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258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 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 32 "Halley's method for root-finding " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada " }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" }}{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 28 "load root-finding procedures" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 269 7 "roots.m" }{TEXT -1 38 " contai ns the code for the procedures " }{TEXT 0 6 "halley" }{TEXT -1 5 " and " }{TEXT 0 11 "halley_step" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 122 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives \+ its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K: \\\\Maple/procdrs/roots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "The iterative formula for Halley's method " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 320 "Newton's friend, the astronomer Edmund Halley, is famous for taking measurements on the comet with his name so that he was abl e to predict future times for the comet's return. He was also able to work backwards to calculate times when it had previously appeared suc h as, for example, at the Battle of Hastings in 1066. " }}{PARA 0 "" 0 "" {TEXT -1 71 "His investigations are behind the root-finding metho d considered here. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 60 " be a function which has first and second order derivatives " } {XPPEDIT 18 0 "`f '`(x);" "6#-%$f~'G6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "`f ''`(x);" "6#-%%f~''G6#%\"xG" }{TEXT -1 18 " and supp ose that " }{TEXT 275 1 "a" }{TEXT -1 32 " is an approximation for a z ero " }{TEXT 276 1 "r" }{TEXT -1 8 " of the " }{XPPEDIT 18 0 "f(x);" " 6#-%\"fG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "The se cond order Taylor polynomial for " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#% \"xG" }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x = a" "6#/%\"xG%\"aG" } {TEXT -1 5 " is: " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " p(x) = f(a)+`f '`(a)*(x-a)+1/2;" "6#/-%\"pG6#%\"xG,(-%\"fG6#%\"aG\"\" \"*&-%$f~'G6#F,F-,&F'F-F,!\"\"F-F-*&F-F-\"\"#F3F-" }{TEXT -1 1 " " } {XPPEDIT 18 0 "`f ''`(a)*(x-a)^2;" "6#*&-%%f~''G6#%\"aG\"\"\"*$,&%\"xG F(F'!\"\"\"\"#F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 "It i s reasonable to suppose that a zero of " }{XPPEDIT 18 0 "p(x)" "6#-%\" pG6#%\"xG" }{TEXT -1 50 " would provide a better approximation to the \+ zero " }{TEXT 277 1 "r" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x);" "6#- %\"fG6#%\"xG" }{TEXT -1 27 ", so consider the equation " }}{PARA 258 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(a)+`f '`(a)*(x-a)+1/2;" "6#,(- %\"fG6#%\"aG\"\"\"*&-%$f~'G6#F'F(,&%\"xGF(F'!\"\"F(F(*&F(F(\"\"#F/F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`f ''`(a)*(x-a)^2 = 0;" "6#/*&-%%f~''G 6#%\"aG\"\"\"*$,&%\"xGF)F(!\"\"\"\"#F)\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 49 "Rather than attempting to solve this equation f or" }{XPPEDIT 18 0 "``(x-a);" "6#-%!G6#,&%\"xG\"\"\"%\"aG!\"\"" } {TEXT -1 47 ", explicitly rewrite the equation in the form: " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(a)+(x-a)*(`f '`(a)+`f ''` (a)/2*(x-a)) = 0;" "6#/,&-%\"fG6#%\"aG\"\"\"*&,&%\"xGF)F(!\"\"F),&-%$f ~'G6#F(F)*(-%%f~''G6#F(F)\"\"#F-,&F,F)F(F-F)F)F)F)\"\"!" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 8 "to give " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = a-f(a)/``(`f '`(a)+`f ''`(a)/2*(x-a));" " 6#/%\"xG,&%\"aG\"\"\"*&-%\"fG6#F&F'-%!G6#,&-%$f~'G6#F&F'*(-%%f~''G6#F& F'\"\"#!\"\",&F$F'F&F8F'F'F8F8" }{TEXT -1 15 " ------- (i). " }} {PARA 0 "" 0 "" {TEXT -1 72 "This formula may be compared with the sim pler Newton iteration formula: " }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x = a-f(a)/`f '`(a);" "6#/%\"xG,&%\"aG\"\"\"*&-%\"fG6#F &F'-%$f~'G6#F&!\"\"F/" }{TEXT -1 14 " ------- (ii)." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "Now make the heurestic \+ assumption that the difference or \"correction\" " }{XPPEDIT 18 0 "``( x-a)" "6#-%!G6#,&%\"xG\"\"\"%\"aG!\"\"" }{TEXT -1 79 " remaining in th e denominator of (i) is not vastly from the Newton correction " } {XPPEDIT 18 0 "x-a = -f(a)/`f '`(a);" "6#/,&%\"xG\"\"\"%\"aG!\"\",$*&- %\"fG6#F'F&-%$f~'G6#F'F(F(" }{TEXT -1 43 " obtained from (ii). In any \+ case, the term " }{XPPEDIT 18 0 "`f ''`(a)/2*(x-a);" "6#*(-%%f~''G6#% \"aG\"\"\"\"\"#!\"\",&%\"xGF(F'F*F(" }{TEXT -1 35 " is, in general, sm all compared to " }{XPPEDIT 18 0 "`f '`(a);" "6#-%$f~'G6#%\"aG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "Substituting " }{XPPEDIT 18 0 "-f(a)/`f '`(a);" "6#,$*&-% \"fG6#%\"aG\"\"\"-%$f~'G6#F(!\"\"F-" }{TEXT -1 4 " for" }{XPPEDIT 18 0 " ``(x-a)" "6#-%!G6#,&%\"xG\"\"\"%\"aG!\"\"" }{TEXT -1 33 " in the d enominator of (i) gives " }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x = a-f(a)/``(`f '`(a)-`f ''`(a)*f(a)/(2*`f '`(a)));" " 6#/%\"xG,&%\"aG\"\"\"*&-%\"fG6#F&F'-%!G6#,&-%$f~'G6#F&F'*(-%%f~''G6#F& F'-F*6#F&F'*&\"\"#F'-F16#F&F'!\"\"F=F=F=" }{TEXT -1 2 " " }}{PARA 259 "" 0 "" {TEXT -1 3 "or " }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x = a-2*f(a)*`f '`(a)/(2*`f '`(a)^2-f(a)*`f ''`(a))" "6 #/%\"xG,&%\"aG\"\"\"**\"\"#F'-%\"fG6#F&F'-%$f~'G6#F&F',&*&F)F'*$-F.6#F &F)F'F'*&-F+6#F&F'-%%f~''G6#F&F'!\"\"F;F;" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Given an initial \+ approximation " }{XPPEDIT 18 0 "x[0]=a" "6#/&%\"xG6#\"\"!%\"aG" } {TEXT -1 12 " for a root " }{TEXT 279 1 "r" }{TEXT -1 4 " of " } {XPPEDIT 18 0 "f(x) = 0;" "6#/-%\"fG6#%\"xG\"\"!" }{TEXT -1 15 ", the \+ formula: " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x[n+1] = x[n]-f(x[n])/``(`f '`(x[n])-f(x[n])*`f ''`(x[n])/(2*`f '`(x[n])));" " 6#/&%\"xG6#,&%\"nG\"\"\"F)F),&&F%6#F(F)*&-%\"fG6#&F%6#F(F)-%!G6#,&-%$f ~'G6#&F%6#F(F)*(-F/6#&F%6#F(F)-%%f~''G6#&F%6#F(F)*&\"\"#F)-F86#&F%6#F( F)!\"\"FLFLFL" }{TEXT -1 15 " ------- (iii) " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{TEXT 280 22 "______________________" }{TEXT -1 18 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "n=0,1,2,` . . . `" "6&/%\"nG\"\"!\"\"\"\"\"#%(~.~.~.~G " }{TEXT -1 37 " gives a sequence of approximations: " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x[0],x[1],x[2],` . . . `,x[n],x[n+ 1],` . . . `;" "6)&%\"xG6#\"\"!&F$6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\" nG&F$6#,&F0F)F)F)F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "fo r the root " }{TEXT 278 1 "r" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 25 "The formula (iii) is the " }{TEXT 266 24 "Halley iteration form ula" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 60 "An alternative derivation of the formula for Halley's method" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "Given the quadratic equation " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a*x^2+b*x+c=0" "6#/,(*&%\"aG\"\"\"*$%\"xG\"\"#F'F'*&%\"bGF'F)F'F'% \"cGF'\"\"!" }{TEXT -1 15 " ------- (i), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a<>0" "6#0%\"aG\"\"!" }{TEXT -1 39 ", th e standard quadratic formula gives " }}{PARA 258 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "x = -b/(2*a)" "6#/%\"xG,$*&%\"bG\"\"\"*&\"\"#F(%\"aG F(!\"\"F," }{TEXT -1 1 " " }{TEXT 310 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(b^2-4*a*c)/(2*a)" "6#*&-%%sqrtG6#,&*$%\"bG\"\"#\"\"\"*(\"\" %F+%\"aGF+%\"cGF+!\"\"F+*&F*F+F.F+F0" }{TEXT -1 16 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 14 "Note that, if " }{XPPEDIT 18 0 "a=0" "6 #/%\"aG\"\"!" }{TEXT -1 51 ", the equation (i) reduces to the linear e quation " }{XPPEDIT 18 0 "b*x+c=0" "6#/,&*&%\"bG\"\"\"%\"xGF'F'%\"cGF '\"\"!" }{TEXT -1 14 ", which gives " }{XPPEDIT 18 0 "x=-c/b" "6#/%\"x G,$*&%\"cG\"\"\"%\"bG!\"\"F*" }{TEXT -1 17 " ( provided that " } {XPPEDIT 18 0 "b<>0" "6#0%\"bG\"\"!" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT -1 81 "We would like to compare the value given by the formul a (ii) with the expression " }{XPPEDIT 18 0 "-c/b" "6#,$*&%\"cG\"\"\"% \"bG!\"\"F(" }{TEXT -1 6 " when " }{TEXT 316 1 "a" }{TEXT -1 17 " is \+ \"close to\" 0." }}{PARA 0 "" 0 "" {TEXT -1 32 "Taking the + sign in ( ii) gives " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=(-b+s qrt(b^2-4*a*c))/(2*a)" "6#/%\"xG*&,&%\"bG!\"\"-%%sqrtG6#,&*$F'\"\"#\" \"\"*(\"\"%F/%\"aGF/%\"cGF/F(F/F/*&F.F/F2F/F(" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x = (-b+sqrt(b^2-4*a*c))*(-b-sqrt(b^2-4*a*c))/((2*a)*(- b-sqrt(b^2-4*a*c)))" "6#/%\"xG*(,&%\"bG!\"\"-%%sqrtG6#,&*$F'\"\"#\"\" \"*(\"\"%F/%\"aGF/%\"cGF/F(F/F/,&F'F(-F*6#,&*$F'F.F/*(F1F/F2F/F3F/F(F( F/*(F.F/F2F/,&F'F(-F*6#,&*$F'F.F/*(F1F/F2F/F3F/F(F(F/F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = (b^2-(b^2-4*a*c))/(2*a*(-b-sqrt(b^2-4*a*c )))" "6#/%\"xG*&,&*$%\"bG\"\"#\"\"\",&*$F(F)F**(\"\"%F*%\"aGF*%\"cGF*! \"\"F1F**(F)F*F/F*,&F(F1-%%sqrtG6#,&*$F(F)F**(F.F*F/F*F0F*F1F1F*F1" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = 2*c/(-b-sqrt(b^2-4*a*c));" "6#/%\"x G*(\"\"#\"\"\"%\"cGF',&%\"bG!\"\"-%%sqrtG6#,&*$F*F&F'*(\"\"%F'%\"aGF'F (F'F+F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 48 "This last e xpression can be written in the form " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = 2*c/(-b-abs(b)*sqrt(1-4*a*c/(b^2)));" "6#/%\"x G*(\"\"#\"\"\"%\"cGF',&%\"bG!\"\"*&-%$absG6#F*F'-%%sqrtG6#,&F'F'**\"\" %F'%\"aGF'F(F'*$F*F&F+F+F'F+F+" }{TEXT -1 17 " ------- (iii). " }} {PARA 0 "" 0 "" {TEXT -1 47 "Similarly, taking the minus sign in (ii) \+ gives " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = 2*c/(-b +sqrt(b^2-4*a*c));" "6#/%\"xG*(\"\"#\"\"\"%\"cGF',&%\"bG!\"\"-%%sqrtG6 #,&*$F*F&F'*(\"\"%F'%\"aGF'F(F'F+F'F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "which can be written in the form " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = \+ 2*c/(-b+abs(b)*sqrt(1-4*a*c/(b^2)));" "6#/%\"xG*(\"\"#\"\"\"%\"cGF',&% \"bG!\"\"*&-%$absG6#F*F'-%%sqrtG6#,&F'F'**\"\"%F'%\"aGF'F(F'*$F*F&F+F+ F'F'F+" }{TEXT -1 16 " ------- (iv). " }}{PARA 0 "" 0 "" {TEXT -1 32 "From (iii) and (iv) we see that " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = 2*c/(-b-b*sqrt(1-4*a*c/(b^2)));" "6#/%\"xG*(\"\"# \"\"\"%\"cGF',&%\"bG!\"\"*&F*F'-%%sqrtG6#,&F'F'**\"\"%F'%\"aGF'F(F'*$F *F&F+F+F'F+F+" }{TEXT -1 13 " ------- (v)" }}{PARA 0 "" 0 "" {TEXT -1 21 "is a solution of (i)." }}{PARA 0 "" 0 "" {TEXT -1 10 "Note that " }{XPPEDIT 18 0 "2*c/(-b-b*sqrt(1-4*a*c/(b^2)));" "6#*(\"\"#\"\"\"% \"cGF%,&%\"bG!\"\"*&F(F%-%%sqrtG6#,&F%F%**\"\"%F%%\"aGF%F&F%*$F(F$F)F) F%F)F)" }{TEXT -1 10 " tends to " }{XPPEDIT 18 0 "-c/b;" "6#,$*&%\"cG \"\"\"%\"bG!\"\"F(" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "a->0" "6#f*6#% \"aG7\"6$%)operatorG%&arrowG6\"\"\"!F*F*F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Consider the eq uation " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(a)+`f '` (a)*(x-a)+1/2;" "6#,(-%\"fG6#%\"aG\"\"\"*&-%$f~'G6#F'F(,&%\"xGF(F'!\" \"F(F(*&F(F(\"\"#F/F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`f ''`(a)*(x-a) ^2 = 0;" "6#/*&-%%f~''G6#%\"aG\"\"\"*$,&%\"xGF)F(!\"\"\"\"#F)\"\"!" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 258 " " 0 "" {XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "`f ''`(a)*(x-a)^2+`f '`(a)*(x-a)+f(a) = 0;" "6#/,(*&-%% f~''G6#%\"aG\"\"\"*$,&%\"xGF*F)!\"\"\"\"#F*F**&-%$f~'G6#F)F*,&F-F*F)F. F*F*-%\"fG6#F)F*\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "Solving for" }{XPPEDIT 18 0 "``(x-a)" "6#-%!G6#,&%\"xG\"\"\"%\"aG!\" \"" }{TEXT -1 29 " using the formula (v) gives " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x-a = 2*f(a)/(-`f '`(a)-`f '`(a)*sqrt(1 -2*`f ''`(a)*f(a)/(`f '`(a)^2)));" "6#/,&%\"xG\"\"\"%\"aG!\"\"*(\"\"#F &-%\"fG6#F'F&,&-%$f~'G6#F'F(*&-F06#F'F&-%%sqrtG6#,&F&F&**F*F&-%%f~''G6 #F'F&-F,6#F'F&*$-F06#F'F*F(F(F&F(F(" }{TEXT -1 16 " ------- (vi). " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The bino mial series for " }{XPPEDIT 18 0 "(1+u)^(1/2)" "6#),&\"\"\"F%%\"uGF%*& F%F%\"\"#!