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"" -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 "T imes" 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" -1 258 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 60 "Evaluation of functions using con tinued fraction expansions " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter S tone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: \+ 25.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 44 "Introductory example of a c ontinued fraction" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 17 "It is clear that " }{XPPEDIT 18 0 "1 < sq rt(2)" "6#2\"\"\"-%%sqrtG6#\"\"#" }{XPPEDIT 18 0 "`` < 2" "6#2%!G\"\"# " }{TEXT -1 15 " and hence that" }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "sqrt(2)=1+1/p" "6#/-%%sqrtG6#\"\"#,&\"\"\"F)*&F)F)%\"pG !\"\"F)" }{TEXT -1 14 " ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 6 "w here " }{XPPEDIT 18 0 "p>1" "6#2\"\"\"%\"pG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1/p=sqrt(2)-1" "6#/*&\"\"\"F%%\"pG!\"\",&-%%sqrtG6#\"\" #F%F%F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p = 1/(sqrt(2)-1)" "6#/%\"p G*&\"\"\"F&,&-%%sqrtG6#\"\"#F&F&!\"\"F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 "Multiplying the numerator and denominator by " } {XPPEDIT 18 0 "sqrt(2)+1" "6#,&-%%sqrtG6#\"\"#\"\"\"F(F(" }{TEXT -1 13 " we see that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " p = (sqrt(2)+1)/((sqrt(2)-1)*(sqrt(2)-1))" "6#/%\"pG*&,&-%%sqrtG6#\"\" #\"\"\"F+F+F+*&,&-F(6#F*F+F+!\"\"F+,&-F(6#F*F+F+F0F+F0" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "p = sqrt(2)+1;" "6#/%\"pG,&-%%sqrtG6#\"\"#\" \"\"F*F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 36 "Substituting \+ this back in (i) gives " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(2) = 1+ 1/(1+sqrt(2))" "6#/-%%sqrtG6#\"\"#,&\"\"\"F)*&F)F), &F)F)-F%6#F'F)!\"\"F)" }{TEXT -1 15 " ------- (ii)." }}{PARA 0 "" 0 " " {TEXT -1 30 "Now substitute the expression " }{XPPEDIT 18 0 "1+1/(1+ sqrt(2))" "6#,&\"\"\"F$*&F$F$,&F$F$-%%sqrtG6#\"\"#F$!\"\"F$" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 41 " in the denominator on the right of (ii)." }}{PARA 0 "" 0 "" {TEXT -1 11 "This gives " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "sqrt(2) = 1+1/(1+1+1/(1+sqrt(2)));" "6#/-%%sqrtG6#\"\"# ,&\"\"\"F)*&F)F),(F)F)F)F)*&F)F),&F)F)-F%6#F'F)!\"\"F)F0F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(2) = 1+1/(2+1/(1+sqrt(2)))" "6#/-%%sqrtG 6#\"\"#,&\"\"\"F)*&F)F),&F'F)*&F)F),&F)F)-F%6#F'F)!\"\"F)F0F)" }{TEXT -1 16 " ------- (iii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 96 "This process can be repeated, substituting for sqrt( 2) on the right of (iii) using (ii) to give " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(2) = 1+1/(2+1/(2+1/(1+sqrt(2))))" "6#/-%%sqrtG6#\"\"#,&\"\"\"F)*&F)F),&F'F)*&F)F),&F'F)*&F)F),&F)F)-F%6# F'F)!\"\"F)F2F)F2F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 10 "a nd so on." }}{PARA 0 "" 0 "" {TEXT -1 13 "We may write " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(2) = 1+1/(2+1/(2+1/(2+1/(2+ ` . . . `))));" "6#/-%%sqrtG6#\"\"#,&\"\"\"F)*&F)F),&F'F)*&F)F),&F'F)* &F)F),&F'F)*&F)F),&F'F)%(~.~.~.~GF)!\"\"F)F3F)F3F)F3F)" }{TEXT -1 1 ", " }}{PARA 0 "" 0 "" {TEXT -1 62 "where we imagine the fraction on the \+ right continuing forever." }}{PARA 0 "" 0 "" {TEXT -1 24 "This is an e xample of a " }{TEXT 259 18 "continued fraction" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 60 "If we st op at some stage and replace the last occurrence of " }{XPPEDIT 18 0 " sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 89 " by some rough approximatio n, the continued fraction produces a better approximation for " } {XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "xx : = 3/2;\n1+1/(2+1/(2+1/(2+1/(2+xx))));\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG#\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6## \"$L\"\"#%*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Evaluating continued fr actions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 43 "A general continued fraction has the form: " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = a[0]+b[1]/(a[1]+b [2]/(a[2]+` . . . `));" "6#/%\"xG,&&%\"aG6#\"\"!\"\"\"*&&%\"bG6#F*F*,& &F'6#F*F**&&F-6#\"\"#F*,&&F'6#F5F*%(~.~.~.~GF*!\"\"F*F:F*" }{TEXT -1 13 " ------- (i)." }}{PARA 0 "" 0 "" {TEXT -1 5 "Let " }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r[n] = a[0]+b[1]/(a[1]+b[2]/(` \+ `[` . `][` . `][` `]/(a[n-1]+b[n]/a[n])));" "6#/&%\"rG 6#%\"nG,&&%\"aG6#\"\"!\"\"\"*&&%\"bG6#F-F-,&&F*6#F-F-*&&F06#\"\"#F-*&& &&%*~~~~~~~~~G6#%$~.~G6#F?6#%+~~~~~~~~~~GF-,&&F*6#,&F'F-F-!\"\"F-*&&F0 6#F'F-&F*6#F'FGF-FGFGF-FGF-" }{TEXT -1 15 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "Then the meaning \+ of equation (i) is that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x=Limit(r[n],n=infinity)" "6#/%\"xG-%&LimitG6$&%\"rG6#% \"nG/F+%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 57 "One way of evaluating a continued fraction is as follows." }}{PARA 0 "" 0 "" {TEXT -1 28 "Set up four initial values: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([p[-1] = 1, p[0] = a[0]],[q[- 1] = 0, q[0] = 1]);" "6#-%*PIECEWISEG6$7$/&%\"pG6#,$\"\"\"!\"\"F,/&F)6 #\"\"!&%\"aG6#F17$/&%\"qG6#,$F,F-F1/&F86#F1F," }{TEXT -1 3 " , " }} {PARA 0 "" 0 "" {TEXT -1 13 "then iterate " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([p[n] = a[n]*p[n-1]+b[n]*p[n-2] ,` ` ],[ q[n] = a[n]*q[n-1]+b[n]*q[n-2] , `` ])" "6#-%*PIECEWISEG6$7$/&% \"pG6#%\"nG,&*&&%\"aG6#F+\"\"\"&F)6#,&F+F1F1!\"\"F1F1*&&%\"bG6#F+F1&F) 6#,&F+F1\"\"#F5F1F1%!G7$/&%\"qG6#F+,&*&&F/6#F+F1&FB6#,&F+F1F1F5F1F1*&& F86#F+F1&FB6#,&F+F1F=F5F1F1F>" }{TEXT -1 15 " ------- (iii) " }}{PARA 0 "" 0 "" {TEXT -1 4 "for " }{XPPEDIT 18 0 "n = 1,2,3,` . . . `" "6&/% \"nG\"\"\"\"\"#\"\"$%(~.~.~.~G" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "Then it can be shown by induction that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r[n] = p[n]/q[n]" "6#/&%\"rG6#%\"n G*&&%\"pG6#F'\"\"\"&%\"qG6#F'!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 8 "so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = Limit(p[n]/q[n],n=infinity)" "6#/%\"xG-%&LimitG6$*&&%\"pG6#% \"nG\"\"\"&%\"qG6#F,!\"\"/F,%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 51 "Inductive justification for the i terative formulas " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 74 "Suppose that the iterative formulas (iii) correctly produce the value for " }{XPPEDIT 18 0 "r[n]" "6#&%\"rG6#% \"nG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "r[n+1]" "6#&%\"rG6#,&%\"nG\"\"\"F(F(" }{TEXT -1 32 " can be obta ined by substituting" }{XPPEDIT 18 0 "``(a[n]+b[n+1]/a[n+1]);" "6#-%!G 6#,&&%\"aG6#%\"nG\"\"\"*&&%\"bG6#,&F*F+F+F+F+&F(6#,&F*F+F+F+!\"\"F+" } {TEXT -1 13 " in place of " }{XPPEDIT 18 0 "a[n]" "6#&%\"aG6#%\"nG" } {TEXT -1 21 " in the formula (ii)." }}{PARA 0 "" 0 "" {TEXT -1 25 "Hen ce we can also obtain " }{XPPEDIT 18 0 "r[n+1]" "6#&%\"rG6#,&%\"nG\"\" \"F(F(" }{TEXT -1 16 " by substituting" }{XPPEDIT 18 0 "``(a[n]+b[n+1] /a[n+1]);" "6#-%!G6#,&&%\"aG6#%\"nG\"\"\"*&&%\"bG6#,&F*F+F+F+F+&F(6#,& F*F+F+F+!\"\"F+" }{TEXT -1 13 " in place of " }{XPPEDIT 18 0 "a[n]" "6 #&%\"aG6#%\"nG" }{TEXT -1 47 " in each of the formulas (iii) to and di viding." }}{PARA 0 "" 0 "" {TEXT -1 8 "That is," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r[n+1]=((a[n]+b[n+1]/a[n+1])*p[n-1] + b [n]*p[n-2])/((a[n]+b[n+1]/a[n+1])*q[n-1] + b[n]*q[n-2])" "6#/&%\"rG6#, &%\"nG\"\"\"F)F)*&,&*&,&&%\"aG6#F(F)*&&%\"bG6#,&F(F)F)F)F)&F/6#,&F(F)F )F)!\"\"F)F)&%\"pG6#,&F(F)F)F9F)F)*&&F36#F(F)&F;6#,&F(F)\"\"#F9F)F)F), &*&,&&F/6#F(F)*&&F36#,&F(F)F)F)F)&F/6#,&F(F)F)F)F9F)F)&%\"qG6#,&F(F)F) F9F)F)*&&F36#F(F)&FR6#,&F(F)FDF9F)F)F9" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 45 "Multiplying the numerator and denominator by " } {XPPEDIT 18 0 "a[n+1]" "6#&%\"aG6#,&%\"nG\"\"\"F(F(" }{TEXT -1 8 " giv es " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "r[n+1] = ((a [n+1]*a[n]+b[n+1])*p[n-1]+a[n+1]*b[n]*p[n-2])/((a[n+1]*a[n]+b[n+1])*q[ n-1]+a[n+1]*b[n]*q[n-2]);" "6#/&%\"rG6#,&%\"nG\"\"\"F)F)*&,&*&,&*&&%\" aG6#,&F(F)F)F)F)&F06#F(F)F)&%\"bG6#,&F(F)F)F)F)F)&%\"pG6#,&F(F)F)!\"\" F)F)*(&F06#,&F(F)F)F)F)&F66#F(F)&F:6#,&F(F)\"\"#F=F)F)F),&*&,&*&&F06#, &F(F)F)F)F)&F06#F(F)F)&F66#,&F(F)F)F)F)F)&%\"qG6#,&F(F)F)F=F)F)*(&F06# ,&F(F)F)F)F)&F66#F(F)&FU6#,&F(F)FGF=F)F)F=" }{TEXT -1 14 " ------- (iv )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "On \+ the other hand, replacing " }{TEXT 262 1 "n" }{TEXT -1 4 " by " } {XPPEDIT 18 0 "n+1" "6#,&%\"nG\"\"\"F%F%" }{TEXT -1 30 " in the formul as (iii) gives " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " r[n+1] = (a[n+1]*p[n]+b[n+1]*p[n-1])/(a[n+1]*q[n]+b[n+1]*q[n-1])" "6#/ &%\"rG6#,&%\"nG\"\"\"F)F)*&,&*&&%\"aG6#,&F(F)F)F)F)&%\"pG6#F(F)F)*&&% \"bG6#,&F(F)F)F)F)&F26#,&F(F)F)!\"\"F)F)F),&*&&F.6#,&F(F)F)F)F)&%\"qG6 #F(F)F)*&&F66#,&F(F)F)F)F)&FC6#,&F(F)F)F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 90 "The following procedure uses the formulas (iii) to eval uate a (finite) continued fraction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 527 "contfrac := proc(a::list(a lgebraic),b::list(algebraic))\n local pp,pc,pf,qp,qc,qf,i,n,m;\n\n \+ n := nops(a);\n m := nops(b);\n if m=0 then\n error \"2nd li st cannot be empty\"\n end if;\n if n<>m+1 then\n error \"1st list must have exactly one more element than 2nd list\"\n end if;\n pp := 1;\n pc := a[1];\n qp := 0;\n qc := 1;\n for i from 2 to n do \n pf := a[i]*pc + b[i-1]*pp;\n qf := a[i]*qc + b[i -1]*qp;\n pp := pc;\n qp := qc;\n pc := pf;\n qc : = qf;\n end do;\n pf/qf;\nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "n := 1;\nunassign('a','b'):\nA := [seq(a[i],i=0..n)]; \nB := [seq(b[i],i=1..n)];\ncontfrac(A,B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG7 $&%\"aG6#\"\"!&F'6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG7#& %\"bG6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&&%\"aG6#\"\"\"F )&F'6#\"\"!F)F)&%\"bGF(F)F)F&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "n := 2;\nunassign('a','b'):\nA := [seq(a[i],i=0..n)];\nB := [seq (b[i],i=1..n)];\ncontfrac(A,B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"nG\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG7%&%\"aG6#\"\"!&F'6 #\"\"\"&F'6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG7$&%\"bG6# \"\"\"&F'6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&&%\"aG6#\"\" #\"\"\",&*&&F'6#F*F*&F'6#\"\"!F*F*&%\"bGF.F*F*F**&&F3F(F*F/F*F*F*,&*&F &F*F-F*F*F5F*!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "n := 15;\nA := [1,2$(n-1)];\nB := [1$(n-1)] ;\ncontfrac(A,B);\nevalf[12](%);\nevalf[12](sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" AG71\"\"\"\"\"#F'F'F'F'F'F'F'F'F'F'F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG70\"\"\"F&F&F&F&F&F&F&F&F&F&F&F&F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6##\"'2eF\"'D]>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"-OiN@99!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-PiN@99!#6" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "Digits := 50:\nn := 65;\nA := [1.,2.$(n-1)]:\nB := [1.$(n-1)]:\nc ontfrac(A,B);\nevalf(sqrt(2));\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#l" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SpPv=n p&y!)p4Us)o,)[]4tBc8UT\"!#\\" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"SpP v=np&y!)p4Us)o,)[]4tBc8UT\"!#\\" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "n := 20;\nA := [1,2$(n-1)]; \nB := [seq((2*j-1)^2,j=1..n-1)];\ncontfrac(A,B);\nevalf(%,12);\nevalf (4/Pi,12);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#?" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"AG76\"\"\"\"\"#F'F'F'F'F'F'F'F'F'F'F'F'F'F'F 'F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG75\"\"\"\"\"*\"#D\"# \\\"#\")\"$@\"\"$p\"\"$D#\"$*G\"$h$\"$T%\"$H&\"$D'\"$H(\"$T)\"$h*\"%*3 \"\"%D7\"%p8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"0DK.3m'p;\"0C&y][\\ !H\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-Qv%=QH\"!#6" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"-uW&RKF\"!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "n := 1000;\nA := [1,2.$( n-1)]:\nB := [seq((2.*j-1)^2,j=1..n-1)]:\ncontfrac(A,B);\nevalf(4/Pi,1 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"%+5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+i\\kt7!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\" -uW&RKF\"!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 50 "Lentz 's method for evaluating continued fractions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 23 "The iterative \+ formulas " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWIS E([p[-1] = 1, p[0] = a[0]],[q[-1] = 0, q[0] = 1]);" "6#-%*PIECEWISEG6$ 7$/&%\"pG6#,$\"\"\"!\"\"F,/&F)6#\"\"!&%\"aG6#F17$/&%\"qG6#,$F,F-F1/&F8 6#F1F," }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([p[n] = a[n]*p[n-1]+b[ n]*p[n-2] ,`` ],[ q[n] = a[n]*q[n-1]+b[n]*q[n-2] , `` ])" "6#-%*PIEC EWISEG6$7$/&%\"pG6#%\"nG,&*&&%\"aG6#F+\"\"\"&F)6#,&F+F1F1!\"\"F1F1*&&% \"bG6#F+F1&F)6#,&F+F1\"\"#F5F1F1%!G7$/&%\"qG6#F+,&*&&F/6#F+F1&FB6#,&F+ F1F1F5F1F1*&&F86#F+F1&FB6#,&F+F1F=F5F1F1F>" }{TEXT -1 15 "------- (iii ) ," }}{PARA 0 "" 0 "" {TEXT -1 29 "for obtaining approximations " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "r[n] = p[n]/q[n]" "6# /&%\"rG6#%\"nG*&&%\"pG6#F'\"\"\"&%\"qG6#F'!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 55 "for a continued fraction date back to J. \+ Wallis (1655)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "An undesirable feature of this method is that the numerat ors and denominators " }{XPPEDIT 18 0 "p[n]" "6#&%\"pG6#%\"nG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q[n]" "6#&%\"qG6#%\"nG" }{TEXT -1 21 " ca n grow very large." }}{PARA 0 "" 0 "" {TEXT -1 77 "An alternative iter ation scheme, due to W. J. Lentz (1976), uses the ratios " }{XPPEDIT 18 0 "s[n] = p[n]/p[n-1]" "6#/&%\"sG6#%\"nG*&&%\"pG6#F'\"\"\"&F*6#,&F' F,F,!\"\"F0" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "t[n] = q[n-1]/q[n]; " "6#/&%\"tG6#%\"nG*&&%\"qG6#,&F'\"\"\"F-!\"\"F-&F*6#F'F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "r[n]=r[n-1]*`.`*s[n]*`.`*t[n]" "6#/&%\"rG6#% \"nG*,&F%6#,&F'\"\"\"F,!\"\"F,%\".GF,&%\"sG6#F'F,F.F,&%\"tG6#F'F," } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p[n] = a[n]*p[n-1]+b[n]*p[n-2]" "6#/&%\"pG6#%\"nG,&*&&%\"aG6#F'\"\"\" &F%6#,&F'F-F-!\"\"F-F-*&&%\"bG6#F'F-&F%6#,&F'F-\"\"#F1F-F-" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "we have " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "p[n]/p[n-1] = a[n]+b[n]*``(p[n-2]/p[n-1 ]);" "6#/*&&%\"pG6#%\"nG\"\"\"&F&6#,&F(F)F)!\"\"F-,&&%\"aG6#F(F)*&&%\" bG6#F(F)-%!G6#*&&F&6#,&F(F)\"\"#F-F)&F&6#,&F(F)F)F-F-F)F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "s[n] = a[n]+b[n]/s[n-1]" "6#/&%\"sG6#% \"nG,&&%\"aG6#F'\"\"\"*&&%\"bG6#F'F,&F%6#,&F'F,F,!\"\"F4F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "Similarly," }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[n] = 1/(a[n]+b[n]*t[n-1])" "6#/&%\"tG 6#%\"nG*&\"\"\"F),&&%\"aG6#F'F)*&&%\"bG6#F'F)&F%6#,&F'F)F)!\"\"F)F)F5 " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "This gives the itera tion scheme " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECE WISE([s[0] = a[0], t[0] = 0],[r[0] = a[0], ``]);" "6#-%*PIECEWISEG6$7$ /&%\"sG6#\"\"!&%\"aG6#F+/&%\"tG6#F+F+7$/&%\"rG6#F+&F-6#F+%!G" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {XPPEDIT 18 0 "PIECEWISE([s[n] = a[n]+b[n]/s[n-1], ``],[t[n] = 1/(a[n] +b[n]*t[n-1]), ``],[r[n] = r[n-1]*s[n]*t[n], ``]);" "6#-%*PIECEWISEG6% 7$/&%\"sG6#%\"nG,&&%\"aG6#F+\"\"\"*&&%\"bG6#F+F0&F)6#,&F+F0F0!\"\"F8F0 %!G7$/&%\"tG6#F+*&F0F0,&&F.6#F+F0*&&F36#F+F0&F=6#,&F+F0F0F8F0F0F8F97$/ &%\"rG6#F+*(&FL6#,&F+F0F0F8F0&F)6#F+F0&F=6#F+F0F9" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "The foll owing procedure implements this algorithm." }}{PARA 0 "" 0 "" {TEXT -1 96 "An advantage of this iteration scheme is that convergence can b e checked by comparing the ratio " }{XPPEDIT 18 0 "Delta[n]=s[n]*t[n] " "6#/&%&DeltaG6#%\"nG*&&%\"sG6#F'\"\"\"&%\"tG6#F'F," }{TEXT -1 9 " wi th 1. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 524 "lentz := proc(a::list(algebraic),b::list(algebraic)) \n local tiny,eps,i,n,m,rp,sp,tp,rc,tc,sc;\n\n n := nops(a);\n m := nops(b);\n if m=0 then\n error \"2nd list cannot be empty\" \n end if;\n if n<>m+1 then\n error \"1st list must have exac tly one more element than 2nd list\"\n end if;\n\n rp := a[1];\n \+ sp := rp;\n tp := 0;\n for i from 2 to n do\n sc := a[i] + b [i-1]/sp; \n tc := 1/(a[i] + b[i-1]*tp);\n rc := rp*sc*tc;\n rp := rc;\n tp := tc;\n sp := sc;\n end do;\n rc; \nend proc:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "We can check that this new procedure works and gives the same \+ results as the procedure given in the previous section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "n := 1; \nunassign('a','b'):\nA := [seq(a[i],i=0..n)];\nB := [seq(b[i],i=1..n) ];\nlentz(A,B);\nnormal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG7$&%\"aG6#\"\"!&F'6#\" \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG7#&%\"bG6#\"\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*(&%\"aG6#\"\"!\"\"\",&&F%6#F(F(*&&%\" bGF+F(F$!\"\"F(F(F*F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&&%\"aG6 #\"\"\"F)&F'6#\"\"!F)F)&%\"bGF(F)F)F&!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "n := 3;\nunassign('a','b'):\nA := [seq(a[i],i=0..n)] ;\nB := [seq(b[i],i=1..n)];\nlentz(A,B);\nans1 := normal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG7&&%\"aG6#\"\"!&F'6#\"\"\"&F'6#\"\"#&F'6#\"\"$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"BG7%&%\"bG6#\"\"\"&F'6#\"\"#&F'6#\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*0&%\"aG6#\"\"!\"\"\",&&F%6#F(F(*&&%\" bGF+F(F$!\"\"F(F(F*F/,&&F%6#\"\"#F(*&&F.F2F(F)F/F(F(,&F1F(*&F5F(F*F/F( F/,&&F%6#\"\"$F(*&&F.F:F(F0F/F(F(,&F9F(*&F=F(F6F/F(F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ans1G*&,,**&%\"aG6#\"\"$\"\"\"&F)6#\"\"#F,&F)6# F,F,&F)6#\"\"!F,F,*(F(F,F-F,&%\"bGF1F,F,*(F(F,&F7F.F,F2F,F,*(&F7F*F,F0 F,F2F,F,*&F;F,F6F,F,F,,(*(F(F,F-F,F0F,F,*&F(F,F9F,F,*&F;F,F0F,F,!\"\" " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "n := 3;\nunassign('a','b'): \nA := [seq(a[i],i=0..n)];\nB := [seq(b[i],i=1..n)];\ncontfrac(A,B);\n ans2 := normal(%);\nis(ans2=ans1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"nG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG7&&%\"aG6#\"\"!&F '6#\"\"\"&F'6#\"\"#&F'6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"B G7%&%\"bG6#\"\"\"&F'6#\"\"#&F'6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&&%\"aG6#\"\"$\"\"\",&*&&F'6#\"\"#F*,&*&&F'6#F*F*&F'6#\"\"!F*F *&%\"bGF3F*F*F**&&F8F.F*F4F*F*F*F**&&F8F(F*F0F*F*F*,&*&F&F*,&*&F-F*F2F *F*F:F*F*F**&F%%ans 2G*&,,**&%\"aG6#\"\"$\"\"\"&F)6#\"\"#F,&F)6#F,F,&F)6#\"\"!F,F,*(F(F,F- F,&%\"bGF1F,F,*(F(F,&F7F.F,F2F,F,*(&F7F*F,F0F,F2F,F,*&F;F,F6F,F,F,,(*( F(F,F-F,F0F,F,*&F(F,F9F,F,*&F;F,F0F,F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "n := 15;\nA := [1,2$(n-1)];\nB := [ 1$(n-1)];\nlentz(A,B);\nevalf[12](%);\nevalf[12](sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"AG71\"\"\"\"\"#F'F'F'F'F'F'F'F'F'F'F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG70\"\"\"F&F&F&F&F&F&F&F&F&F&F&F&F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6##\"'2eF\"'D]>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"-OiN@99!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-PiN@99!#6" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 47 "Argument reduction for \+ the arctangent function " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 0 "" 0 "" {TEXT -1 13 "The formula " }{XPPEDIT 18 0 "tan(2 *theta)=2*tan(theta)/(1-tan(theta)^2)" "6#/-%$tanG6#*&\"\"#\"\"\"%&the taGF)*(F(F)-F%6#F*F),&F)F)*$-F%6#F*F(!\"\"F2" }{TEXT -1 83 " can be u sed to obtain a method of argument reduction for the arctangent functi on." }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " }{XPPEDIT 18 0 "y=a rctan(x)" "6#/%\"yG-%'arctanG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[0] = arctan(x[0])" "6#/&%\"yG6#\"\"!-%'arctanG6#&%\"xG6#F'" } {TEXT -1 7 " where " }{XPPEDIT 18 0 "y=2*y[0]" "6#/%\"yG*&\"\"#\"\"\"& F$6#\"\"!F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " } {XPPEDIT 18 0 "x = tan(y)" "6#/%\"xG-%$tanG6#%\"yG" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "x[0]=tan(y[0])" "6#/&%\"xG6#\"\"!-%$tanG6#&%\"yG6#F' " }{TEXT -1 9 " so that " }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x=tan(2*y[0])" "6#/%\"xG-%$tanG6#*&\"\"#\"\"\"&%\"yG6# \"\"!F*" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*tan(y[0])/(1-tan(y[0])^2) " "6#*(\"\"#\"\"\"-%$tanG6#&%\"yG6#\"\"!F%,&F%F%*$-F'6#&F*6#F,F$!\"\"F 3" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 10 "that is, " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=2*x[0]/(1-x[0]^2)" "6#/%\"xG*(\"\"#\"\"\"&F$6#\"\"!F',&F'F'*$&F$6#F*F&!\"\"F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 164 "p1 := plot(2*x0/(1-x0^2),x0=-4..4,x=-4..4,discont=tr ue):\np2 := plots[implicitplot](\{x0=-1,x0=1\},x0=-4..4,x=-4..4,color= black,linestyle=3):\nplots[display]([p1,p2]);\n" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6&7gn7$$!\"%\"\"!$\"3EL LLLLLL`!#=7$$!3NxVnP&3Y$R!#<$\"3W?[![s7TV&F-7$$!3s,6&fx6x(QF1$\"31Chd4 [7DbF-7$$!30Knu;as8QF1$\"3()p'o%o,SJcF-7$$!3S$\\=W\"\\J\\PF1$\"3I`?7,m $Gu&F-7$$!3[QW'Ra5_o$F1$\"3uqf F-7$$!3aJqy,#QUc$F1$\"3#\\7/1$)R24'F-7$$!3$*[n&>3%f+NF1$\"3]*)[$=ey4A' F-7$$!3SAfipS:PMF1$\"3K,0+QK%oN'F-7$$!3/Q\\<>#)*=P$F1$\"3]Cxjm%oL]'F-7 $$!3!3@y0$3U9LF1$\"3)p]*3*fV&QmF-7$$!31g2]/\\r\\KF1$\"3%4M6Gp1\")z'F-7 $$!3w&QmP(GF1$\"3Q$3u\\A7$=zF-7$$ !3Y^0;)=$z9GF1$\"33L1@IQkJ\")F-7$$!3=#4e*[/4]FF1$\"3qn?(\\t#f!Q)F-7$$! 3S\"Rs.9y%)o#F1$\"3j&[]AFQOj)F-7$$!3r?)=*f@>CEF1$\"3(elVNaUh\"*)F-7$$! 3cY7&>cd^c#F1$\"3G(\\[%G#eS>*F-7$$!32^gus2[,DF1$\"35A,<>l-;&*F-7$$!3UI 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DF17$7$F_gn$!3y+++++++CF1Fain7$Fein7$F_gn$!3%>++++++?#F17$7$F_gn$!3'4+ +++++3#F1Fiin7$F]jn7$F_gn$!3m,+++++!)=F17$7$F_gn$!37,+++++g6F1Fi^o7$F]_o7$F_gn$\"3t)** *********>8F17$7$F_gn$\"3]************R9F1Fa_o7$Fe_o7$F_gn$\"3z)****** *****R;F17$7$F_gn$\"3c************fF17$7$F_gn$\"3i************z?F1Fa`o7$Fe`o7$F_gn$\"3[)***********zA F17$7$F_gn$\"3Y*************R#F1Fi`o7$F]ao7$F_gn$\"3u)************f#F1 7$7$F_gn$\"3J************>FF1Faao7$Feao7$F_gn$\"3:)***********>HF17$7$ F_gn$\"3:************RIF1Fiao7$F]bo7$F_gn$\"3W)***********RKF17$7$F_gn $\"3U************fLF1Fabo7$Febo7$F_gn$\"3%y***********fNF17$7$F_gn$\"3 #))***********zOF1Fibo7$F]co7$F_gn$\"37)***********zQF17$7$F_gn$\"3m)* ************RF1Faco-Fdfn6&FffnF*F*F*-%*LINESTYLEG6#\"\"$-F$6V7$7$$\"\" \"F*F(7$Fbdo$!3M++++++!)QF17$7$FbdoFfgnFddo7$Fhdo7$Fbdo$!3]++++++gNF17 $7$FbdoF^hnFjdo7$F^eo7$Fbdo$!35,+++++SKF17$7$Fbdo$!3/++++++SIF1F`eo7$7 $FbdoFfhn7$Fbdo$!3#3++++++#HF17$7$FbdoF^inFieo7$F]fo7$Fbdo$!3U,++++++E F17$7$Fbdo$!3M+++++++CF1F_fo7$7$FbdoFfin7$Fbdo$!39,+++++!G#F17$7$FbdoF ^jnFhfo7$F\\go7$Fbdo$!3I,+++++g>F17$7$Fbdo$!3o++++++g*F-7$7$FbdoF^_oF [[p7$F_[p7$Fbdo$\"3)))***********R7F17$7$FbdoFf_oFa[p7$Fe[p7$Fbdo$\"3s )***********f:F17$7$Fbdo$\"3M************f#F17$7$FbdoF^aoFh\\p7$F\\]p7$Fbdo$\"3p)***********>DF17 $7$Fbdo$\"3'))***********>FF1F^]p7$7$FbdoFfao7$Fbdo$\"3_)***********RG F17$7$FbdoF^boFg]p7$F[^p7$Fbdo$\"3O)***********fJF17$7$Fbdo$\"3)*)**** *******fLF1F]^p7$Fa^p7$Fbdo$\"3A)***********zMF17$7$Fbdo$\"3Q)******** ***zOF1Fe^p7$7$FbdoF^co7$Fbdo$\"3/)************z$F17$7$Fbdo$\"3A)***** ********RF1F^_pFhcoFjco-%+AXESLABELSG6%Q#x06\"Q\"xFi_p-%%FONTG6#%(DEFA ULTG-%%VIEWG6$;F(F_fnFb`p" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Now suppose that " }{XPPEDIT 18 0 "-1=0" "6#1\"\"!*&%\"xG\" \"\"&F&6#F$F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " } }{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*x[0]=sqrt(x^2+1)-1 " "6#/*&%\"xG\"\"\"&F%6#\"\"!F&,&-%%sqrtG6#,&*$F%\"\"#F&F&F&F&F&!\"\" " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*x[0] = (sqrt(x^2+1)-1)*(sqrt(x^2 +1)+1)/(1+sqrt(x^2+1));" "6#/*&%\"xG\"\"\"&F%6#\"\"!F&*(,&-%%sqrtG6#,& *$F%\"\"#F&F&F&F&F&!\"\"F&,&-F-6#,&*$F%F1F&F&F&F&F&F&F&,&F&F&-F-6#,&*$ F%F1F&F&F&F&F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }} {PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x*x[0] = x^2/(1+sqrt( x^2+1));" "6#/*&%\"xG\"\"\"&F%6#\"\"!F&*&F%\"\"#,&F&F&-%%sqrtG6#,&*$F% F+F&F&F&F&!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " } }{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x[0] = x/(1+sqrt(x^2 +1));" "6#/&%\"xG6#\"\"!*&F%\"\"\",&F)F)-%%sqrtG6#,&*$F%\"\"#F)F)F)F)! \"\"" }{TEXT -1 13 " ------- (i)." }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{TEXT 273 10 "__________" }{TEXT -1 14 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "The formula for the \+ inverse function is " }{XPPEDIT 18 0 "g^(-1)" "6#)%\"gG,$\"\"\"!\"\"" }{TEXT -1 2 "( " }{TEXT 272 1 "x" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 " x/(1+sqrt(x^2+1));" "6#*&%\"xG\"\"\",&F%F%-%%sqrtG6#,&*$F$\"\"#F%F%F%F %!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "plot([x /(1+sqrt(x^2+1)),1,-1],x=-8..8,color=[red,black$2],linestyle=[1,3$2]); " }}{PARA 13 "" 1 "" {GLPLOT2D 459 182 182 {PLOTDATA 2 "6'-%'CURVESG6 %7S7$$!\")\"\"!$!3_(=t`=Ay#))!#=7$$!3OLLLLbC^w!#<$!3I0Ic252y()F-7$$!3? 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Then " }{XPPEDIT 18 0 "y=2*y[0]" "6#/%\"yG* &\"\"#\"\"\"&F$6#\"\"!F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 81 "Using formula this method of argum ent reduction in conjunction with the formulas " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([ arctan(x) = -arctan(-x),``] ,[arctan(x) = Pi/2-arctan(1/x) ,``])" "6#-%*PIECEWISEG6$7$/-%'arctanG 6#%\"xG,$-F)6#,$F+!\"\"F0%!G7$/-F)6#F+,&*&%#PiG\"\"\"\"\"#F0F9-F)6#*&F 9F9F+F0F0F1" }{TEXT -1 15 " ------- (ii), " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "it is sufficient to find an app roximation for " }{XPPEDIT 18 0 "arctan(x)" "6#-%'arctanG6#%\"xG" } {TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-tan(Pi/8), tan(Pi/8) ];" "6#7$,$-%$tanG6#*&%#PiG\"\"\"\"\")!\"\"F,-F&6#*&F)F*F+F," }{TEXT -1 34 ", or approximately [-0.414,0.414]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "To illustrate this idea, we define a function " }{TEXT 0 5 "atan1" }{TEXT -1 52 " using Maple's arctange nt function on the interval " }{XPPEDIT 18 0 "[-tan(Pi/8), tan(Pi/8)] ;" "6#7$,$-%$tanG6#*&%#PiG\"\"\"\"\")!\"\"F,-F&6#*&F)F*F+F," }{TEXT -1 58 ", but ensure that it gives no value outside this interval." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "atan1 := x -> if abs(x)<=.4142135625 then arctan(x) else FAIL end if;\nplot('atan1(x)',x=-0.6..0.6,thickness=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&atan1Gf*6#%\"xG6\"6$%)operatorG%&arrowGF(@%1-%$absG6 #9$$\"+Dc8UT!#5-%'arctanGF0%%FAILGF(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 410 290 290 {PLOTDATA 2 "6&-%'CURVESG6$7L7$$!3!******p2?w8%! 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+Ms$z#)*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+Ks$z#)*!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "plot(['atan2(x)',Pi/2,-Pi/2],x=-8..8,color=[red,black$2],\n \+ linestyle=[1,4$2],thickness=[2,1$2]); " }}{PARA 13 "" 1 "" {GLPLOT2D 498 177 177 {PLOTDATA 2 "6'-%'CURVESG6&7W7$$!\")\"\"!$!35N\" [ALTkW\"!#<7$$!3OLLLLbC^wF-$!3&zkhSIN3W\"F-7$$!3?mmmOhzZtF-$!3Np*=%H@` N9F-7$$!3LLLL`b`1qF-$!3V0$oP')H!H9F-7$$!3#HLLL(G,jmF-$!3*>NYZ$f#=U\"F- 7$$!30nmm'*G7@jF-$!3Ic,Xzo*QT\"F-7$$!3XLLLBr9/gF-$!3sxtYe'fdS\"F-7$$!3 !)******pq$fn&F-$!3%eK3Hm.kR\"F-7$$!3fLLLj<]O`F-$!31gY8Afb&Q\"F-7$$!3Q +++I]:)*\\F-$!3u-ZP\"yHLP\"F-7$$!3YmmmEQ7]YF-$!3%o#3y_U(*e8F-7$$!3HLLL `xdVVF-$!3v$)*f\\H9XM\"F-7$$!3I+++![z%)*RF-$!3\\RTI*=GdK\"F-7$$!35++++ U'>l$F-$!3r!**e=gANI\"F-7$$!3/+++?D.=LF-$!33VWC?32y7F-7$$!3SLLLj0z9IF- $!3q*3OQH=0D\"F-7$$!3!pmmma1Ul#F-$!3@B3wfJ[57F-7$$!3=nmm'eW([BF-$!3+iF enEGo6F-7$$!3S+++5(>M*>F-$!3z_3S(=He5\"F-7$$!3Unmm')p*)y;F-$!37(3`%yof L5F-7$$!3l******4d\"QL\"F-$!3g(=3SL)ou#*!#=7$$!3*)******Hn@05F-$!3+7lF\\r$\"3KpBr`:PxdF\\r7$$\" 3AemmmK\"f$)*F\\r$\"3)H='eu9ErxF\\r7$$\"3W******f0AE8F-$\"3A'4u3`esC*F \\r7$$\"3M)*****>kTh;F-$\"3nK'o'HQ)*G5F-7$$\"3u)*****\\ct&)>F-$\"3J0Q_ %[zU5\"F-7$$\"3e)*****fo$eM#F-$\"3#)*3Tl*f$y;\"F-7$$\"3?KLL8QSpEF-$\"3 ![L^&4GO77F-7$$\"3p*******f!)[,$F-$\"3y+/(y@F0D\"F-7$$\"3%fmmm\"R$zK$F -$\"3))QyVXI*)y7F-7$$\"3s******zQ=qOF-$\"3$4!R\")fvy/8F-7$$\"3mJLLBW@# *RF-$\"3K1x4[)e`K\"F-7$$\"3.******H\"H)GVF-$\"393K:8&pPM\"F-7$$\"3mKLL L:$zl%F-$\"3!**e(Q7)=$f8F-7$$\"3E******\\7Z-]F-$\"31DC()pd\\t8F-7$$\"3 2nmmYRIM`F-$\"3?47BN8[&Q\"F-7$$\"3?mmm13ltcF-$\"39_zO1[L'R\"F-7$$\"33L LL.x=5gF-$\"3pFKdQD#fS\"F-7$$\"3d******f,V>jF-$\"3ST\\jQb&QT\"F-7$$\"3 ?LLL8p&Qn'F-$\"3$*Q]0LW1A9F-7$$\"33mmmE/'3*pF-$\"3p8u`YirG9F-7$$\"3Q++ +!H_)GtF-$\"3Nu'p'fn=N9F-7$$\"3O+++ION_wF-$\"3p#y')=\"R&3W\"F-7$$\"\") F*$\"35N\"[ALTkW\"F--%'COLOURG6&%$RGBG$\"*++++\"F)$F*F*Fd\\l-%*THICKNE SSG6#\"\"#-%*LINESTYLEG6#\"\"\"-F$6&7S7$F($\"3c'*[zEjzq:F-7$F/Fa]l7$F4 Fa]l7$F9Fa]l7$F>Fa]l7$FCFa]l7$FHFa]l7$FMFa]l7$FRFa]l7$FWFa]l7$FfnFa]l7 $F[oFa]l7$F`oFa]l7$FeoFa]l7$FjoFa]l7$F_pFa]l7$FdpFa]l7$FipFa]l7$F^qFa] l7$FcqFa]l7$FhqFa]l7$F^rFa]l7$FcrFa]l7$F]sFa]l7$FgsFa]l7$FbtFa]l7$F\\u Fa]l7$FauFa]l7$FfuFa]l7$F[vFa]l7$F`vFa]l7$FevFa]l7$FjvFa]l7$F_wFa]l7$F dwFa]l7$FiwFa]l7$F^xFa]l7$FcxFa]l7$FhxFa]l7$F]yFa]l7$FbyFa]l7$FgyFa]l7 $F\\zFa]l7$FazFa]l7$FfzFa]l7$F[[lFa]l7$F`[lFa]l7$Fe[lFa]l7$Fj[lFa]l-F_ \\l6&Fa\\lF*F*F*-Ff\\lF[]l-Fj\\l6#\"\"%-F$6&7S7$F($!3c'*[zEjzq:F-7$F/F ]al7$F4F]al7$F9F]al7$F>F]al7$FCF]al7$FHF]al7$FMF]al7$FRF]al7$FWF]al7$F fnF]al7$F[oF]al7$F`oF]al7$FeoF]al7$FjoF]al7$F_pF]al7$FdpF]al7$FipF]al7 $F^qF]al7$FcqF]al7$FhqF]al7$F^rF]al7$FcrF]al7$F]sF]al7$FgsF]al7$FbtF]a l7$F\\uF]al7$FauF]al7$FfuF]al7$F[vF]al7$F`vF]al7$FevF]al7$FjvF]al7$F_w F]al7$FdwF]al7$FiwF]al7$F^xF]al7$FcxF]al7$FhxF]al7$F]yF]al7$FbyF]al7$F gyF]al7$F\\zF]al7$FazF]al7$FfzF]al7$F[[lF]al7$F`[lF]al7$Fe[lF]al7$Fj[l F]alFc`lFe`lFf`l-%+AXESLABELSG6$Q\"x6\"Q!Fcdl-%%VIEWG6$;F(Fj[l%(DEFAUL TG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 89 "A procedure for evaluating the arctangent function using a continu ed fraction expansion: " }{TEXT 0 6 "atanCF" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 15 "" 0 "" {TEXT -1 15 " This procedure " }{TEXT 0 6 "atanCF" }{TEXT -1 39 " uses the continued fraction expansion " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x/arctan(x)=1+x^2/(3+4*x^2/(5+9*x^2/(7+16*x^2/(9+25/(11+` . . . `)* x^2))))" "6#/*&%\"xG\"\"\"-%'arctanG6#F%!\"\",&F&F&*&F%\"\"#,&\"\"$F&* (\"\"%F&*$F%F-F&,&\"\"&F&*(\"\"*F&*$F%F-F&,&\"\"(F&*(\"#;F&*$F%F-F&,&F 6F&*(\"#DF&,&\"#6F&%(~.~.~.~GF&F*F%F-F&F*F&F*F&F*F&F*F&" }{TEXT -1 2 " . " }}{PARA 15 "" 0 "" {TEXT -1 82 "The evaluation of the continued fr action expansion is performed by Lentz's method." }}{PARA 15 "" 0 "" {TEXT -1 14 "The formula: " }{XPPEDIT 18 0 "arctan(x)=Pi/2-arctan(1/x )" "6#/-%'arctanG6#%\"xG,&*&%#PiG\"\"\"\"\"#!\"\"F+-F%6#*&F+F+F'F-F-" }{TEXT -1 37 " is used for argument reduction when " }{XPPEDIT 18 0 "a bs(x) > 1" "6#2\"\"\"-%$absG6#%\"xG" }{TEXT -1 1 "." }}{PARA 15 "" 0 " " {TEXT -1 56 "Further argument reduction is provided by the formula: \+ " }{XPPEDIT 18 0 "arctan(x) = 2*arctan(x/(1+sqrt(1+x^2)));" "6#/-%'ar ctanG6#%\"xG*&\"\"#\"\"\"-F%6#*&F'F*,&F*F*-%%sqrtG6#,&F*F**$F'F)F*F*! \"\"F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "atanCF: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 20 "Calling Sequenc e: " }{TEXT -1 12 "atanCF( 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 proce dure " }{TEXT 0 6 "atanCF" }{TEXT 266 44 " calculates the arctangent o f a real number." }}{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].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 22 "atanCF: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1755 "atanCF := proc(xx:: realcons)\n local x,z,ai,bi,tiny,eps,i,rp,sp,tp,rc,tc,sc,\n val ,del,saveDigits,maxit,invert,isneg,ct,bdry,\n mag_x;\n\n tiny : = Maple_floats(MIN_FLOAT);\n eps := Float(5,-Digits);\n maxit := D igits*2; \n\n\011\011 # Increase precision for the computation by a f ew digits\n\011\011 saveDigits := Digits;\n Digits := Digits + lengt h(Digits)+2; \n x := evalf(xx);\n\n # arctan is an odd function\n \+ isneg := false;\n if x<0 then\n isneg := true;\n x := -x ;\n end if;\n \n # Handle special cases of large and small magnit ude\n mag_x := ilog10(x);\n if 2*mag_x<-saveDigits-1 then\n r eturn evalf(x,saveDigits);\n elif saveDigits<4*mag_x then # use 2 t erms of Maclaurin series\n val := evalf(Pi)*0.5-(1-1/(3*x*x))/x; \+ \n if isneg then val := -val end if;\n return evalf(val ,saveDigits);\n end if;\n\n # use arctan(x)=Pi/2-arctan(1/x)\n i nvert := false;\n if x>1.0 then\n invert := true;\n x := 1 /x;\n end if;\n \n # apply arctan(x)=2*arctan(x/(1+sqrt(1+x*x))\n ct := 1;\n bdry := 7.5;\n while bdry " 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 6 "atanCF" }{TEXT -1 10 ": 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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "evalf(atanCF(0.99),100);\nevalf(arc tan(0.99),100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q]%p+e+c1T%z5-4 _g9()Qlx3#o:rR:,!f^$>qM]DF<:(y*))*ejm+3t.y!$+\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q]%p+e+c1T%z5-4_g9()Qlx3#o:rR:,!f^$>qM]DF<:(y*))*ej m+3t.y!$+\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example \+ 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot(['atanCF(x)',1/x],x=-2..2,y=-2..2,discont=true); " }}{PARA 13 "" 1 "" {GLPLOT2D 322 291 291 {PLOTDATA 2 "6&-%'CURVESG6$ 7S7$$!\"#\"\"!$!+=([r5\"!\"*7$$!+$Q6G\">F-$!+OG3*3\"F-7$$!+M!\\p$=F-$! 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B_\\9oe&Fi[l7$$\"3c-#)\\9@BM=F]\\l$\"3SY9=0F(=X&Fi[l7$$\"3M=ce`v&Q(=F] \\l$\"3^U\"4*R\\eO`Fi[l7$$\"3%pyF]\\l$\"3*G*pD!\\;*=_Fi[l7$$\" 3k6p$Q?Wl&>F]\\l$\"3 " 0 "" {MPLTEXT 1 0 42 "evalf(fsolve('atanCF(x)'=1/x,x=0.1.2),50);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"S)\\y/r\">)z7GpSW/,$R?y[yK)RB;\"! #\\" }}}{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 82 "A procedure for evaluating the l og function using a continued fraction expansion: " }{TEXT 0 4 "lnCF" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 15 "" 0 "" {TEXT -1 15 "This procedure " }{TEXT 0 4 "lnCF" } {TEXT -1 39 " uses the continued fraction expansion " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln((1+z)/(1-z)) = 2*z/(1-z^2/(3-4* z^2/(5-9*z^2/(7-16*z^2/(9+` . . . `)))));" "6#/-%#lnG6#*&,&\"\"\"F)%\" zGF)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),&F9F)%(~.~.~.~GF )F,F,F,F,F,F,F,F,F," }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 82 " The evaluation of the continued fraction expansion is performed by Len tz's method." }}{PARA 15 "" 0 "" {TEXT -1 35 "Argument reduction to th e interval " }{XPPEDIT 18 0 "[1/sqrt(10), sqrt(10)];" "6#7$*&\"\"\"F%- %%sqrtG6#\"#5!\"\"-F'6#F)" }{TEXT -1 15 " is performed. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "lnCF: usag e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 274 20 "Calling Sequence: " }{TEXT -1 10 "lnCF( x ) " }{TEXT 275 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 15 "Parameters: " }{TEXT 276 21 "x - a real constant" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 279 12 "Description:" }{TEXT -1 1 " " }{TEXT 278 14 "The procedure " }{TEXT 0 4 "lnCF" }{TEXT 277 51 " calculates the natural logarithm of a real number." }}{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 ma ke the procedure active open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the s ubsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "lnCF: implementatio n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8768 "lnCF := proc(x::realcons)\n local xx,u,z,zs,ai,bi ,tiny,eps,i,rp,sp,tp,rc,tc,sc,\n del,saveDigits,maxit,invert,isne g,piBy2,scale,\n ct,r,k; \n \n tiny := evalhf(DBL_MIN);\n \+ eps := Float(1,-Digits);\n maxit := Digits*2;\n\n saveDigits := Di gits;\n Digits := Digits+length(Digits)+2;\n xx := evalf(x);\n i f xx<=0 then return 'procname(args)' end if;\n\n # reduce argument r ange to 1<=u<=10\n k := ilog10(xx);\n u := xx*Float(1,-k); # u=z*1 0^(-k)\n\n # reduce argument range to 1/sqrt(10)<=u<=sqrt(10)\n if u>0.31622776601683793320 then\n u := u*evalf(root10inv);\n \+ k := k + 0.5;\n end if;\n\n z := (u-1.0)/(u+1.0);\n zs := z*z;\n sp := 3;\n tp := 1/(3-zs);\n rp := 6*z*tp;\n ai := 3;\n for i from 2 to maxit do\n ai := ai+2;\n bi := -i*i*zs; \n \+ tc := ai + bi*tp;\n if tc=0 then tc := tiny end if;\n sc : = ai + bi/sp;\n if sc=0 then sc := tiny end if;\n tc := 1/tc ;\n del := sc*tc;\n rc := rp*del;\n if abs(del-1)0 then rc := rc + k*evalf('ln10') end if; \+ \n Digits := saveDigits;\n return evalf(rc);\nend proc:\n\n`evalf/ constant/root10inv` := proc()\nlocal d,r;\nglobal _root10inv;\n if D igits<=55 then evalf(.316227766016837933199889354443271853371955513932 5216827)\n elif Digits<=length(op(1,_root10inv)) then evalf(_root10i nv)\n else\n d := length(op(1,_root10inv));\n r := _root10 inv;\n while d " 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 4 "lnCF" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Ex ample 1" }}{PARA 0 "" 0 "" {TEXT -1 33 "Here are some numerical exampl es." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "evalf(lnCF(5),100);\nevalf(ln(5),100);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"_q6'zJ4yQ8IYwdwq()*yTZ\"*yk7>s<&oUN,c_Rw=EKLf 2gu.5MC\"zV4;!#**" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_q6'zJ4yQ8IYwd wq()*yTZ\"*yk7>s<&oUN,c_Rw=EKLf2gu.5MC\"zV4;!#**" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "xx := 0.0000 1;\nevalf(lnCF(xx),100);\nevalf(ln(xx),100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG$\"\"\"!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$! _q\\\"*zWDg)*z6SinQ[I'y$[]Rm;!)['Q9Vu]0!Q5#=Uts&**3?%G-(\\YDH^6!#)*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#$!_q\\\"*zWDg)*z6SinQ[I'y$[]Rm;!)['Q9 Vu]0!Q5#=Uts&**3?%G-(\\YDH^6!#)*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "xx := 999999;\nevalf(lnCF(xx),100);\nevalf(ln(xx),100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xxG\"'******" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qtgV9om:#GET!fj!y!4P=0**f%*)**p`\"*)y3m57>lsZ9:Yx.T xjzb4b\"Q\"!#)*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qtgV9om:#GET!fj !y!4P=0**f%*)**p`\"*)y3m57>lsZ9:Yx.Txjzb4b\"Q\"!#)*" }}}{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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "We solve the equation ln( x)=arctan(x) using the procedures atanCF and lnCF." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(['lnCF( x)','atanCF(x)'],x=0..5,y=-3..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 322 291 291 {PLOTDATA 2 "6&-%'CURVESG6$7^o7$$\"+S`!eS$!#7$!+]QF#o&!\"*7$$ \"+!o5;\"oF*$!+qm7*)\\F-7$$\"+-;u@5!#6$!+i:m$e%F-7$$\"+O@Ki8F6$!+*[zfH %F-7$$\"+qE!Hq\"F6$!+Qf$G2%F-7$$\"+/K[V?F6$!+\"Q90*QF-7$$\"+QP1%Q#F6$! 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Crandall, Springer-Verlag, 1996." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Investigate the efficie ncy of Lagrange's continued fraction: " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "k*x/(1-(1+x)^(-k)) = 1+[x*(1+k)/(1*`.`*2)]/(1+[x *(1-k)/(2*`.`*3)]/(1+[2*x*(2+k)/(3*`.`*4)]/(1+[2*x*(2-k)/(4*`.`*5)]/(1 +[3*x*(3+k)/(5*`.`*6)]/(1+[3*x*(3-k)/(6*`.`*7)]/(1+` . . . `))))));" " 6#/*(%\"kG\"\"\"%\"xGF&,&F&F&),&F&F&F'F&,$F%!\"\"F,F,,&F&F&*&7#*(F'F&, &F&F&F%F&F&*(F&F&%\".GF&\"\"#F&F,F&,&F&F&*&7#*(F'F&,&F&F&F%F,F&*(F4F&F 3F&\"\"$F&F,F&,&F&F&*&7#**F4F&F'F&,&F4F&F%F&F&*(F;F&F3F&\"\"%F&F,F&,&F &F&*&7#**F4F&F'F&,&F4F&F%F,F&*(FBF&F3F&\"\"&F&F,F&,&F&F&*&7#**F;F&F'F& ,&F;F&F%F&F&*(FIF&F3F&\"\"'F&F,F&,&F&F&*&7#**F;F&F'F&,&F;F&F%F,F&*(FPF &F3F&\"\"(F&F,F&,&F&F&%(~.~.~.~GF&F,F&F,F&F,F&F,F&F,F&F,F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 34 "for the approximate evaluation of " }{XPPEDIT 18 0 "k*x/(1-(1+x)^(-k))" "6#*(%\"kG\"\"\"%\"xGF%,&F%F%), &F%F%F&F%,$F$!\"\"F+F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 18 "To start with let " }{XPPEDIT 18 0 "k=11/2" "6#/%\"kG*&\"#6\"\"\"\"\" #!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Using the procedure " }{TEXT 0 8 "contfrac" }{TEXT -1 17 " from above with " }{XPPEDIT 18 0 "n=6" "6#/%\"nG\"\"'" }{TEXT -1 75 " yields a rational expression consisting of a cubic divided by \+ a quadratic." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "n := 6;\nk := 11/2;\nA := [1$n];\nB := [seq(floor ((j+1)/2)*x*(floor((j+1)/2)-(-1)^j*k)/(j*(j+1)),j=1..n-1)];\ncontfrac( A,B);\ng := unapply(normal(%),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"nG\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG#\"#6\"\"#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG7(\"\"\"F&F&F&F&F&" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"BG7',$%\"xG#\"#8\"\"%,$F'#!\"$F*,$F'#\"\"&F* ,$F'#!\"(\"#?,$F'#\"#%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,$*&,*\"#k\"\"\"*&\"$s#F09$F0F0*&\"$3%F0)F3\"\"#F0F0*&\"$@#F0)F3\" \"$F0F0F0,(\"#;F0*&F=F0F3F0F0*&\"#6F0F6F0F0!\"\"#F0\"\"%F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "On the o ther hand, we obtain a rational approximation with combined degree 6 b y forming a binomial expansion of " }{XPPEDIT 18 0 "(1+x)^(-k)" "6#),& \"\"\"F%%\"xGF%,$%\"kG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "k := 11/2;\nserie s(1-(1+x)^(-k),x,8);\nconvert(%,polynom):\nnormal(k*x/%):\nfactor(%); \nh := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG#\"#6\" \"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+3%\"xG#\"#6\"\"#\"\"\"#!$V\"\" \")F'#\"$:(\"#;\"\"$#!&b@\"\"$G\"\"\"%#\"&*=Y\"$c#\"\"&#!'BLK\"%C5\"\" '#\"(ZB1\"\"%[?\"\"(-%\"OG6#F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$ *&\"\"\"F%,0\"%C5F%*&\"%GLF%%\"xGF%!\"\"*&\"%?$)F%)F*\"\"#F%F%*&\"&!o< F%)F*\"\"$F%F+*&\"&#fLF%)F*\"\"%F%F%*&\"&'yeF%)F*\"\"&F%F+*&\"&xl*F%)F *\"\"'F%F%F+F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6#%\"xG6\"6$ %)operatorG%&arrowGF(,$*&\"\"\"F.,0\"%C5F.*&\"%GLF.9$F.!\"\"*&\"%?$)F. )F3\"\"#F.F.*&\"&!o " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 36 "Compare the approximating functions " }{XPPEDIT 18 0 "g (x)" "6#-%\"gG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "h(x)" "6#-% \"hG6#%\"xG" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "f(x)=k*x/(1-(1+x)^(- k))" "6#/-%\"fG6#%\"xG*(%\"kG\"\"\"F'F*,&F*F*),&F*F*F'F*,$F)!\"\"F/F/ " }{TEXT -1 31 " graphically over the interval " }{XPPEDIT 18 0 "[-1,1 ]" "6#7$,$\"\"\"!\"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 97 "Increase the number of terms in the continued fraction to 8 to for m a rational approximation for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x G" }{TEXT -1 195 " having the form of a rational function consisting o f a quartic divided by a cubic, and compare this approximation with a \+ second rational function obtained by extending the binomial expansion \+ of " }{XPPEDIT 18 0 "(1+x)^(-k)" "6#),&\"\"\"F%%\"xGF%,$%\"kG!\"\"" } {TEXT -1 47 " so that the denominator (after cancelling the " }{TEXT 282 1 "x" }{TEXT -1 34 " from the numerator) has degree 8." }}{PARA 0 "" 0 "" {TEXT -1 52 "Perform similar investigations with other values \+ of " }{TEXT 283 1 "k" }{TEXT -1 3 ". \n" }{TEXT 259 4 "Note" }{TEXT -1 7 ": When " }{TEXT 284 1 "k" }{TEXT -1 55 " is a positive integer, \+ the continued fraction of with " }{XPPEDIT 18 0 "2*k" "6#*&\"\"#\"\"\" %\"kGF%" }{TEXT -1 42 " terms or more appears to be identical to " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 40 "________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }