*&%\"sGF&F%F&F&F*F &" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "s=(x-1)/(x+1)" "6#/%\"sG*&,&%\" xG\"\"\"F(!\"\"F(,&F'F(F(F(F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " }{XPPEDIT 18 0 "ln(x)=ln((1+s)/(1-s))" "6#/-%#lnG6 #%\"xG-F%6#*&,&\"\"\"F,%\"sGF,F,,&F,F,F-!\"\"F/" }{XPPEDIT 18 0 "``=f( s)" "6#/%!G-%\"fG6#%\"sG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 34 "The Maclaurin series expansion of " }{XPPEDIT 18 0 "f(x)" "6#-%\"f G6#%\"xG" }{TEXT -1 4 " is " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(x)=2*x+2*x^3/3+2*x^5/5+2*x^7/7+` . . . `" "6#/-%\"fG6 #%\"xG,,*&\"\"#\"\"\"F'F+F+*(F*F+*$F'\"\"$F+F.!\"\"F+*(F*F+*$F'\"\"&F+ F2F/F+*(F*F+*$F'\"\"(F+F5F/F+%(~.~.~.~GF+" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 68 "Fewer terms of this series are needed than those o f the series for " }{XPPEDIT 18 0 "ln(1+x)" "6#-%#lnG6#,&\"\"\"F'%\"x GF'" }{TEXT -1 64 " to evaluate the natural logarithm function to a gi ven accuracy." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 22 "In order to calculate " }{XPPEDIT 18 0 "ln(x)" "6#-%#lnG6 #%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[sqrt(2)/2,sq rt(2)]" "6#7$*&-%%sqrtG6#\"\"#\"\"\"F(!\"\"-F&6#F(" }{TEXT -1 23 ", we need to calculate " }{XPPEDIT 18 0 "ln((1+x)/(1-x))" "6#-%#lnG6#*&,& \"\"\"F(%\"xGF(F(,&F(F(F)!\"\"F+" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[2*sqrt(2)-3,3-2*sqrt(2)]" "6#7$,&*&\"\"#\"\"\"-%%sqrtG 6#F&F'F'\"\"$!\"\",&F+F'*&F&F'-F)6#F&F'F," }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 36 "This follows from the fact that, if " }{XPPEDIT 18 0 "g(x) = (x+1)/(x-1);" "6#/-%\"gG6#%\"xG*&,&F'\"\"\"F*F*F*,&F'F*F* !\"\"F," }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "g(sqrt(2)/2) = 2*sqrt(2 )-3;" "6#/-%\"gG6#*&-%%sqrtG6#\"\"#\"\"\"F+!\"\",&*&F+F,-F)6#F+F,F,\" \"$F-" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(sqrt(2)) = 3-2*sqrt(2);" "6#/-%\"gG6#-%%sqrtG6#\"\"#,&\"\"$\"\"\"*&F*F--F(6#F*F-!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Argument reduction" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 19 "The Maple function \+ " }{TEXT 0 6 "ilog10" }{TEXT -1 20 " gives the exponent " }{XPPEDIT 18 0 "k = ilog10(x);" "6#/%\"kG-%'ilog10G6#%\"xG" }{TEXT -1 11 " such \+ that " }{XPPEDIT 18 0 "10^k <= x;" "6#1)\"#5%\"kG%\"xG" }{XPPEDIT 18 0 "`` < 10^(k+1);" "6#2%!G)\"#5,&%\"kG\"\"\"F)F)" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 10 "Then, if " }{XPPEDIT 18 0 "u = x*`.`*10^ (-k);" "6#/%\"uG*(%\"xG\"\"\"%\".GF')\"#5,$%\"kG!\"\"F'" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "1<=u" "6#1\"\"\"%\"uG" }{XPPEDIT 18 0 "`` <10" "6#2%!G\"#5" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 80 "Since Maple's software floating point arithmetic base is 10, the computatio n of " }{TEXT 267 9 "ilog10(x)" }{TEXT -1 75 " is very efficient, and \+ does not require the computation of any logarithms." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 283 8 "Examples" }{TEXT -1 2 " : " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "ilog10(0.012345);\nilog10(0.12345);\nilog10(1.2345); \nilog10(12.345);\nilog10(123.45);\nilog10(1234.5);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Note that, if " }{XPPEDIT 18 0 "x = u*`.`*10^k;" "6#/%\"xG*(%\"uG\"\"\"%\" .GF')\"#5%\"kGF'" }{TEXT -1 7 ", then " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x) = ln(u)+k*ln(10);" "6#/-%#lnG6#%\"xG,&-F%6 #%\"uG\"\"\"*&%\"kGF,-F%6#\"#5F,F," }{TEXT -1 13 " ------- (i)." }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 284 12 "____________" }{TEXT -1 16 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "This formula can be used to reduce the argument ran ge of the natural logarithm function to the interval " }{XPPEDIT 18 0 "[1,10]" "6#7$\"\"\"\"#5" }{TEXT -1 37 " at the cost of storing the co nstant " }{XPPEDIT 18 0 "ln(10)" "6#-%#lnG6#\"#5" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The argum ent range can be further reduced to the interval " }{XPPEDIT 18 0 "[1/ R, R];" "6#7$*&\"\"\"F%%\"RG!\"\"F&" }{TEXT -1 9 " , where " } {XPPEDIT 18 0 "R = 10^(1/8);" "6#/%\"RG)\"#5*&\"\"\"F(\"\")!\"\"" } {TEXT -1 13 ", as follows." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "Divide " }{TEXT 292 1 "u" }{TEXT -1 17 " success ively by " }{XPPEDIT 18 0 "T=R^2" "6#/%\"TG*$%\"RG\"\"#" }{XPPEDIT 18 0 "``=10^(1/4)" "6#/%!G)\"#5*&\"\"\"F(\"\"%!\"\"" }{TEXT -1 11 " to ob tain " }{TEXT 285 1 "v" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "1<=v" "6# 1\"\"\"%\"vG" }{XPPEDIT 18 0 "`` < T;" "6#2%!G%\"TG" }{TEXT -1 11 " su ch that " }{XPPEDIT 18 0 "u = v*`.`*T^p;" "6#/%\"uG*(%\"vG\"\"\"%\".GF ')%\"TG%\"pGF'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "0 <= p;" "6#1\"\"! %\"pG" }{XPPEDIT 18 0 "`` < 3;" "6#2%!G\"\"$" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "u = v*`.`*R^(2*p); " "6#/%\"uG*(%\"vG\"\"\"%\".GF')%\"RG*&\"\"#F'%\"pGF'F'" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 7 "Divide " }{TEXT 288 1 "v" }{TEXT -1 4 " by " }{TEXT 289 1 "R" }{TEXT -1 11 " to obtain " }{XPPEDIT 18 0 "w = v/R;" "6#/%\"wG*&%\"vG\"\"\"%\"RG!\"\"" }{TEXT -1 7 " where " } {XPPEDIT 18 0 "1/R<=w" "6#1*&\"\"\"F%%\"RG!\"\"%\"wG" }{XPPEDIT 18 0 " `` " 0 "" {MPLTEXT 1 0 113 "fn1 := x -> if abs(x)<=.142 9262348 then ln((1+x)/(1-x)) else FAIL end if;\nplot('fn1'(x),x=-0.2.. 0.2,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$fn1Gf*6#%\"xG6 \"6$%)operatorG%&arrowGF(@%1-%$absG6#9$$\"+[BEH9!#5-%#lnG6#*&,&F1\"\" \"F:F:F:,&F:F:F1!\"\"F<%%FAILGF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7L7$$!3'******>$3CH9!#=$!3uH'Gl V(=yGF*7$$!3()******\\D6C9F*$!3!*4:1sxrnGF*7$$!30+++oU)*=9F*$!35dnQo' \\s&GF*7$$!3++++a[bw8F*$!3WH\\=A(*pqFF*7$$!3-+++Ta7M8F*$!3CTs?sFD%o#F* 7$$!3-+++e(Q&\\7F*$!3>cRphr?7DF*7$$!3/+++d4`i6F*$!3atS*>\">iNBF*7$$!3/ +++QW*e3\"F*$!3K5]F@hQ!=#F*7$$!3A++++()>'***!#>$!3DQe<=!Rf+#F*7$$!3a++ ++0\"*H\"*FS$!3!G&erL63J=F*7$$!3#)*******H\"3&H)FS$!3K<6_(=PGm\"F*7$$! 3-+++5k(p`(FS$!3S=L.a$f-^\"F*7$$!3s******pj^NmFS$!35CmZ%=c!H8F*7$$!3y* *****p9'=(eFS$!3Y%FS$!3))>j*[j>%*R)FS7$$!3$)******z#RXL$FS$!3s9[.F?brmFS 7$$!3()******H=/8DFS$!3sd%[_6Ur-&FS7$$!3++++?a*el\"FS$!3eC&[je$47LFS7$ $!3a*******Ruwo)!#?$!3.$)*eefyvt\"FS7$$!37++++!eV(>!#@$!31)R480;([RF[r 7$$\"3\\********e`@')Feq$\"39Ki*\\!*\\Vs\"FS7$$\"3<+++qw%)H;FS$\"3w\"z T:-%)*fKFS7$$\"3)*******>$y*eCFS$\"3$[e?hA[*=\\FS7$$\"3r*******R^bJ$FS $\"38z#GZBMNj'FS7$$\"3;+++]5a`TFS$\"35\"eo,:k=J)FS7$$\"3y******>\"RV' \\FS$\"3n]?V>m%o$**FS7$$\"3M+++]@fkeFS$\"3Gqqx,fEu6F*7$$\"3a+++I&4Nn'F S$\"3;kE\"\\h)oO8F*7$$\"31++++:?PvFS$\"3s]8BHYI5:F*7$$\"3q*******yM)>$ )FS$\"3y\")4m*GAym\"F*7$$\"3M*******pfa<*FS$\"3TB-@LzES=F*7$$\"3Q+++gg `!)**FS$\"3'=$eyA\\x-?F*7$$\"3/+++#G2A3\"F*$\"3!H)RC=T#H<#F*7$$\"3%*** ***H)G[k6F*$\"3bl6fP$z&RBF*7$$\"3++++7yh]7F*$\"3#pxP(H&*R9DF*7$$\"3'** ****p)fdL8F*$\"3Mx3veR8$o#F*7$$\"3)******RM%*fP\"F*$\"3])=J'eqbpFF*7$$ \"33+++-FT=9F*$\"3-$G9f0$3cGF*7$$\"3++++\"4rOU\"F*$\"33Hpt#e;o'GF*7$$ \"3#*******z%H*G9F*$\"357>+] " 0 "" {MPLTEXT 1 0 643 "ln2 := proc(xx)\n local x,s,R,T,L,k,r;\n\n x := evalf(xx);\n \n # reduce the argument range to 10^(-1/8)<=z<=10^(1/8)\n k := 0; \n r := 0;\n if x<.7498942093 or x>1.333521432 then \n k := \+ ilog10(x);\n x := Scale10(x,-k); # x*10^(-k); \n if x>1.3335 21432 then\n R := evalf(10^(1/8));\n T := R^2;\n \+ while x>1.778279410 do\n x := x/T;\n r := r+2 ;\n end do;\n x := x/R;\n r := r+1;\n end if;\n end if;\n\n # transformation to use approx for ln((1+x)/(1- x))\n s := (x-1)/(x+1);\n if k<>0 or r<> 0 then L := evalf(ln(10)/ 8) end if;\n fn1(s)+(8*k+r)*L;\nend proc; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ln2Gf*6#%#xxG6)%\"xG%\"sG%\"RG%\"TG%\"LG%\"kG%\"rG6 \"F0C)>8$-%&evalfG6#9$>8)\"\"!>8*F:@$52F3$\"+$4U*)\\(!#52$\"+K9_L8!\"* F3C%>F9-%'ilog10G6#F3>F3-%(Scale10G6$F3,$F9!\"\"@$FCC'>8&-F56#*$)\"#5# \"\"\"\"\")Ffn>8'*$)FU\"\"#Ffn?(F0FfnFfnF02$\"+5%z#yF3*&F3FfnF inFQ>F<,&FF3*&F3FfnFUFQ>F<,&F8%*&,&F3FfnFfnFQFf n,&F3FfnFfnFfnFQ@$50F9F:0F8(-F56#,$*&FenFfn-%#lnG6#FZFfnFfn,&-%$fn 1G6#F[pFfn*&,&*&FgnFfnF9FfnFfnF " 0 "" {MPLTEXT 1 0 26 "plot('l n2'(x),x=0.01..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7gn7$$\"\"\"!\"#$!+'=q^g%!\"*7$$\"+1*z/o\"! #6$!+t24'3%F-7$$\"+7)f4O#F1$!+\\>5YPF-7$$\"+=(R9/$F1$!+\">RG\\$F-7$$\" +D'>>s$F1$!+B1$4H$F-7$$\"+P%zG3&F1$!+mAHzHF-7$$\"+]#RQW'F1$!+mc/UFF-7$ $\"+v)ed;*F1$!+0bp*Q#F-7$$\"+]yw)=\"!#5$!+UxmH@F-7$$\"+v<:LsTFU$!+)G@9 u)FU7$$\"+RO%HI'FU$!+OKo:YFU7$$\"3Q+++sL\"yW)!#=$!3?RtcwXx'o\"F^p7$$\" 33+++\"*)[#e5!#<$\"3=)\\\"=Y_bhc!#>7$$\"31+++S1;c7Fdp$\"3sa\"p&z&*f!G# F^p7$$\"+#z'3h9F-$\"+s`!=z$FU7$$\"+7<-t;F-$\"+)*RJY^FU7$$\"+`pF%)=F-$ \"+\"oTaL'FU7$$\"+O*y:5#F-$\"+i$*)oU(FU7$$\"+S'yHH#F-$\"+#)o^)H)FU7$$ \"+v$\\%3DF-$\"+Qyk'>*FU7$$\"+`[![s#F-$\"+7oR-5F-7$$\"+XMILHF-$\"+V#Hh 2\"F-7$$\"+U,kAJF-$\"+W)y'Q6F-7$$\"+(yzxM$F-$\"+wtH37F-7$$\"+*f-&QNF-$ \"+WOqj7F-7$$\"+p&e.w$F-$\"+ sTF-$\"+\">V%G9F-7$$\"+/yOxVF-$\"+'eZkZ\"F-7$$\"+?,W\"f%F-$\"+EP>C:F-7 $$\"+J`-)y%F-$\"+y!=hc\"F-7$$\"+S!p++&F-$\"+?cF-$\"+=oF-7$$\"+R(yG3(F-$\"+E/od>F-7$$\"+g5d'H(F-$\"+?XS()>F-7$$\"+\"))Qw\\ (F-$\"+c\")e9?F-7$$\"+)o7yq(F-$\"+\\WBU?F-7$$\"+,gH8zF-$\"+#QW&o?F-7$$ \"+()zTG\")F-$\"+:jO&4#F-7$$\"+FggN$)F-$\"+Ai`?@F-7$$\"+Ad[Z&)F-$\"+sr jX@F-7$$\"+Z4hd()F-$\"+XJ#*p@F-7$$\"+1Up]*)F-$\"+%4J<>#F-7$$\"+5%*)><* F-$\"+5U:;AF-7$$\"+z%=*p$*F-$\"+'R/vB#F-7$$\"+\\@&4e*F-$\"+wpxfAF-7$$ \"+)HQHy*F-$\"+z)R1G#F-7$$\"#5\"\"!$\"+$4&e-BF--%'COLOURG6&%$RGBG$F\\^ l!\"\"$F]^lF]^lFf^l-%+AXESLABELSG6$Q\"x6\"Q!F[_l-%%VIEWG6$;F(F[^l%(DEF AULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve \+ 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Th e following example illustrates how the procedure " }{TEXT 0 3 "ln2" } {TEXT -1 8 " works. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "xx := 90;\nprintlevel := 10:\nln2(xx);\np rintlevel := 1:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"#!*" }} {PARA 9 "" 1 "" {TEXT -1 25 "\{--> enter ln2, args = 90" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"xG$\"#!*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"#!*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"RG$\"+K9_L8!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG$\"+5%z#y%\"xG$\"+F>2h]!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"+%*)\\g%G!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\" \"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"+p9X+;!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"+*Gp,?\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\" (" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG$\"+mR!z4*!#6" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"LG$\"+m8ByG!#5" }}{PARA 9 "" 1 "" {TEXT -1 37 "\{--> enter fn1, args = .9097903966e-1" }}{PARA 9 "" 1 "" {TEXT -1 40 "<-- exit fn1 (now in ln2) = .1824626210\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+q'4)*\\%!\"*" }}{PARA 9 "" 1 "" {TEXT -1 46 "<-- ex it ln2 (now at top level) = 4.499809670\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+q'4)*\\%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(ln(xx));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+q'4)*\\%!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 75 "An arbitrary precision procedure to evaluate the n atural logarithm function" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "lnAP: implementation" }} {PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 16 ": The procedure " } {TEXT 0 4 "lnAP" }{TEXT -1 20 " uses two constants " }{XPPEDIT 18 0 "l n(10);" "6#-%#lnG6#\"#5" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "10^(-1/8) ;" "6#)\"#5,$*&\"\"\"F'\"\")!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 43 "This limits the precision of the procedure " }{TEXT 0 4 "lnAP" }{TEXT -1 32 " when x is not in the interval " }{XPPEDIT 18 0 "[10^(-1/8),10^(1/8)]" "6#7$)\"#5,$*&\"\"\"F(\"\")!\"\"F*)F%*&F(F(F) F*" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9186 "lnAP := proc(xx::realcons)\n local x, term,eps,k,i,maxit,saveDigits,\n extraDigits,sum,s,t,z,R,T,L,r,m,pow ,even;\n \n # Increase precision\n saveDigits := Digits;\n Dig its := Digits+length(Digits)+1;\n x := evalf(xx);\n \n # Re duce the range to 10^(-1/8)<=x<10^(1/8) approximately\n k := 0;\n \+ r := 0;\n if x>1.3335214321633240257 or x<.74989420933245582730 then \n k := ilog10(x);\n x := Scale10(x,-k); # x*10^(-k)\n \+ if x>1.3335214321633240257 then\n R := evalf(root8of10inv);\n \+ # multiply successively by 10^(-1/4)\n if x>3.16227766 01683793320 then\n T := R^2;\n while x>1.7782794 100389228012 do\n x := x*T;\n r := r+2;\n \+ end do;\n end if;\n x := x*R;\n r : = r+1;\n end if;\n end if; \n\n t := x-1;\n if k=0 and \+ r=0 then\n extraDigits := -ilog10(t);\n # avoid subtraction \+ error when x is close to 1\n if extraDigits>1 then\n t := evalf[Digits+extraDigits](xx-1);\n end if;\n end if;\n s := \+ t/(x+1);\n\n # Initialisation for Maclaurin series loop\n eps := F loat(5,-Digits);\n maxit := Digits*3;\n\n pow := s;\n z := s*s; \n sum := s; \n for i from 3 to maxit by 2 do\n pow := pow*z; \n term := pow/i;\n sum := sum+term;\n if abs(term)<=ep s*abs(sum) then break end if;\n end do;\n sum := 2*sum;\n\n if k <>0 or r<>0 then\n L := evalf(Ln10)*0.125;\n sum := sum+(8*k +r)*L\n end if;\n\n Digits := saveDigits;\n evalf(sum);\nend pro c: # of lnAP\n\n`evalf/constant/root8of10inv` := proc()\nlocal d,r,r2, r4;\nglobal _root8of10inv;\n if Digits<=55 then evalf(.7498942093324 558273021842756151364384418679181649710146)\n elif Digits<=length(op (1,_root8of10inv)) then evalf(_root8of10inv)\n else\n d := leng th(op(1,_root8of10inv));\n r := _root8of10inv;\n while d " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Testing \+ the procedure lnAP" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 49 "plot('lnAP(x)',x=0.01..10,color=red,thicknes s=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 450 266 266 {PLOTDATA 2 "6'-%'CURV ESG6#7gn7$$\"\"\"!\"#$!+'=q^g%!\"*7$$\"+1*z/o\"!#6$!+v24'3%F-7$$\"+7)f 4O#F1$!+]>5YPF-7$$\"+=(R9/$F1$!+\">RG\\$F-7$$\"+D'>>s$F1$!+B1$4H$F-7$$ \"+P%zG3&F1$!+nAHzHF-7$$\"+]#RQW'F1$!+nc/UFF-7$$\"+v)ed;*F1$!+2bp*Q#F- 7$$\"+]yw)=\"!#5$!+UxmH@F-7$$\"+v<:LsTFU$!+%G@9u)FU7$$\"+RO%HI'FU$!+WK o:YFU7$$\"+sL\"yW)FU$!+xXx'o\"FU7$$\"+\"*)[#e5F-$\"+Y_bhcF17$$\"+S1;c7 F-$\"+!e*f!G#FU7$$\"+#z'3h9F-$\"+p`!=z$FU7$$\"+7<-t;F-$\"+)*RJY^FU7$$ \"+`pF%)=F-$\"+!oTaL'FU7$$\"+O*y:5#F-$\"+l$*)oU(FU7$$\"+S'yHH#F-$\"+&) o^)H)FU7$$\"+v$\\%3DF-$\"+Myk'>*FU7$$\"+`[![s#F-$\"+7oR-5F-7$$\"+XMILH F-$\"+W#Hh2\"F-7$$\"+U,kAJF-$\"+V)y'Q6F-7$$\"+(yzxM$F-$\"+wtH37F-7$$\" +*f-&QNF-$\"+VOqj7F-7$$\"+p&e.w$F-$\"+sTF-$\"+\">V%G9F-7$$\"+/yOxVF-$\"+'eZkZ\"F-7$$\"+?,W\"f%F-$ \"+EP>C:F-7$$\"+J`-)y%F-$\"+y!=hc\"F-7$$\"+S!p++&F-$\"+?cF-$\"+=oF-7$$\"+R(yG3(F-$\"+F/od>F-7$$\"+g5d'H(F-$\"+?XS()>F-7$ $\"+\"))Qw\\(F-$\"+b\")e9?F-7$$\"+)o7yq(F-$\"+\\WBU?F-7$$\"+,gH8zF-$\" +$QW&o?F-7$$\"+()zTG\")F-$\"+:jO&4#F-7$$\"+FggN$)F-$\"+Ai`?@F-7$$\"+Ad [Z&)F-$\"+srjX@F-7$$\"+Z4hd()F-$\"+XJ#*p@F-7$$\"+1Up]*)F-$\"+%4J<>#F-7 $$\"+5%*)><*F-$\"+5U:;AF-7$$\"+z%=*p$*F-$\"+'R/vB#F-7$$\"+\\@&4e*F-$\" +wpxfAF-7$$\"+)HQHy*F-$\"+y)R1G#F-7$$\"#5\"\"!$\"+$4&e-BF--%'COLOURG6& %$RGBG$\"*++++\"!\")$Fj]lFj]lFd^l-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q \"x6\"Q!F]_l-%%VIEWG6$;F(Fh]l%(DEFAULTG" 1 2 0 1 10 2 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "xx:= 9;\nDigits := 100:\nprintlevel := 1:\nlnAP(xx); \nprintlevel := 1:\nevalf(ln(xx));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qyTY F'4:BZP$z@Pke))\\Fn')QpM!*)\\Xc6\")\\H490XQZ!\\!z#Q>iLxXA(>#!#**" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qyTYF'4:BZP$z@Pke))\\Fn')QpM!*)\\X c6\")\\H490XQZ!\\!z#Q>iLxXA(>#!#**" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "xx:= 0.0012345:\nDigits := \+ 100:\nlnAP(xx);\nevalf(ln(xx));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!_qAUpT(4!RK2!z_C#yV#)=jZ![(>N;c(>ha&31\\'zX'>ZE1?-\"* R!z\"\\#*3(p'!#**" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!_qAUpT(4!RK2!z_ C#yV#)=jZ![(>N;c(>ha&31\\'zX'>ZE1?-\"*R!z\"\\#*3(p'!#**" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "xx:= 1.000000001234567895:\nDigits := 10:\nlnAP(xx);\nevalf(ln(xx));\neval f(evalf(ln(xx),20));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+%*ycM7!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+&*********!#>" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%*ycM7!#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Speed comparisons " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Maple's function " } {TEXT 0 2 "ln" }{TEXT -1 16 " is faster than " }{TEXT 0 4 "lnAP" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 157 "st := time():\nfor i to 500 do t := lnAP(rand() *Float(1,-11)) end do:\ntime()-st;\nst := time():\nfor i to 500 do t : = ln(rand()*Float(1,-11)) end do:\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%a:!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$5#!\" $" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 185 "st := time():\nDigits := 50:\nfor i to 100 do t := l nAP(rand()*Float(1,-11)) end do:\ntime()-st;\nst := time():\nfor i to \+ 100 do t := ln(rand()*Float(1,-11)) end do:\ntime()-st;\nDigits := 10: " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%z:!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$/\"!\"$" }}}{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 70 "A fixed pr ecision procedure to evaluate the natural logarithm function" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 97 "In this section we construct a fixed precision version of the e xponential function which can use " }{TEXT 259 34 "hardware floating p oint arithmetic" }{TEXT -1 1 "." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "A base 2 integer logarithm function " }}{PARA 0 "" 0 "" {TEXT -1 33 "The variable precision procedure " }{TEXT 0 4 "lnAP" }{TEXT -1 32 " made use of the Maple function " }{TEXT 0 6 "ilog10" }{TEXT -1 68 " \+ which cannot be evaluated using hardware floating point arithmetic." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "evalhf(ilog10(12.567));" }}{PARA 8 "" 1 "" {TEXT -1 63 "Error, u nable to evaluate built-in function `ilog10` in evalhf\n" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 197 "When using hardwa re floating point arithmetic, it is more appropriate to work with a ba se 2 integer logarithm function in order to obtain the argument reduct ion for the natural logarithm function. " }}{PARA 0 "" 0 "" {TEXT -1 33 "The standard C library function \"" }{TEXT 267 5 "frexp" }{TEXT -1 137 "\" performs this task by extracting the binary exponent of a b inary (double precision) floating point number. However, the Maple fun ction " }{TEXT 0 5 "ilog2" }{TEXT -1 29 " appears to be unrelated to \+ \"" }{TEXT 267 5 "frexp" }{TEXT -1 35 "\", as it cannot be evaluated u sing " }{TEXT 0 6 "evalhf" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "evalhf(ilog2(12.567)) ;" }}{PARA 8 "" 1 "" {TEXT -1 62 "Error, unable to evaluate built-in f unction `ilog2` in evalhf\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{TEXT 0 7 "intlog2" }{TEXT -1 93 " given below does the job, however, by the simply searching throug h an array of powers of 2. " }}{PARA 0 "" 0 "" {TEXT -1 198 "A binary \+ search would be inefficient under general usage where any input argume nt is likely to be closer to 0 than to the largest and smallest double precision floating point numbers which are near " }{XPPEDIT 18 0 "2^1 020" "6#*$\"\"#\"%?5" }{TEXT -1 1 " " }{TEXT 271 1 "~" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "10^308" "6#*$\"#5\"$3$" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "2^(-1020)" "6#)\"\"#,$\"%?5!\"\"" }{TEXT -1 1 " " } {TEXT 272 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-307)" "6#)\"#5,$ \"$2$!\"\"" }{TEXT -1 16 " respectively. \n" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 41 "evalhf(DBL_MAX_EXP);\nevalhf(DBL_MIN_EXP);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"%C5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!%@5\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 91 "Co nsequently, it is better to use a method which gradually broadens the \+ search away from 0." }}{PARA 0 "" 0 "" {TEXT -1 24 "The following proc edure " }{TEXT 0 4 "hunt" }{TEXT -1 75 " is adapted from a program in \+ the book \"Numerical Recipies in C\", page 118." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "hunt: u sage " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 275 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 276 4 " " }{TEXT -1 21 "hunt( A, m, n, j, x) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 274 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 113 " A - an array or hfarray wi th bounds m..n and entries which are floating point numbers in ascendi ng order" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 23 4 "j - " }{TEXT -1 69 " an index between m and n \+ inclusive from which to start the search\n" }}{PARA 0 "" 0 "" {TEXT 23 6 " x - " }{TEXT -1 31 " a real floating point number" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 116 ": No type checking is performed. It is the reponsibility of th e user to ensure that sensible arguments are supplied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 273 12 "Description:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 4 "hunt" }{TEXT -1 48 " searches the array A for the i ndex i such that " }{XPPEDIT 18 0 "A[i]<=x" "6#1&%\"AG6#%\"iG%\"xG" } {XPPEDIT 18 0 "`` " 0 "" {MPLTEXT 1 0 949 "hunt := proc(A,m,n,j,x)\n \+ local jLo,jHi,jM,inc;\n\n jLo := j;\n if jLon then\n \+ jLo := m;\n jHi := n;\n else \n inc := 1;\n if x>= A[jLo] then \n if jLo=n then return jLo end if;\n jHi \+ := jLo+1;\n while x>=A[jHi] do \n jLo := jHi;\n \+ inc := inc+inc;\n jHi := jLo+inc;\n if j Hi>n then \n jHi := n+1;\n break;\n \+ end if;\n end do;\n else\n if jLo=m then ret urn jLo end if;\n jHi := jLo-1;\n while x 1 do\n \+ jM := trunc((jHi+jLo)/2);\n if x>=A[jM] then jLo := jM else jHi := jM end if;\n end do;\n return jLo;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 4 "hun t" }{TEXT -1 160 " can be used in conjunction with an hfarray, with en tries which are powers of 2 in ascending order, to give a reasonably f ast base 2 integer logarithm function." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "POW2 := hfarray(-1019..1 019):\nfor i from -1019 to 1019 do\n POW2[i] := evalhf(2^i):\nend do :" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 7 "Exa mple" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "xx := 1234;\nprintlevel := 10:\nevalf(hun t(POW2,-1019,1019,0,xx));\nprintlevel := 1:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"%M7" }}{PARA 9 "" 1 "" {TEXT -1 132 "\{--> ente r hunt, args = Array(-1019..1019, [...], datatype = float[8], storage \+ = rectangular, order = C_order), -1019, 1019, 0, 1234" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$jLoG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $incG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$jHiG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$jLoG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$incG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$jHiG\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$jLoG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$incG\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$jHiG \"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$jLoG\"\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$incG\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% $jHiG\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#jMG\"#6" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$jHiG\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %#jMG\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$jLoG\"\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#jMG\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$jLoG\"#5" }}{PARA 9 "" 1 "" {TEXT -1 38 "<-- exit hunt (now at to p level) = 10\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#5\"\"!" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "ilog2(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "It is convenient to \+ set up a function " }{TEXT 0 7 "intlog2" }{TEXT -1 139 " which uses th is method to calculate base 2 integer logarithms with the specific int ention of using hardware floating point arithmetic via " }{TEXT 0 6 "e valhf" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "intlog2: usage " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 279 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 280 4 " " }{TEXT -1 12 "intlog2( x) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT 278 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT 23 6 " x - " }{TEXT -1 31 " a rea l floating point number" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 116 ": No type checking is performed . It is the reponsibility of the user to ensure that a sensible argume nt is supplied." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT 277 12 "Description:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 7 "intlog2" }{TEXT -1 24 " determines the integer " }{TEXT 267 12 "k=intlog2(x)" }{TEXT -1 11 " such that " } {XPPEDIT 18 0 "2^k <= x;" "6#1)\"\"#%\"kG%\"xG" }{XPPEDIT 18 0 "`` < 2 ^(k+1);" "6#2%!G)\"\"#,&%\"kG\"\"\"F)F)" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 25 "intlog2: \+ implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1193 "POW2 := hfarray(-1019..1019):\nfor i from -1 019 to 1019 do\n POW2[i] := evalhf(2^i):\nend do:\n\nintlog2 := proc (x)\n if x>POW2[1019] or xn then\n jLo := m;\n jHi := n;\n else \n inc := 1;\n if x>=A[jLo] then \n if jLo=n then r eturn jLo end if;\n jHi := jLo+1;\n while x>=A[jHi] do \n jLo := jHi;\n inc := inc+inc;\n j Hi := jLo+inc;\n if jHi>n then \n jHi := n+1; \n break;\n end if;\n end do;\n \+ else\n if jLo=m then return jLo end if;\n jHi := jLo-1 ;\n while x 1 do\n jM := trunc((jHi+jLo)/2);\n if \+ x>=A[jM] then jLo := jM else jHi := jM end if;\n end do;\n return \+ jLo;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 282 8 "Ex amples" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "xx := 123.45678:\nintlog2(xx);\nilo g2(xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "xx := 12345;\nevalhf(intlog2(xx));\nilog2 (xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"&XB\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#8\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#8 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "xx := 65535.999;\nevalhf(intlog2(xx));\nilog2(xx);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\")**f`l!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#:\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "xx := evalhf(2^(-1019));\nevalhf(intlog2(xx));\nilog2(xx);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"3;hd!o3f+y\"!$C$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!%>5\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#! %>5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "xx := evalhf(2^(-1020));\nevalhf(intlog2(xx));\nilog2 (xx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"!F&" }}{PARA 8 "" 1 "" {TEXT -1 32 "Error, argument is out of range\n" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$%)infinityG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "xx := 65535.999;\nst := t ime():\nfor i to 100 do yy := intlog2(xx) end do:\nyy;\ntime()-st;\nst := time():\nfor i to 100 do yy := ilog2(xx) end do:\nyy;\ntime()-st; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\")**f`l!\"$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#V!\" $" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#S!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 22 "As is to be expected, " }{TEXT 0 7 "intlog2" } {TEXT -1 43 " is a bit slow when handling large numbers." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "xx : = 5e300;\nst := time():\nfor i to 100 do yy := intlog2(xx) end do:\nyy ;\ntime()-st;\nst := time():\nfor i to 100 do yy := ilog2(xx) end do: \nyy;\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"&\"$+$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$)**" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$3$!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"$)**" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#R!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Argument reduction " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "The function " }{TEXT 0 7 "intlog2" }{TEXT -1 44 " of the previous section gives t he exponent " }{XPPEDIT 18 0 "k = intlog2(x);" "6#/%\"kG-%(intlog2G6#% \"xG" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "2^k <= x;" "6#1)\"\"# %\"kG%\"xG" }{XPPEDIT 18 0 "`` < 2^(k+1);" "6#2%!G)\"\"#,&%\"kG\"\"\"F )F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Then, if " } {XPPEDIT 18 0 "u = x*`.`*2^(-k);" "6#/%\"uG*(%\"xG\"\"\"%\".GF')\"\"#, $%\"kG!\"\"F'" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "1<=u" "6#1\" \"\"%\"uG" }{XPPEDIT 18 0 "`` < 2;" "6#2%!G\"\"#" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Note that , if " }{XPPEDIT 18 0 "x = u*`.`*2^k;" "6#/%\"xG*(%\"uG\"\"\"%\".GF') \"\"#%\"kGF'" }{TEXT -1 7 ", then " }}{PARA 257 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "ln(x) = ln(u)+k*ln(2);" "6#/-%#lnG6#%\"xG,&-F%6#%\"u G\"\"\"*&%\"kGF,-F%6#\"\"#F,F," }{TEXT -1 13 " ------- (i)." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 269 12 "____________" }{TEXT -1 16 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "This formula can be used to reduce the argument range of the natural logarithm function to the interval " }{XPPEDIT 18 0 "[1,2 ]" "6#7$\"\"\"\"\"#" }{TEXT -1 37 " at the cost of storing the constan t " }{XPPEDIT 18 0 "ln(2)" "6#-%#lnG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 58 "The argument ra nge can be further reduced to the interval " }{XPPEDIT 18 0 "[sqrt(2)/ 2, sqrt(2)];" "6#7$*&-%%sqrtG6#\"\"#\"\"\"F(!\"\"-F&6#F(" }{TEXT -1 18 " , by dividing by " }{XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"#" } {TEXT -1 16 ", if necessary. " }}{PARA 0 "" 0 "" {TEXT -1 52 "If this \+ is done then the formula (i) is replaced by " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln(x) = ln (u)+(k+1/2)*ln(2);" "6#/-%#lnG6#%\"xG,&-F%6#%\"uG\"\"\"*&,&%\"kGF,*&F, F,\"\"#!\"\"F,F,-F%6#F1F,F," }{TEXT -1 15 " ------- (ii). " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 270 15 "_______________" }{TEXT -1 18 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "To illustrate this idea, we define a function " } {TEXT 0 3 "ln1" }{TEXT -1 58 " using Maple's natural logarithm functio n on the interval " }{XPPEDIT 18 0 "[sqrt(2)/2, sqrt(2)];" "6#7$*&-%%s qrtG6#\"\"#\"\"\"F(!\"\"-F&6#F(" }{TEXT -1 59 ", but ensure that it gi ves no value outside this interval.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "evalf(evalf(sqrt(2)/2,13));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7y1rq!#5" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 116 "ln1 := x -> if x>=.70710678 12 and x<=1.414213562 then ln(x) else FAIL end if;\nplot('ln1(x)',x=0. 6..1.6,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ln1Gf*6#%\" xG6\"6$%)operatorG%&arrowGF(@%31$\"+7y1rq!#59$1F2$\"+iN@99!\"*-%#lnG6# F2%%FAILGF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6$7O7$$\"3!*******>>1uq!#=$!3#3/%o!\\-:Y$F*7$$\"3e*****fy Vk3(F*$!3)p([[*f9SW$F*7$$\"3W+++`c#))4(F*$!3UH%HhBdlU$F*7$$\"3$)***** \\Q*eBrF*$!3]()z#\\nL(F*$!3q\"*oxPZ)zG$F*7$$\"3E+++[!3uC(F*$!39&4.n)>T>KF*7$$\"37+++!pt* \\tF*$!3IQan$f$))yIF*7$$\"3))*****4LRDX(F*$!3))f,!))o-.%HF*7$$\"3=+++) R'okwF*$!3YQ=5]\\hfEF*7$$\"33+++1J:wyF*$!3_B)G%G\\X(Q#F*7$$\"3))*****z gsO4)F*$!3Oh*e3'\\-:@F*7$$\"3!)*****R!RE&G)F*$!3!>'4%R*e1\")=F*7$$\"3f *****\\K]4])F*$!3<:31G82C;F*7$$\"3r*****\\PAvr)F*$!3qWW`n-]s8F*7$$\"3J +++v'Hi#*)F*$!3CKZ2h*4f8\"F*7$$\"3m*****p*ev:\"*F*$!3KVQMRf2e#*!#>7$$ \"3;+++347T$*F*$!3MyO4RO)e\"oFgp7$$\"3a+++LY.K&*F*$!3k=BQU+p#z%Fgp7$$ \"31+++\"o7Tv*F*$!3AUqm[$3'*[#Fgp7$$\"3%)*****HQ*o]**F*$!3f8=++&fK%\\! #?7$$\"35+++=lj;5!#<$\"3I'*z_3Z'*\\;Fgp7$$\"3,+++aR>Y29\"F`r$\"3*Q!3lEg#oJ\"F*7$$\"3/+++eWZh6 F`r$\"3aQ/dCG!p\\\"F*7$$\"3/+++&y))G=\"F`r$\"3k(Gzep&fz;F*7$$\"3'***** *f_QQ?\"F`r$\"3C&>XFB_^&=F*7$$\"3++++y%3TA\"F`r$\"39@4!)f!G@-#F*7$$\"3 -+++/[hY7F`r$\"3%ojM#3sJ/AF*7$$\"3++++Qx$oE\"F`r$\"3/ntg_#Q_O#F*7$$\"3 #******p.I%)G\"F`r$\"3er8:;XCMDF*7$$\"31+++qe*zI\"F`r$\"3+umPb4'\\o#F* 7$$\"3.+++#\\'QH8F`r$\"3'[X))f^vr%GF*7$$\"31+++,M^\\8F`r$\"3G))=rP3W(* HF*7$$\"3)*******>=bq8F`r$\"3'ehP1ZM@:$F*7$$\"31+++@27\"R\"F`r$\"3,_'* yjp4,LF*7$$\"30+++(e()=S\"F`r$\"3#*HGT\\g>yLF*7$$\"3.+++`Wl79F`r$\"3O1 7<`_qaMF*7$$\"35+++kDI89F`r$\"3)[`Se4#HfMF*7$$\"32+++u1&RT\"F`r$\"3Wk% *[Go(QY$F*7$%*undefinedGFby-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F[zFjy-%* THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6\"Q!Fdz-%%VIEWG6$;$\"\"'Fiy$\"#; Fiy%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 557 "ln2 := proc(xx)\n local x,r2i,log2,k,r,val;\n x := evalf(xx);\n\n # Reduce the ar gument range to 1/sqrt(2)<=z<=sqrt(2)\n k := 0;\n r := 0;\n if x >1.414213562 or x<.7071067812 then\n if x>2 or x<1 then\n \+ k := ilog2(x);\n x := x*2.^(-k);\n end if;\n if x>1 .414213562 then\n r2i := evalf(sqrt(2))*0.5; \n x := x *r2i;\n r := 1;\n end if;\n end if;\n\n val := ln1(x) ;\n\n if k<>0 or r<>0 then \n log2 := evalf(ln(2));\n retu rn val + (k+r*0.5)*log2;\n else\n return val\n end if;\nend p roc; " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$ln2Gf*6#%#xxG6(%\"xG%$r2 iG%%log2G%\"kG%\"rG%$valG6\"F/C(>8$-%&evalfG6#9$>8'\"\"!>8(F9@$52$\"+i N@99!\"*F22F2$\"+7y1rq!#5C$@$52\"\"#F22F2\"\"\"C$>F8-%&ilog2G6#F2>F2*& F2FL)$FJF9,$F8!\"\"FL@$F>C%>8%,$*&$\"\"&FWFL-F46#-%%sqrtG6#FJFLFL>F2*& FenFLF2FL>F;FL>8)-%$ln1GFQ@%50F8F90F;F9C$>8&-F46#-%#lnGF^oO,&FcoFL*&,& F8FL*&FhnFLF;FLFLFLF\\pFLFLOFcoF/F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "plot('ln2(x)',x=0.01..10 );" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG 6$7gn7$$\"\"\"!\"#$!+'=q^g%!\"*7$$\"+1*z/o\"!#6$!+w24'3%F-7$$\"+7)f4O# F1$!+\\>5YPF-7$$\"+=(R9/$F1$!+#>RG\\$F-7$$\"+D'>>s$F1$!+B1$4H$F-7$$\"+ P%zG3&F1$!+nAHzHF-7$$\"+]#RQW'F1$!+mc/UFF-7$$\"+v)ed;*F1$!+2bp*Q#F-7$$ \"+]yw)=\"!#5$!+VxmH@F-7$$\"+v<:LsTFU$!+(G@9u)FU7$$\"+RO%HI'FU$!+VKo:Y FU7$$\"3Q+++sL\"yW)!#=$!3?RtcwXx'o\"F^p7$$\"33+++\"*)[#e5!#<$\"3=)\\\" =Y_bhc!#>7$$\"31+++S1;c7Fdp$\"3mc\"p&z&*f!G#F^p7$$\"3#******>z'3h9Fdp$ \"3m=[!*o`!=z$F^p7$$\"3%******>r@In\"Fdp$\"3ab^/)*RJY^F^p7$$\"33+++`pF %)=Fdp$\"3QzE/!oTaL'F^p7$$\"+O*y:5#F-$\"+l$*)oU(FU7$$\"+S'yHH#F-$\"+') o^)H)FU7$$\"+v$\\%3DF-$\"+Qyk'>*FU7$$\"+`[![s#F-$\"+7oR-5F-7$$\"+XMILH F-$\"+V#Hh2\"F-7$$\"+U,kAJF-$\"+W)y'Q6F-7$$\"+(yzxM$F-$\"+wtH37F-7$$\" +*f-&QNF-$\"+WOqj7F-7$$\"+p&e.w$F-$\"+;V^C8F-7$$\"+%pQn&RF-$\"+C,Uv8F- 7$$\"+;))>sTF-$\"+\">V%G9F-7$$\"+/yOxVF-$\"+'eZkZ\"F-7$$\"+?,W\"f%F-$ \"+EP>C:F-7$$\"+J`-)y%F-$\"+y!=hc\"F-7$$\"+S!p++&F-$\"+?cF-$\"+=oF-7$$\"+R(yG3(F-$\"+G/od>F-7$$\"+g5d'H(F-$\"+@XS()>F-7$ $\"+\"))Qw\\(F-$\"+b\")e9?F-7$$\"+)o7yq(F-$\"+]WBU?F-7$$\"+,gH8zF-$\"+ $QW&o?F-7$$\"+()zTG\")F-$\"+:jO&4#F-7$$\"+FggN$)F-$\"+Ai`?@F-7$$\"+Ad[ Z&)F-$\"+srjX@F-7$$\"+Z4hd()F-$\"+XJ#*p@F-7$$\"+1Up]*)F-$\"+&4J<>#F-7$ $\"+5%*)><*F-$\"+5U:;AF-7$$\"+z%=*p$*F-$\"+'R/vB#F-7$$\"+\\@&4e*F-$\"+ xpxfAF-7$$\"+)HQHy*F-$\"+y)R1G#F-7$$\"#5\"\"!$\"+$4&e-BF--%'COLOURG6&% $RGBG$F\\^l!\"\"$F]^lF]^lFf^l-%+AXESLABELSG6$Q\"x6\"Q!F[_l-%%VIEWG6$;F (F[^l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The following example illustrates how the procedure " } {TEXT 0 3 "ln2" }{TEXT -1 8 " works. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "xx := 15;\nprintlevel := 10:\nln2(xx);\nprintlevel := 1:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #xxG\"#:" }}{PARA 9 "" 1 "" {TEXT -1 25 "\{--> enter ln2, args = 15" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"#:\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"xG$\"++++v=!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%$r2iG$\"+5y1rq!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG$\"+9_# eK\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG\"\"\"" }}{PARA 9 " " 1 "" {TEXT -1 34 "\{--> enter ln1, args = 1.325825214" }}{PARA 9 "" 1 "" {TEXT -1 40 "<-- exit ln1 (now in ln2) = .2820350686\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$valG$\"+'o].#G!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%log2G$\"+1=ZJp!#5" }}{PARA 9 "" 1 "" {TEXT -1 46 "<- - exit ln2 (now at top level) = 2.708050201\}" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,-03F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(ln(xx));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+,-03F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "A polynomial approximation to calculate " }{XPPEDIT 18 0 "ln(x)" "6#-%#lnG6#%\"xG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "We can calcula te " }{XPPEDIT 18 0 "ln(x);" "6#-%#lnG6#%\"xG" }{TEXT -1 26 " by means of the function " }{XPPEDIT 18 0 "f(x) = ln((1+x)/(1-x))" "6#/-%\"fG6 #%\"xG-%#lnG6#*&,&\"\"\"F-F'F-F-,&F-F-F'!\"\"F/" }{XPPEDIT 18 0 "``=2* arctanh(x)" "6#/%!G*&\"\"#\"\"\"-%(arctanhG6#%\"xGF'" }{TEXT -1 4 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "x=(1+s)/(1-s) " "6#/%\"xG*&,&\"\"\"F'%\"sGF'F',&F'F'F(!\"\"F*" }{TEXT -1 10 ", we ha ve " }{XPPEDIT 18 0 "x-s*x=1+s" "6#/,&%\"xG\"\"\"*&%\"sGF&F%F&!\"\",&F &F&F(F&" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "x-1=s*x+s" "6#/,&% \"xG\"\"\"F&!\"\",&*&%\"sGF&F%F&F&F*F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "s=(x-1)/(x+1)" "6#/%\"sG*&,&%\"xG\"\"\"F(!\"\"F(,&F'F(F(F(F)" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 22 "In order to calculate " }{XPPEDIT 18 0 "ln(x)" "6#-%#lnG6 #%\"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[sqrt(2)/2,sq rt(2)]" "6#7$*&-%%sqrtG6#\"\"#\"\"\"F(!\"\"-F&6#F(" }{TEXT -1 23 ", we need to calculate " }{XPPEDIT 18 0 "ln((1+x)/(1-x))" "6#-%#lnG6#*&,& \"\"\"F(%\"xGF(F(,&F(F(F)!\"\"F+" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[2*sqrt(2)-3,3-2*sqrt(2)]" "6#7$,&*&\"\"#\"\"\"-%%sqrtG 6#F&F'F'\"\"$!\"\",&F+F'*&F&F'-F)6#F&F'F," }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 36 "This follows from the fact that, if " }{XPPEDIT 18 0 "g(x) = (x-1)/(x+1);" "6#/-%\"gG6#%\"xG*&,&F'\"\"\"F*!\"\"F*,&F'F *F*F*F+" }{TEXT -1 7 ", then " }{XPPEDIT 18 0 "g(sqrt(2)/2) = 2*sqrt(2 )-3;" "6#/-%\"gG6#*&-%%sqrtG6#\"\"#\"\"\"F+!\"\",&*&F+F,-F)6#F+F,F,\" \"$F-" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(sqrt(2)) = 3-2*sqrt(2);" "6#/-%\"gG6#-%%sqrtG6#\"\"#,&\"\"$\"\"\"*&F*F--F(6#F*F-!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "g := x -> (x-1)/(x+1);\ng(sqrt(2)/2);\nexpand(rationa lize(%));\ng(sqrt(2));\nexpand(rationalize(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,&9$\"\"\"F/ !\"\"F/,&F.F/F/F/F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$-%% sqrtG6#\"\"#\"\"\"#F*F)F*!\"\"F*,&F%F+F*F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&!\"$\"\"\"*&\"\"#F%-%%sqrtG6#F'F%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*$-%%sqrtG6#\"\"#\"\"\"F*F*!\"\"F*,&F%F*F*F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"$\"\"\"*&\"\"#F%-%%sqrtG6#F'F% !\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "We construct a polynom ial approximation for the function " }{XPPEDIT 18 0 "f(x)=ln((1+x)/(1- x))" "6#/-%\"fG6#%\"xG-%#lnG6#*&,&\"\"\"F-F'F-F-,&F-F-F'!\"\"F/" } {TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-0.1717,0.1717]" "6#7 $,$-%&FloatG6$\"%<0],[infinity,x=0]" "6#/-%\"wG6#%\"xG-%*PIECEWISEG6$7$\" \"\"0F'\"\"!7$%)infinityG/F'F." }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 91 "is the theoretical weight function needed. However a cont inuous approximation of the form " }{XPPEDIT 18 0 "w(x)=1+M*exp(-M*x^ 2)" "6#/-%\"wG6#%\"xG,&\"\"\"F)*&%\"MGF)-%$expG6#,$*&F+F)*$F'\"\"#F)! \"\"F)F)" }{TEXT -1 8 ", where " }{TEXT 293 1 "M" }{TEXT -1 27 " is la rge, will do the job." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "wt := x -> 1+1e300*exp(-1e300*x^2);\n'wt' (0)=wt(0);\n'wt'(.1*10^(-35))=wt(.1*10^(-35));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#wtGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"\"F-*&$F -\"$+$F--%$expG6#,$*$)9$\"\"#F-$!\"\"F0F-F-F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#wtG6#\"\"!$\"\"\"\"$+$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#wtG6#$\"+++++5!#X$\"\"\"\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "evalf(remez(ln((1+x)/(1-x))/x,x=-0.1717..0.1717,1 4,\n 'maxerr',weight=wt(x),errtype=absolute,type=even,info=true),3 0):\nr := unapply(%,x);\nmaxerr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%] 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0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "f := x -> ln((1+x)/(1-x));\n evalf(numapprox[infnorm]((1-h(x)/f(x)),x=0..3-2*sqrt(2)),35);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(-%#lnG6#*&,&9$\"\"\"F2F2F2,&F2F2F1!\"\"F4F(F(F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"359e@&>LiC\"!#N" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "xx := evalf(3-2*sqrt(2),20 );\nevalf(evalf(h(xx),20),16);\nevalf(evalf(ln((1+xx)/(1-xx)),20),16); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"4C!*4QDvGdr\"!#>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1F(*z-ftlM!#;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1F(*z-ftlM!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 46 "We can test the accuracy of the approxima tion " }{XPPEDIT 18 0 "h(x)" "6#-%\"hG6#%\"xG" }{TEXT -1 5 " for " } {XPPEDIT 18 0 "ln(1+x);" "6#-%#lnG6#,&\"\"\"F'%\"xGF'" }{TEXT -1 35 " \+ with random numbers between 0 and " }{XPPEDIT 18 0 "3-2*sqrt(2)" "6#,& \"\"$\"\"\"*&\"\"#F%-%%sqrtG6#F'F%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 92 "The printout occurs only when the relative error is \+ greater than or equal to the specified \"" }{TEXT 267 3 "eps" }{TEXT -1 3 "\".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 401 "randomize(): \neps := Float(5,-16);\nfor i from 1 to 100 do\n xx := evalf(rand()/ Float(1,12)*.1715728752538099024,20);\n axx := evalf(ln((1+xx)/(1-x x)),20);\n hxx := evalf(h(xx),16);\n e := evalf(abs((axx-hxx)/axx) ,20);\n if e>=eps then \n printf(\" trial no. %d, x = %.16f, \\n\",i,xx);\n printf(\" ln((1+x)/(1-x)) = %.16f, h(x) = %.16f, \+ rel error = %.2e\\n\\n\",axx,hxx,e); \n end if;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$epsG$\"\"&!#;" }}{PARA 6 "" 1 "" {TEXT -1 38 " trial no. 3, x = .1420009109083362," }}{PARA 6 "" 1 "" {TEXT -1 85 " ln((1+x)/(1-x)) = .2859341500359927, h(x) = .2859341500 359926, rel error = 5.04e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 12, x = .0509969795705326," }} {PARA 6 "" 1 "" {TEXT -1 85 " ln((1+x)/(1-x)) = .1020825156557756, h( x) = .1020825156557755, rel error = 6.64e-16" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 44, x = .120486285 6142576," }}{PARA 6 "" 1 "" {TEXT -1 85 " ln((1+x)/(1-x)) = .24214889 61977107, h(x) = .2421488961977109, rel error = 6.22e-16" }}{PARA 6 " " 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " trial no. 51, x = .0103135104699110," }}{PARA 6 "" 1 "" {TEXT -1 85 " ln((1+x)/(1-x) ) = .0206277523415814, h(x) = .0206277523415814, rel error = 5.52e-16 " }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 39 " tri al no. 54, x = .1567666978292659," }}{PARA 6 "" 1 "" {TEXT -1 85 " l n((1+x)/(1-x)) = .3161403908151009, h(x) = .3161403908151007, rel erro r = 5.38e-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 4 "ln16" }}{PARA 0 "" 0 "" {TEXT -1 147 "He re is the code for the fixed precision version of the natural logarith m function, which can be evaluated with hardware floating point arithm etic." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2191 "ln16 := proc(xx)\n local a1,a2,a3,a4,a5,a6,a7,x,z ,u,s,\n p,k,r,ln2,r2i;\n\n ln2 := .69314718055994530942;\n \+ r2i := .70710678118654752440;\n a1 := .66666666666667351301;\n a2 := .39999999999409354093;\n a3 := .28571428743686413264;\n a4 := \+ .22222198428962873094;\n a5 := .18183572356621292090;\n a6 := .153 13832102577005190;\n a7 := .14798259574717589604;\n\n x := evalf(x x);\n\n # Reduce the argument range to 1/sqrt(2)<=z<=sqrt(2)\n k : = 0;\n r := 0;\n if x>1.414213562 or x<.7071067812 then\n if \+ x>2 or x<1 then\n k := intlog2(x);\n x := x*2.^(-k);\n end if;\n if x>1.414213562 then \n x := x*r2i;\n \+ r := 1;\n end if;\n end if;\n\n # transformation to use approx for ln((1+x)/(1-x))\n s := (x-1)/(x+1);\n\n z := s*s; \n # evaluate the polynomial approximation\n p := (2+(a1+(a2+(a3+(a4 +(a5+(a6+a7*z)*z)*z)*z)*z)*z)*z)*s;\n if k<>0 or r<>0 then p+ln2*(k+ r*0.5) else p end if;\nend proc: # of ln16\n\nPOW2 := hfarray(-1019..1 019):\nfor i from -1019 to 1019 do\n POW2[i] := evalhf(2^i):\nend do :\n\nintlog2 := proc(x)\n if x>POW2[1019] or xn then\n jLo := m;\n jHi := n;\n else \n inc := 1;\n if x>=A[jLo] then \n if jLo=n then return jLo end if;\n jHi := jLo +1;\n while x>=A[jHi] do \n jLo := jHi;\n \+ inc := inc+inc;\n jHi := jLo+inc;\n if jHi>n t hen \n jHi := n+1;\n break;\n e nd if;\n end do;\n else\n if jLo=m then return jL o end if;\n jHi := jLo-1;\n while x 1 do\n jM : = trunc((jHi+jLo)/2);\n if x>=A[jM] then jLo := jM else jHi := jM end if;\n end do;\n return jLo;\nend proc: # of hunt " }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "Testing the procedure l n16" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot('ln16'(x),x=0.01..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7gn7$$\"3-+++++++5!#> $!3\"44))f=q^g%!#<7$$\"33D1k1*z/o\"F*$!3G[4lu24'3%F-7$$\"3'*\\7G8)f4O# F*$!3G>z3\\>5YPF-7$$\"3&[(=#*>(R9/$F*$!39J(*f!>RG\\$F-7$$\"3s*\\ili>>s $F*$!3u)>a@iI4H$F-7$$\"3?]P%)R%zG3&F*$!3Ud>6mAHzHF-7$$\"3&***\\7`#RQW' F*$!3(>Y&=mc/UFF-7$$\"3[*\\(oz)ed;*F*$!3;1bp*Q#F-7$$\"3/+]i]yw)=\"! #=$!3'GpI>un'H@F-7$$\"33+v$fx^Jt\"FT$!3-?B@1Nk_sTFT$!3]0 p$[G@9u)FT7$$\"3U+]()QO%HI'FT$!3TpHbWKo:YFT7$$\"3],]7sL\"yW)FT$!3iq$>k duno\"FT7$$\"3+]PR\"*)[#e5F-$\"35&=*Q$Gb:m&F*7$$\"3%*\\(o)R1;c7F-$\"3' \\IC&y&*f!G#FT7$$\"3-](=>z'3h9F-$\"3;D([$o`!=z$FT7$$\"3+](=:r@In\"F-$ \"3UD'o^*RJY^FT7$$\"3)*\\P9`pF%)=F-$\"3Cqb!3oTaL'FT7$$\"35+]sN*y:5#F-$ \"3o*o$oj$*)oU(FT7$$\"3k*\\i-kyHH#F-$\"3S=SR')o^)H)FT7$$\"32+]nu$\\%3D F-$\"3;\\mDKyk'>*FT7$$\"3s**\\i_[![s#F-$\"3WE_$=\"oR-5F-7$$\"3)***\\KX MILHF-$\"3)>RqOCHh2\"F-7$$\"3t\\(=?9SE7$F-$\"3pQ:OV)y'Q6F-7$$\"3#***\\ 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1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "We can use Maple's natural log function to check the accuracy of t he procedure ln16." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 50 "evalf(plot(ln(x)-'ln16(x)',x=1..2,color=blue ),20);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CUR VESG6#7c]m7$$\"\"\"\"\"!$F*F*7$$\"5nmm;a)G\\a+\"!#>$!\"\"!#A7$$\"5MLLL 3x&)*3,\"F/$F1!#@7$$\"5+++]ilyM;5F/$!\"%F77$$\"5nmmm;arz@5F/$!\"(F77$$ \"5+++D\"y%*z7.\"F/$F7F77$$\"5LLL$e9ui2/\"F/$!#QF77$$\"5+++voMrU^5F/$! #gF77$$\"5nmmm\"z_\"4i5F/$!#uF77$$\"5nmmT&))HvZ1\"F/$!#xF77$$\"5nmm;zp !fu1\"F/$!#yF77$$\"5nm;/Eb4!)o5F/$!#!)F77$$\"5nmm\"H2%G9q5F/$!#zF77$$ \"5nm;z>EZ[r5F/Fgn7$$\"5nmmmm6m#G2\"F/$!#vF77$$\"5nmmTg#Q5b2\"F/FS7$$ \"5nmm;a`T>y5F/$!#rF77$$\"5nmm\"zW#z(33\"F/$!#nF77$$\"5nmmmT&phN3\"F/F N7$$\"5+++v=ddC%4\"F/$F2F77$$\"5LLL$e*=)H\\5\"F/$\"#GF77$$\"5+++](=JN[ 6\"F/$\"\")!#?7$$\"5nmm;z/3uC6F/$\"#8Fiq7$$\"5+++](o3p)H6F/$\"#:Fiq7$$ 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p$>F/$\"#PFiq7$$\"5++]7`**)4A%>F/Feel7$$\"5LLLeRA5\\Z>F/$\"#VFiq7$$\"5 ++D\"GQeJ,&>F/Fjel7$$\"5mm;/EX@x_>F/$\"#NFiq7$$\"5Le*[=cGKM&>F/Fecq7$$ \"5+]il(fU#4a>F/Feel7$$\"5mTNYLmDva>F/Fhgl7$$\"5LL3Fp1FTb>F/Fecq7$$\"5 +D\"y]q%G2c>F/F_ap7$$\"5m;a)3u)Htc>F/Fcz7$$\"5L3FpwFJRd>F/Fcz7$$\"5+++ ]7oK0e>F/$\"#LFiq7$$\"5+](ozm5&oe>F/F_ap7$$\"5++vVBXpJf>F/F^[l7$$\"5+] i!*y$y[*f>F/Fcz7$$\"5++]PMA1eg>F/F^`p7$$\"5++DJX*HW='>F/Fb\\l7$$\"5+++ Dcwz5j>F/Fhhl7$$\"5++v=n`;Pk>F/Fbr7$$\"5++]7yI`jl>F/Fj\\l7$$\"5++D1*y+ **o'>F/F`hn7$$\"5+++++&oi\"o>F/Fgs7$$\"5++](=#R+pq>F/F+7$$\"5+++vV$RF/Fcem7$$\"5+](=#*>B\\Q(>F/F^_n7$$\"5++voaq5[u>F/F`v7$$\"5+]i:54H6v> F/Fjbl7$$\"5++]ilZZuv>F/F^u7$$\"5++DcwC%3q(>F/Fg]m7$$\"5+++](=5s#y>F/F _eo7$$\"5+]7y]&4I'z>F/F[`l7$$\"5++D19*3))4)>F/Fj\\m7$$\"5+DJq&f3n;)>F/ F[al7$$\"5+]PMx#3YB)>F/$!#RFiq7$$\"5+vV)*ez]-$)>F/$!#SFiq7$$\"5++]iSwS q$)>F/Fcjq7$$\"5+]i!R+2i])>F/Fe\\m7$$\"5++v=nj+U')>F/F]\\m7$$\"5+D\"G) [g!*4()>F/F]\\m7$$\"5+](o/t0yx)>F/$!#WFiq7$$\"5+v$4@T0d%))>F/F]\\m7$$ \"5+++v$40O\"*)>F/Fe[m7$$\"5+]7.dWS\\!*>F/Fhjq7$$\"5++DJ?Q?&=*>F/F``l7 $$\"5+DJ&>].JD*>F/F_]m7$$\"5+]Pf$=.5K*>F/F^bl7$$\"5+vVBlG!*)Q*>F/Fg]m7 $$\"5++](oa-oX*>F/Ffbl7$$\"5+Dc^GAqC&*>F/F[v7$$\"5+]i:5>g#f*>F/F^_n7$$ \"5+voz\"f,0m*>F/Fdcl7$$\"5++vVt7SG(*>F/Fg]l7$$\"5+D\"y]&4I'z*>F/F]w7$ $\"5+](=nj+U')*>F/Fgr7$$\"5+v$f$=.5K**>F/F`hl7$$Fh]lF*Fhgl-%'COLOURG6& %$RGBGF+F+$\"*++++\"Fet-%+AXESLABELSG6$Q\"x6\"Q!Fd_r-%%VIEWG6$;F(Fi^r% (DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu rve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "xx := 14.98;\nevalf(ln16(xx),16);\nevalf(evalf(ln(xx) ,20),16);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"%)\\\"!\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1u!*3yfr1F!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"1t!*3yfr1F!#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "xx := 100;\nevalf(evalhf(ln1 6(xx)),15);\nevalf(evalhf(ln(xx)),15);\nevalf(ln(xx),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"04))f=q^g%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"04))f=q^g%!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"04))f=q^g%!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "xx := 0.0 1;\nevalf(evalhf(ln16(xx)),15);\nevalf(evalhf(ln(xx)),15);\nevalf(ln(x x),15);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"\"!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!04))f=q^g%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!04))f=q^g%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!04 ))f=q^g%!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "xx := evalhf(DBL_MAX/18);\nevalf(evalhf(ln16(xx)), 15);\nevalf(evalhf(ln(xx)),15);\nevalf(ln(xx),15);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#xxG$\"3+++++bbbb\"$*G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0qU(o$eI1(!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" 0qU(o$eI1(!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"0qU(o$eI1(!#7" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "Translating from Maple \+ code to C code " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "We can translate this program into C using the Maple pro cedure C in the code generating package codegen. \n see " }{HYPERLNK 17 "codegen[C]" 2 "codegen[C]" "" }{TEXT -1 5 " and " }{HYPERLNK 17 "c odegen/C/procedure" 2 "codegen/C/procedure" "" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 27 ": The lines of code . . . " }}{PARA 6 "" 1 "" {TEXT -1 19 " k = intlog2(x);" }}{PARA 6 "" 1 "" {TEXT -1 26 " x = x*pow( 2.0,-1.0*k);" }}{PARA 0 "" 0 "" {TEXT -1 37 " . . . should be replace d by . . . " }}{PARA 6 "" 0 "" {TEXT -1 44 " x = frexp(x,&k);\n \+ x += x;\n k -= 1;" }}{PARA 0 "" 0 "" {TEXT -1 61 "In addition, the type of the variable k must be changed from " }{TEXT 267 6 "double" } {TEXT -1 4 " to " }{TEXT 267 3 "int" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "with(codeg en,C):\nC(ln16,ansi);" }}{PARA 6 "" 1 "" {TEXT -1 17 "#include " }}{PARA 6 "" 1 "" {TEXT -1 22 "double ln16(double xx)" }}{PARA 6 " " 1 "" {TEXT -1 1 "\{" }}{PARA 6 "" 1 "" {TEXT -1 12 " double a1;" }} {PARA 6 "" 1 "" {TEXT -1 12 " double a2;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double a3;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double a4;" }} {PARA 6 "" 1 "" {TEXT -1 12 " double a5;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double a6;" }}{PARA 6 "" 1 "" {TEXT -1 12 " double a7;" }} {PARA 6 "" 1 "" {TEXT -1 11 " double k;" }}{PARA 6 "" 1 "" {TEXT -1 13 " double ln2;" }}{PARA 6 "" 1 "" {TEXT -1 11 " double p;" }} {PARA 6 "" 1 "" {TEXT -1 8 " int r;" }}{PARA 6 "" 1 "" {TEXT -1 13 " \+ double r2i;" }}{PARA 6 "" 1 "" {TEXT -1 11 " double s;" }}{PARA 6 " " 1 "" {TEXT -1 11 " double u;" }}{PARA 6 "" 1 "" {TEXT -1 11 " doub le x;" }}{PARA 6 "" 1 "" {TEXT -1 11 " double z;" }}{PARA 6 "" 1 "" {TEXT -1 3 " \{" }}{PARA 6 "" 1 "" {TEXT -1 29 " ln2 = 0.693147180 5599453;" }}{PARA 6 "" 1 "" {TEXT -1 29 " r2i = 0.7071067811865475; " }}{PARA 6 "" 1 "" {TEXT -1 28 " a1 = 0.6666666666666735;" }} {PARA 6 "" 1 "" {TEXT -1 28 " a2 = 0.3999999999940935;" }}{PARA 6 " " 1 "" {TEXT -1 28 " a3 = 0.2857142874368641;" }}{PARA 6 "" 1 "" {TEXT -1 28 " a4 = 0.2222219842896287;" }}{PARA 6 "" 1 "" {TEXT -1 28 " a5 = 0.1818357235662129;" }}{PARA 6 "" 1 "" {TEXT -1 28 " a 6 = 0.1531383210257701;" }}{PARA 6 "" 1 "" {TEXT -1 28 " a7 = 0.147 9825957471759;" }}{PARA 6 "" 1 "" {TEXT -1 17 " x = 0.1E1*xx;" }} {PARA 6 "" 1 "" {TEXT -1 12 " k = 0.0;" }}{PARA 6 "" 1 "" {TEXT -1 10 " r = 0;" }}{PARA 6 "" 1 "" {TEXT -1 48 " if( 0.1414213562E1 \+ < x || x < 0.7071067812 )" }}{PARA 6 "" 1 "" {TEXT -1 7 " \{" }} {PARA 6 "" 1 "" {TEXT -1 32 " if( 2.0 < x || x < 1.0 )" }} {PARA 6 "" 1 "" {TEXT -1 11 " \{" }}{PARA 6 "" 1 "" {TEXT -1 27 " k = intlog2(x);" }}{PARA 6 "" 1 "" {TEXT -1 34 " \+ x = x*pow(2.0,-1.0*k);" }}{PARA 6 "" 1 "" {TEXT -1 11 " \+ \}" }}{PARA 6 "" 1 "" {TEXT -1 32 " if( 0.1414213562E1 < x )" }}{PARA 6 "" 1 "" {TEXT -1 11 " \{" }}{PARA 6 "" 1 "" {TEXT -1 22 " x = x*r2i;" }}{PARA 6 "" 1 "" {TEXT -1 18 " \+ r = 1;" }}{PARA 6 "" 1 "" {TEXT -1 11 " \}" }}{PARA 6 "" 1 "" {TEXT -1 7 " \}" }}{PARA 6 "" 1 "" {TEXT -1 24 " s = (x-1 .0)/(x+1.0);" }}{PARA 6 "" 1 "" {TEXT -1 12 " z = s*s;" }}{PARA 6 " " 1 "" {TEXT -1 63 " p = (2.0+(a1+(a2+(a3+(a4+(a5+(a6+a7*z)*z)*z)*z )*z)*z)*z)*s;" }}{PARA 6 "" 1 "" {TEXT -1 30 " if( k != 0.0 || r != 0.0 )" }}{PARA 6 "" 1 "" {TEXT -1 30 " return(p+ln2*(k+0.5*r)); " }}{PARA 6 "" 1 "" {TEXT -1 9 " else " }}{PARA 6 "" 1 "" {TEXT -1 16 " return(p);" }}{PARA 6 "" 1 "" {TEXT -1 3 " \}" }}{PARA 6 " " 1 "" {TEXT -1 1 "\}" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 294 13 "Modified code" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 2 "" 1 "" {TEXT -1 17 "#include " }}{PARA 2 "" 1 "" {TEXT -1 22 "double \+ ln16(double xx)" }}{PARA 2 "" 1 "" {TEXT -1 1 "\{" }}{PARA 2 "" 1 "" {TEXT -1 12 " double a1;" }}{PARA 2 "" 1 "" {TEXT -1 12 " double a2; " }}{PARA 2 "" 1 "" {TEXT -1 12 " double a3;" }}{PARA 2 "" 1 "" {TEXT -1 12 " double a4;" }}{PARA 2 "" 1 "" {TEXT -1 12 " double a5; " }}{PARA 2 "" 1 "" {TEXT -1 12 " double a6;" }}{PARA 2 "" 1 "" {TEXT -1 12 " double a7;" }}{PARA 2 "" 1 "" {TEXT -1 8 " int k;" }} {PARA 2 "" 1 "" {TEXT -1 13 " double ln2;" }}{PARA 2 "" 1 "" {TEXT -1 11 " double p;" }}{PARA 2 "" 1 "" {TEXT -1 8 " int r;" }}{PARA 2 "" 1 "" {TEXT -1 13 " double r2i;" }}{PARA 2 "" 1 "" {TEXT -1 11 " d ouble s;" }}{PARA 2 "" 1 "" {TEXT -1 11 " double u;" }}{PARA 2 "" 1 " " {TEXT -1 11 " double x;" }}{PARA 2 "" 1 "" {TEXT -1 11 " double z; " }}{PARA 2 "" 1 "" {TEXT -1 3 " \{" }}{PARA 2 "" 1 "" {TEXT -1 29 " \+ ln2 = 0.6931471805599453;" }}{PARA 2 "" 1 "" {TEXT -1 29 " r2i = 0.7071067811865475;" }}{PARA 2 "" 1 "" {TEXT -1 28 " a1 = 0.666666 6666666735;" }}{PARA 2 "" 1 "" {TEXT -1 28 " a2 = 0.399999999994093 5;" }}{PARA 2 "" 1 "" {TEXT -1 28 " a3 = 0.2857142874368641;" }} {PARA 2 "" 1 "" {TEXT -1 28 " a4 = 0.2222219842896287;" }}{PARA 2 " " 1 "" {TEXT -1 28 " a5 = 0.1818357235662129;" }}{PARA 2 "" 1 "" {TEXT -1 28 " a6 = 0.1531383210257701;" }}{PARA 2 "" 1 "" {TEXT -1 28 " a7 = 0.1479825957471759;" }}{PARA 2 "" 1 "" {TEXT -1 17 " x = 0.1E1*xx;" }}{PARA 2 "" 1 "" {TEXT -1 12 " k = 0.0;" }}{PARA 2 " " 1 "" {TEXT -1 10 " r = 0;" }}{PARA 2 "" 1 "" {TEXT -1 48 " if( 0.1414213562E1 < x || x < 0.7071067812 )" }}{PARA 2 "" 1 "" {TEXT -1 7 " \{" }}{PARA 2 "" 1 "" {TEXT -1 32 " if( 2.0 < x || x < 1.0 )" }}{PARA 2 "" 1 "" {TEXT -1 11 " \{" }}{PARA 2 "" 1 " " {TEXT -1 11 " \}" }}{PARA 2 "" 1 "" {TEXT -1 75 " \+ x = frexp(x,&k);\n x += x;\n k -= 1; " } }{PARA 2 "" 0 "" {TEXT -1 36 " if( 0.1414213562E1 < x )" }} {PARA 2 "" 1 "" {TEXT -1 11 " \{" }}{PARA 2 "" 1 "" {TEXT -1 22 " x = x*r2i;" }}{PARA 2 "" 1 "" {TEXT -1 18 " \+ r = 1;" }}{PARA 2 "" 1 "" {TEXT -1 11 " \}" }}{PARA 2 "" 1 " " {TEXT -1 7 " \}" }}{PARA 2 "" 1 "" {TEXT -1 24 " s = (x-1.0) /(x+1.0);" }}{PARA 2 "" 1 "" {TEXT -1 12 " z = s*s;" }}{PARA 2 "" 1 "" {TEXT -1 63 " p = (2.0+(a1+(a2+(a3+(a4+(a5+(a6+a7*z)*z)*z)*z)* z)*z)*z)*s;" }}{PARA 2 "" 1 "" {TEXT -1 30 " if( k != 0.0 || r != 0 .0 )" }}{PARA 2 "" 1 "" {TEXT -1 30 " return(p+ln2*(k+0.5*r));" } }{PARA 2 "" 1 "" {TEXT -1 9 " else " }}{PARA 2 "" 1 "" {TEXT -1 16 " return(p);" }}{PARA 2 "" 1 "" {TEXT -1 3 " \}" }}{PARA 2 "" 1 "" {TEXT -1 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 44 "More fixed precision pr ocedures to evaluate " }{XPPEDIT 18 0 "ln((1+x)/(1-x))" "6#-%#lnG6#*&, &\"\"\"F(%\"xGF(F(,&F(F(F)!\"\"F+" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "A 2 5 digit rational approximation for " }{XPPEDIT 18 0 "f(x)=ln((1+x)/(1- x))" "6#/-%\"fG6#%\"xG-%#lnG6#*&,&\"\"\"F-F'F-F-,&F-F-F'!\"\"F/" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "We can calculate " }{XPPEDIT 18 0 "ln(x);" "6#- %#lnG6#%\"xG" }{TEXT -1 26 " by means of the function " }{XPPEDIT 18 0 "f(x) = ln((1+x)/(1-x))" "6#/-%\"fG6#%\"xG-%#lnG6#*&,&\"\"\"F-F'F-F- ,&F-F-F'!\"\"F/" }{XPPEDIT 18 0 "``=2*arctanh(x)" "6#/%!G*&\"\"#\"\"\" -%(arctanhG6#%\"xGF'" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "x=(1+s)/(1-s)" "6#/%\"xG*&,&\"\"\"F'%\"sGF'F' ,&F'F'F(!\"\"F*" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "x-s*x=1+s" "6#/,&%\"xG\"\"\"*&%\"sGF&F%F&!\"\",&F&F&F(F&" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "x-1=s*x+s" "6#/,&%\"xG\"\"\"F&!\"\",&*&%\"sGF&F%F&F &F*F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "s=(x-1)/(x+1)" "6#/%\"sG*&, &%\"xG\"\"\"F(!\"\"F(,&F'F(F(F(F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 22 "In order to calculate \+ " }{XPPEDIT 18 0 "ln(x)" "6#-%#lnG6#%\"xG" }{TEXT -1 17 " on the inter val " }{XPPEDIT 18 0 "[10^(-1/8), 10^(1/8)];" "6#7$)\"#5,$*&\"\"\"F(\" \")!\"\"F*)F%*&F(F(F)F*" }{TEXT -1 23 ", we need to calculate " } {XPPEDIT 18 0 "ln((1+x)/(1-x))" "6#-%#lnG6#*&,&\"\"\"F(%\"xGF(F(,&F(F( F)!\"\"F+" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[(1-10^(1/ 8))/(10^(1/8)+1), (10^(1/8)-1)/(10^(1/8)+1)];" "6#7$*&,&\"\"\"F&)\"#5* &F&F&\"\")!\"\"F+F&,&)F(*&F&F&F*F+F&F&F&F+*&,&)F(*&F&F&F*F+F&F&F+F&,&) F(*&F&F&F*F+F&F&F&F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 " We construct a rational approximation " }{XPPEDIT 18 0 "r(x)" "6#-%\"r G6#%\"xG" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "ln((1+x)/(1-x))" "6#-%#l nG6#*&,&\"\"\"F(%\"xGF(F(,&F(F(F)!\"\"F+" }{TEXT -1 47 " on the slight ly wider interval [-0.143,0.143]." }}{PARA 0 "" 0 "" {TEXT -1 37 "We u se a weight function of the form " }{XPPEDIT 18 0 "w(x)=1+M*exp(-M*x^2 )" "6#/-%\"wG6#%\"xG,&\"\"\"F)*&%\"MGF)-%$expG6#,$*&F+F)*$F'\"\"#F)!\" \"F)F)" }{TEXT -1 8 ", where " }{TEXT 295 1 "M" }{TEXT -1 11 " is larg e. " }}{PARA 0 "" 0 "" {TEXT -1 65 "This provides a continuous approxi mation for the weight function " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "w(x)=PIECEWISE([1,x<>0],[infinity,x=0])" "6#/-%\"wG6#% \"xG-%*PIECEWISEG6$7$\"\"\"0F'\"\"!7$%)infinityG/F'F." }{TEXT -1 3 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 62 "and will produce a minimax approxima tion with zero error when " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalf((10^(1/8)-1)/(10^(1/8)+1));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+[BEH9!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 184 "w := x -> 1+1e300*exp(-1e3 00*x^2);\nevalf(remez(ln((1+x)/(1-x))/x,x=-0.143..0.143,[10,8],\n \+ 'maxerr',weight=w(x),errtype=absolute,type=even,info=true),50):\nr := \+ unapply(%,x);\nmaxerr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wGf*6#% \"xG6\"6$%)operatorG%&arrowGF(,&\"\"\"F-*&$F-\"$+$F--%$expG6#,$*$)9$\" \"#F-$!\"\"F0F-F-F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~al gorithm:~calculating~minimax~error~estimate~by~solving~a~rational~equa tionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%M--~minimising~the~weighted~ ABSOLUTE~error~--G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,iteration~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G--------------------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Uerror~graph~dr awn~for~the~right~half~of~the~intervalG" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6;-%'CURVESG6%7_w7$$\"\"!F)%%FAILG7$$\"gnILLL LLLLLLLLLLLLLLLLL$eR(\\rQM](*!#k$!kh!#f7$$\"gnmmmmmmmmmmm mmmmmmmmmm;z%*Hu(o+&>!#j$!=:a'QeV1**fh:WbY#F17$$\"gn****************** ***********************\\(=#\\9;.^#HF5$!=#3vc#*z`1'pi*opa&F17$$\"gnKLL LLLLLLLLLLLLLLLLLLLe*)f[v8+RF5$!=qBU@rX&y8[R?+')*F17$$\"gn)*********** ******************************\\P%)*GK1-&eF5$!>'*yoJ>GGcJd;'p(4v-!yF5$!>sPFD)yw0pv%\\f0%RF17$$\"gn++++++ +++++++++++++++](ozXET+<\"!#i$!>eh'>%G\"e++B3oL`))F17$$\"gnLLLLLLLLLLL LLLLLLLLLLL$eR%>]0g:FO$!?O1[*o[yS!3ZzUrq:F17$$\"gn++++++++++++++++++++ ++v$f\"HD3SBFO$!?1$4YGh+1#fHgI^8NF17$$\"gnmmmmmmmmmmmmmmmmmmmmmmm\"z)Q +6?JFO$!?6E.^,\"\\At[**3P_>'F17$$\"gn********************************* *********\\(oC(pb]xWFO$!@$eM:)3*z#3DM(*zK1D\"F17$$\"gnLLLLLLLLLLLLLLLL LLLLLL3-d+6!\\$eFO$!@()GM,ZLgrS%>>Wsm?F17$$\"gn*********************** *******************\\7GdSVVhtFO$!@(oFmy3:\"Rsz/*RohJF17$$\"gnmmmmmmmmm mmmmmmmmmmmmm;ad!enz)))FO$!@fROwC\\;-%3B>-@'Q%F17$$\"gnnmmmmmmmmmmmmmm mmmmmmmm,Qn>YU5!#h$!@+sDb#>t:1S)*zQ$3o&F17$$\"gnnmmmmmmmmmmmmmmmmmmmmm \"z-n\"Fbp$!@^T'3CBn%))*48vySjpF17$$\"gn+++++++++++++++++++++DJ!4@ /i!\\8Fbp$!@2?mB=0aGc&42)eF:)F17$$\"gnLLLLLLLLLLLLLLLLLLLLL$3F:v!p*>] \"Fbp$!@J1Q()eUy'pMHUhb#=*F17$$\"gn+++++++++++++++++++++]7Bx/ryV;Fbp$! @t\\-TF]'y)eIfG*[O**F17$$\"gnmmmmmmmmmmmmmmmmmmmmm;a$H?Ixby\"Fbp$!A115 ND,\\sC!oodi\\/\"F17$$\"gnLLLLLLLLLLLLLLLLLLLLLL$3Uh=\"GA=Fbp$!A4VXkfY '=4uXM$4%Q0\"F17$$\"gn+++++++++++++++++++++]7[Dq])*e=Fbp$!Au$e`G%RH)Q' )R`\"fzg5F17$$\"gnmmmmmmmmmmmmmmmmmmmmmmTvOa*)o&*=Fbp$!A!4?-Mh(y0T%RKh yd1\"F17$$\"gnLLLLLLLLLLLLLLLLLLLLL$3F![QGRK>Fbp$!A/!p*[t7T*f&**oY_uo5 F17$$\"gn+++++++++++++++++++++++IfAn4p>Fbp$!Az)H\"*=h\"fw7l5F17$$\"gn+++++++++++ ++++++++++](=J\\P3#z?Fbp$!A/k4pS!oVL5L!4Kwf5F17$$\"gnLLLLLLLLLLLLLLLLL LLLL$3_]F@X5B#Fbp$!AF2u8\\$Gu7vL)\\IN95F17$$\"gnmmmmmmmmmmmmmmmmmmmmm; a)p00#)GQ#Fbp$!@jXgamT(zZ:!o$4l@$*F17$$\"gnLLLLLLLLLLLLLLLLLLLLL$ekCd% >BMDFbp$!@$o8$eTssGEGR8KJ:)F17$$\"gn+++++++++++++++++++++]P%z3%=e&o#Fb p$!@O;1H'3h*f`K-fK'pmF17$$\"gnmmmmmmmmmmmmmmmmmmmmmT5IJ6QETGFbp$!@cf'f F\\J$[?r#>CWn[F17$$\"gnLLLLLLLLLLLLLLLLLLLLLL$eYrKFbp$\" ?ZfO'*HVhB?=Ky;)[*F17$$\"gnLLLLLLLLLLLLLLLLLLLLL$3-GqJlbU$Fbp$\"@S)G.? 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.666666666666666666666667294110;\n \+ a2 := -1.14797224113924323579879814202;\n a3 := .5903503053269873614 65764591737;\n a4 := -.865664194907786855930604193839e-1;\n a5 := \+ +.464652936012806969997236389703e-3;\n b1 := -2.32195836170886485369 636882863;\n b2 := 1.85012904644437138176470533310;\n b3 := -.5781 35378275181709679450917850;\n b4 := .559246057352437269125330586694e -1;\n\n z := x*x;\n num := (a1+(a2+(a3+(a4+a5*z)*z)*z)*z)*z;\n d en := 1+(b1+(b2+(b3+b4*z)*z)*z)*z;\n (2+num/den)*x;\nend proc: " }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "xx := evalf((10^(1/8)-1)/(10^(1/8)+1),30);\nevalf(evalf(ln25(xx), 27),25);\nevalf(evalf(ln((1+xx)/(1-xx)),27),25);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"?H&e4['pkGw;[BEH9!#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\":*[A]5dDCm8ByG!#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\":*[A]5dDCm8ByG!#D" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 38 "A 55 digit rational approximation for " }{XPPEDIT 18 0 "f (x)=ln((1+x)/(1-x))" "6#/-%\"fG6#%\"xG-%#lnG6#*&,&\"\"\"F-F'F-F-,&F-F- F'!\"\"F/" }{TEXT -1 1 " " }}{PARA 258 "" 0 "" {TEXT -1 22 "In order t o calculate " }{XPPEDIT 18 0 "ln(x)" "6#-%#lnG6#%\"xG" }{TEXT -1 17 " \+ on the interval " }{XPPEDIT 18 0 "[10^(-1/8), 10^(1/8)];" "6#7$)\"#5,$ *&\"\"\"F(\"\")!\"\"F*)F%*&F(F(F)F*" }{TEXT -1 23 ", we need to calcul ate " }{XPPEDIT 18 0 "ln((1+x)/(1-x))" "6#-%#lnG6#*&,&\"\"\"F(%\"xGF(F (,&F(F(F)!\"\"F+" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[(1 -10^(1/8))/(10^(1/8)+1), (10^(1/8)-1)/(10^(1/8)+1)];" "6#7$*&,&\"\"\"F &)\"#5*&F&F&\"\")!\"\"F+F&,&)F(*&F&F&F*F+F&F&F&F+*&,&)F(*&F&F&F*F+F&F& F+F&,&)F(*&F&F&F*F+F&F&F&F+" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1870 "Digits := 70:\ns:=x ->x*(2.-9.604349065314314194557274068472460361204716153843962980723975 3235915237070*x^2+19.5595922643361522694410486762713661172942052088128 18885980508842669296277*x^4-22.000738589585228071496526736599230901377 778525646846381944471694846786446*x^6+14.91453960546429569149789883484 2263433033152568352679574780683132536421390*x^8-6.23890848079620708040 03825863785633568729685945017131289372295061711040425*x^10+1.583421603 4834474824653924574696548882492633438840592199357593195102318498*x^12- .229935969516128337214096147228428356074237595592833467865746367814802 84292*x^14+.1688174548118253209257390966571152083038492281609521671844 6305570832990988e-1*x^16-.47832490265106489325195721812345175804951512 927201035476805633829019817527e-3*x^18+.222301164280680102674863824625 37890811667242065408523772891816049225619707e-5*x^20)/(1.-5.1355078659 904904306119703675695635139356914102553148236766358594049007990*x^2+11 .291632087498239611591181127325537563292333074491514222907290663900135 389*x^4-13.88000222695107201063245348008469145947538581512618325604946 0498446761302*x^6+10.4481436639608638297607013886211764621819878400996 02482540845288119338718*x^8-4.9595557536745810616886028108431422190910 917363379041820690248087345770550*x^10+1.47344026819822124028258428849 72791768107845017861061340605048124166144212*x^12-.2627126804287995740 1897999790378476533007151915832720907897769702829052285*x^14+.25701788 105497411683686102895039682894666274681509430306350961915350305290e-1* x^16-.1150722174405968665060027117554182891822215227143746759528150161 8172070098e-2*x^18+.15141274808079134683912703617159960450555252344891 849450930985438931645770e-4*x^20):\nh := unapply(evalf(convert(2*x+nor mal(s(x)-2*x),horner),61),x);\nDigits := 10:\nf := x -> ln((1+x)/(1-x) );\nevalf(plot(10^30*(1-h(x)/f(x)),x=0..(10^(1/8)-1)/(10^(1/8)+1)\n \+ ,color=COLOR(RGB,.4,0,.9)),65);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&9$$\"\"#\"\"!**$\"\"\"F0 F3,6$!hn]HHmmmmmmmmmmmmmmmmmmmmmmmmmmm'!#hF3*&$\"hnqS$)f&4-nBI!#gF3)F-F/F3F3*&$\"hn]W:I,_0Y5$*HdVdCsZ<)fF;F3) F-\"\"'F3F3*&$\"hn?3?N_4u\"y[@483@xINI#o(H/bHbEI?!o$F;F3)F-\"\")F3FB*& $\"hn^_=[I:)ofcIs`Y.\\_>hx*4)*\\*H\"H$*eMO\"F;F3)F-\"#5F3F3*&$\"hn!4AH ]4#QsUa!feu69z&[Q'Q#3\"3ZT8R*[&HF7F3)F-\"#7F3FB*&$\"hnhbU*QkBpaEw%*e\\ %yOChH)zu7H7)H2$=_M!#iF3)F-\"#9F3F3*&$\"hnW#)GkJ[:]K:\\fDS\"\\)p,(4ooV s3;Y%>J#=!#jF3)F-\"#;F3FB*&$\"hn!GdCl%GC$[!yV*>=8m!))*on2T$o9Ntz`f!G!# lF3)F-\"#=F3F3F3F-\"\"$,8$F3F0F3*&$\"hnOmnB[Jb-T\"pNR^j&pvOq>hI/\\!*f' y]N^F;F3F=ik<@')Q,2wHQ'3'RmV\"[/\"F\\pF3FKF3F3*&$\"hnD!p?=/zLO<4\"4>A9V3 \"Gg)oh5euOvb&f\\F;F3FPF3FB*&$\"hn001Mh5'y,Xy5oo-Wt9F; 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co$!$'fF87$$\"\\ozd-Jg'4NyOoAv!>vr,'*[V\"Hj\"fE7yg4pCU\"Ffco$!%)Q#F87$ $\"\\oKB&4bi\\`O1b#*f(=J)*G1y-%zx-!yLi1cRMU\"Ffco$!%!4%F87$$\"\\o&))y3 2f*=Zft5@&p36::-9y\"*z2r`1=I@OA$) \\Ta4NE9Ffco$!%GmF87$$\"\\oW&e1j[4Fp%=?()\\***=9KF9Ffco $!%YcF87$$\"\\o'4&e]^%\\XFa)=M=<6JldB*pI]e]8Me$)=HG9Ffco$!%sIF87$$\"\\ o\\;^q;%*QcQ_<\"oorwDzE&e4['pkGw;[BEH9Ffco$\"&Gh\"F1-%+AXESLABELSG6$Q \"x6\"Q!Fh^s-%&COLORG6&%$RGBG$\"\"%!\"\"F)$\"\"*F`_s-%%VIEWG6$;F($\"\\ od;^q;%*QcQ_<\"oorwDzE&e4['pkGw;[BEH9Ffco%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "h := unapply(evalf(convert(2*x+normal(s(x)-2*x),ho rner),60),x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$% )operatorG%&arrowGF(,&9$$\"\"#\"\"!**$\"\"\"F0F3,6$!gn?$Hmmmmmmmmmmmmm mmmmmmmmmmmmmm'!#gF3*&$\"gn5M)f&4-nBI!#fF3)F- F/F3F3*&$\"gn]a,8?bg/J*HdVdCsZ<)fF;F3)F-\"\"'F3F3*&$\"gn\"3?N_4 u\"y[@483@xINI#o(H/bHbEI?!o$F;F3)F-\"\")F3FB*&$\"gnC&=[I:)ofcIs`Y.\\_> hx*4)*\\*H\"H$*eMO\"F;F3)F-\"#5F3F3*&$\"gn5AH]4#QsUa!feu69z&[Q'Q#3\"3Z T8R*[&HF7F3)F-\"#7F3FB*&$\"gndD%*QkBpaEw%*e\\%yOChH)zu7H7)H2$=_M!#hF3) F-\"#9F3F3*&$\"gnC)GkJ[:]K:\\fDS\"\\)p,(4ooVs3;Y%>J#=!#iF3)F-\"#;F3FB* &$\"gnGdCl%GC$[!yV*>=8m!))*on2T$o9Ntz`f!G!#kF3)F-\"#=F3F3F3F-\"\"$,8$F 3F0F3*&$\"gnkwO#[Jb-T\"pNR^j&pvOq>hI/\\!*f'y]N^F;F3F=ik<@')Q,2wHQ'3'RmV\"[ /\"F\\pF3FKF3F3*&$\"gn-p?=/zLO<4\"4>A9V3\"Gg)oh5euOvb&f\\F;F3FPF3FB*&$ \"gn]gS81hy,Xy5oo-Wt9F;F3FUF3F3*&$\"gny*y!4sKe\">:2I`w %y.z**z*=Sd*zG/o7FEF7F3FenF3FB*&$\"gn5N1.V4:ouim%*GoR]*G5'o$o6u\\0\")y ,d#FZF3F[oF3F3*&$\"gn:G&fnuVrA:A#=*G=av6F+1l'ofSu@s]6FjnF3FaoF3FB*&$\" gn5$4X\\=*[M__b]/'*frh.F\"RoM\"z!3[FT^\"F`oF3)F-\"#?F3F3FBFBF(F(F(" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "ln55" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1794 "ln55 := \+ proc(x)\n local a1,a2,a3,a4,a5,a6,a7,a8,a9,a10,\n b1,b2,b3,b4,b 5,b6,b7,b8,b9,b10,num,den,z;\n\n a1:=.666666666666666666666666666666 666666666666666666666666629320;\n a2:=-3.023671910660326953741313578 37970900929046094017020955983410;\n a3:=5.75926586431691594976838022 357015201757299310460552013015450;\n a4:=-5.981747722457431968023503 94240008949133082311184652539030090;\n a5:=3.68020302655295504297682 303530772108130921487817409523520081;\n a6:=-1.363458932912994998099 77611952490346537230565968815304818524;\n a7:=.295489391341470810823 863848579141174585905442723820950292210;\n a8:=\n -.3452183072981 22912747982961243678449589476265469236438942557e-1;\n a9:=\n .182 311944616087243686809701698491402559491532501548316428824e-2;\n a10: =\n -.280595379733514683410767689880661318199437804832428465245728e -4; \n b1:=-5.135507865990490430611970367569563513935691410255314823 67664;\n b2:=11.2916320874982396115911811273255375632923330744915142 229073;\n b3:=-13.88000222695107201063245348008469145947538581512618 32560495;\n b4:=10.4481436639608638297607013886211764621819878400996 024825408;\n b5:=-4.959555753674581061688602810843142219091091736337 90418206902;\n b6:=1.47344026819822124028258428849727917681078450178 610613406050;\n b7:=-.2627126804287995740189799979037847653300715191 58327209078978;\n b8:=\n .257017881054974116836861028950396828946 662746815094303063510e-1;\n b9:=\n -.1150722174405968665060027117 55418289182221522714374675952815e-2;\n b10:=\n .15141274808079134 6839127036171599604505552523448918494509310e-4;\n\n z := x*x;\n nu m := (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+(b8+\n \+ (b9+b10*z)*z)*z)*z)*z)*z)*z)*z)*z)*z;\n (2+num/den)*x;\nend pro c: " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "xx := evalf((10^(1/8)-1)/(10^(1/8)+1),60);\nevalf(ev alf(ln55(xx),58),55);\nevalf(evalf(ln((1+xx)/(1-xx)),58),55);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"gn0nT*QcQ_<\"oorwDzE&e4['pkGw ;[BEH9!#g" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"X?i'fygoP,&f_Xb$=$*[A] 5dDCm8ByG!#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"X?i'fygoP,&f_Xb$=$* [A]5dDCm8ByG!#b" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 90 "xx := evalf(0.14,60);\nevalf(evalf(ln55(xx),58 ),55);\nevalf(evalf(ln((1+xx)/(1-xx)),58),55);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"#9!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"X ^&GKSUjLdK1A`ka:I9$zs()49_6&=G!#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"X^&GKSUjLdK1A`ka:I9$zs()49_6&=G!#b" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "st := time():\nfor i to 5000 do\n xx := evalf(10^(1/8)-10^(-7)*i,60);\n evalf(evalf(ln55(xx ),58),55);\nend do:\ntime()-st; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"&'p=!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 109 "st := time():\nfor i to 5000 do\n xx := evalf(10^ (1/8)-10^(-7)*i,60);\n evalf(ln(xx),55);\nend do:\ntime()-st; " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&Ga\"!\"$" }}}{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 59 "A procedure for evaluating the natural logarithm functi on: " }{TEXT 0 3 "ln_" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "ln_: usage" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 262 20 "Calling Sequence: " }{TEXT -1 9 "ln_( x ) " }{TEXT 263 1 "\n" }{TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 15 "Parameters: " }{TEXT 264 21 "x \+ - a real constant" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 266 12 "Description:" }{TEXT -1 1 " " }{TEXT 265 14 "The pr ocedure " }{TEXT -1 3 "ln_" }{TEXT 268 38 " calculates ln(x) for a rea l 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 anywhere after the prompt [ > and press [Enter].\n You can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "ln_: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15628 "ln_ := proc(x::algebraic) \n local u;\n\n if nargs <> 1 then\n error \"expecting 1 argu ment, got %1\", nargs;\n end if;\n if type(x,'float') then evalf(' ln_'(x))\n elif type(x,'function') and nops(x)=1 then\n u := op (1,x);\n if op(0,x)='exp' or op(0,x)='exp_' then u\n else 'l n_'(x)\n end if;\n elif x=0 then -'infinity'\n elif x='infini ty' then 'infinity'\n else 'ln_'(x)\n end if;\nend proc:\n\n`evalf /ln_` := proc(xx)\n local x,term,eps,k,i,maxit,saveDigits,hfDigits, \n extraDigits,sum,s,t,z,R,T,L,r,m,pow,val,d;\n\n if not type(xx,r ealcons) then return 'ln_'(xx) end if;\n\n if xx=0 then\n retur n Float(-infinity)\n elif xx=1. then\n return 0.\n else\n \+ x := evalf(xx);\n if x<0 then\n error \"cannot evaluate for a negative argument\"\n end if;\n end if;\n\n # Use a fi xed precision procedure for low precision values\n hfDigits := trunc (evalhf(Digits));\n if Digits<=hfDigits then\n # avoid subtract ion error when x is close to 1\n d := abs(x-1);\n if d<0.01 \+ then\n if d=0 then return 0. end if;\n saveDigits := D igits;\n Digits := max(hfDigits+2,Digits-ilog10(d));\n \+ t := evalf(xx)-1;\n s := t/(x+1);\n z := s*s; \+ \n val := (2+(0.66666666666666666667+0.4*z)*z)*s;\n Di gits := saveDigits;\n return evalf(val);\n else \n \+ x := evalf(xx,hfDigits+2);\n val := traperror(evalhf(ln16(x) ));\n if val<>lasterror then\n return evalf(val);\n end if;\n end if;\n end if;\n \n # Increase precis ion\n saveDigits := Digits;\n Digits := Digits+length(Digits)+1;\n x := evalf(xx);\n \n # Reduce the range to 10^(-1/8)<=x<10 ^(1/8) approximately\n k := 0;\n r := 0;\n if x>1.33352143216332 40257 or x<.74989420933245582730 then\n k := ilog10(x);\n x \+ := Scale10(x,-k); # x*10^(-k)\n if x>1.3335214321633240257 then\n R := evalf(root8of10inv);\n # multiply successively b y 10^(-1/4)\n if x>1.7782794100389228012 then\n T : = R^2;\n while x>1.7782794100389228012 do\n x := x*T;\n r := r+2;\n end do;\n end \+ if;\n x := x*R;\n r := r+1;\n end if;\n end if; \n\n if saveDigits<=hfDigits then\n val := traperror(evalhf(ln1 6(x)));\n if val<>lasterror then\n if k<>0 or r<>0 then\n L := evalf(Ln10)*0.125;\n val := val+(8*k+r)*L \n end if;\n return evalf[saveDigits](val);\n end if;\n end if; \n\n t := x-1;\n if k=0 and r=0 then\n e xtraDigits := -ilog10(abs(t));\n # avoid subtraction error when x is close to 1\n if extraDigits>1 then\n t := evalf[Digit s+extraDigits](xx-1);\n end if;\n end if;\n\n s := t/(x+1);\n \n if saveDigits<27 then\n val := ln25(s)\n elif saveDigits<5 7 then\n val := ln55(s)\n else # use Maclaurin series\n ep s := Float(5,-Digits);\n maxit := Digits*3;\n\n pow := s;\n \+ z := s*s;\n sum := s; \n for i from 3 to maxit by 2 do \n pow := pow*z;\n term := pow/i;\n sum := sum + term;\n if abs(term)<=eps*abs(sum) then break end if;\n \+ end do;\n val := 2*sum;\n end if;\n\n if k<>0 or r<>0 then \n L := evalf(Ln10)*0.125;\n val := val+(8*k+r)*L\n end if ;\n\n Digits := saveDigits;\n evalf(val);\nend proc: # of `evalf/l n_`\n\n`evalf/constant/root8of10inv` := proc()\nlocal d,r,r2,r4;\nglob al _root8of10inv;\n if Digits<=55 then evalf(.7498942093324558273021 842756151364384418679181649710146)\n elif Digits<=length(op(1,_root8 of10inv)) then evalf(_root8of10inv)\n else\n d := length(op(1,_ root8of10inv));\n r := _root8of10inv;\n while d1.414213562 or x<.7071067812 then\n if x>2 or x< 1 then\n k := intlog2(x);\n x := x*2.^(-k);\n end if;\n if x>1.414213562 then \n x := x*r2i;\n r : = 1;\n end if;\n end if;\n\n # transformation to use approx f or ln((1+x)/(1-x))\n s := (x-1)/(x+1);\n\n z := s*s; \n # eval uate the polynomial approximation\n p := (2+(a1+(a2+(a3+(a4+(a5+(a6+ a7*z)*z)*z)*z)*z)*z)*z)*s;\n if k<>0 or r<>0 then p+ln2*(k+r*0.5) el se p end if;\nend proc: # of ln16\n\nPOW2 := hfarray(-1019..1019):\nfo r i from -1019 to 1019 do\n POW2[i] := evalhf(2^i):\nend do:\n\nintl og2 := proc(x)\n if x>POW2[1019] or xn then\n jLo := m;\n \+ jHi := n;\n else \n inc := 1;\n if x>=A[jLo] then \n \+ if jLo=n then return jLo end if;\n jHi := jLo+1;\n \+ while x>=A[jHi] do \n jLo := jHi;\n inc := \+ inc+inc;\n jHi := jLo+inc;\n if jHi>n then \n \+ jHi := n+1;\n break;\n end if;\n \+ end do;\n else\n if jLo=m then return jLo end if; \n jHi := jLo-1;\n while x 1 do\n jM := trunc(( jHi+jLo)/2);\n if x>=A[jM] then jLo := jM else jHi := jM end if; \n end do;\n return jLo;\nend proc: # of hunt" }}}{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 gi ven 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 3 "ln_" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Testing the procedure" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 61 "xx := 123.456;\nevalf(ln_(xx),20);\nevalf(evalf(ln( xx),25),20);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"'cM7!\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5J)QE$G<[)e\"[!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5J)QE$G<[)e\"[!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "xx := 5e309;\nevalf(ln_ (xx),100);\nevalf(ln(xx),100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x xG$\"\"&\"$4$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qQ0z2DqKbdTd4!y`6 a/Hp4@;tOf0MhfE)yFZpI)=,;On@%fZ;B3Jr!#(*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qQ0z2DqKbdTd4!y`6a/Hp4@;tOf0MhfE)yFZpI)=,;On@%fZ;B3Jr!#(*" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "plot(ln_(x),x=0.01..10);" }}{PARA 13 "" 1 "" {GLPLOT2D 431 290 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