\"\"" }{TEXT -1 12 " shows that " }{XPPEDIT 18 0 "(1+u)^(1/2 )" "6#),&\"\"\"F%%\"uGF%*&F%F%\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 311 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1+ u/2" "6#,&\"\"\"F$*&%\"uGF$\" \"#!\"\"F$" }{TEXT -1 6 " when " }{TEXT 312 1 "u" }{TEXT -1 13 " is \" small\". " }}{PARA 0 "" 0 "" {TEXT -1 78 "Using this approximation for the square root in (vi) with the assumption that " }{XPPEDIT 18 0 "2* `f ''`(a)*f(a)/(`f '`(a)^2);" "6#**\"\"#\"\"\"-%%f~''G6#%\"aGF%-%\"fG6 #F)F%*$-%$f~'G6#F)F$!\"\"" }{TEXT -1 18 " is \"small\" gives " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x-a" "6#,&%\"xG\"\"\"%\"aG!\"\"" }{TEXT -1 1 " " } {TEXT 313 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*f(a)/(-`f '`(a)-`f ' `(a)*(1-`f ''`(a)*f(a)/(`f '`(a)^2)));" "6#*(\"\"#\"\"\"-%\"fG6#%\"aGF %,&-%$f~'G6#F)!\"\"*&-F,6#F)F%,&F%F%*(-%%f~''G6#F)F%-F'6#F)F%*$-F,6#F) F$F.F.F%F.F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "This giv es " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{TEXT 315 1 "x" }{TEXT -1 1 " " }{TEXT 314 1 "~" }{TEXT -1 2 " " }{XPPEDIT 18 0 "a-f(a)/``(`f '`(a )-`f ''`(a)*f(a)/(2*`f '`(a)));" "6#,&%\"aG\"\"\"*&-%\"fG6#F$F%-%!G6#, &-%$f~'G6#F$F%*(-%%f~''G6#F$F%-F(6#F$F%*&\"\"#F%-F/6#F$F%!\"\"F;F;F;" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 27 "which is Halley's form ula. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Introductory example " }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " } {XPPEDIT 18 0 "f(x) = x^5-4;" "6#/-%\"fG6#%\"xG,&*$F'\"\"&\"\"\"\"\"%! \"\"" }{TEXT -1 35 ", and consider the calculation of " }{XPPEDIT 18 0 "``[5]*sqrt(4);" "6#*&&%!G6#\"\"&\"\"\"-%%sqrtG6#\"\"%F(" }{TEXT -1 14 " as a zero of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 20 " by Halley's method." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The general iterative formula for Halley's meth od is: " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x[n+1] = x [n]-f(x[n])/(`f '`(x[n])-f(x[n])*`f ''`(x[n])/(2*`f '`(x[n])));" "6#/& %\"xG6#,&%\"nG\"\"\"F)F),&&F%6#F(F)*&-%\"fG6#&F%6#F(F),&-%$f~'G6#&F%6# F(F)*(-F/6#&F%6#F(F)-%%f~''G6#&F%6#F(F)*&\"\"#F)-F56#&F%6#F(F)!\"\"FIF IFI" }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 40 "which, with the \+ starting approximation " }{XPPEDIT 18 0 "x[0]=1.5" "6#/&%\"xG6#\"\"!- %&FloatG6$\"#:!\"\"" }{TEXT -1 33 ", will give a sequence of values:" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x[0],x[1],x[2],` . \+ . . `,x[n],x[n+1],` . . . `;" "6)&%\"xG6#\"\"!&F$6#\"\"\"&F$6#\"\"#%(~ .~.~.~G&F$6#%\"nG&F$6#,&F0F)F)F)F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 23 "converging to the root " }{XPPEDIT 18 0 "r = ``[5]*sqrt(4 );" "6#/%\"rG*&&%!G6#\"\"&\"\"\"-%%sqrtG6#\"\"%F*" }{TEXT -1 17 " of t he equation " }{XPPEDIT 18 0 "g(x)=0" "6#/-%\"gG6#%\"xG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 "We can calculate successive a pproximations for " }{TEXT 281 1 "r" }{TEXT -1 33 " with the starting \+ approximation " }{XPPEDIT 18 0 "x[0]=1.5" "6#/&%\"xG6#\"\"!-%&FloatG6$ \"#:!\"\"" }{TEXT -1 13 " as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 223 "f := x -> x^5-4:\nDf := D(f):\nD2f := D(Df):\n`f '`(x) = Df(x);\n`f ''`(x) = D2f(x);\nnextapp rox := x -> evalf(x - f(x)/(Df(x)-1/2*f(x)*D2f(x)/Df(x)));\nx0 := 1.5; \nx1 := nextapprox(x0);\nx2 := nextapprox(x1);\nx3 := nextapprox(x2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$f~'G6#%\"xG,$*&\"\"&\"\"\")F' \"\"%F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%f~''G6#%\"xG,$*&\"#? \"\"\")F'\"\"$F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+nextapproxGf* 6#%\"xG6\"6$%)operatorG%&arrowGF(-%&evalfG6#,&9$\"\"\"*&-%\"fG6#F0F1,& -%#DfGF5F1*&#F1\"\"#F1*(F3F1-%$D2fGF5F1F7!\"\"F1F?F?F?F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"#:!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+'4t[K\"!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#x2G$\"+(33&>8!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$\"+6z ]>8!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "The last value gives the value of " }{XPPEDIT 18 0 "``[5]*sqrt(4); " "6#*&&%!G6#\"\"&\"\"\"-%%sqrtG6#\"\"%F(" }{TEXT -1 22 " correct to 1 0 digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(4^(1/5));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+6z]>8!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 56 "Oscul ating hyperbolas and the Halley iteration formula " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 118 "Referenc e: Scavo, T. R. and Thoo, J. B. \"On the Geometry of Halley's Method. \" Amer. Math. Monthly 102, 417-426, 1995. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{URLLINK 17 "http://ms.yuba.cc.ca.us/~jb2/cvandpubs/halley.p df" 4 "http://ms.yuba.cc.ca.us/~jb2/cvandpubs/halley.pdf" "" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 135 "It would be nice to have geometrical pictures for each step of Ha lley's method similar to those which can be drawn for Newton's method. " }}{PARA 0 "" 0 "" {TEXT -1 124 "To achieve this we consider an alter native derivation of Halley's iteration formula to that considered in \+ the first section." }}{PARA 0 "" 0 "" {TEXT -1 13 "Suppose that " } {TEXT 283 1 "a" }{TEXT -1 32 " is an approximation for a root " } {TEXT 282 1 "r" }{TEXT -1 4 " of " }{XPPEDIT 18 0 "f(x) = 0;" "6#/-%\" fG6#%\"xG\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 23 "Consid er approximating " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 23 " in a neighbourhood of " }{XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" } {TEXT -1 46 " by means of a rational function of the form: " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x) = (``(x-a)+h)/(p*(x-a) +q);" "6#/-%\"gG6#%\"xG*&,&-%!G6#,&F'\"\"\"%\"aG!\"\"F.%\"hGF.F.,&*&% \"pGF.,&F'F.F/F0F.F.%\"qGF.F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "We suppose that: " }} {PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECEWISE([g(a) = f(a ), ``],[g*`'`(a) = `f '`(a), ``],[g*`''`(a) = `f ''`(a), ``]);" "6#-%* PIECEWISEG6%7$/-%\"gG6#%\"aG-%\"fG6#F+%!G7$/*&F)\"\"\"-%\"'G6#F+F3-%$f ~'G6#F+F/7$/*&F)F3-%#''G6#F+F3-%%f~''G6#F+F/" }{TEXT -1 3 ", " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "that is, \+ the value of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 31 " \+ and its first two derivatives " }{TEXT 266 5 "match" }{TEXT -1 10 " th ose of " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 4 " at " } {XPPEDIT 18 0 "x=a" "6#/%\"xG%\"aG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "The curve " }{XPPEDIT 18 0 "y=g(x)" "6#/%\"yG-%\"gG6#%\"x G" }{TEXT -1 19 " is then called an " }{TEXT 266 20 "osculating hyperb ola" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{XPPEDIT 18 0 "p<> 0" "6#0%\"pG\"\"!" }{TEXT -1 14 " we can write " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 14 " in the form: " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "g(x)=1/p+(p*h-q)/(p*(p*(x-a )+q))" "6#/-%\"gG6#%\"xG,&*&\"\"\"F*%\"pG!\"\"F**&,&*&F+F*%\"hGF*F*%\" qGF,F**&F+F*,&*&F+F*,&F'F*%\"aGF,F*F*F1F*F*F,F*" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`g '`(x) = -(p*h-q)/((p*(x-a)+q)^2);" "6#/-%$g~'G6# %\"xG,$*&,&*&%\"pG\"\"\"%\"hGF-F-%\"qG!\"\"F-*$,&*&F,F-,&F'F-%\"aGF0F- F-F/F-\"\"#F0F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " } }{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`g ''`(x) = 2*(p*h-q )/((p*(x-a)+q)^3)*p;" "6#/-%%g~''G6#%\"xG**\"\"#\"\"\",&*&%\"pGF*%\"hG F*F*%\"qG!\"\"F**$,&*&F-F*,&F'F*%\"aGF0F*F*F/F*\"\"$F0F-F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 " The requirements " }{XPPEDIT 18 0 "g(a) = f(a),`g '`(a) = `f '`(a);" " 6$/-%\"gG6#%\"aG-%\"fG6#F'/-%$g~'G6#F'-%$f~'G6#F'" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "`g ''`(a) = `f ''`(a);" "6#/-%%g~''G6#%\"aG-%%f~''G6 #F'" }{TEXT -1 27 ", give rise the equations: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "PIECE WISE([f(a) = h/q, ``],[`f '`(a) = (q-p*h)/(q^2), ``],[`f ''`(a) = 2*p* (p*h-q)/(q^3), ``]);" "6#-%*PIECEWISEG6%7$/-%\"fG6#%\"aG*&%\"hG\"\"\"% \"qG!\"\"%!G7$/-%$f~'G6#F+*&,&F/F.*&%\"pGF.F-F.F0F.*$F/\"\"#F0F17$/-%% f~''G6#F+**F " 0 "" {MPLTEXT 1 0 157 "unassign('a','h','p','q','f','g'):\ng := x -> (x-a+h )/(p*(x-a)+q);\neqns := normal(\{f(a)=g(a),D(f)(a)=D(g)(a),(D@@2)(f)(a )=(D@@2)(g)(a)\});\nsolve(eqns,\{p,q,h\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,(9$\"\"\"% \"aG!\"\"%\"hGF/F/,&*&%\"pGF/,&F.F/F0F1F/F/%\"qGF/F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<%/-%\"fG6#%\"aG*&%\"hG\"\"\"%\"qG! \"\"/---%#@@G6$%\"DG\"\"#6#F(F),$**F7F-%\"pGF-,&F.F/*&F,F-F;F-F-F-F.! \"$F-/--F6F8F),$*&F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The os culating hyperbola " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 11 " meets the " }{TEXT 285 1 "x" }{TEXT -1 12 " axis where " }} {PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(x-a)+h=0" "6#/,&-% !G6#,&%\"xG\"\"\"%\"aG!\"\"F*%\"hGF*\"\"!" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 15 "that is, where " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=a-h" "6#/%\"xG,&%\"aG\"\"\"%\"hG!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "such that " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "h = 2*f(a)*`f '`(a)/(2*`f '`(a)^2-f(a)* `f ''`(a));" "6#/%\"hG**\"\"#\"\"\"-%\"fG6#%\"aGF'-%$f~'G6#F+F',&*&F&F '*$-F-6#F+F&F'F'*&-F)6#F+F'-%%f~''G6#F+F'!\"\"F:" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 50 "This is essentially the Halley iteration \+ formula. " }}{PARA 0 "" 0 "" {TEXT -1 92 "Thus Halley's method replace s the tangent lines of Newton's method by osculating hyperbolas." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 66 "An example to illustrat e Halley's method by osculating hyperbolas " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 55 "Consider the prob lem of calculating the single zero of " }{XPPEDIT 18 0 "f(x) = arctan( x)-exp(-x^2);" "6#/-%\"fG6#%\"xG,&-%'arctanG6#F'\"\"\"-%$expG6#,$*$F' \"\"#!\"\"F3" }{TEXT -1 52 " by Halley's method with the starting appr oximation " }{XPPEDIT 18 0 "x[0] = 3/2;" "6#/&%\"xG6#\"\"!*&\"\"$\"\" \"\"\"#!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "f := x -> arctan(x)-exp(-x^2 );\nplot(f(x),x=-3..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#% \"xG6\"6$%)operatorG%&arrowGF(,&-%'arctanG6#9$\"\"\"-%$expG6#,$*$)F0\" \"#F1!\"\"F9F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 480 228 228 {PLOTDATA 2 "6%-%'CURVESG6$7V7$$!\"$\"\"!$!3?TB?#=p\"\\7!#<7$$!3!***** *\\2<#pGF-$!3:k9ZO,qN7F-7$$!3#)***\\7bBav#F-$!3Z)4L#>>;B7F-7$$!36++]K3 XFEF-$!3%zesU)>837F-7$$!3%)****\\F)H')\\#F-$!3;A(3AfW?>\"F-7$$!3#**** \\i3@/P#F-$!3[K-6k=@v6F-7$$!3;++Dr^b^AF-$!3yGM8IR6f6F-7$$!3$****\\7Sw% G@F-$!38w2%3UfB9\"F-7$$!3*****\\7;)=,?F-$!3$fM1\"4]hD6F-7$$!3/++DO\"3V (=F-$!34,NSv@\\56F-7$$!3#******\\V'zV>bi!4\"F-7$$!3!******\\!)H%*\\\"F-$!3yL3Jry>)3\"F-7$$!3 /+++vl[p8F-$!3k*fg$))pO$4\"F-7$$!3\"******\\>iUC\"F-$!3BJOy)yUk5\"F-7$ $!3-++DhkaI6F-$!3'y0;2f^^7\"F-7$$!3s******\\XF`**!#=$!3MTH^Q#z2))Ffp$!3MDS#f1UC=\"F-7$$!3S++]7RKvuFfp$!3*\\vF(>A\"Q@\"F- 7$$!3s,+++P'eH'Ffp$!3F`3bfAkM7F-7$$!3A++D1j$)[cFfp$!3z?'[V!f+T7F-7$$!3 q)***\\7*3=+&Ffp$!3.-fh.AXU7F-7$$!3g)****\\#eo&Q%Ffp$!3>\\)\\'G8LQ7F-7 $$!3[)***\\PFcpPFfp$!3rRnM'fA!G7F-7$$!3K)**\\7$HqEJFfp$!3D5`MpS!*47F-7 $$!3;)****\\7VQ[#Ffp$!3=At1`si$=\"F-7$$!32)***\\i6:.8Ffp$!3=Q)*>Biu76F -7$$!3Wb+++v`hH!#?$!3Z()e&Ffp7$$\"3!z******4FL(\\Ffp$!3S%p_t'yj$>$Ffp7$$\"3A)****\\d6.B'Ffp $!3dP*z%RB;67Ffp7$$\"3s****\\(o3lW(Ffp$\"3GO_DI?Brl!#>7$$\"35*****\\A) )oz)Ffp$\"3>#RYPwmCg#Ffp7$$\"3e******Hk-,5F-$\"3wLoo5\\'y=%Ffp7$$\"36+ ++D-eI6F-$\"3%[K/d'>v!o&Ffp7$$\"3u***\\(=_(zC\"F-$\"3G@*=_)=!f%oFfp7$$ \"3M+++b*=jP\"F-$\"3vZQao]D?zFfp7$$\"3g***\\(3/3(\\\"F-$\"3%3z$QYjob() Ffp7$$\"33++vB4JB;F-$\"3E'H?d%*z'p%*Ffp7$$\"3u*****\\KCnu\"F-$\"3'=nGU LLN+\"F-7$$\"3s***\\(=n#f(=F-$\"3_:34\"4=90\"F-7$$\"3P+++!)RO+?F-$\"3w PPV8D$*)3\"F-7$$\"30++]_!>w7#F-$\"3vmr>wCh?6F-7$$\"3O++v)Q?QD#F-$\"3%) Q>KO&zp9\"F-7$$\"3G+++5jypBF-$\"3Wq=![rZy;\"F-7$$\"3<++]Ujp-DF-$\"3muL RCpv)=\"F-7$$\"3++++gEd@EF-$\"3VImw#yY`?\"F-7$$\"39++v3'>$[FF-$\"3^z6s pXI@7F-7$$\"37++D6EjpGF-$\"3PQ78\")R@N7F-7$$\"\"$F*$\"3onTfiB#*[7F--%' COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fa\\l-%+AXESLABELSG6$Q\"x6\"Q!Ff\\l-%%VI EWG6$;F(Ff[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 68 "In order to illustrate the first step we calculate the coefficients " }{XPPEDIT 18 0 "p,q" "6$%\"pG%\"qG" }{TEXT -1 5 " \+ and " }{TEXT 286 1 "h" }{TEXT -1 59 " in the osculating hyperbola whic h passes through the point" }{XPPEDIT 18 0 "``(x[0],f(x[0]));" "6#-%!G 6$&%\"xG6#\"\"!-%\"fG6#&F'6#F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 251 "f := x -> a rctan(x)-exp(-x^2);\nx0 := 1.5;\nh := 'h': p := 'p': q := 'q': \ng := \+ x -> (x-x0+h)/(p*(x-x0)+q);\neqns := normal(\{f(x0)=g(x0),D(f)(x0)=D(g )(x0),\n (D@@2)(f)(x0)=(D@@2)(g)(x0)\});\nsols := solve(eq ns,\{p,q,h\});\nassign(sols);\nx1 := x0-h;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%'arctanG6# 9$\"\"\"-%$expG6#,$*$)F0\"\"#F1!\"\"F9F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"#:!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,(9$\"\"\"%#x0G!\"\"%\"hGF/F/ ,&*&%\"pGF/,&F.F/F0F1F/F/%\"qGF/F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<%/$\"+')\\%Rx)!#5*&%\"hG\"\"\"%\"qG!\"\"/$!+T#==-\"!\"* ,$**\"\"#F,,&*&%\"pGF,F+F,F,F-F.F,F-!\"$F8F,F,/$\"+:)**)QiF),$*&F6F,F- !\"#F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG<%/%\"hG$\"+*=Kg`'!# 5/%\"qG$\"+l`O\\uF*/%\"pG$\"+PBN+hF*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#x1G$\"+6y'RY)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "We may now illustrate the first step which gives the new improved approximation " }{XPPEDIT 18 0 "x[1] = x[0]-h;" "6#/&%\" xG6#\"\"\",&&F%6#\"\"!F'%\"hG!\"\"" }{TEXT -1 1 " " }{TEXT 287 1 "~" } {TEXT -1 15 " 0.8463967811. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 308 "p1 := plot(f(x),x=0..2.5,y= -0.5..1.5,color=red):\np2 := plot(g(x),x=0.5..2.5,y=-0.5..1.5,color=gr een,thickness=2):\np3 := plot([[x0,0],[x0,f(x0)]],color=blue,linestyle =2):\np4 := plot([[[x0,0],[x0,f(x0)],[x1,0]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=black):\nplots[display]([p1,p2,p3 ,p4]);" }}{PARA 13 "" 1 "" {GLPLOT2D 334 281 281 {PLOTDATA 2 "6*-%'CUR VESG6$7S7$$\"\"!F)$!\"\"F)7$$\"3WmmmT&)G\\a!#>$!36talq)efU*!#=7$$\"3PL $ek`o!>5F2$!3q\"[0SjC6)))F27$$\"3omm\"z>)G_:F2$!3g`hj-E#>A)F27$$\"3-nm T&QU!*3#F2$!3C*3y2+^N^(F27$$\"3HL$eRZXKi#F2$!3G(=G#o&f&pnF27$$\"3xm;z> ,_=JF2$!3qkb$)=?L]gF27$$\"3v**\\7G$[8j$F2$!3E-S'\\j'H\"G&F27$$\"35n;z% *frhTF2$!3qXVb**>,mWF27$$\"3A+]ilFQ!p%F2$!3\"fF27$$\"3=MLL$e'3I))F2$\"3I[O^'Gj![EF27$$\"3?+]7.-%F27$$\"3;+DJ&H\"fT 5!#<$\"3q!yQ:mf$yYF27$$\"35+v$f)[$H4\"Fjq$\"3/sHJiy>p_F27$$\"3cL$ek`1l 9\"Fjq$\"3Ynxx8CJ\\eF27$$\"3OLe*[.-d>\"Fjq$\"3Wy+ty$H\"\\jF27$$\"3km;/ Egw[7Fjq$\"3;2*)Q3%[J&oF27$$\"3zm\"z%*f%)QI\"Fjq$\"3f)G2!\\dzQtF27$$\" 3/+voza'=N\"Fjq$\"3wT8sVC*4t(F27$$\"3(om\"zWho.9Fjq$\"3o3c\\r\")zB\")F 27$$\"3-++]i>Ad9Fjq$\"3!p`K*z>a(\\)F27$$\"32+]i:jf4:Fjq$\"36YBwSoML))F 27$$\"39+DJ&>r-c\"Fjq$\"3WpD'Hk&)=8*F27$$\"3++]P4q`;;Fjq$\"3=V.lHt2N%* F27$$\"3;LL$eM%4n;Fjq$\"3;\\F@s2-%o*F27$$\"37++v$4v5s\"Fjq$\"36@1\"*zG IF**F27$$\"3cm\"zWn*)*p1K.\"Fjq7$$\"3CLek.Nyt=Fjq$\"3aS-skVq]5Fjq7$$\"3=+Dc^&zj#>Fjq$\"39# >3$oW_n5Fjq7$$\"3CLL3-=!y(>Fjq$\"3U<8UsGm#3\"Fjq7$$\"3))*\\7G8O;.#Fjq$ \"3[zqF\"Rus4\"Fjq7$$\"3!pmm;*\\[$3#Fjq$\"3#ou(yu+G56Fjq7$$\"3*pmT&Qz] O@Fjq$\"3_0Xb,whA6Fjq7$$\"3iLekG=4*=#Fjq$\"3QS$oFY)*R8\"Fjq7$$\"3F++]i 4TPAFjq$\"3raPNA\")yV6Fjq7$$\"3qL$3F9!z#H#Fjq$\"3k)\\k3@3V:\"Fjq7$$\"3 'pmm;%>KUBFjq$\"3+3(okO_J;\"Fjq7$$\"3/+DJqJ8&R#Fjq$\"3-L!*o,&e?<\"Fjq7 $$\"3G+voa-oXCFjq$\"3Um,K\"HH,=\"Fjq7$$\"3++++++++DFjq$\"3#RIYb\\f$)= \"Fjq-%'COLOURG6&%$RGBG$\"*++++\"!\")F(F(-F$6%7Y7$$\"3++++++++]F2$!3Yd lQi$zxc#Fjq7$$\"30LL$3x&)*3^F2$!3MYs2gv#Fjq7$$\"3E++]7t&pK&F2$!3zfRUI%ze-#Fjq7$$\"3JLLL$3VfV&F2$!3cUYw- <*\\(=Fjq7$$\"3t***\\i&*)fDcF2$!3Rd?J@!f+k\"Fjq7$$\"3Dnm;H[D:eF2$!3BCC o'GpXV\"Fjq7$$\"3&3+]PpU&GgF2$!3/Hd&[a:AB\"Fjq7$$\"3LLLLe0$=C'F2$!3?\\ ]'pne[0\"Fjq7$$\"39MLLLA`ckF2$!39fj]7Eer*)F27$$\"3'QLL$3RBrmF2$!3mIcXB +**ovF27$$\"3=mm;zjf)4(F2$!3c#*eNb_/$>&F27$$\"3=LL$e4;[\\(F2$!378uUz5t vLF27$$\"3C++]i'y]!zF2$!3Mvb#[+:1z\"F27$$\"3,LL$ezs$H$)F2$!3O(o>U\"[/# )RF/7$$\"3_****\\7iI_()F2$\"3a+xH/hiDzF/7$$\"3#pmmm@Xt=*F2$\"3/i'HkryJ &=F27$$\"3QLLL3y_q&*F2$\"3=]aY]+5Fjq$\"3;c[6d \\-&\\$F27$$\"3*******\\Z/N/\"Fjq$\"3E$yQ[5CcA%F27$$\"35+++NfC&3\"Fjq$ \"33\"HC$>zUb[F27$$\"3LLLez6:B6Fjq$\"35&z\"[/tRt`F27$$\"3_mmm\"=C#o6Fj q$\"3Ll%*4uu%=$fF27$$\"3gmmmEpS17Fjq$\"3Q#[s%4YYijF27$$\"3%)***\\i`A3D \"Fjq$\"3]\"H#=Gn!3#oF27$$\"3Ymmmwy8!H\"Fjq$\"3k\\7'eQDH>(F27$$\"3/++D OIFL8Fjq$\"3aYX>eKFpvF27$$\"3!****\\(3zMu8Fjq$\"3(\\OlZK(4+zF27$$\"3em m;H_?<9Fjq$\"3N/\"e.sE)>#)F27$$\"3mmm\"zihlX\"Fjq$\"3cHhnb9%H\\)F27$$ \"3KLL$3#G,*\\\"Fjq$\"39.As!4\"yn()F27$$\"3 P.*F27$$\"3!****\\PQ#\\\"e\"Fjq$\"37y$))G\"4c]#*F27$$\"3BLL$e\"*[Hi\"F jq$\"31^s9cV%3Z*F27$$\"3#*******pvxl;Fjq$\"3IsOm@')e%o*F27$$\"3z****\\ _qn2Fjq$ \"3q$e1>k!)42\"Fjq7$$\"3'******\\)Hxe>Fjq$\"3w0M)>S_a3\"Fjq7$$\"3Ymm\" H!o-**>Fjq$\"3I6vo&*=T)4\"Fjq7$$\"3))***\\7k.6/#Fjq$\"3tit1AaK66Fjq7$$ \"3emmmT9C#3#Fjq$\"3&=C/VqtL6x^>\"Fjq7$$\"30++DOl5;CFjq$\"3 " 0 "" {MPLTEXT 1 0 240 "f := x -> arctan(x)-exp(-x^2);\nh := 'h': p := 'p': q := 'q': \ng := x -> (x -x1+h)/(p*(x-x1)+q);\neqns := normal(\{f(x1)=g(x1),D(f)(x1)=D(g)(x1), \n (D@@2)(f)(x1)=(D@@2)(g)(x1)\});\nsols := solve(eqns,\{p ,q,h\});\nassign(sols);\nx2 := x1-h;" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%'arctanG6#9$\"\"\"-%$ex pG6#,$*$)F0\"\"#F1!\"\"F9F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,(9$\"\"\"%#x1G!\"\"%\"hGF/F/ ,&*&%\"pGF/,&F.F/F0F1F/F/%\"qGF/F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<%/$!+%HLW(**!#5,$**\"\"#\"\"\",&*&%\"pGF-%\"hGF-F-%\"qG !\"\"F-F2!\"$F0F-F-/$\"+p?d49!\"*,$*&F.F-F2!\"#F3/$\"+Sm%)Q@F)*&F1F-F2 F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG<%/%\"pG$\"+Px;#Q#!#5/% \"qG$\"+!*z)Gt'F*/%\"hG$\"+&[h+W\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#x2G$\"+Ej!R-(!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "The second step which gives the new approximation " }{XPPEDIT 18 0 "x[2] = x[1]-h;" "6#/&%\"xG6#\"\"#,&&F%6#\"\"\"F+%\"hG! \"\"" }{TEXT -1 1 " " }{TEXT 288 1 "~" }{TEXT -1 45 " 0.7023906326 can be illustrated as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 310 "p1 := plot(f(x),x=0.6..0.9, y=-0.1..0.3,color=red):\np2 := plot(g(x),x=0.6..0.9,y=-0.1..0.3,color= green,thickness=2):\np3 := plot([[x1,0],[x1,f(x1)]],color=blue,linesty le=2):\np4 := plot([[[x1,0],[x1,f(x1)],[x2,0]]$3],style=point,\n \+ symbol=[circle,diamond,cross],color=black):\nplots[display]([p1,p2, p3,p4]);" }}{PARA 13 "" 1 "" {GLPLOT2D 414 323 323 {PLOTDATA 2 "6*-%'C URVESG6$7S7$$\"3w**************f!#=$!3wpW+e#oDd\"F*7$$\"3Q****\\i9RlgF *$!3WS'zet'zp9F*7$$\"3')**\\PC#)GAhF*$!35Q^6-PZ!Q\"F*7$$\"3%****\\Peui ='F*$!3'GwPYOQ,G\"F*7$$\"3k***\\i3&o]iF*$!3UkHVHKFz6F*7$$\"3f**\\(oX*y 9jF*$!3Kt[*f)=.z5F*7$$\"3E**\\P9CAujF*$!3at5GuJKi)*!#>7$$\"3P**\\P*zhd V'F*$!3f;%)Ge%3H!*)FK7$$\"3y**\\P>fS*\\'F*$!3Y<@#))\\TB\"zFK7$$\"3))** \\(=$f%Gc'F*$!3p(=!>]5sEpFK7$$\"3$*****\\#y,\"GmF*$!3_OdsHh#[\"fFK7$$ \"37++Dr\"zbo'F*$!3n\"=\"\\#f]_-&FK7$$\"3]****\\(4&G]nF*$!3]/eU'f!zDSF K7$$\"3l****\\7nD:oF*$!3-Z2E5rVCIFK7$$\"3[****\\-*oy(oF*$!3ge*eD:G;1#F K7$$\"3K+]PpnsMpF*$!3_>]#G=G#*=\"FK7$$\"3Y++]siL-qF*$!3aH*RJ#GFV:!#?7$ $\"3e*******Q5'fqF*$\"3Ndy%3!G.-sFhp7$$\"3a**\\P/QBErF*$\"3=mx#3)p!\\t \"FK7$$\"3O+++:o?&=(F*$\"3K\\2L\"H.2j#FK7$$\"3G+]Pa&4*\\sF*$\"3:2!okGv 3h$FK7$$\"33+]7j=_6tF*$\"3@Pr))RAdTXFK7$$\"3m***\\P%y!eP(F*$\"3_Io/o] \")4bFK7$$\"3K+](=WU[V(F*$\"3?#=]:uTjR'FK7$$\"3U++DJ#>&)\\(F*$\"3#Q79w !\\p\\tFK7$$\"3)***\\P>:mkvF*$\"3]g')p#[7nL)FK7$$\"3&***\\iv&QAi(F*$\" 3QCbkkk:$>*FK7$$\"31++vtLU%o(F*$\"3.9&>]@>:,\"F*7$$\"37+++bjm[xF*$\"3$ *o&\\.GYk5\"F*7$$\"38++vyb^6yF*$\"3+Qcb.V)*)>\"F*7$$\"3W+]PMaKsyF*$\"3 c!e4i3.#)G\"F*7$$\"3'****\\7TW)RzF*$\"3ot8<]M*oQ\"F*7$$\"3B+++:K^+!)F* $\"3]\"Hn!*3J_Z\"F*7$$\"3#)****\\7,Hl!)F*$\"3e3'e#>&)=p:F*7$$\"3k**\\P 4w)R7)F*$\"3'e>`+q**Rl\"F*7$$\"31,+]x%f\")=)F*$\"3))=AX)pdju\"F*7$$\"3 -+]P/-a[#)F*$\"3yFS]zw!H$=F*7$$\"3[+](=Yb;J)F*$\"3nk5\"y<2I#>F*7$$\"3) )****\\i@Ot$)F*$\"3wE#e(*pC2,#F*7$$\"3I+]PfL'zV)F*$\"3CV7vGy:-@F*7$$\" 3u*******)>=+&)F*$\"3?\")*\\1cF)*=#F*7$$\"3C++DE&4Qc)F*$\"37h`hu+3zAF* 7$$\"3S+]P%>5pi)F*$\"3#pn0kO'=nBF*7$$\"3q*****\\:$*[o)F*$\"3c3i78]yZCF *7$$\"3k***\\7<[8v)F*$\"3dl[@h(G(RDF*7$$\"3++++Ijy5))F*$\"3$=HX.`p:i#F *7$$\"3I+]P/)fT())F*$\"3hna:`NT3FF*7$$\"3i**\\i0j\"[$*)F*$\"3?R;6n'G6z #F*7$$\"3A+++++++!*F*$\"3+bcjb.dzGF*-%'COLOURG6&%$RGBG$\"*++++\"!\")$ \"\"!Fa[lF`[l-F$6%7S7$F($!3?Db,)*4*fm\"F*7$F.$!3#ob(px+lb:F*7$F3$!3gM% f,&f4g9F*7$F8$!3eB07J_8`8F*7$F=$!3IKhp(='*fC\"F*7$FB$!3+E0[/6*)R6F*7$F G$!3)RN)Q;C)>/\"F*7$FM$!3LTf1.5r5%*FK7$FR$!33LdktB0s$)FK7$FW$!3LFIiicr TtFK7$Ffn$!3c6u>k([qG'FK7$F[o$!3W2T7cBSi`FK7$F`o$!3TheT!))fiK%FK7$Feo$ !3iXA\">3K4H$FK7$Fjo$!3W&HpC[tzH#FK7$F_p$!3=snPS9I+9FK7$Fdp$!3KO(oH$fR yLFhp7$Fjp$\"3I1Yn1;?!e&Fhp7$F_q$\"3gMGc-HK&f\"FK7$Fdq$\"3tf,n`!Q#4DFK 7$Fiq$\"3E-t/cKO2NFK7$F^r$\"3IG^h-iQ`WFK7$Fcr$\"3!zt%>***))eV&FK7$Fhr$ \"3`ykfGy/MjFK7$F]s$\"3y%>J\"o&y%)H(FK7$Fbs$\"3!*=@Fi&)\\&H)FK7$Fgs$\" 3%*yTcl;\\f\"*FK7$F\\t$\"3:f]#e\"F*7$F[u$\"3ek%pf``qG\"F*7$F`u$\"3e!\\En#z4'Q\"F*7$Feu$\"3Erns aLou9F*7$Fju$\"3CxTv*RT)o:F*7$F_v$\"3)f1ngI&y`;F*7$Fdv$\"3OY'p;uViu\"F *7$Fiv$\"3;0iW/O&G$=F*7$F^w$\"3S5**y\\\"))H#>F*7$Fcw$\"3KqHu52s5?F*7$F hw$\"3t$oaPtd@5#F*7$F]x$\"3!oKh=\"y#)*=#F*7$Fbx$\"3!36]wL&3zAF*7$Fgx$ \"3'e!p)R94sO#F*7$F\\y$\"33q$\\5bTyW#F*7$Fay$\"3d4')R4E&)RDF*7$Ffy$\"3 C8Wfviy@EF*7$F[z$\"3v<4#G\\q(3FF*7$F`z$\"3O!f=`7m;z#F*7$Fez$\"3!fx$Qd& f.)GF*-Fjz6&F\\[lF`[lF][lF`[l-%*THICKNESSG6#\"\"#-F$6%7$7$$\"3]+++6y'R Y)F*F`[l7$Fbel$\"3-+++Sm%)Q@F*-Fjz6&F\\[lF`[lF`[lF][l-%*LINESTYLEGF\\e l-F$6&7%FaelFdel7$$\"34+++Ej!R-(F*F`[l-%'SYMBOLG6#%'CIRCLEG-Fjz6&F\\[l Fa[lFa[lFa[l-%&STYLEG6#%&POINTG-F$6&F]fl-Fbfl6#%(DIAMONDGFeflFgfl-F$6& F]fl-Fbfl6#%&CROSSGFeflFgfl-%+AXESLABELSG6%Q\"x6\"Q\"yFigl-%%FONTG6#%( DEFAULTG-%%VIEWG6$;$\"\"'!\"\"$\"\"*Fehl;$FehlFehl$\"\"$Fehl" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "The zero of " }{XPPEDIT 18 0 "f(x)=ar ctan(x) - exp(-x^2)" "6#/-%\"fG6#%\"xG,&-%'arctanG6#F'\"\"\"-%$expG6#, $*$F'\"\"#!\"\"F3" }{TEXT -1 42 " ( which is the solution of the equat ion " }{XPPEDIT 18 0 "arctan(x)=exp(-x^2)" "6#/-%'arctanG6#%\"xG-%$ex pG6#,$*$F'\"\"#!\"\"" }{TEXT -1 72 " ) can be calculated correct to 10 digits by Halley's method as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 284 "f := x -> arctan(x)-exp (-x^2);\nDf := D(f):\nD2f := D(Df):\nnextapprox := x -> evalf(x - 2*f( x)*Df(x)/(2*Df(x)^2-f(x)*D2f(x)));\nDigits := 13:\nx0 := 1.5;\nx1 := n extapprox(x0);\nx2 := nextapprox(x1);\nx3 := nextapprox(x2);\nx4 := ne xtapprox(x3);\nx5 := nextapprox(x4);\nDigits := 10:\nevalf(%%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,&-%'arctanG6#9$\"\"\"-%$expG6#,$*$)F0\"\"#F1!\"\"F9F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%+nextapproxGf*6#%\"xG6\"6$%)operator G%&arrowGF(-%&evalfG6#,&9$\"\"\"**\"\"#F1-%\"fG6#F0F1-%#DfGF6F1,&*&F3F 1)F7F3F1F1*&F4F1-%$D2fGF6F1!\"\"F?F?F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"#:!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# x1G$\".%34y'RY)!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x2G$\".?dK1R- (!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x3G$\"..5zLC,(!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x4G$\".x^yLC,(!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x5G$\".x^yLC,(!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+&yLC,(!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 63 "A p rocedure for graphing successive stages of Halley's method: " }{TEXT 0 11 "halley_step" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 11 "halley_step" }{TEXT -1 69 " \+ enables the progress of Halley's method to be observed graphically." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "halley_step: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 270 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 271 2 " " }{TEXT -1 67 " halley_step( eqn, approxroot ) or ha lleystep( eqn, approxroot ) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 4 " \+ " }}{PARA 0 "" 0 "" {TEXT 23 11 " eqn - " }{TEXT -1 66 " an \+ equation or expression involving a single variable, say x," }}{PARA 0 "" 0 "" {TEXT -1 26 " " }{TEXT 266 2 "OR" } {TEXT -1 34 " a function of the form x -> f(x)," }}{PARA 0 "" 0 "" {TEXT -1 96 " where f(x) evaluates to a real or complex floating point number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 23 15 "approxroot - " } {TEXT 272 109 "an initial approximation for the root in the form of a \+ real constant a, when the1st argument is a procedure, " }}{PARA 0 "" 0 "" {TEXT 274 115 " and in the form of a n equation x=a when the1st argument is an expression or equation." }} {PARA 0 "" 0 "" {TEXT -1 10 " " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 11 "halley_step" }{TEXT -1 70 " \+ performs a single Halley iteration and returns the new approximation \+ " }}{PARA 258 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "b = a - 2*phi (a)*phi*`'`(a)/(2*phi*`'`(a)^2-phi*`''`(a)*phi(a))" "6#/%\"bG,&%\"aG\" \"\"*,\"\"#F'-%$phiG6#F&F'F+F'-%\"'G6#F&F',&*(F)F'F+F'-F.6#F&F)F'*(F+F '-%#''G6#F&F'-F+6#F&F'!\"\"F:F:" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "for a root of " }{XPPEDIT 18 0 "phi(x)" "6#-%$phiG6#%\"xG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 "A picture is drawn t o illustrate the step. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 8 "Options:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "draw=true/false" }}{PARA 0 "" 0 "" {TEXT -1 80 "t his options determines whether to draw the picture. The default is \"d raw=true\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "color=c or colour=c" }}{PARA 0 "" 0 "" {TEXT -1 229 "If c is a \+ list of up to 4 colours, these colours will be applied in respective o rder to the curve, the tangent line, the ordinate of the initial appro ximation, and the 3 points shown. A single colour is applied to the cu rve only." }}{PARA 0 "" 0 "" {TEXT -1 46 "The default is \"colour=[red ,green,blue,navy]\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 11 "thickness=t" }}{PARA 0 "" 0 "" {TEXT -1 153 "If t is is a list of 1 or 2 positive integers, then they will be applied in resp ective order to specify the thickness of the curve and the tangent lin e. " }}{PARA 0 "" 0 "" {TEXT -1 103 "A single thickness is applied to \+ both the curve and the tangent line. The default is \"thickness=[1,2] \"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 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 27 " halley_step: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "halley_step" {MPLTEXT 1 0 6977 "# to allow for differe nt spellings\nhalleystep := proc() halley_step(args[1..nargs]) end:\n \nhalley_step := proc(ff,approx)\n local x,x1,y1,x2,w2,h2,xL,xR,yT,y B,xrange,yrange,df,graphs,\n pts,ord,f,fn,approxroot,lmr,sf,proct ype,vars,Options,i,\n clr,colr,thk,thik,drawpic,lft,rgt,xx2,p,q,d n,nm,t,d1,d2,\n h,d2f,disc,xDisc,p1,p2,p3,xrange2,xrange3;\n\n \+ if nargs<2 then\n error \"at least 2 arguments are required; the \+ basic syntax is: 'halley_step(f(x),x=a)'.\"\n end if;\n\n if type( ff,procedure) then\n if nops([op(1,eval(ff))])<>1 then\n \+ error \"the 1st argument, %1, is invalid .. it should be a procedure w ith a single argument\",ff;\n end if;\n proctype := true;\n \+ if type(approx,complexcons) then\n approxroot := approx; \n else\n error \"the 2nd argument, %1, is invalid .. whe n the 1st argument is a procedure, the 2nd argument should be a comple x constant\",approx;\n end if;\n elif type(ff,algebraic) or typ e(ff,equation) then\n if type(ff,equation) then\n lmr := \+ lhs(ff)-rhs(ff);\n sf := traperror(simplify(lmr));\n i f sf<>lasterror then\n f := sf;\n else\n \+ f := lmr;\n end if;\n else\n f := ff;\n end if;\n vars := indets(f,name) minus indets(f,complexcons);\n \+ if nops(vars)<>1 then \n if not has(indets(f),\{Int,Sum\}) th en\n error \"the 1st argument, %1, is invalid .. it should \+ be an expression or an equation which depends only on a single variabl e\",ff;\n end if;\n end if;\n if type(approx,name=co mplexcons) then\n proctype := false;\n x := op(1,appro x);\n if not member(x,vars) then\n error \"the 1st \+ argument, %1, is invalid .. it should be an expression or an equation \+ which depends only on the variable %2\",ff,x;\n end if;\n \+ approxroot := op(2,approx);\n else\n error \"the 2nd \+ argument, %1, is invalid .. it should have the form 'x=a', to provide \+ a starting approximation for a root\",approx;\n end if;\n else \n error \"the 1st argument, %1, is invalid .. it should be an al gebraic expression in a single variable, an equation in a single varia ble, or a procedure with a single argument\",ff;\n end if;\n\n # G et the options.\n # Set the default value to start with.\n drawpic := false;\n clr := [COLOR(RGB,1,0,0),COLOR(RGB,0,1,0),COLOR(RGB,0,0 ,1), COLOR(RGB,.137,.137,.557)];\n thk := [1,2];\n if nargs>2 then \n Options:=[args[3..nargs]];\n if not type(Options,list(equ ation)) then\n error \"each optional argument must be an equat ion\"\n end if;\n if hasoption(Options,'draw','drawpic','Opt ions') then\n if drawpic<>true then drawpic := false end if;\n end if;\n if hasoption(Options,'color','colr','Options') or \n hasoption(Options,'colour','colr','Options') then\n \+ if type(colr,list) then\n for i from 1 to min(nops(colr),4 ) do\n clr[i] := `plot/color`(colr[i]);\n end do;\n else\n clr[1] := `plot/color`(colr);\n \+ end if;\n end if;\n if hasoption(Options,'thickness','thi k','Options') then\n if type(thik,list) then\n for \+ i from 1 to min(nops(thik),2) do\n thk[i] := thik[i];\n \+ end do;\n else\n thk := [thik,thik];\n \+ end if;\n end if;\n if nops(Options)>0 then\n \+ error \"%1 is not a valid option for %2 .. the recognised options are \\\"draw\\\", \\\"colour\\\", or (\\\"color\\\") and \\\"thickness\\ \"\",op(1,Options),procname;\n end if;\n end if;\n\n if proct ype then\n fn := ff;\n else\n # Evaluate any real constant s in f\n fn := unapply(evalf(f),x);\n end if;\n\n x1 := evalf (approxroot);\n df := D(fn);\n d2f := D(df);\n \n y1 := \+ traperror(evalf(fn(x1)));\n if y1=lasterror or not type(y1,numeric) \+ then\n error \"failed to evaluate function at %1\",x1;\n end if ;\n d1 := traperror(evalf(df(x1)));\n if d1=lasterror or not type( d1,numeric) then\n error \"failed to evaluate derivative at %1\", x1;\n end if;\n d2 := traperror(evalf(d2f(x1)));\n if d2=lasterr or or not type(d2,complex(numeric)) then\n error \"failed to eval uate 2nd derivative at %1\",x1;\n end if;\n\n dn := d1-1/2*y1*d2/d 1;\n if dn=0 then\n error \"zero denominator obtained in Halley formula\"\n end if;\n h := y1/dn;\n x2 := x1 - h;\n\n if draw pic then\n # recalulate for the picture\n Digits := max(Digi ts,15);\n t := traperror(evalf(fn(x1)));\n if t<>lasterror a nd type(t,numeric) then\n y1 := t\n end if;\n t := t raperror(evalf(df(x1)));\n if t<>lasterror and type(t,numeric) th en\n d1 := t \n end if;\n t := traperror(evalf( d2f(x1)));\n if t<>lasterror and type(t,numeric) then\n d 2 := t \n end if;\n t := 2*d1^2-y1*d2;\n if t<>0 t hen dn := t end if;\n nm := 2*y1*d1;\n p := -d2/dn;\n q := 2*d1/dn;\n h := q*y1;\n xx2 := x1 - h;\n xL := min( x1,xx2);\n xR := max(x1,xx2);\n w2 := (xR-xL)/2;\n lft \+ := xL-w2;\n rgt := xR+w2;\n if lft<>rgt then\n xrang e := lft..rgt;\n disc := false;\n if p<>0 then\n \+ xDisc := x1-q/p;\n if xDisclft then\n \+ disc := true;\n xrange2 := lft..(.999*xDis c+.001*lft);\n xrange3 := (.999*xDisc+.001*rgt)..rgt;\n \+ end if;\n end if;\n yT := max(0,y1);\n \+ yB := min(0,y1);\n h2 := (yT-yB)/2;\n yrange := y B-h2..yT+h2;\n if disc then\n p1 := CURVES(op(1,op (1,plot('fn'(x),x=xrange))),\n THICKNESS(thk[1 ]),clr[1]);\n p2 := CURVES(op(1,op(1,plot(((x-x1)+h)/(p*(x -x1)+q),\n x=xrange2))),THICKNESS(thk[2]),clr[2]); \n p3 := CURVES(op(1,op(1,plot(((x-x1)+h)/(p*(x-x1)+q),\n \+ x=xrange3))),THICKNESS(thk[2]),clr[2]);\n \+ ord := CURVES([[x1,0],[x1,y1]],LINESTYLE(2),op(3,clr));\n \+ pts := POINTS([x1,0],[x1,y1],[xx2,0],\n \+ SYMBOL(CIRCLE),op(4,clr)),\n POINTS([x1,0],[x1,y1],[xx2 ,0],SYMBOL(CROSS),op(4,clr)),\n POINTS([x1,0],[x1,y1],[xx2 ,0],SYMBOL(DIAMOND),op(4,clr));\n print(PLOT(pts,ord,p1,p2, p3,VIEW(xrange,yrange)));\n else \n graphs := plot (['fn'(x),((x-x1)+h)/(p*(x-x1)+q)],\n x=xrange,yrange,co lor=[op(1,clr),op(2,clr)],\n thickness=thk);\n \+ ord := CURVES([[x1,0],[x1,y1]],LINESTYLE(2),op(3,clr));\n \+ pts := POINTS([x1,0],[x1,y1],[xx2,0],SYMBOL(CIRCLE),\n \+ op(4,clr)),\n POINTS([x1,0],[x1,y1],[xx2,0],SYMBO L(CROSS),op(4,clr)),\n POINTS([x1,0],[x1,y1],[xx2,0],SYMBO L(DIAMOND),op(4,clr));\n print(PLOT(pts,ord,op(graphs)));\n end if;\n else\n WARNING(\"the range for the plo t is empty\");\n end if;\n end if;\n x2;\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Examples are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 11 "halley_step" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 266 4 "N ote" }{TEXT -1 53 ": The examples in this section require the procedur e " }{TEXT 0 11 "halley_step" }{TEXT -1 3 " - " }{HYPERLNK 17 "halley_ step" 1 "" "halley_step" }{TEXT -1 3 " . " }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 30 "In this example \+ the procedure " }{TEXT 0 11 "hayley_step" }{TEXT -1 71 " is used to il lustrate steps in the calculation of the single zero of " }{XPPEDIT 18 0 "x^3-1/100;" "6#,&*$%\"xG\"\"$\"\"\"*&F'F'\"$+\"!\"\"F*" }{TEXT -1 21 " by Hayley's method." }}{PARA 0 "" 0 "" {TEXT -1 45 "A single \+ step can be illustrated as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "f := x->x^3-1/100;\nx0 := \+ 1\nhalley_step(f(x),x=0,true);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 50 "The following loop illustrates a number o f steps. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "f := x->x^3-1/100;\nxin := 1:\nprint(x[0]=xin);\nfor i from 1 to 6 do\n xout := halley_step(f(x),x=xin,draw=is(i<=4));\n print(x[i]=xout);\n xin := xout;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&*$)9$\"\"$\" \"\"F1#F1\"$+\"!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"xG6 #\"\"!\"\"\"" }}{PARA 13 "" 1 "" {GLPLOT2D 365 365 365 {PLOTDATA 2 "6* -%'POINTSG6'7$$\"\"\"\"\"!F)7$F'$\",++++!**!#67$$\"0jrc'oiu]!#:F)-%'SY MBOLG6#%'CIRCLEG-%&COLORG6&%$RGBG$\"$P\"!\"$F:$\"$d&F<-F$6'F&F*F.-F36# %&CROSSGF6-F$6'F&F*F.-F36#%(DIAMONDGF6-%'CURVESG6%7$F&F*-%*LINESTYLEG6 #\"\"#-F76&F9F)F)F(-FJ6%7S7$$\"0X2&)HS>h#F1$\"/+v2IE>y!#;7$$\"03#eL&em #GF1$\"0`tXt*\\e7Fen7$$\"0NG=s$[8IF1$\"0-kP#*plt\"Fen7$$\"0vZa=)eBKF1$ \"0,TXO)z\\BFen7$$\"0\"e$yN&3NMF1$\"0J,>%fL`IFen7$$\"03#3@tdXOF1$\"0D \"onv0XQFen7$$\"0?eeJI2%QF1$\"0sE\")pTbm%Fen7$$\"0EY#Q**zUSF1$\"0(*zau Xwg&Fen7$$\"0e$yI1y^UF1$\"0t'>O`@'o'Fen7$$\"0b*e>64gWF1$\"0WPaO(>syFen 7$$\"0IS>1kVn%F1$\"/Ea2cL8#*F17$$\"0.%>'y&4j[F1$\"0$ed%f2,0\"F17$$\"0U ;n+jb2&F1$\"0G2Z.Lv?\"F17$$\"0qf!QE!*)G&F1$\"0j00#yVz8F17$$\"0NJP!\\\\ %\\&F1$\"0$=(R7f(e:F17$$\"0C_qP#>\"o&F1$\"0oZ1jeOt\"F17$$\"0-(fIP>.fF1 $\"0?t&=q7d>F17$$\"0$\\,(*zD\"4'F1$\"0mD2Gl+;#F17$$\"0],sW@+J'F1$\"0m/ %G:U7CF17$$\"0sc'oTm.lF1$\"03gB&p*3l#F17$$\"0W8[#47;nF1$\"0ep!GBRHHF17 $$\"0\"4K)**H%=pF1$\"0ZB)RR[6KF17$$\"0Y5HH=&HrF1$\"0SyZOOR_$F17$$\"0ix >CjLK(F1$\"0Pr;[Sw#QF17$$\"0d*327XKvF1$\"0X!*op[P<%F17$$\"0#)Q(R[j\\xF 1$\"0#f&y!e=aXF17$$\"0Y5/$QpQzF1$\"0ort(=>.\\F17$$\"09w(>F)G9)F1$\"0$y L0PE*H&F17$$\"03)QA)HQN)F1$\"0)R_oJ%)HdF17$$\"0%e$yS+-c)F1$\"0-A%32msh F17$$\"0NM1cu)f()F1$\"0=fm)\\#>i'F17$$\"0*f!Qmy:)*)F1$\"0*e&QrF`9(F17$ $\"00%>h'*y!=*F1$\"0&\\0\"*G?QwF17$$\"0\"=k@**[$R*F1$\"0(\\+L]f)=)F17$ $\"06#3rzA'e*F1$\"0PNHpP$4()F17$$\"0Z5zcTpz*F1$\"0db_z5JI*F17$$\"0PJ7k 1_***F1$\"0Eza\"oi&))*F17$$\"0eLq2X--\"!#9$\"0&RC=J(>0\"F]z7$$\"0NJ(yp 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{PARA 4 "" 0 "" {TEXT -1 59 "A procedure implementing Halley's method for root-finding : " }{TEXT 0 6 "halley" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "halley: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }{TEXT 259 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 260 2 " " }{TEXT -1 29 " halley( eqn, approxroot ) " }{TEXT 261 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters: " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT 23 11 " \+ eqn - " }{TEXT -1 66 " an equation or expression involving a s ingle variable, say x," }}{PARA 0 "" 0 "" {TEXT -1 26 " \+ " }{TEXT 266 2 "OR" }{TEXT -1 34 " a function of the form x -> f(x)," }}{PARA 0 "" 0 "" {TEXT -1 96 " \+ where f(x) evaluates to a real or complex floating point number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 23 15 "approxroot - " }{TEXT 262 68 "an initial approximation \+ for the root, which may be real or complex:" }}{PARA 0 "" 0 "" {TEXT 265 98 " in the form of a constant a when the1st argument is a procedure, and" }}{PARA 0 "" 0 "" {TEXT 264 111 " in the form of an equation x=a when the 1st argument is an expression or equation." }}{PARA 0 "" 0 "" {TEXT -1 10 " " }}{PARA 256 "" 0 "" {TEXT -1 12 "Description:" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The proce dure " }{TEXT 0 6 "halley" }{TEXT -1 28 " attempts to find a root of \+ " }{XPPEDIT 18 0 "f(x) = 0;" "6#/-%\"fG6#%\"xG\"\"!" }{TEXT -1 51 " by Halley's method given an initial approximation." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Options:" }{TEXT -1 1 "\n " }}{PARA 0 "" 0 "" {TEXT -1 135 "maxiterations=n\nThis option can be \+ used to override the default value of Digits*5 for the maximum number \+ of iterations to be performed." }}{PARA 0 "" 0 "" {TEXT -1 50 "The abr eviated form \"maxiter=n\" may also be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "precision=fixed/variable " }}{PARA 0 "" 0 "" {TEXT -1 309 "If the computed value of the functio n exhibits a loss of significant digits as the successive approximatio ns converge the root then the working precision is increased to compen sate for this. This feature can be turned off via the option \"precisi on=fixed\". The default for this option is \"precision=variable\". " } }{PARA 0 "" 0 "" {TEXT -1 52 "The abreviated form \"prcsn=fixed\" may \+ also be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "info=true/false/0/1/2/3\n\"info=0\" is the same as \"inf o=false\" and \"info=1\" is the same as \"info=true\"." }}{PARA 0 "" 0 "" {TEXT -1 123 "This option allows the progress of the computation \+ to be monitored by printing the result of each Halley step as it occur s." }}{PARA 0 "" 0 "" {TEXT -1 96 "With the option \"info= 2\" the exp ressions for function and derivative being used are also given." }} {PARA 0 "" 0 "" {TEXT -1 253 "The option \"info= 3\" provides addition al information regarding the value of the function, its 1st and 2nd de rivatives and the correction term at each step, together with informat ion regarding any change in the working precision used in the computat ion. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 266 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 "halley: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8751 "halley := proc(ff,approx)\n loc al Options,x,df,d2f,fx,dfx,d2fx,dn,h,eps,saveDigits,\n i,maxit,pr ntflg,f,fn,approxroot,xx,lmr,sf,proctype,\n complexround,vars,wor kingDigits,extraDigits,\n adjustDigits,eps2,prsn,dfn,d2fn,triedze ro,small,f0;\n\n if nargs<2 then\n error \"at least 2 arguments are required; the basic syntax is: 'halley(f(x),x=a)'.\"\n end if; \+ \n\n if type(ff,procedure) then\n if nops([op(1,eval(ff))])<> 1 then\n error \"the 1st argument, %1, is invalid .. it should be a procedure with a single argument\",ff;\n end if;\n pro ctype := true;\n if type(approx,complexcons) then\n appro xroot := approx;\n else\n error \"the 2nd argument, %1, i s invalid .. when the 1st argument is a procedure, the 2nd argument sh ould be a complex constant\",approx;\n end if;\n elif type(ff,a lgebraic) or type(ff,equation) then\n if type(ff,equation) then\n lmr := lhs(ff)-rhs(ff);\n sf := traperror(simplify(lm r));\n if sf<>lasterror then\n f := sf;\n e lse\n f := lmr;\n end if;\n else\n f : = ff;\n end if;\n vars := indets(f,name) minus indets(f,comp lexcons);\n if nops(vars)<>1 then \n if not has(indets(f) ,\{Int,Sum\}) then\n error \"the 1st argument, %1, is inval id .. it should be an expression or an equation which depends only on \+ a single variable\",ff;\n end if;\n end if;\n if typ e(approx,name=complexcons) then\n proctype := false;\n \+ x := op(1,approx);\n if not member(x,vars) then\n \+ error \"the 1st argument, %1, is invalid .. it should be an expression or an equation which depends only on the variable %2\",ff,x;\n \+ end if;\n approxroot := op(2,approx);\n else\n \+ error \"the 2nd argument, %1, is invalid .. it should have the form 'x =a', to provide a starting approximation for a root\",approx;\n e nd if;\n else\n error \"the 1st argument, %1, is invalid .. it \+ should be an algebraic expression in a single variable, an equation in a single variable, or a procedure with a single argument\",ff;\n en d if;\n \n # Get the options \"maxiterations\" and \"info\".\n # Set the default values to start with.\n maxit := Digits*5;\n prnt flg := 0;\n prsn := 1;\n if nargs>2 then\n Options:=[args[3.. nargs]];\n if not type(Options,list(equation)) then\n err or \"each optional argument must be an equation\"\n end if;\n \+ if hasoption(Options,'maxiterations','maxit','Options') then\n \+ if not type(maxit,posint) then\n error \"\\\"maxiteratio ns\\\" must be a positive integer\"\n end if;\n elif haso ption(Options,'maxiter','maxit','Options') then\n if not type( maxit,posint) then\n error \"\\\"maxiter\\\" must be a posi tive integer\"\n end if;\n end if;\n if hasoption(Op tions,'precision','prsn','Options') then\n if not member(prsn, \{'fixed','variable'\}) then\n error \"\\\"precision\\\" mu st be 'fixed' or 'variable'\"\n end if;\n if prsn='fix ed' then prsn := 0 else prsn := 1 end if;\n elif hasoption(Option s,'prcsn','prsn','Options') then\n if not member(prsn,\{'fixed ','variable'\}) then\n error \"\\\"prcsn\\\" must be 'fixed ' or 'variable'\"\n end if;\n if prsn='fixed' then prs n := 0 else prsn := 1 end if;\n end if;\n if hasoption(Optio ns,'info','prntflg','Options') then\n if not member(prntflg,\{ true,false,0,1,2,3\}) then\n error \"\\\"info\\\" must be f alse <-> 0, true <-> 1,2 or 3\"\n end if;\n if prntflg =false then prntflg := 0\n elif prntflg=true then prntflg := 1 end if; \n end if;\n if nops(Options)>0 then\n erro r \"%1 is not a valid option for %2 .. the recognised options are \\\" maxiterations\\\",(or \\\"maxiter\\\"),\\\"precision\\\",(or \\\"prcsn \\\") and \\\"info\\\"\",op(1,Options),procname;\n end if;\n en d if;\n\n # local procedure\n complexround := proc(zz)\n loca l re,im,eps;\n re := Re(zz);\n im := Im(zz);\n if im=0 \+ then return Re(zz) end if;\n if re=0 then return Im(zz) end if;\n if not type(re,float) or not type(im,float) then\n retur n zz\n end if;\n eps := Float(1,-Digits);\n if abs(re)< =eps*abs(im) then return im*I\n elif abs(im)<=eps*abs(re) then re turn re\n else return zz end if;\n end proc: # of complexround \n\n # Increase precision for the computation\n saveDigits := Digi ts;\n extraDigits := min(iquo(iquo(Digits,5)+1,2)+3,8);\n workingD igits := Digits + extraDigits;\n Digits := workingDigits;\n\n if p roctype then\n fn := ff;\n dfn := D(fn);\n d2fn := D(df n);\n else\n # Evaluate any real constants in f\n fn := un apply(evalf(f),x);\n df := diff(f,x);\n d2f := diff(df,x);\n dfn := unapply(evalf(df),x);\n d2fn := unapply(evalf(d2f),x );\n if prntflg>1 then\n print(`Attempting to calculate a zero of`);\n print(f); \n print(`by Halley's method, \+ using the derivative`);\n print(df);\n print(`and the \+ second derivative`);\n print(d2f);\n print(``);\n \+ end if;\n end if;\n if prntflg>2 then\n print(`** working pr ecision is `||Digits||` digits **`);\n end if;\n \n xx := evalf( approxroot);\n\n eps := Float(1,-saveDigits-min(iquo(Digits,10),2)); \n eps2 := Float(1,-iquo(saveDigits,2));\n small := abs(xx)*Float( 1,-trunc(saveDigits*.75)-1);\n triedzero := false;\n\n h := xx;\n \+ \n for i to maxit do\n fx := traperror(evalf(fn(xx))); \n if fx=lasterror or not type(fx,complex(numeric)) then\n \+ error \"failed to evaluate function at %1\",evalf[saveDigits](xx);\n end if;\n if prntflg>2 then\n print(`value`=evalf[w orkingDigits](fx))\n end if;\n if prsn=1 and fx<>0 then\n \+ adjustDigits := extraDigits-\n max(length(SFloatMan tissa(Re(fx))),\n length(SFloatMantissa(Im(fx))));\n \+ if adjustDigits>0 and (abs(h)<=eps2*abs(xx) or abs(fx)2 then\n print(`** in creasing working precision to `||Digits||` digits **`); \n \+ end if;\n if not proctype then\n fn : = unapply(evalf(f),x);\n dfn := unapply(evalf(df),x);\n \+ d2fn := unapply(evalf(d2f),x);\n end if;\n \+ fx := traperror(evalf(fn(xx)));\n if fx=lasterror \+ or not type(fx,complex(numeric)) then\n error \"failed t o evaluate function at %1\",evalf[saveDigits](xx);\n end if ;\n if prntflg>2 then\n print(`value`=evalf[w orkingDigits](fx))\n end if;\n end if;\n end i f;\n\n dfx := traperror(evalf(dfn(xx)));\n if dfx=lasterror \+ or not type(dfx,complex(numeric)) then\n error \"failed to eva luate derivative at %1\",evalf[saveDigits](xx);\n end if;\n \+ if dfx=0 then\n error \"zero derivative obtained, so cannot ca lculate Halley correction\"\n end if;\n if prntflg>2 then\n \+ print(`derivative`=evalf[workingDigits](dfx))\n end if;\n \n d2fx := traperror(evalf(d2fn(xx)));\n if d2fx=lasterror o r not type(d2fx,complex(numeric)) then\n error \"failed to eva luate 2nd derivative at %1\",evalf[saveDigits](xx);\n end if;\n \+ if prntflg>2 then\n print(`2nd derivative`=evalf[workingDi gits](d2fx))\n end if;\n\n dn := dfx-1/2*fx*d2fx/dfx;\n \+ if dn=0 then\n error \"zero denominator obtained in Halley fo rmula\"\n end if;\n if prntflg>2 then\n print(`modif ied derivative -> `,`derivative`-`value times 2nd derivative`/`deri vative times 2`=evalf[workingDigits](dn));\n end if;\n h := \+ fx/dn;\n if prntflg>2 then\n print(`correction -> `,-` value`/`modified derivative`=evalf[workingDigits](-h));\n prin t(``);\n end if;\n xx := xx - h;\n if prntflg>0 then\n \+ print(`approximation `||i||` -> `,evalf[workingDigits](xx)) \n end if;\n if prntflg>2 then print(``) end if;\n if a bs(h)<=eps*abs(xx) then\n Digits := saveDigits;\n retu rn evalf(complexround(xx));\n end if;\n if i>6 and not tried zero \n and abs(xx)0 then\n print(`The values appear to be converging to \+ 0`);\n print(``);\n end if;\n retu rn 0.0\n end if;\n triedzero := true;\n end if;\n end do;\n\n Digits := saveDigits;\n print(`last iteration gives `,evalf(xx));\n error \"reached max, %1, iterations without converg ence\",maxit;\nend proc:" }}}{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 48 "Examples are given in the sections wh ich follow." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 6 "halley" } {TEXT -1 11 ": examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }{TEXT 294 34 ". . comparison with Newton's method" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "In this example the positive zero of the function " } {XPPEDIT 18 0 "f(x)=1-2*exp(-x*arctan(x))" "6#/-%\"fG6#%\"xG,&\"\"\"F) *&\"\"#F)-%$expG6#,$*&F'F)-%'arctanG6#F'F)!\"\"F)F4" }{TEXT -1 55 " is calculated correct to10 digits by Halley's method. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "f := x -> \+ 1-2*exp(-x*arctan(x));\nplot(f(x),x=-3..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&\"\"\"F-*&\" \"#F--%$expG6#,$*&9$F--%'arctanG6#F5F-!\"\"F-F9F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 452 226 226 {PLOTDATA 2 "6%-%'CURVESG6$7in7$$!\"$\"\"!$ \"3dnw$on'HG&*!#=7$$!3!******\\2<#pG!#<$\"3=\")4`r5ZA%*F-7$$!3#)***\\7 bBav#F1$\"3([.^\"\\vZ6$*F-7$$!36++]K3XFEF1$\"3qvEtB(G8;*F-7$$!3%)**** \\F)H')\\#F1$\"3!o&pb1Ikx*)F-7$$!3#****\\i3@/P#F1$\"3idL/*4$ob()F-7$$! 3;++Dr^b^AF1$\"3D21mID13&)F-7$$!3$****\\7Sw%G@F1$\"3X%*)=1;%3,#)F-7$$! 3*****\\7;)=,?F1$\"3ZyF-7$$!3/++DO\"3V(=F1$\"3'Qxk9@V:O(F-7$$!3 #******\\V'zViUC\"F1$\"3\\ba<#)4(GU$F-7$$!3-++DhkaI6F1$\"39Q VDj.-?BF-7$$!3s******\\XF`**F-$\"3#o@SvZ[NE)!#>7$$!3u*******>#z2))F-$! 3O)R#y%)*31)eFip7$$!3S++]7RKvuF-$!39><&4X]uP#F-7$$!3s,+++P'eH'F-$!3CNy _rw)3/%F-7$$!3q)***\\7*3=+&F-$!3Y$eks#)Q#feF-7$$!3[)***\\PFcpPF-$!3=)H D\\@E)euF-7$$!3;)****\\7VQ[#F-$!3ZT=#p<@k#))F-7$$!35)**\\P9(\\$*=F-$!3 &e]2\"\\At.$*F-7$$!32)***\\i6:.8F-$!3QtmUG$)4l'*F-7$$!3%p)*\\i:sw%)*Fi p$!3_O@4c9g2)*F-7$$!39$***\\(oKQm'Fip$!38;BC0T^6**F-7$$!3C'*\\7`H\">2& Fip$!3]Zm9]9m[**F-7$$!3O***\\(=K**zMFip$!3OZi*Hr.e(**F-7$$!3W-]P%[t!)) =Fip$!3Y3KRuC(G***F-7$$!3Wb+++v`hH!#?$!3/,[>(eC)****F-7$$\"3c%*\\7`MSd 8Fip$!3uv9ZxaJ'***F-7$$\"3l%**\\ilg4,$Fip$!3?6^!)>>)=)**F-7$$\"3w%*\\P fy^kYFip$!3')[\"H@Ijl&**F-7$$\"3%[***\\i]2=jFip$!3UuB*HaG/#**F-7$$\"3/ &**\\(o%*=D'*Fip$!3@:hI048;)*F-7$$\"3]****\\(QIKH\"F-$!3u?HPj#3,n*F-7$ $\"3K******\\4+p=F-$!3,b\\(4([7@$*F-7$$\"38****\\7:xWCF-$!3G!*o_@`Jh)) F-7$$\"3E,++vuY)o$F-$!3#y$oNE4)fb(F-7$$\"3!z******4FL(\\F-$!3g9d)*QcA) *eF-7$$\"3A)****\\d6.B'F-$!3^.MtE15MTF-7$$\"3s****\\(o3lW(F-$!3tqR\"30 ruT#F-7$$\"35*****\\A))oz)F-$!3\"=@kP[$H@gFip7$$\"3e******Hk-,5F1$\"3v @7\"[THE$*)Fip7$$\"36+++D-eI6F1$\"3tbp#fco.K#F-7$$\"3u***\\(=_(zC\"F1$ \"3_udxn2bcMF-7$$\"3M+++b*=jP\"F1$\"39H,s/VkLXF-7$$\"3g***\\(3/3(\\\"F 1$\"3Q3c%y=t8S&F-7$$\"33++vB4JB;F1$\"3%[dI(Hx\"H<'F-7$$\"3u*****\\KCnu \"F1$\"3Q9)p=rH%4oF-7$$\"3s***\\(=n#f(=F1$\"3u>.&=cCzO(F-7$$\"3P+++!)R O+?F1$\"3*pvLW_$f;yF-7$$\"30++]_!>w7#F1$\"3mI\"yM'Qu)>)F-7$$\"3O++v)Q? QD#F1$\"3/')*\\fg.K^)F-7$$\"3G+++5jypBF1$\"3I^KfgUZa()F-7$$\"3<++]Ujp- DF1$\"3I:RCv`+%)*)F-7$$\"3++++gEd@EF1$\"3)QG.N*[q`\"*F-7$$\"39++v3'>$[ FF1$\"3=@_Bow)QI*F-7$$\"37++D6EjpGF1$\"3\\L%G(>?%GU*F-7$$\"\"$F*F+-%'C OLOURG6&%$RGBG$\"#5!\"\"$F*F*F\\_l-%+AXESLABELSG6$Q\"x6\"Q!Fa_l-%%VIEW G6$;F(Fc^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 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "halley(f(x),x=1,info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/9S(RzEF*! #9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G$\"/mG! =t9F*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$ \"/0G!=t9F*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!=t9F*!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Convergen ce requires one less step than Newton's method via the procedure " } {TEXT 0 6 "newton" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 260 "How ever the evaluation of the function, its derivative and its second der vative are needed for each step of Halley's method. This means that a \+ total of 9 evaluations of some function are required for Halley's meth od as opposed to 8 when using Newton's method. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "newton(f(x), x=1,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~ ->~~~G$\"/Ft.0<[#*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximati on~2~~->~~~G$\"/*R9._9F*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7appro ximation~3~~->~~~G$\"/IE!=t9F*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% 7approximation~4~~->~~~G$\"/1G!=t9F*!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!=t9F*!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }{TEXT 299 63 ".. different levels of information via \+ the option \"info=1/2/3\" " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 300 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 94 "Use H alley's method to find (correct to 10 digits) the two positive solutio ns of the equation " }{XPPEDIT 18 0 "exp(x/2)-1=tan(x*sqrt(x))" "6#/,& -%$expG6#*&%\"xG\"\"\"\"\"#!\"\"F*F*F,-%$tanG6#*&F)F*-%%sqrtG6#F)F*" } {TEXT -1 29 " which have least magnitude. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 65 "It is a bit difficult to draw a \"nice\" \+ picture because the graph " }{XPPEDIT 18 0 "y=tan(x*sqrt(x))" "6#/%\"y G-%$tanG6#*&%\"xG\"\"\"-%%sqrtG6#F)F*" }{TEXT -1 30 " has vertical asy mpotes where " }{XPPEDIT 18 0 "x=(Pi/2)^(2/3),(3*Pi/2)^(2/3),(5*Pi/2)^ (2/3),` . . . `" "6&/%\"xG)*&%#PiG\"\"\"\"\"#!\"\"*&F)F(\"\"$F*)*(F,F( F'F(F)F**&F)F(F,F*)*(\"\"&F(F'F(F)F**&F)F(F,F*%(~.~.~.~G" }{TEXT -1 51 " and these cannot be \"discovered\" with the option \"" }{TEXT 269 12 "discont=true" }{TEXT -1 6 "\" for " }{TEXT 0 4 "plot" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 492 "dc1 := evalf((Pi/2)^(2/3)): dc2 := evalf((3*Pi/2)^(2 /3)):\na := dc1-.001: b := dc1+.001:\nc := dc2-.001: d := dc2+.001:\np 1 := plot(tan(x*sqrt(x)),x=0..a,color=blue):\np2 := plot(tan(x*sqrt(x) ),x=b..c,color=blue):\np3 := plot(tan(x*sqrt(x)),x=d..3.3,color=blue): \np4 := plots[implicitplot](\{x=dc1,x=dc2\},x=0..3.3,y=-3..3.3,\n \+ color=COLOR(RGB,.3,.3,.3),linestyle=3):\np5 := plot(exp(x/2)-1,x= 0..3,color=red):\nplots[display]([p1,p2,p3,p4,p5],\n view=[0. .3.3,-3..3.3],labels=[`x`,`y`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 477 477 477 {PLOTDATA 2 "6*-%'CURVESG6$7hn7$$\"\"!F)F(7$$\"3vVC#Q^MK%H!#>$ \"3COqzXTT\\]!#?7$$\"3cC*4p7FT]&F-$\"3iBNCK#)Q\"H\"F-7$$\"3[^Ts0(=TQ)F -$\"3y#=-kKF\"GCF-7$$\"3kb$>Q2?$G6!#=$\"37A&GKl'*=z$F-7$$\"3EA@)RR]oT \"F>$\"3D!p6$[!R#Q`F-7$$\"3-h[AN\\N%o\"F>$\"3glt*HJvP#pF-7$$\"3kyR@$RS 8'>F>$\"3n\"HSBd/\"3()F-7$$\"3w;\")4]\"*zZAF>$\"3m78W`gvp5F>7$$\"3`Vf6 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" }}{PARA 0 "" 0 "" {TEXT -1 13 "We see th at " }{TEXT 303 1 "x" }{TEXT -1 1 " " }{TEXT 302 1 "~" }{TEXT -1 45 " 2.7 is a positive solution of the equation " }{XPPEDIT 18 0 "exp(x/2 )-1=tan(x*sqrt(x))" "6#/,&-%$expG6#*&%\"xG\"\"\"\"\"#!\"\"F*F*F,-%$tan G6#*&F)F*-%%sqrtG6#F)F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 92 "This solution can be calculated correct to 10 digits by Halley's m ethod using the procedure " }{TEXT 0 6 "halley" }{TEXT -1 28 ". Incorp orating the option \"" }{TEXT 269 6 "info=2" }{TEXT -1 14 "\" instead \+ of \"" }{TEXT 269 9 "info=true" }{TEXT -1 27 "\" (which is equivalent \+ to \"" }{TEXT 269 6 "info=1" }{TEXT -1 199 "\") provides some addition al information apart from the successive approximations. Specifically \+ the single function with which the procedure works is given along with the first and second derivatives." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "x2 := halley(exp(x/2)-1=tan( x*sqrt(x)),x=2.7,info=2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%BAttempt ing~to~calculate~a~zero~ofG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(-%$ex pG6#,$*&\"\"#!\"\"%\"xG\"\"\"F,F,F,F*-%$tanG6#*$)F+#\"\"$F)F,F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%Iby~Halley's~method,~using~the~deriva tiveG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&#\"\"\"\"\"#F&-%$expG6#,$ *&F'!\"\"%\"xGF&F&F&F&*&#\"\"$F'F&*&,&F&F&*$)-%$tanG6#*$)F.#F1F'F&F'F& F&F&F.F%F&F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%:and~the~second~deriv ativeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&#\"\"\"\"\"%F&-%$expG6#, $*&\"\"#!\"\"%\"xGF&F&F&F&*&#\"\"*F-F&*(-%$tanG6#*$)F/#\"\"$F-F&F&,&F& F&*$)F4F-F&F&F&F/F&F&F.*&#F:F'F&*&F;F&F/#F.F-F&F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~ 1~~->~~~G$\"/d)zVzHn#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxim ation~2~~->~~~G$\"/,#GWFHn#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7ap proximation~3~~->~~~G$\"/'>GWFHn#!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#x2G$\"+Vu#Hn#!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "There is positive solution of smaller magnitude for which an approximate value can be obtained by drawing a suitable \"zo omed\" picture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 61 "plot([exp(x/2)-1,tan(x*sqrt(x))],x=0..0.35,col or=[red,blue]);" }}{PARA 13 "" 1 "" {GLPLOT2D 440 309 309 {PLOTDATA 2 "6&-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3[KLLeR+Hw!#?$\"3%3?26X'y@QF-7$$\"3l m;/^fpE9!#>$\"33D1kHO)*erF-7$$\"3?LL3xM?t@F3$\"35\\m,'pED4\"F37$$\"3FL LeR$fY#HF3$\"3Mg[#)=S2t9F37$$\"3Sm;ajOasOF3$\"3'=D$pt\\B`=F37$$\"3#GL3 x;GfO%F3$\"3anr&42lp?#F37$$\"3=**\\Pfw)Q3&F3$\"3!o2-crEXd#F37$$\"3\\K$ 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{XPPMATH 20 "6$%2correction~~->~~~G/,$*&%&value G\"\"\"%4modified~derivativeG!\"\"F*$\"/+2`/0hl!#?" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximatio n~2~~->~~~G$\"/Av#=mD%G!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"\"(!#9" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%P**~increasing~working~precision~to~17~digits~**G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%&valueG$\"%$*\\!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%+derivativeG$!/+i5AF?C!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%/2nd~derivativeG$!/J3C(f7N\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%;modified~derivative~~->~~~G/,&%+derivativeG\"\"\"*&%; value~times~2nd~derivativeGF'%3derivative~times~2G!\"\"F+$!/9i5AF?C!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2correction~~->~~~G/,$*&%&valueG \"\"\"%4modified~derivativeG!\"\"F*$\"/yc(3\"*H1#!#E" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximati on~3~~->~~~G$\"/Vv#=mD%G!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$\"+$=mD%G!#5" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "The two positive soluti ons of least magnitude are (correct to 10 digits):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "x=x1,x=x2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"xG$\"+$=mD%G!#5/F$$\"+Vu#Hn#!\"* " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 \+ " }{TEXT 295 42 ".. careful choice of initial approximation" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 42 "In this example the zeros of t he function " }{XPPEDIT 18 0 " f(x)=x^2+cos(64*x)/10-2" "6#/-%\"fG6#% \"xG,(*$F'\"\"#\"\"\"*&-%$cosG6#*&\"#kF+F'F+F+\"#5!\"\"F+F*F3" }{TEXT -1 56 " are calculated correct to10 digits by Halley's method. 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11 "" 1 "" {XPPMATH 20 "6$%8approximation~19~~ ->~~~G$\"/kmQ14*Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximat ion~20~~->~~~G$\"/\"HHe)*GQ\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8 approximation~21~~->~~~G$\"/IM@Bm/9!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~22~~->~~~G$\"/kFP-;\"R\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~23~~->~~~G$\"/D'pf'z$Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~24~~->~~~G$\"/07CWZw8!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~25~~->~~~G$\"/&z3:$H\" Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~26~~->~~~G$ \"/y#H?N5Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~27~ ~->~~~G$\"/RFjo?!Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approxima tion~28~~->~~~G$\"/8t))))*QP\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% 8approximation~29~~->~~~G$\"/V3'4$yz8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~30~~->~~~G$\"/,0I!p'\\N!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 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11 "" 1 "" {XPPMATH 20 "6$%8approximation~45~~- >~~~G$!/l<+zk`()!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation ~46~~->~~~G$!/a]P&*3L%)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approx imation~47~~->~~~G$!/gL.*)*yp)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% 8approximation~48~~->~~~G$!/We*o\\j'z!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~49~~->~~~G$!/:K0%eQ9)!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~50~~->~~~G$!/`D2\\#)Gv!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%6last~iteration~gives~G$!+2\\#)Gv!#5" }}{PARA 8 "" 1 "" {TEXT -1 67 "Error, (in halley) reached max, 50, iterations \+ without convergence\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "f := x -> x^2+cos(64*x)/10-2;\nhall ey(f(x),x=1.38,info=true,maxiter=70);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(*$)9$\"\"#\"\"\"F1*&#F1 \"#5F1-%$cosG6#,$*&\"#kF1F/F1F1F1F1F0!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"/~~~G$\"/!eQGUjP\"!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"/\")3;i>\" Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\" /#4m'ft!Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~- >~~~G$\"/K$4k#**y8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximati on~6~~->~~~G$\"/^rwV'eQ\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7appr oximation~7~~->~~~G$\"/!Qx&3(4Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%7approximation~8~~->~~~G$\"/IC'zx*z8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~9~~->~~~G$\"/=MBy&QO\"!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~10~~->~~~G$\"/$p9)Gfv8!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~11~~->~~~G$\"/Dh&f,2Q \"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~12~~->~~~G$\" /t$Q+?)y8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~13~~-> ~~~G$\"/]*pJw\\Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximati on~14~~->~~~G$\"/`aEb**z8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8appr oximation~15~~->~~~G$\"/,Uy.Sl8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %8approximation~16~~->~~~G$\"/'e]%Q?w8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~17~~->~~~G$\"/Pa#H)4\"Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~18~~->~~~G$\"/f7(\\?/Q\"!#8 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~19~~->~~~G$\"/VE.( ooP\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~20~~->~~~G $\"/dRWog\"Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~2 1~~->~~~G$\"/%eDX&)>Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approx imation~22~~->~~~G$\"/'Q^cHLQ\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ %8approximation~23~~->~~~G$\"/Su#HL#o8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~24~~->~~~G$\"/&pS(GLx8!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~25~~->~~~G$\"/p*R-T?Q\"!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~26~~->~~~G$\"/lSWNd$Q \"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~27~~->~~~G$\" /Z%[R6VP\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~28~~- >~~~G$\"/iiR.**z8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximatio n~29~~->~~~G$\"/nR.k(\\O\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8app roximation~30~~->~~~G$\"/$e'[f.w8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%8approximation~31~~->~~~G$\"/b6jR)4Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~32~~->~~~G$\"/2Z)\\D+Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~33~~->~~~G$\"/yT>e\\n8!#8" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~34~~->~~~G$\"/[))=p.x 8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~35~~->~~~G$\"/ 1@,Xv\"Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~36~~- >~~~G$\"/DKQJY#Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximati on~37~~->~~~G$\"/$)QMw_'Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8ap proximation~38~~->~~~G$\"/Jqwh[\"Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~39~~->~~~G$\"/$)R&H;;Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~40~~->~~~G$\"/v>s\\,#Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~41~~->~~~G$\"/Pb=pX$Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~42~~->~~~G$\"/x5a:Hs 8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~43~~->~~~G$\"/ jVtK.z8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~44~~->~~ ~G$\"/T!z$o7'Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation ~45~~->~~~G$\"/*\\*p()=\"Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8a pproximation~46~~->~~~G$\"/S)y_72Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~47~~->~~~G$\"/Pv`f()y8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~48~~->~~~G$\"/O>*yI_Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~49~~->~~~G$\"/e~~~G$\"/$opJTeP \"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~51~~->~~~G$\" /T/rn&3Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~52~~- >~~~G$\"/iFQY`z8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation ~53~~->~~~G$\"/jlLOz&R\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8appro ximation~54~~->~~~G$\"/!H5=vcQ\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%8approximation~55~~->~~~G$\"/w9N*)z!Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~56~~->~~~G$\"/z3xjGz8!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~57~~->~~~G$\"/vr!GY')Q\"!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~58~~->~~~G$\"/U\\%=)o# Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~59~~->~~~G$ \"/PM]I`!R\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~60~ ~->~~~G$\"/_F/Z`$Q\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approxima tion~61~~->~~~G$\"/H]OHut8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8app roximation~62~~->~~~G$\"/TM)o1(z8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$%8approximation~63~~->~~~G$\"/(=jxpoU\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~64~~->~~~G$\"/K76c\\N9!#8" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8approximation~65~~->~~~G$\"/V)H#\\,P9!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximation~66~~->~~~G$\"/(=Yu~~~G$\" /(=Yu " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }{TEXT 304 29 ".. a case of slow convergence" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 305 8 "Question" }{TEXT -1 79 ": \nUse a numerical method to fin d the single real zero of the cubic polynomial " }{XPPEDIT 18 0 "x^3-3 *Pi*x^2+78422406/2648617*x-19349653/624056" "6#,**$%\"xG\"\"$\"\"\"*(F &F'%#PiGF'F%\"\"#!\"\"*(\")1CUyF'\"(<'[EF+F%F'F'*&\")`'\\$>F'\"'cSiF+F +" }{TEXT -1 24 " correct to 10 digits. " }}{PARA 0 "" 0 "" {TEXT 306 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "f(x)=x^3-3*Pi*x^2+78422406/2648617*x-19349653/624056" "6#/-%\"fG6#%\"xG,**$F'\"\"$\"\"\"*(F*F+%#PiGF+F'\"\"#!\"\"*(\")1CUyF+ \"(<'[EF/F'F+F+*&\")`'\\$>F+\"'cSiF/F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "Since f ''(" }{TEXT 309 1 "x" }{TEXT -1 1 ")" }{XPPEDIT 18 0 "``=6*x-6*Pi" "6#/%!G,&*&\"\" '\"\"\"%\"xGF(F(*&F'F(%#PiGF(!\"\"" }{TEXT -1 15 ", the graph of " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 33 " has a point of i nflection where " }{XPPEDIT 18 0 "x=Pi" "6#/%\"xG%#PiG" }{TEXT -1 40 " . The single real zero is close to this " }{TEXT 307 1 "x" }{TEXT -1 51 " value and the point of inflection is close to the " }{TEXT 308 1 "x" }{TEXT -1 219 " axis. Root-finding procedures like Newton's method , the secant method and Halley's method will exhibit slow convergence \+ to the real zero. However, Halley's method will converge more rapidly \+ that the other two methods. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "f := x -> x^3-3*Pi*x^2+78422 406/2648617*x-19349653/624056:\n'f(x)'=f(x);\nplot(f(x),x=0..6);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG,**$)F'\"\"$\"\"\"F,*(F+ F,%#PiGF,)F'\"\"#F,!\"\"*&#\")1CUy\"(<'[EF,F'F,F,#\")`'\\$>\"'cSiF1" } }{PARA 13 "" 1 "" {GLPLOT2D 416 475 475 {PLOTDATA 2 "6%-%'CURVESG6$7S7 $$\"\"!F)$!3E#zH!owi+J!#;7$$\"3%*******\\#HyI\"!#=$!3i\\Jre:HHFF,7$$\" 33++]([kdW#F0$!3\"RK$zF'z8V#F,7$$\"3++++v;\\DPF0$!38'QN&*G#>B@F,7$$\"3 W+++D)FV7$$\"33+++lN?c7FV$!3o%f!3Rv(>q'FV7$$\"3-++]U$e6P\"FV$!3)o^C8'fJ \\bFV7$$\"36+++&>q0]\"FV$!3j&f5%e\"*>>WFV7$$\"3'******\\U80j\"FV$!3aWY #[_Q.X$FV7$$\"35+++0ytb@FV$!3Ov/4?$G'o5 FV7$$\"3'****\\(3wY_AFV$!3_Qc!R()=*GqF07$$\"3#)******HOTqBFV$!3m3o>?ML 'e%F07$$\"37++v3\">)*\\#FV$!3rY\"=ww$HVEF07$$\"3:++DEP/BEFV$!33&GbbrTV R\"F07$$\"3=++](o:;v#FV$!34,/fC!\\3$f!#>7$$\"3=++v$)[opGFV$!3!peKqU=., #Fgr7$$\"3%*****\\i%Qq*HFV$!3J**f8n>f?I!#?7$$\"3&****\\(QIKHJFV$!3?,[) 4d3r%=!#B7$$\"3#****\\7:xWC$FV$\"3q_nMH]0*3\"Fbs7$$\"37++]Zn%)oLFV$\"3 w#=:,mSO<\"Fgr7$$\"3y******4FL(\\$FV$\"3haaq^b#>]%Fgr7$$\"3#)****\\d6. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "Digits := 10:\nf := x -> x^3-3*Pi*x^2+78422406/2648617*x-1934965 3/624056:\nfsolve(f(x),x=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%y +W9$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 128 "However, if we increase precision for the calculation, the result agrees with that given by Halley's method using the procedure " } {TEXT 0 6 "halley" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "f := x -> x^3-3*Pi*x^2+78422 406/2648617*x-19349653/624056:\nevalf[20](fsolve(f(x),x=3));\nevalf(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5\"Q.oMBcw99$!#>" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"+ilZTJ!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "Since cubic equations can be solved an alytically, the real zero can be obtained using the Maple procedure " }{TEXT 0 5 "solve" }{TEXT -1 88 ". However, there are problems involve d in evaluating the analytical expression given by " }{TEXT 0 5 "solve " }{TEXT -1 126 ". It apprears to be necessary to increase precision t o a minimum of 31 digits in order for the analytical expression given \+ by " }{TEXT 0 5 "solve" }{TEXT -1 41 " to be evaluated correctly to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "f := x -> x^3-3*Pi*x^2+78422406/2648617*x-19349653/6 24056:\nsolve(f(x)):\nop(remove(has,[%],Complex(1)));\nevalf[10](evalf [31](%));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(*&\"=KhZ2uio=K42!\\c\"! \"\",*\"^p#\\)3(4dE$QJg3b%>)z%40eTzn])Qq(4f%GG?.x#odu7&z7%fF&*&\"^p7tJ l2biDZ/\"fA7r,[>(G,qRB`QCS]BnKXx`!\\Y;S]tc\"\"\"%#PiGF+F+*&\"]po>x$o([ ^0dl-R?f!R_xE*QU)>!=r!fXt(yT>>\"*=l:wIKQF+)F,\"\"$F+F&*&\"VowPVCii&4\" )*fk](o/!\\$*GUNfXW8g$F+,*\"inx>=D7!*42*e*\\BcH&z+Mcpvy[a:hJ*)>3O\"F+* &\"gn_ny^VtAc%=YA7#QWu?d1b_G))y%z$Hzs#)F+)F,\"\"#F+F&*&\"hn#>_-A%oG'p] GR;,t\"oZ%QM`1/A]q.\"\\z>&F+F,F+F&*&\"gn)GkMgx%p@j8llt&pO!>RGyll*[L8P# 36NF+F/F+F+#F+F8F+#F+F0F&*(\" " 0 "" {MPLTEXT 1 0 53 "st := time():\nevalf[75](halley(f(x),x=3));\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"fo$zXtWM_x/\\c.u#HjKy)Qu.pX1%)=u%Qh-x/Bcw 99$!#u" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#M!\"$" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "st := time ():\nevalf[85](fsolve(f(x),x=3)):\nevalf[75](%);\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"fo$zXtWM_x/\\c.u#HjKy)Qu.pX1%)=u%Qh-x/Bcw 99$!#u" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"#f!\"$" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }{TEXT 296 25 ".. a p athological example" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "Th e function " }{XPPEDIT 18 0 "f(x)=tanh(x^7-1)+x/24" "6#/-%\"fG6#%\"xG, &-%%tanhG6#,&*$F'\"\"(\"\"\"F/!\"\"F/*&F'F/\"#CF0F/" }{TEXT -1 38 " ha s a single real zero and the graph " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG -%\"fG6#%\"xG" }{TEXT -1 15 " has the lines " }{XPPEDIT 18 0 "y = x/24 ;" "6#/%\"yG*&%\"xG\"\"\"\"#C!\"\"" }{TEXT -1 1 " " }{TEXT 292 1 "+" } {TEXT -1 26 " 1 as oblique asymptotes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "f := x -> tanh(x^7-1)+ x/24;\nplot([f(x),x/24+1,x/24-1],x=-4..4,color=[red,black$2],linestyle =[1,3$2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)op eratorG%&arrowGF(,&-%%tanhG6#,&*$)9$\"\"(\"\"\"F5F5!\"\"F5*&#F5\"#CF5F 3F5F5F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 420 234 234 {PLOTDATA 2 "6'- %'CURVESG6%7io7$$!\"%\"\"!$!3ummmmmmm6!#<7$$!3ommmmFiDQF-$!3?66h[4Sf6F -7$$!35LLLo!)*Qn$F-$!3ab0='3zI:\"F-7$$!3nmmmwxE.NF-$!3UWWp!\\pf9\"F-7$ $!3YmmmOk]JLF-$!3WWW>oF\")Q6F-7$$!3_LLL[9cgJF-$!3>AsMg+pJ6F-7$$!3smmmh N2-IF-$!3pxFS)R'3D6F-7$$!3!******\\`oz$GF-$!3SL$e*)o[#=6F-7$$!3!omm;)3 DoEF-$!3ZW%p+7x66\"F-7$$!3?+++:v2*\\#F-$!3em;zH#GT5\"F-7$$!3BLLL8>1DBF -$!3KAAszv(o4\"F-7$$!3kmmmw))yr@F-$!3[WW>.7\\!4\"F-7$$!3;+++S(R#**>F-$ !3ELL$el,L3\"F-7$$!30++++@)f#=F-$!3'*****\\(e#3w5F-7$$!3-+++gi,f;F-$!3 )*****\\xc7p5F-7$$!3qmmm\"G&R2:F-$!336htO\"3G1\"F-7$$!3XLLLtK5F8F-$!3] 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$!33+++]?X$o)Fdr7$F^bl$!3ebb0VJh4')Fdr7$Fcbl$!3>WWWW2dV&)Fdr7$Fhbl$!3K nm\"HxbJZ)Fdr7$F]cl$!3LnmTg'fdS)Fdr7$Fbcl$!3qLLLLLLL$)FdrF[[mF][m-%+AX ESLABELSG6$Q\"x6\"Q!Fjam-%%VIEWG6$;F(Fbcl%(DEFAULTG" 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 -1 125 "For this rather patholog ical function it can be verified by trial and error that the interval \+ for convergence to the zero of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"x G" }{TEXT -1 30 " by Newton's method is roughly" }}{PARA 258 "" 0 "" {TEXT -1 4 " 0." }{XPPEDIT 18 0 "8622255<=x" "6#1\"(bAi)%\"xG" } {XPPEDIT 18 0 "``<=1.130341" "6#1%!G-%&FloatG6$\"(T.8\"!\"'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 59 "The interval for convergence b y Halley's method is roughly " }}{PARA 258 "" 0 "" {TEXT -1 3 " 0." } {XPPEDIT 18 0 "34876917<=x" "6#1\") " 0 "" {MPLTEXT 1 0 52 "f := x \+ -> tanh(x^7-1)+x/24:\nnewton(f(x),x=1.130341);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+z>sR**!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "newton(f(x),x=.8622255);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+z>sR**!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "halley(f(x), x=1.3300023);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+z>sR**!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "halley(f(x),x=0.34876917);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+ z>sR**!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 143 "Taking a starting value for each method outside the respective interval of convergence leads eventually to a cycling between the val ues 24 and " }{XPPEDIT 18 0 "-24" "6#,$\"#C!\"\"" }{TEXT -1 15 " which are the " }{TEXT 293 1 "x" }{TEXT -1 33 " intercepts of the two asymp otes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "f := x -> tanh(x^7-1)+x/24:\nhalley(f(x),x=1.3300024, info=true,maxiter=10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximat ion~1~~->~~~G$!.F\\7\"\\>%*!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7ap proximation~2~~->~~~G$!/$4LXbaV%!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%7approximation~3~~->~~~G$!/`&)[(p/'))!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"/zmLUUV8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~5~~->~~~G$!/O_w~~~G$\"/***********R#!# 7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~7~~->~~~G$!/,++++ +C!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~8~~->~~~G$\"/ ***********R#!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~9~ ~->~~~G$!/,+++++C!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8approximatio n~10~~->~~~G$\"/***********R#!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6 last~iteration~gives~G$\"+++++C!\")" }}{PARA 8 "" 1 "" {TEXT -1 67 "Er ror, (in halley) reached max, 10, iterations without convergence\n" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 29 "The osculating hyperbolas at " }{XPPEDIT 18 0 "x=24" "6#/%\"xG\"#C" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=-24" "6#/%\"xG,$\"#C!\"\"" }{TEXT -1 33 " degenerate into the asymptotes. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "f := x -> tanh(x^7-1)+x/24;\nx0 := 24.;\nh := 'h': p := 'p': \+ q := 'q': \ng := x -> (x-x0+h)/(p*(x-x0)+q);\neqns := normal(\{f(x0)=g (x0),D(f)(x0)=D(g)(x0),\n (D@@2)(f)(x0)=(D@@2)(g)(x0)\}); \nsols := solve(eqns,\{p,q,h\});\nassign(sols);\nx1 := x0-h;\ng(x);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arro wGF(,&-%%tanhG6#,&*$)9$\"\"(\"\"\"F5F5!\"\"F5*&#F5\"#CF5F3F5F5F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"#C\"\"!" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&,(9$\"\"\" %#x0G!\"\"%\"hGF/F/,&*&%\"pGF/,&F.F/F0F1F/F/%\"qGF/F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eqnsG<%/$\"\"!F(,$**\"\"#\"\"\",&*&%\"pGF ,%\"hGF,F,%\"qG!\"\"F,F1!\"$F/F,F,/$\"+++++?!\"**&F0F,F1F2/$\"+nmmmT!# 6,$*&F-F,F1!\"#F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%solsG<%/%\"hG$ \"+++++[!\")/%\"qG$\"+++++CF*/%\"pG$\"\"!F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x1G$!+++++C!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, &*&$\"+nmmmT!#6\"\"\"%\"xGF(F($\"+++++5!\"*F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "We can see what is happen ing with the procedure " }{TEXT 0 11 "halley_step" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "f := x -> tanh(x^7-1)+x/24;\nxin := 1.34:\nprint(x[0]=xin);\nfor \+ i from 1 to 4 do\n xout := halley_step(f(x),x=xin,draw=is(i<=3));\n \+ print(x[i]=xout);\n xin := xout;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,&-%%tanhG6#,& *$)9$\"\"(\"\"\"F5F5!\"\"F5*&#F5\"#CF5F3F5F5F(F(F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/&%\"xG6#\"\"!$\"$M\"!\"#" }}{PARA 13 "" 1 "" {GLPLOT2D 380 317 317 {PLOTDATA 2 "6*-%'POINTSG6'7$$\"$M\"!\"#\"\"!7$F '$\"0&RQL1$e0\"!#97$$!0;4@>!)=$GF.F*-%'SYMBOLG6#%'CIRCLEG-%&COLORG6&%$ RGBG$\"$P\"!\"$F:$\"$d&F<-F$6'F&F+F/-F36#%&CROSSGF6-F$6'F&F+F/-F36#%(D IAMONDGF6-%'CURVESG6%7$F&F+-%*LINESTYLEG6#\"\"#-F76&F9F*F*\"\"\"-FJ6%7 do7$$!0uj\")G?y\"\\F.$!0))*oy\"4\\?\"F.7$$!0@Rn0]ft%F.$!0#*zBDJt>\"F.7 $$!0cJ3u0xd%F.$!0d;#Rxt!>\"F.7$$!02PtXV(*R%F.$!0QqSkAL=\"F.7$$!0/3V_,1 A%F.$!0CgoRee<\"F.7$$!0+;;-6B/%F.$!099fiH%o6F.7$$!0_ZG!H,xQF.$!0C)H/Aa h6F.7$$!0Z.`Fceq$F.$!0+]\"y1Ta6F.7$$!0Kbi%e%)GNF.$!0t0OCNq9\"F.7$$!00) *=4.CN$F.$!0ft@Y$oR6F.7$$!0Fq9.54<$F.$!0$=K^77K6F.7$$!0uc]%306IF.$!0S( )=Xga7\"F.7$$!0W+PE(3JGF.$!0HI$p>'z6\"F.7$$!0b)eMZQ]EF.$!0O$G(pK/6\"F. 7$$!0$[UkVCwCF.$!0'f_[o<.6F.7$$!0dq`A3\"=BF.$!0]JF%ye'4\"F.7$$!0A!yg!p +8#F.$!0hRv(Gv)3\"F.7$$!0OMKAv2(>F.$!0+3Vj:@3\"F.7$$!0?$R@'yay\"F.$!0+ kD%\\Ru5F.7$$!0)GYJ(f9i\"F.$!0p\\@#3cn5F.7$$!0q8O,0:W\"F.$!0!Q!*3F1g5F .7$$!0aPmcX,F\"F.$!0(37>k\"H0\"F.7$$!0'4djxu!=\"F.$!0:R#eyu[5F.7$$!0P/ 0'*\\84\"F.$!0J]:LA(Q5F.7$$!0W:.([D45F.$!0(Gose\\55F.7$$!/]E,yfr#*F.$! 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" }}{PARA 0 "" 0 "" {TEXT -1 57 "The following numerical calculation of the posi tive zero " }{XPPEDIT 18 0 "1/3=0" "6#/*&\"\"\"F%\"\"$!\"\"\"\"!" } {TEXT -1 7 ".33333 " }{TEXT 291 5 ". . ." }{TEXT -1 5 " of " } {XPPEDIT 18 0 "f(x)=1/x^2-9" "6#/-%\"fG6#%\"xG,&*&\"\"\"F**$F'\"\"#!\" \"F*\"\"*F-" }{TEXT -1 19 " illustrates this. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "f := x -> 1/x^2-9;\nevalf(halley(f(x),x=.3,info= true),100);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)o peratorG%&arrowGF(,&*&\"\"\"F.*$)9$\"\"#F.!\"\"F.\"\"*F3F(F(F(" }} {PARA 12 "" 1 "" {XPPMATH 20 "6$%7approximation~1~~->~~~G$\"gq&>J^obju 3!z.4$e+'e$Rbq1Ri-P6F\\~~~G$\"gqhjcT*y9'yk'*yjwcq `;3l\"\\b%G&4`)))*e+E?OH<92t5mMwvd*G_tEJLLLLL!$3\"" }}{PARA 12 "" 1 " " {XPPMATH 20 "6$%7approximation~3~~->~~~G$\"gqQ,Y!=wyEbq3JH!)y?wKZVO/ !GRdn*Gdw*H\"=mU#\\8LLLLLLLLLLLLLLLLL!$3\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$%7approximation~4~~->~~~G$\"gqgdJLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLL!$3\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$ %7approximation~5~~->~~~G$\"gqLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL LLLLLLLLLLLLLL!$3\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"_qLLLLLLLLLL LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL!$+\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 7 " }{TEXT 298 29 ".. calculati ng complex roots" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 6 "halley" } {TEXT -1 63 " can be used to find complex numerical solutions to equat ions. " }}{PARA 0 "" 0 "" {TEXT -1 13 "The equation " }{XPPEDIT 18 0 " ln(x) = x;" "6#/-%#lnG6#%\"xGF'" }{TEXT -1 70 " has no real solutions, but it has two solutions in the complex plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "z := halley( ln(x)=x,x=.5+I,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approx imation~1~~->~~~G^$$\"/Gxd/:WJ!#9$\"/CA.(oyK\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~2~~->~~~G^$$\"/^wc^J\"=$!#9$\"/q`4eBP8! #8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approximation~3~~->~~~G^$$\"/v /_]J\"=$!#9$\"/2V,dBP8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%7approxi mation~4~~->~~~G^$$\"/w/_]J\"=$!#9$\"/2V,dBP8!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"zG^$$\"+_]J\"=$!#5$\"+,dBP8!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "We can check that " } {XPPEDIT 18 0 "ln(z);" "6#-%#lnG6#%\"zG" }{TEXT -1 27 " is approximate ly equal to " }{TEXT 290 1 "z" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ln(z);\nz := 'z':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"+\\]J\"=$!#5$\"+,dBP8!\" *" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "The conjugate of the first solution is also a solution. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "halley(ln (x)=x,x=.5-I);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#^$$\"+_]J\"=$!#5$!+, dBP8!\"*" }}}{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 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "T asks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 69 "Use Halley's \+ method to calculate the solutions of the cubic equation " }{XPPEDIT 18 0 "x^3+8*x^2-5*x-1=0" "6#/,**$%\"xG\"\"$\"\"\"*&\"\")F(*$F&\"\"#F(F (*&\"\"&F(F&F(!\"\"F(F/\"\"!" }{TEXT -1 23 " correct to 10 digits. " } }{PARA 0 "" 0 "" {TEXT -1 30 "______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 30 "__ ____________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 59 "Use Halley's method to calculate the zeros of the functio n " }{XPPEDIT 18 0 "f(x)=x^2+sin(64*x)/12-2" "6#/-%\"fG6#%\"xG,(*$F'\" \"#\"\"\"*&-%$sinG6#*&\"#kF+F'F+F+\"#7!\"\"F+F*F3" }{TEXT -1 23 " corr ect to 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 30 "___________________ ___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 30 "______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Use a num erical method to find the single real zero of the quintic polynomial \+ " }}{PARA 258 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^5-5*exp(1)*x^4+ 10*exp(2)*x^3-16050363812861/79910056048*x^2+38381789784980/1405974003 21*x-13224083397672/89103173045" "6#,.*$%\"xG\"\"&\"\"\"*(F&F'-%$expG6 #F'F'F%\"\"%!\"\"*(\"#5F'-F*6#\"\"#F'F%\"\"$F'*(\"/hG\"QO]g\"F'\",[g05 *zF-F%F2F-*(\"/!)\\y*y\"QQF'\"-@.S(fS\"F-F%F'F'*&\"/swR$3CK\"F'\",XI<. \"*)F-F-" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "correct to 1 0 digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 30 "____________________ __________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 18 "Find th e zeros of " }{XPPEDIT 18 0 "f(x)=1-2/(1+x^127*sinh(x))" "6#/-%\"fG6#% \"xG,&\"\"\"F)*&\"\"#F),&F)F)*&F'\"$F\"-%%sinhG6#F'F)F)!\"\"F2" } {TEXT -1 42 " correct to 10 digits by Halley's method. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 "The interval of con vergence when using Newton's method to calculate the positive solution can be found by trial and error to be approximately " }}{PARA 258 "" 0 "" {TEXT -1 2 " 0" }{XPPEDIT 18 0 "0.9818106454<=x" "6#1-%&FloatG6$ \"+ak5=)*!#5%\"xG" }{XPPEDIT 18 0 "``<=1.015704578" "6#1%!G-%&FloatG6$ \"+yXq:5!\"*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 31 "Check th is with a few examples." }}{PARA 0 "" 0 "" {TEXT -1 81 "What is the in terval of convergence to the positive solution for Halley's method?" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 4 "Hint" } {TEXT -1 110 ": Try some starting values close to zero and some starti ng values between 2 and 10, making use of the option \"" }{TEXT 269 16 "maxiterations=??" }{TEXT -1 63 "\" if necessary. What happens if y ou take 0 as a starting value?" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 30 "______________________________" }}{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 30 "__ ____________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 40 "Find the real solution of the equation " }{XPPEDIT 18 0 "ln(x)=sin(x)" "6#/-%#lnG6#%\"xG-%$sinG6#F'" }{TEXT -1 41 " correct to 10 digits by Halley's method. " }}{PARA 0 "" 0 "" {TEXT -1 89 "Use Hal ley's method to find three distinct complex (non-real) solutions of th e equation " }{XPPEDIT 18 0 "ln(x)=sin(x)" "6#/-%#lnG6#%\"xG-%$sinG6# F'" }{TEXT -1 62 " with the real and imaginary parts given correct to1 0 digits. " }}{PARA 0 "" 0 "" {TEXT -1 30 "___________________________ ___" }}{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 30 "______________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }