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{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 "Subtraction round-off error " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Consider the numerical evaluati on of the expression " }{XPPEDIT 18 0 "sqrt(67080)-259" "6#,&-%%sqrtG6 #\"&!3n\"\"\"\"$f#!\"\"" }{TEXT -1 34 " using floating point arithmeti c. " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "a = sqrt(6708 0);" "6#/%\"aG-%%sqrtG6#\"&!3n" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "b = 259;" "6#/%\"bG\"$f#" }{TEXT -1 7 ". Then " }{TEXT 272 1 "a" } {TEXT -1 5 " and " }{TEXT 273 1 "b" }{TEXT -1 18 " are nearly equal." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Digits:= 10:\na := evalf(sqrt(67080));\nb := 259;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"aG$\"+%p!)**e#!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"$f#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "In performing the following subtraction with 1 0 digit precision, we only obtain 5 digits of the result." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Digit s:= 10:\nb - a;\nans1 := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"&1$>!\"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ans1G$\"&1$>!\"(" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "The value obtained here has onl y 5 " }{TEXT 259 18 "significant digits" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 82 "If you can't see what has happened, do the subtrac tion 259 - 258.9980694 by hand." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 21 " 259.0000000" }{XPPEDIT 18 0 " ``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 269 13 " 258.9980694 " }}{PARA 256 "" 0 "" {TEXT -1 17 " 0.0019306 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 139 "If we want 10 digits of the answer, that is, an answer which is correct to 10 digits, we have to increase the \+ precision of the computation." }}{PARA 0 "" 0 "" {TEXT -1 133 "By usin g a few guard digits, and then rounding afterwards, we can obtain a re sult which we can be confident is correct to 10 digits. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "Digi ts := 20:\na := evalf(sqrt(67080));\nb := 259;\nd := b-a;\nDigits := 1 0:\nans2 := evalf(d); # round to 10 digits" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"51xZ(3\\p!)**e#!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG\"$f#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG$ \"0%HAD\"40$>!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ans2G$\"+D\"40$ >!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Again you can see what \+ the subtraction looks like when performed by hand." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 30 " 259.0000000000 0000000" }{XPPEDIT 18 0 "``-``" "6#,&%!G\"\"\"F$!\"\"" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 270 23 " 258.998069490874 77706 " }}{PARA 256 "" 0 "" {TEXT -1 27 " 0.00193050912522294 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "\nThe a bsolute error and relative error in the first result are:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "abserr := abs(ans1-ans2);\nrelerr : = abserr/abs(ans2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"&v 3*!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"+duI2Z!#9" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 176 "Ideally, we would like the res ult of a numerical computation to differ from the most accurate 10 dig it result, that is, a result which is correct to 10 digits, by no more than " }{TEXT 259 24 "1 unit in the last place" }{TEXT -1 11 " ( 1 ul p). " }}{PARA 0 "" 0 "" {TEXT -1 15 "For the number:" }}{PARA 256 "" 0 "" {TEXT -1 23 " 0.001930509125, " }}{PARA 0 "" 0 "" {TEXT -1 9 "1 ulp is " }}{PARA 256 "" 0 "" {TEXT -1 23 " 0.000000000001. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Given a number " }{TEXT 282 1 "x" }{TEXT -1 20 ", 1 ulp relative to " } {TEXT 283 1 "x" }{TEXT -1 20 " is the Maple value " }{TEXT 262 29 "Flo at(1, -Digits+1+ilog10(x))" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 6 ": The " } {TEXT 262 9 "ilog10(x)" }{TEXT -1 35 " function returns the integer ba se " }{TEXT 262 2 "10" }{TEXT -1 14 " logarithm of " }{TEXT 262 1 "x" }{TEXT -1 23 ", that is, the integer " }{TEXT 262 1 "r" }{TEXT -1 11 " such that " }{TEXT 262 25 "10^r <= abs(x) < 10^(r+1)" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ulp_ans2 := Float(1,-Digits+1+ilog10(ans2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)ulp_ans2G$\"\"\"!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "The absolute error in the first r esult, which exhibits a severe round-off error, is many times greater \+ than a unit in the last place. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "abserr/ulp_ans2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++](3*!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "The relative error provides a simi lar comparison with the machine epsilon for 10 digit arithmetic. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "mach_eps := Float(5,-Digits);\nrelerr/mach_eps;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%)mach_epsG$\"\"&!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+9\\h9%*!\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The phenomena which has occurred here is " }{TEXT 259 27 "subtraction round-off error" }{TEXT -1 123 ", resulting from t he subtraction of nearly equal quantities. It should be avoided if at \+ all possible. It is also sometimes " }{TEXT 259 18 "cancellation error " }{TEXT -1 95 " because of the fact that a number of digits in the su btracted numbers may be \"cancelled out\". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "A procedure for solving quadratic equatio ns: " }{TEXT 0 16 "solve_quadratic " }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 23 "solve_quadratic: usage " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 276 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT 277 4 " " }{TEXT -1 53 "solve_quadratic( q, x ) or solve_quadratic( q )" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 275 11 "Parameters:" }} {PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 87 " q \+ - an expression involving a single variable, say x, which simpli fies to a " }}{PARA 0 "" 0 "" {TEXT -1 62 " quadratic p olynomial, or a quadratic equation\n" }}{PARA 0 "" 0 "" {TEXT -1 4 " \+ " }{TEXT 23 4 "x - " }{TEXT -1 36 " the variable in the quadrati c q" }}{PARA 0 "" 0 "" {TEXT -1 91 " This argument may be omitted if the quadratic contains a single variable.\n" }}{PARA 258 "" 0 "" {TEXT 274 12 "Description:" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 0 15 "solve_quadratic" }{TEXT -1 40 " attempts to solve a quadratic equation " }{XPPEDIT 18 0 "a*x^2+b*x+c=0" "6#/,( *&%\"aG\"\"\"*$%\"xG\"\"#F'F'*&%\"bGF'F)F'F'%\"cGF'\"\"!" }{TEXT -1 120 " (or an equation which simplifies to a quadratic) by replacing on e of the expressions in the standard quadratic formula:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "x = (-b+sqrt(b^2-4*a*c))/(2*a);" "6#/%\"xG*&,&%\"bG!\"\"-%%sqrtG6#, &*$F'\"\"#\"\"\"*(\"\"%F/%\"aGF/%\"cGF/F(F/F/*&F.F/F2F/F(" }{TEXT -1 9 " or " }{XPPEDIT 18 0 "(-b-sqrt(b^2-4*a*c))/(2*a);" "6#*&,&%\"b G!\"\"-%%sqrtG6#,&*$F%\"\"#\"\"\"*(\"\"%F-%\"aGF-%\"cGF-F&F&F-*&F,F-F0 F-F&" }{TEXT -1 3 " " }}{PARA 256 "" 0 "" {TEXT -1 6 "with " } {XPPEDIT 18 0 "x = 2*c/(-b+sqrt(b^2-4*a*c));" "6#/%\"xG*(\"\"#\"\"\"% \"cGF',&%\"bG!\"\"-%%sqrtG6#,&*$F*F&F'*(\"\"%F'%\"aGF'F(F'F+F'F+" } {TEXT -1 9 " or " }{XPPEDIT 18 0 "2*c/(-b-sqrt(b^2-4*a*c));" "6#* (\"\"#\"\"\"%\"cGF%,&%\"bG!\"\"-%%sqrtG6#,&*$F(F$F%*(\"\"%F%%\"aGF%F&F %F)F)F)" }{TEXT -1 3 " ,\n" }}{PARA 0 "" 0 "" {TEXT -1 25 "depending o n the sign of " }{TEXT 278 1 "b" }{TEXT -1 128 ", in order to minimize any subtraction error which arises on subsequent evaluation of the so lutions as floating point numbers. \n" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT 259 16 "How to activate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure active open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "solve_quadratic: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "solve_quadratic" {MPLTEXT 1 0 1271 "solve_quadr atic := proc(ff)\n local fx,x,xx,vars,a,b,c,q;\n\n vars := indets( ff);\n\n if type(ff,`=`) then \n fx := lhs(ff)-rhs(ff) \n els e \n fx := ff \n end if;\n\n fx := numer(normal(fx));\n if \+ nargs>1 then\n x := args[2];\n if not type(x,name) then\n \+ error \"the optional 2nd argument must be of type name\"\n \+ end if;\n if nops(vars)=1 and type(vars,list(name)) then\n \+ if x<>op(1,vars) then\n error \"expecting 1st argument to depend on the variable %1\",x;\n end if;\n end if;\n e lse\n if nops(vars)=1 then\n x := op(1,vars)\n else \n error \"please give a 2nd argument to specify the variable \+ in the 1st argument\"\n end if;\n end if; \n \n if not type(fx,polynom(anything,x)) or degree(fx,x)<>2 then\n error \"e xpecting the 1st argument to be a quadratic or a quadratic equation in the variable %1\",x;\n end if;\n a := coeff(fx,x,2);\n b := coe ff(fx,x,1);\n c := coeff(fx,x,0);\n if sign(a)<0 then\n a := \+ -a; b := -b; c := -c;\n end if;\n\n if type([a,b,c],list(realcons) ) and signum(b^2-4*a*c)>=0 then\n q := -(b+sign(b)*sqrt(b*b-4*a*c ))/2;\n return q/a,c/q;\n else\n q := sqrt(b^2-4*a*c);\n \+ return (-b+q)/2/a,(-b-q)/2/a;\n end if;\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 40 "An examp le is 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 -1 91 "Avoiding subtraction error by changing the method of ca lculation or by increasing precision" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 112 "Sometimes it is po ssible to change the method of calculation so as to avoid excessive su braction roundoff error." }}{PARA 0 "" 0 "" {TEXT -1 102 "The usual wa y to solve a quadratic using the quadratic formula can lead to subtrac tion rounding error." }}{PARA 0 "" 0 "" {TEXT -1 16 "Using the Maple \+ " }{TEXT 0 5 "solve" }{TEXT -1 41 " command to solve the quadratic equ ation " }{XPPEDIT 18 0 "x^2-54*x+1=0" "6#/,(*$%\"xG\"\"#\"\"\"*&\"#aF( F&F(!\"\"F(F(\"\"!" }{TEXT -1 23 " shows what can happen." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "solve (x^2-54*x+1=0);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&\"#F\" \"\"*$-%%sqrtG6#\"$#=F%\"\"#,&F$F%F&!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+7v9)R&!\")$\"()[_=F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 75 "The second of the solutions exhibits a lo ss of about 3 digits of precision." }}{PARA 0 "" 0 "" {TEXT -1 27 "The problem arises because " }{XPPEDIT 18 0 "2*sqrt(182)" "6#*&\"\"#\"\" \"-%%sqrtG6#\"$#=F%" }{TEXT -1 16 " is close to 27." }}{PARA 0 "" 0 " " {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(2* sqrt(182));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7v9)p#!\")" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Excessive subtraction error can be avoided when solving the quadratic equation" }}{PARA 256 "" 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 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 36 "by replacing one of the expressions:" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "(-b+sqrt(b^2-4*a*c))/(2*a);" "6#*&,&%\"bG!\"\"-%%sqrtG6#,&*$F%\" \"#\"\"\"*(\"\"%F-%\"aGF-%\"cGF-F&F-F-*&F,F-F0F-F&" }{TEXT -1 8 " or " }{XPPEDIT 18 0 "(-b-sqrt(b^2-4*a*c))/(2*a);" "6#*&,&%\"bG!\"\"-%% sqrtG6#,&*$F%\"\"#\"\"\"*(\"\"%F-%\"aGF-%\"cGF-F&F&F-*&F,F-F0F-F&" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 26 "in the quadratic formul a: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x = -b/(2*a)" "6#/%\"xG,$*&%\"bG\"\"\"*&\"\"#F(%\"aGF(!\"\"F," }{TEXT -1 1 " " } {TEXT 281 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 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 7 "with " }{XPPEDIT 18 0 "2*c/(-b-sqrt(b^2-4*a* c));" "6#*(\"\"#\"\"\"%\"cGF%,&%\"bG!\"\"-%%sqrtG6#,&*$F(F$F%*(\"\"%F% %\"aGF%F&F%F)F)F)" }{TEXT -1 5 " or " }{XPPEDIT 18 0 "2*c/(-b+sqrt(b^ 2-4*a*c));" "6#*(\"\"#\"\"\"%\"cGF%,&%\"bG!\"\"-%%sqrtG6#,&*$F(F$F%*( \"\"%F%%\"aGF%F&F%F)F%F)" }{TEXT -1 2 ",\n" }}{PARA 0 "" 0 "" {TEXT -1 25 "depending on the sign of " }{TEXT 271 1 "b" }{TEXT -1 16 ". The procedure " }{TEXT 0 15 "solve_quadratic" }{TEXT -1 3 " - " } {HYPERLNK 17 "" 1 "" "" }{TEXT -1 1 " " }{HYPERLNK 17 "solve_quadratic " 1 "" "solve_quadratic" }{TEXT -1 49 " given in the preceding section implements this. " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 46 ": The quadratic may be given to the procedure " }{TEXT 0 15 "solve_qu adratic" }{TEXT -1 35 " as a polynomial or as an equation." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "so lve_quadratic(x^2-54*x+1=0,x);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&\"#F\"\"\"*&\"\"#F%-%%sqrtG6#\"$#=F%F%*&F%F%F#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+7v9)R&!\")$\"+at[_=!#6" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 252 "Instead \+ of resorting to clever analytical schemes to minimise rounding error, \+ there is a \"quick and dirty\" way of getting round such problems when working with Maple. (This may not be possible, or so easy to achieve, in other programming environments.)" }}{PARA 0 "" 0 "" {TEXT 259 100 "Increase the precision used for the computation, and then round the r esult to the required precision" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 7 "Example" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 22 "The calculation . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "solve(x ^2-54*x+1=0);\nevalf(%,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&\"#F \"\"\"*$-%%sqrtG6#\"$#=F%\"\"#,&F$F%F&!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+7v9)R&!\")$\"()[_=F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 " . . . involves loss of precision \+ due to subtraction roundoff error." }}{PARA 0 "" 0 "" {TEXT -1 60 "Thi s can be avoided by increasing precision as follows . . ." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "evalf (solve(x^2-54*x+1=0),15);\nevalf(%,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"0SYE^Z\")R&!#8$\"-g`t[_=F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $$\"+8v9)R&!\")$\"+at[_=!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "A disadavantage in this approach is that in lo ng and complex computations the time penalty involved in increasing pr ecision may become large." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 259 "" 0 "" {TEXT 306 70 "Subtraction round-off error in the evaluation of integra ls of the form" }{TEXT 261 1 " " }{XPPEDIT 18 0 "Int(x^n/(x+10),x = 0 \+ .. 1)" "6#-%$IntG6$*&)%\"xG%\"nG\"\"\",&F(F*\"#5F*!\"\"/F(;\"\"!F*" } {TEXT 261 2 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 117 "Acknowledgement : The idea for this exa mple comes from a Maple worksheet by Bruno Guerrieri, Florida A&M Univ ersity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "Consider the definite integral " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "Int(x^n/(x+10),x = 0 .. 1);" "6#-%$IntG6$*&)%\"xG%\" nG\"\"\",&F(F*\"#5F*!\"\"/F(;\"\"!F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{TEXT 263 1 "n" }{TEXT -1 24 " is a positive i nteger. " }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "0<=x" "6 #1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "G[n](x) = x^n/(x+10);" "6#/-&%\"GG6#%\"nG6#%\"xG*&)F*F( \"\"\",&F*F-\"#5F-!\"\"" }{TEXT -1 36 " is an increasing function for \+ each " }{TEXT 264 1 "n" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "G[n](x)>= 0" "6#1\"\"!-&%\"GG6#%\"nG6#%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "G[ n](0)=0" "6#/-&%\"GG6#%\"nG6#\"\"!F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "G[n](1) = 1/11;" "6#/-&%\"GG6#%\"nG6#\"\"\"*&F*F*\"#6!\"\"" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "For example, the graph of " }{XPPEDIT 18 0 "G[3](x) = x^3/(x+10);" "6#/-&%\"GG6#\"\"$6#%\"x G*&F*F(,&F*\"\"\"\"#5F-!\"\"" }{TEXT -1 15 " is as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "G \+ := (n,x) -> x^n/(x+10);\n'G(3,x)'=G(3,x);\nplot(G(3,x),x=0..1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GGf*6$%\"nG%\"xG6\"6$%)operatorG%& arrowGF)*&)9%9$\"\"\",&F/F1\"#5F1!\"\"F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"GG6$\"\"$%\"xG*&F(F',&F(\"\"\"\"#5F+!\"\"" }} {PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$ $\"\"!F)F(7$$\"3emmm;arz@!#>$\"3,\\&G]2lL.\"!#B7$$\"3[LL$e9ui2%F-$\"3Q s?esWkXnF07$$\"3nmmm\"z_\"4iF-$\"3-())Ho_y!zB!#A7$$\"3[mmmT&phN)F-$\"3 EcI')**3R'y&F;7$$\"3CLLe*=)H\\5!#=$\"3;@!)y'f3L9\"!#@7$$\"3gmm\"z/3uC \"FD$\"31'\\/\"4l3<>FG7$$\"3%)***\\7LRDX\"FD$\"3xW9@5F`o 7$$\"3)******\\nHi#HFD$\"3.$3r.^SWV#F`o7$$\"3jmm\"z*ev:JFD$\"3OT;Q.#fL $HF`o7$$\"3?LLL347TLFD$\"3>MvOQt84OF`o7$$\"3,LLLLY.KNFD$\"3=Lab@_)fD%F `o7$$\"3w***\\7o7Tv$FD$\"3KjV7%yq$*4&F`o7$$\"3'GLLLQ*o]RFD$\"3YB\"z2Kl =$fF`o7$$\"3A++D\"=lj;%FD$\"3j'*R:@s&H%pF`o7$$\"31++vV&RY2aFD$\"3#*>ete91+:F-7$$\"39mm;zXu9cFD$\"3)*G=0 !*p'fn\"F-7$$\"3l******\\y))GeFD$\"3iXMD72Mr=F-7$$\"3'*)***\\i_QQgFD$ \"37(*py.UMw?F-7$$\"3@***\\7y%3TiFD$\"35T')fqr;)G#F-7$$\"35****\\P![hY 'FD$\"3^)ecH^m$RDF-7$$\"3kKLL$Qx$omFD$\"3#H'R))Q@()zFF-7$$\"3!)*****\\ P+V)oFD$\"3aYZjw\"pD0$F-7$$\"3?mm\"zpe*zqFD$\"3^pZ2v,C9LF-7$$\"3%)**** *\\#\\'QH(FD$\"3\"p?I\\_\"e;OF-7$$\"3GKLe9S8&\\(FD$\"3y!>[^Rip\"RF-7$$ \"3R***\\i?=bq(FD$\"3M-xp+Y$yC%F-7$$\"3\"HLL$3s?6zFD$\"3wX*pX#[S)e%F-7 $$\"3a***\\7`Wl7)FD$\"3Y+b2y.Zj\\F-7$$\"3#pmmm'*RRL)FD$\"3j%p=,6 " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "For " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 9 " we have " }{XPPEDIT 18 0 "1/11 <= 1/(x+10);" "6#1*&\"\"\"F%\"#6 !\"\"*&F%F%,&%\"xGF%\"#5F%F'" }{XPPEDIT 18 0 "`` <= 1/10;" "6#1%!G*&\" \"\"F&\"#5!\"\"" }{TEXT -1 6 ", so " }{XPPEDIT 18 0 "x^n/11 <= x^n/(x +10);" "6#1*&)%\"xG%\"nG\"\"\"\"#6!\"\"*&)F&F'F(,&F&F(\"#5F(F*" } {XPPEDIT 18 0 "`` <= x^n/10;" "6#1%!G*&)%\"xG%\"nG\"\"\"\"#5!\"\"" } {TEXT -1 5 " and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " 1/11;" "6#*&\"\"\"F$\"#6!\"\"" }{XPPEDIT 18 0 "Int(x^n,x = 0 .. 1) <= \+ Int(x^n/(x+10),x = 0 .. 1);" "6#1-%$IntG6$)%\"xG%\"nG/F(;\"\"!\"\"\"-F %6$*&)F(F)F-,&F(F-\"#5F-!\"\"/F(;F,F-" }{XPPEDIT 18 0 "`` <= 1/10;" "6 #1%!G*&\"\"\"F&\"#5!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^n,x = \+ 0 .. 1)" "6#-%$IntG6$)%\"xG%\"nG/F';\"\"!\"\"\"" }{TEXT -1 1 "," }} {PARA 0 "" 0 "" {TEXT -1 10 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1/(11*(n+1)) <= Int(x^n/(x+10),x = 0 .. 1);" "6 #1*&\"\"\"F%*&\"#6F%,&%\"nGF%F%F%F%!\"\"-%$IntG6$*&)%\"xGF)F%,&F0F%\"# 5F%F*/F0;\"\"!F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= 1/(10*(n+1)); " "6#1%!G*&\"\"\"F&*&\"#5F&,&%\"nGF&F&F&F&!\"\"" }{TEXT -1 13 " ------ - (i)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Example 1 - " }{XPPEDIT 18 0 "n=3" "6#/% \"nG\"\"$" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " ;" }}}{PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "n=3" "6#/%\"nG \"\"$" }{TEXT -1 16 ", (i) becomes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/44 <= Int(x^3/(x+10),x = 0 .. 1);" "6#1*&\"\"\"F %\"#W!\"\"-%$IntG6$*&%\"xG\"\"$,&F,F%\"#5F%F'/F,;\"\"!F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= 1/40;" "6#1%!G*&\"\"\"F&\"#S!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Let's see what happens if we evaluate the integral with Maple." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Int(x^3/(x+10),x=0..1);\nvalue(%);\ncombine(%,ln);\nans1 := evalf( %);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"$,&F'\"\"\" \"#5F*!\"\"/F';\"\"!F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"%+5\" \"\"-%#lnG6#\"#6F&!\"\"*&F%F&-F(6#\"\"#F&F&*&F%F&-F(6#\"\"&F&F&#\"$'G \"\"$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"$'G\"\"$\"\"\"*&\"%+5F '-%#lnG6##\"#6\"#5F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ans1G$ \"(``J#!\")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 "There seems to be a bit of a problem here because only 7 significant digits are give n in the numerical value when we would expect to see 10 (assuming that Digits = 10). " }}{PARA 0 "" 0 "" {TEXT -1 30 "The problem arises bec ause of " }{TEXT 259 27 "subtraction round-off error" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 47 "The numerical value is obtained by subt racting " }{XPPEDIT 18 0 "1000*ln(11/10);" "6#*&\"%+5\"\"\"-%#lnG6#*& \"#6F%\"#5!\"\"F%" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "286/3;" "6#*& \"$'G\"\"\"\"\"$!\"\"" }{TEXT -1 43 ", and these expressions are \"nea rly equal\"." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "286/3;\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$'G\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+LLLL&*!\")" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "1000*ln(11/10);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$-% #lnG6##\"#6\"#5\"%+5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!)z,J&*!\" )" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "To \+ obtain a value for " }{XPPEDIT 18 0 "Int(x^3/(x+10),x = 0 .. 1);" "6#- %$IntG6$*&%\"xG\"\"$,&F'\"\"\"\"#5F*!\"\"/F';\"\"!F*" }{TEXT -1 119 " \+ which is accurate to 10 digits, we need to increase the number of digi ts used for the evaluation of these expressions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Int(x^3/(x+1 0),x=0..1);\nvalue(%);\ncombine(%,ln);\nevalf(%,15);\nans2 := evalf(%) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"$,&F'\"\"\"\" #5F*!\"\"/F';\"\"!F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"%+5\"\" \"-%#lnG6#\"#6F&!\"\"*&F%F&-F(6#\"\"#F&F&*&F%F&-F(6#\"\"&F&F&#\"$'G\" \"$F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"$'G\"\"$\"\"\"*&\"%+5F'- %#lnG6##\"#6\"#5F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"-%3!HN:B !#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ans2G$\"+,HN:B!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "This value agre es with the value given by numerical integration. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Int(x^3/(x+1 0),x=0..1);\nevalf(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$* &%\"xG\"\"$,&F'\"\"\"\"#5F*!\"\"/F';\"\"!F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+,HN:B!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "The absolute error in the first result is:\n" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "abserr := abs(ans1-ans2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"#**!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "The relative error i s:\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "relerr := abserr/ans 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"+vg!eF%!#<" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 176 "Ideally, we would like the res ult of a numerical computation to differ from the most accurate 10 dig it result, that is, a result which is correct to 10 digits, by no more than " }{TEXT 259 24 "1 unit in the last place" }{TEXT -1 11 " ( 1 ul p). " }}{PARA 0 "" 0 "" {TEXT -1 15 "For the number:" }}{PARA 256 "" 0 "" {TEXT -1 22 " 0.02315352901, " }}{PARA 0 "" 0 "" {TEXT -1 9 "1 ulp is " }}{PARA 256 "" 0 "" {TEXT -1 22 " 0.00000000001. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Given a number " }{TEXT 284 1 "x" }{TEXT -1 20 ", 1 ulp relative to " } {TEXT 285 1 "x" }{TEXT -1 20 " is the Maple value " }{TEXT 262 29 "Flo at(1, -Digits+1+ilog10(x))" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ulp_ans2 := Float( 1,-Digits+1+ilog10(ans2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)ulp_a ns2G$\"\"\"!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 140 "The absolute error in the first result, which exhibits a severe round-off error, is nearly 100 times greater than a unit in th e last place. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "abserr/ulp_ans2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++++**!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 141 "A similar comparison can be made between the relative error and the machine epsilon for Maple's 10 digit software floating \+ point arithmetic. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 42 "mach_eps := Float(5,-10);\nrelerr/mach_eps; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)mach_epsG$\"\"&!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]@h^&)!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 12 "Example 2 - " }{XPPEDIT 18 0 "n = 11;" "6#/%\"nG\" #6" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 3 "If " }{XPPEDIT 18 0 "n = 11;" "6#/%\"nG\"# 6" }{TEXT -1 16 ", (i) becomes " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/132 <= Int(x^11/(x+10),x = 0 .. 1);" "6#1*&\"\"\"F% \"$K\"!\"\"-%$IntG6$*&%\"xG\"#6,&F,F%\"#5F%F'/F,;\"\"!F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` <= 1/120;" "6#1%!G*&\"\"\"F&\"$?\"!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Let's see what happens if we evaluate the integral with M aple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "Int(x^11/(x+10),x=0..1);\nvalue(%);\ncombine(%,ln);\n evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"#6,&F' \"\"\"\"#5F*!\"\"/F';\"\"!F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%#l nG6#\"#6!-+++++5*&\"-+++++5\"\"\"-F%6#\"\"#F+F+*&F*F+-F%6#\"\"&F+F+#\" .X/Y&*\\g'\"$$pF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\".X/Y&*\\g'\" $$p\"\"\"*&\"-+++++5F'-%#lnG6##\"#6\"#5F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"\"!F$" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 13 "In this case " }{TEXT 259 27 "subtraction round-of f error" }{TEXT -1 44 " has led to a complete loss of significance." } }{PARA 0 "" 0 "" {TEXT -1 47 "The numerical value is obtained by subtr acting " }{XPPEDIT 18 0 "100000000000*ln(11/10)" "6#*&\"-+++++5\"\"\"- %#lnG6#*&\"#6F%\"#5!\"\"F%" }{TEXT -1 7 " from " }{XPPEDIT 18 0 "6604 995460445/693" "6#*&\".X/Y&*\\g'\"\"\"\"$$p!\"\"" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 107 "These expressions have numerical values \+ which agree to the last digit when evaluated correct to 10 digits. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "6604995460445/693;\nevalf(%,15);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\".X/Y&*\\g'\"$$p" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"07S/)z,J&*!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+!)z,J&*\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "100000000000*ln(11/10);\nevalf(%,15);\nevalf(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$-%#lnG6##\"#6\"#5\"-+++++5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"0\\K/)z,J&*!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #$\"+!)z,J&*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 260 "If we ev aluated just using 10 digit arithmetic we may possibly get some variat ion in the last digit of either (or both) of these values. We could no t regard the subsequent difference between the values as bearing any r elation to the true value of the integral." }}{PARA 0 "" 0 "" {TEXT -1 94 "In fact a change of just 1 in the last digit of either of these numbers amounts to a change of" }}{PARA 256 "" 0 "" {TEXT -1 14 " 0.0 000000001 " }{TEXT 266 2 ". " }{XPPEDIT 18 0 "10^10 = 1;" "6#/*$\"#5F% \"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 99 "This magnitude of change is considerably greater than the value of the integral, whi ch lies between" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1/132;" "6#*&\"\"\"F$\"$K\"!\"\"" } {TEXT -1 1 " " }{TEXT 267 1 "~" }{TEXT -1 20 " 0.007575757576 and " } {XPPEDIT 18 0 "1/120;" "6#*&\"\"\"F$\"$?\"!\"\"" }{TEXT -1 1 " " } {TEXT 268 1 "~" }{TEXT -1 17 " 0.008333333333. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "To obtain a numerical val ue for the integral " }{XPPEDIT 18 0 "Int(x^11/(x+10),x = 0 .. 1);" " 6#-%$IntG6$*&%\"xG\"#6,&F'\"\"\"\"#5F*!\"\"/F';\"\"!F*" }{TEXT -1 117 " which is accurate to 10 digits, we need to increase the number of di gits used for the calculation considerably. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Int(x^11/(x+ 10),x=0..1);\nvalue(%);\ncombine(%,ln);\nevalf(%,25);\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"#6,&F'\"\"\"\"#5F*! \"\"/F';\"\"!F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%#lnG6#\"#6!-+++ ++5*&\"-+++++5\"\"\"-F%6#\"\"#F+F+*&F*F+-F%6#\"\"&F+F+#\".X/Y&*\\g'\"$ $pF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\".X/Y&*\\g'\"$$p\"\"\"*&\" -+++++5F'-%#lnG6##\"#6\"#5F'!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \".G-sN%Hw!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+?dVHw!#7" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "This valu e agrees with the value given by numerical integration. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Int(x ^11/(x+10),x=0..1);\nevalf(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% $IntG6$*&%\"xG\"#6,&F'\"\"\"\"#5F*!\"\"/F';\"\"!F*" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+?dVHw!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 259 "" 0 "" {TEXT 313 13 "Evaluation of" } {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x^n/(x+10),x = 0 .. 1)" "6#-%$IntG6 $*&)%\"xG%\"nG\"\"\",&F(F*\"#5F*!\"\"/F(;\"\"!F*" }{TEXT -1 1 " " } {TEXT 312 32 "by upward and downward recursion" }{TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 9 " Define " }{XPPEDIT 18 0 "A[n];" "6#&%\"AG6#%\"nG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(x^n/(x+10),x = 0 .. 1);" "6#-%$IntG6$*&)%\"x G%\"nG\"\"\",&F(F*\"#5F*!\"\"/F(;\"\"!F*" }{TEXT -1 32 ", for each non -negative integer " }{TEXT 307 1 "n" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " Then " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "A[n]+10*A[n-1];" "6#,&&%\"AG6#%\" nG\"\"\"*&\"#5F(&F%6#,&F'F(F(!\"\"F(F(" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "Int(x^n/(x+10),x = 0 .. 1)+10*Int(x^(n-1)/(x+10),x = 0 .. 1);" " 6#,&-%$IntG6$*&)%\"xG%\"nG\"\"\",&F)F+\"#5F+!\"\"/F);\"\"!F+F+*&F-F+-F %6$*&)F),&F*F+F+F.F+,&F)F+F-F+F./F);F1F+F+F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 5 " = " }{XPPEDIT 18 0 "Int(``(x)*x^(n-1)/(x+10 ),x = 0 .. 1)+Int(10*x^(n-1)/(x+10),x = 0 .. 1);" "6#,&-%$IntG6$*(-%!G 6#%\"xG\"\"\")F+,&%\"nGF,F,!\"\"F,,&F+F,\"#5F,F0/F+;\"\"!F,F,-F%6$*(F2 F,)F+,&F/F,F,F0F,,&F+F,F2F,F0/F+;F5F,F," }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "Int((x+10)*x^(n-1)/(x+10),x = \+ 0 .. 1);" "6#-%$IntG6$*(,&%\"xG\"\"\"\"#5F)F))F(,&%\"nGF)F)!\"\"F),&F( F)F*F)F./F(;\"\"!F)" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " = " }{XPPEDIT 18 0 "Int(x^(n-1),x = 0 .. 1);" "6#-%$IntG6$)%\"xG,&%\"n G\"\"\"F*!\"\"/F';\"\"!F*" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "1/n;" "6 #*&\"\"\"F$%\"nG!\"\"" }{TEXT -1 1 "." }}{PARA 257 "" 0 "" {TEXT -1 12 "The formula " }{XPPEDIT 18 0 "A[n]-10*A[n-1] = 1/n;" "6#/,&&%\"AG6 #%\"nG\"\"\"*&\"#5F)&F&6#,&F(F)F)!\"\"F)F/*&F)F)F(F/" }{TEXT -1 29 " c an be arranged in the form " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "A[n] = 1/n-10*A[n-1];" "6#/&%\"AG6#%\"nG,&*&\"\"\"F*F'! \"\"F**&\"#5F*&F%6#,&F'F*F*F+F*F+" }{TEXT -1 15 " ------- (ii). " }} {PARA 257 "" 0 "" {TEXT -1 43 "We can now attempt to compute the seque nce " }{XPPEDIT 18 0 "A[0],A[1],A[2],` . . . `,A[n],` . . . `;" "6(&% \"AG6#\"\"!&F$6#\"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#%\"nGF-" }{TEXT -1 36 " recursively starting with the value" }}{PARA 0 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "A[0] = Int(1/(x+10),x = 0 .. 1);" "6#/&%\"AG6#\"\"!- %$IntG6$*&\"\"\"F,,&%\"xGF,\"#5F,!\"\"/F.;F'F," }{TEXT -1 3 " = " } {XPPEDIT 18 0 "ln(11/10);" "6#-%#lnG6#*&\"#6\"\"\"\"#5!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 " Then " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "A[1] = 1/1-10*A[0];" "6#/&%\"AG6#\"\" \",&*&F'F'F'!\"\"F'*&\"#5F'&F%6#\"\"!F'F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1-10*ln(11/10);" "6#/%!G,&\"\"\"F&*&\"#5F&-%#lnG6#*&\"#6F&F (!\"\"F&F." }{TEXT -1 5 ",\n " }{XPPEDIT 18 0 "A[2] = 1/2-10*A[1];" "6#/&%\"AG6#\"\"#,&*&\"\"\"F*F'!\"\"F**&\"#5F*&F%6#F*F*F+" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "1/2-10*(1-10*ln(11/10)) = -19/2+100*ln(11/10); " "6#/,&*&\"\"\"F&\"\"#!\"\"F&*&\"#5F&,&F&F&*&F*F&-%#lnG6#*&\"#6F&F*F( F&F(F&F(,&*&\"#>F&F'F(F(*&\"$+\"F&-F.6#*&F1F&F*F(F&F&" }{TEXT -1 7 ", \+ etc. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 " The following loop can be used to obtain an analytical expression for \+ " }{XPPEDIT 18 0 "A[n] = Int(x^n/(x+10),x = 0 .. 1);" "6#/&%\"AG6#%\"n G-%$IntG6$*&)%\"xGF'\"\"\",&F-F.\"#5F.!\"\"/F-;\"\"!F." }{TEXT -1 25 " using exact arithmetic. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "A := 'A':\nA[0] := ln(11/10);\nn := 11;\nfor k from 1 to n do\n \+ A[k] := 1/k - 10*A[k-1];\nend do:\nA[n]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"AG6#\"\"!-%#lnG6##\"#6\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\".X/Y &*\\g'\"$$p\"\"\"*&\"-+++++5F'-%#lnG6##\"#6\"#5F'!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "This is the same exp ression as that given by Maple's integration procedure." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "n := 'n ':\nn := 11;\nInt(x^n/(x+10),x=0..1);\nvalue(%);\ncombine(%,ln);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"#6,&F'\"\"\"\"#5F*!\"\"/F';\"\"!F*" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,*-%#lnG6#\"#6!-+++++5*&\"-+++++5\"\" \"-F%6#\"\"#F+F+*&F*F+-F%6#\"\"&F+F+#\".X/Y&*\\g'\"$$pF+" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&#\".X/Y&*\\g'\"$$p\"\"\"*&\"-+++++5F'-%#lnG6## \"#6\"#5F'!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 170 "When the calculations are performed using floating point arithmetic, we need t o increase the number of digits used substantially in order to obtain \+ a numerical value for " }{XPPEDIT 18 0 "A[11]=Int(x^11/(x+9),x = 0 .. \+ 1)" "6#/&%\"AG6#\"#6-%$IntG6$*&%\"xGF',&F,\"\"\"\"\"*F.!\"\"/F,;\"\"!F ." }{TEXT -1 32 " which is correct to 10 digits. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "A := 'A': N := 'n':\nDigits := 25:\nn := 11; \nA[0] := evalf(ln(11/10)):\nprint(`A`||0 = evalf(A[0],10));\nfor k fr om 1 to n do\n A[k] := 1/k - 10*A[k-1];\n print(`A`||k = evalf(A[k ],10));\nend do:\nDigits := 10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"nG\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#A0G$\"+!)z,J&*!#6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%#A1G$\"+'>?)*o%!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#A2G$\"+V!)z,J!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#A3G$\"+,HN:B!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#A4G$\"+#*4 ZY=!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#A5G$\"+&3!HN:!#6" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%#A6G$\"+>ew88!#6" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%#A7G$\"+#4c![6!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%#A8G$\"+x!R%>5!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#A9G$\"+[M? n\"*!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A10G$\"+>b'zK)!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A11G$\"+?dVHw!#7" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 90 "If this calculation is \+ peformed using 10 digit arithmetic throughout, the final value for " } {XPPEDIT 18 0 "A[11]" "6#&%\"AG6#\"#6" }{TEXT -1 21 " is completely wr ong." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "A := 'A': N := 'n':\nDigits := 10:\nn := 11;\nA[0] : = evalf(ln(11/10)):\nprint(`A`||0 = evalf(A[0],10));\nfor k from 1 to \+ n do\n A[k] := 1/k - 10*A[k-1];\n print(`A`||k = evalf(A[k],10)); \nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#6" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%#A0G$\"+!)z,J&*!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#A1G$\"*??)*o%!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%#A2G$\"*+)z,J!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#A3G$\"*L``J#! #5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#A4G$\"*qmk%=!#5" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/%#A5G$\"*+L``\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#A6G$\"*nOLJ\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%#A7G$\"*fZB:\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#A8G$\")5Cl(* !#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#A9G$\"*6qeM\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A10G$!*5,(eM!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A11G$\"+4?znV!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "One way of explaining what has happened is as follows." } }{PARA 0 "" 0 "" {TEXT -1 27 "Suppose that the value for " }{XPPEDIT 18 0 "A[0] = ln(11/10);" "6#/&%\"AG6#\"\"!-%#lnG6#*&\"#6\"\"\"\"#5!\" \"" }{TEXT -1 44 " is out by 1 in the last digit. Then, since " } {XPPEDIT 18 0 "ln(11/10)" "6#-%#lnG6#*&\"#6\"\"\"\"#5!\"\"" }{TEXT -1 1 " " }{TEXT 309 1 "~" }{TEXT -1 53 " 0.09531017980, this amounts to a n absolute error of " }{XPPEDIT 18 0 "10^(-11)" "6#)\"#5,$\"#6!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 105 "Now, since the recursi on formula (ii) involves a multiplication by 10 in each application, t his error is " }{TEXT 259 9 "amplified" }{TEXT -1 194 " ten-fold with \+ each iteration, so that after 11 iterations the error will be about 1. This error is considerably larger in magnitude than the value we are \+ trying to compute, which lies between " }{XPPEDIT 18 0 "1/132;" "6#*& \"\"\"F$\"$K\"!\"\"" }{TEXT -1 1 " " }{TEXT 310 1 "~" }{TEXT -1 20 " 0 .007575757576 and " }{XPPEDIT 18 0 "1/120;" "6#*&\"\"\"F$\"$?\"!\"\"" }{TEXT -1 1 " " }{TEXT 311 1 "~" }{TEXT -1 16 " 0.008333333333." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 174 "An inter esting variation of the previous calculation is to use the recursion f ormula in the downward (or backward) direction, starting with an essen tially arbitrary value of " }{XPPEDIT 18 0 "A[m];" "6#&%\"AG6#%\"mG" } {TEXT -1 16 " for a value of " }{XPPEDIT 18 0 "n=m" "6#/%\"nG%\"mG" } {TEXT -1 61 " somewhat greater than the the index of the desired value of " }{XPPEDIT 18 0 "A[n]" "6#&%\"AG6#%\"nG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 74 "For downward iteration the recursion form ula (ii) is arranged in the form " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[n-1] = 1/10;" "6#/&%\"AG6#,&%\"nG\"\"\"F)!\"\"*&F)F) \"#5F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(1/n-A[n]);" "6#-%!G6#,&*&\" \"\"F(%\"nG!\"\"F(&%\"AG6#F)F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 221 "The idea now is that, since each application of this rec ursion formula involves division by 10, the error is reduced by a fact or of 10 with each iteration. This way even a very crude, almost arbit rary, starting value for " }{XPPEDIT 18 0 "A[m]" "6#&%\"AG6#%\"mG" } {TEXT -1 22 " leads to a value for " }{XPPEDIT 18 0 "A[n]" "6#&%\"AG6# %\"nG" }{TEXT -1 96 " which is correct to 10 digits after around 10 it erations or so, even using 10 digit arithmetic." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Try different values of \+ " }{TEXT 308 1 "m" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "A[m]" "6#&%\"AG 6#%\"mG" }{TEXT -1 18 " in the following." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 269 "A := 'A': N := 'n': \nDigits := 10:\nn := 11;\nm := 28; # somewhat larger than n\nA[m] := \+ 0: # arbitrary starting value\nprint(`A`||m = evalf(A[m]));\nfor k fro m m to n+1 by -1 do # iterate downwards\n A[k-1] := (1/k-A[k])/10;\n print(`A`||(k-1) = evalf(A[k-1],10));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"# G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A28G$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A27G$\"+r&G9d$!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A26G$\"+Z3cYL!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A25G$ \"+hx\\6N!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A24G$\"+C-&)[O!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A23G$\"+W;y,Q!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A22G$\"+BzknR!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A21G$\"+`(*o[T!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A20G$ \"+(yNqM%!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A19G$\"+@kHlX!#7" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A18G$\"+`#Gm![!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A17G$\"+IF*[2&!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A16G$\"+oO'[P&!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A15G$ \"+LO^7d!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A14G$\"+.`T&4'!#7" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A13G$\"+8cJLl!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A12G$\"+Jh(*Qq!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$A11G$\"+?dVHw!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 259 "" 0 "" {TEXT 316 47 "Details concerning the analytical evaluation of" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Int(x^n/(x+10),x = 0 .. 1)" "6#-%$IntG6$*&)% \"xG%\"nG\"\"\",&F(F*\"#5F*!\"\"/F(;\"\"!F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 30 "The standard way to determine " }{XPPEDIT 18 0 "Int(x^11/(x+10),x)" "6#-%$IntG6$*&%\"xG\"#6,&F'\"\"\"\"#5F*!\"\"F'" } {TEXT -1 26 " analytically is to divide" }{XPPEDIT 18 0 "``(x+10)" "6# -%!G6#,&%\"xG\"\"\"\"#5F(" }{TEXT -1 6 " into " }{XPPEDIT 18 0 "x^11" "6#*$%\"xG\"#6" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 61 "We can \+ perform the division using Maple to obtain a quotient " }{TEXT 314 2 " qx" }{TEXT -1 15 " and remainder " }{TEXT 315 2 "rx" }{TEXT -1 13 " as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "qx := quo(x^11,x+10,x,'rx');\nrx;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#qxG,8*$)%\"xG\"#5\"\"\"F**&F)F*)F(\"\"*F*!\"\"*& \"$+\"F*)F(\"\")F*F**&\"%+5F*)F(\"\"(F*F.*&\"&++\"F*)F(\"\"'F*F**&\"'+ +5F*)F(\"\"&F*F.*&\"(+++\"F*)F(\"\"%F*F**&\")+++5F*)F(\"\"$F*F.*&\"*++ ++\"F*)F(\"\"#F*F**&\"+++++5F*F(F*F.\",+++++\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#!-+++++5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 5 "Then " }{XPPEDIT 18 0 "x^11/(x+10)" "6#*&%\"xG\"#6, &F$\"\"\"\"#5F'!\"\"" }{TEXT -1 10 " is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "qx+rx/(x+10) ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,:*$)%\"xG\"#5\"\"\"F(*&F'F()F&\" \"*F(!\"\"*&\"$+\"F()F&\"\")F(F(*&\"%+5F()F&\"\"(F(F,*&\"&++\"F()F&\" \"'F(F(*&\"'++5F()F&\"\"&F(F,*&\"(+++\"F()F&\"\"%F(F(*&\")+++5F()F&\" \"$F(F,*&\"*++++\"F()F&\"\"#F(F(*&\"+++++5F(F&F(F,\",+++++\"F(*&\"-+++ ++5F(,&F&F(F'F(F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 56 "We can obtain the same result using the Maple procedure " }{TEXT 260 21 "convert(..,parfrac,x)" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "gx := convert(x^11/(x+10),parfrac,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% #gxG,:*$)%\"xG\"#5\"\"\"F**&F)F*)F(\"\"*F*!\"\"*&\"$+\"F*)F(\"\")F*F** &\"%+5F*)F(\"\"(F*F.*&\"&++\"F*)F(\"\"'F*F**&\"'++5F*)F(\"\"&F*F.*&\"( +++\"F*)F(\"\"%F*F**&\")+++5F*)F(\"\"$F*F.*&\"*++++\"F*)F(\"\"#F*F**& \"+++++5F*F(F*F.\",+++++\"F**&\"-+++++5F*,&F(F*F)F*F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "It would be easy to integrate this expr ession by hand. " }}{PARA 0 "" 0 "" {TEXT -1 56 "In particular, an ind efinite integral of the last term " }{XPPEDIT 18 0 "-100000000000/(x+ 10)" "6#,$*&\"-+++++5\"\"\",&%\"xGF&\"#5F&!\"\"F*" }{TEXT -1 5 " is \+ " }{XPPEDIT 18 0 "-100000000000*ln(x+10)" "6#,$*&\"-+++++5\"\"\"-%#lnG 6#,&%\"xGF&\"#5F&F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "Integrating the whole expression gives the same analytical formula for " }{XPPEDIT 18 0 "Int(x^11/(x+10),x=0..1)" "6#-%$IntG6$*&%\"xG\"# 6,&F'\"\"\"\"#5F*!\"\"/F';\"\"!F*" }{TEXT -1 41 " as was obtained by t he previous methods." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "int(gx,x=0..1);\ncombine(%,ln);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*#\".X/Y&*\\g'\"$$p\"\"\"*&\"-+++++5F'-%#ln G6#\"#6F'!\"\"*&\"-+++++5F'-F+6#\"\"#F'F'*&F0F'-F+6#\"\"&F'F'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\".X/Y&*\\g'\"$$p\"\"\"*&\"-+++++5F '-%#lnG6##\"#6\"#5F'!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 345 68 "Subtracti on round-off error in the evaluation of specific functions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 259 "" 0 "" {TEXT 445 50 "Subtraction round-off error in the evaluation of " } {XPPEDIT 18 0 "f(x) = ln(1-x)/x" "6#/-%\"fG6#%\"xG*&-%#lnG6#,&\"\"\"F- F'!\"\"F-F'F." }{TEXT 412 7 " when " }{TEXT 410 1 "x" }{TEXT 411 10 " is near 0" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 79 "Acknowledgement: The idea for thi s example comes from some notes by W. Kahan. " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 47 ": A knowledge of Maclaurin series is \+ required. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "This example is concerned with the evaluation of the function \+ " }{XPPEDIT 18 0 "f(x)=ln(1-x)/x" "6#/-%\"fG6#%\"xG*&-%#lnG6#,&\"\"\"F -F'!\"\"F-F'F." }{TEXT -1 6 " when " }{TEXT 402 1 "x" }{TEXT -1 12 " i s near 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 207 "f := x -> ln(1-x)/x:\n'f(x)'=f(x);\np1 := plot(f(x ),x=-1.2..1.2,y=-5..1):\np2 := plots[implicitplot](x=1,x=-1.2..1.2,y=- 5..1,color=black,linestyle=2):\nplots[display]([p1,p2],font=[HELVETICA ,9],tickmarks=[4,3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"x G*&-%#lnG6#,&\"\"\"F-F'!\"\"F-F'F." }}{PARA 13 "" 1 "" {GLPLOT2D 452 312 312 {PLOTDATA 2 "6(-%'CURVESG6$7]o7$$!3%**************>\"!#<$!3#ee NI+y/d'!#=7$$!3/+++IooZ6F*$!3C%o<*3$y-m'F-7$$!32+++?%p@5\"F*$!3t`sDyp( 4u'F-7$$!3/+++L.)40\"F*$!3k%p[G]TZ$oF-7$$!3g******4$>X***F-$!3\\FV;!fI D$pF-7$$!3!******\\M%o\"[*F-$!3od=\")o9XLqF-7$$!3R+++&o?i+*F-$!3tFE@C' =/8(F-7$$!3\\*****\\g0R^)F-$!3NBK/+D[MsF-7$$!3]*****\\k_Z+)F-$!3waWWZ! piM(F-7$$!3!******\\aKs\\(F-$!3YC04Q')=iuF-7$$!3[******Rd=vpF-$!3[\"\\ BI7Hke(F-7$$!3;+++ImO:lF-$!3XIse7?N+xF-7$$!3c******>#>x*fF-$!3\\Sesvd* R$yF-7$$!3]*******HYzZ&F-$!3[X/w,*eV(zF-7$$!3=+++!y[q(\\F-$!3EFdh/7$f6 )F-7$$!3@+++Xe=AXF-$!3%*3K\\gyE]#)F-7$$!3A+++?)48)RF-$!3@SHhn0u<%)F-7$ $!37+++!)o6BNF-$!3!zu)>X!=nc)F-7$$!3!)*****\\cH,*HF-$!3/E^e]B%*[()F-7$ $!3\"*******zaM=DF-$!3'y!p?Vb&*=*)F-7$$!3$******\\cB2+#F-$!3]A,*zm%z: \"*F-7$$!34++++^#y]\"F-$!3H0of#\\?UJ*F-7$$!3R++++DPN**!#>$!30ikf3\"F*7$$\"35+++S3L*)>F-$!3!) yetT?+:6F*7$$\"3%*******HY7#\\#F-$!3%e%ys\\M:]6F*7$$\"3'*******zMgyHF- $!33iQi#f5s=\"F*7$$\"3?+++!Hb(=NF-$!3;\"f5\\+gCB\"F*7$$\"3?+++?d5/SF-$ !3'H)y9MTYx7F*7$$\"3w********3KAXF-$!35^1HF<'4L\"F*7$$\"3w******z3!>* \\F-$!3%zjx$y7I&Q\"F*7$$\"3g******>eF0bF-$!3uId00Arz#F*7$$\"3?+++?$zJO*F-$!3Iw/FI3:THF*7$$ \"3!)******R_9z%*F-$!3c=n9M;BSs`R@&QKF* 7$$\"3\\******p_07'*F-$!3e`\"H#Rri!Q$F*7$$\"3u*******G5&y'*F-$!31))pnH Ob^NF*7$$\"3d*****\\Il\\u*F-$!3y%*p9G.'\\w$F*7$$\"3S+++5G>y(*F-$!3?dk$ z4C\\*QF*7$$\"3`+++?.U6)*F-$!3/mmp'fSr/%F*7$$\"3b******HykW)*F-$!3aO![ CTl.B%F*7$$\"3o******R`(y()*F-$!3!eHglui(fWF*7$$\"3u*****\\4*[%*)*F-$! 3!*Q21F<1+YF*7$$\"3!)******\\G56**F-$!3)GtZk\\'>Rc'\\F*7$$\"3^*****\\NIV%**F-$!3,/!o[qk*>_F*7$$\"3e****** 4T%4'**F-$!3[N&p!Q%*3nbF*7$$\"3M+++gybx**F-$!3+=!oy2>J6'F*7$$\"3S+++:; <%***F-$!3'>G=^qV>X(F*7$%*undefinedGFf_l-%'COLOURG6&%$RGBG$\"#5!\"\"$ \"\"!F_`lF^`l-F$6V7$7$$\"\"\"F_`l$!\"&F_`l7$$\"2))***************F*$!3 a************z\\F*7$7$Fi`l$!3!)************fZF*Fh`l7$7$Fd`lF_al7$Fi`l$ !3A++++++SZF*7$7$Fi`l$!3d************>XF*Fcal7$7$Fd`lFhal7$Fi`l$!37*** **********\\%F*7$7$Fi`l$!3O************zUF*F\\bl7$7$Fd`lFabl7$Fi`l$!3! *)***********fUF*7$7$Fi`l$!3:************RSF*Febl7$7$Fd`lFjbl7$Fi`l$!3 p)***********>SF*7$7$Fi`l$!3[)************z$F*F^cl7$7$Fd`l$!3#*)****** ******z$F*7$Fi`l$!3[)***********zPF*7$7$Fi`l$!3G)***********fNF*Ficl7$ 7$Fd`l$!3s)***********fNF*7$Fi`l$!3E)***********RNF*7$7$Fi`l$!31)***** ******>LF*Fddl7$7$Fd`l$!3^)***********>LF*7$Fi`l$!3/)************H$F*7 $7$Fi`l$!3%y***********zIF*F_el7$7$Fd`l$!3I)***********zIF*7$Fi`l$!3%y ***********fIF*7$7$Fi`l$!3j(***********RGF*Fjel7$7$Fd`l$!33)********** *RGF*7$Fi`l$!3i(***********>GF*7$7$Fi`l$!3U(************f#F*Fefl7$7$Fd `l$!3'y************f#F*7$Fi`l$!3S(***********zDF*7$7$Fi`l$!3?(******** ***fBF*F`gl7$7$Fd`l$!3m(***********fBF*7$Fi`l$!3>(***********RBF*7$7$F i`l$!3W(***********>@F*F[hl7$7$Fd`lF`hl7$Fi`l$!3U(************4#F*7$7$ Fi`l$!3W(***********z=F*Fdhl7$7$Fd`lFihl7$Fi`l$!3W(***********f=F*7$7$ Fi`l$!3Y(***********R;F*F]il7$7$Fd`lFbil7$Fi`l$!3W(***********>;F*7$7$ Fi`l$!3Y(************R\"F*Ffil7$7$Fd`lF[jl7$Fi`l$!3W(***********z8F*7$ 7$Fi`l$!3[(***********f6F*F_jl7$7$Fd`lFdjl7$Fi`l$!3Y(***********R6F*7$ 7$Fi`l$!3'[(***********>*F-Fhjl7$7$Fd`lF][m7$Fi`l$!3ou*************)F- 7$7$Fi`l$!3%Q(***********z'F-Fa[m7$7$Fd`l$!3%\\(***********z'F-7$Fi`l$ !3yu***********f'F-7$7$Fi`l$!3/v***********R%F-F\\\\m7$7$Fd`lFa\\m7$Fi `l$!3'[(***********>%F-7$7$Fi`l$!37v************>F-Fe\\m7$7$Fd`lFj\\m7 $Fi`l$!3%\\(***********z\"F-7$7$Fi`l$\"3!yC++++++%FdrF^]m7$7$Fd`lFc]m7 $Fi`l$\"3]Z-+++++gFdr7$7$Fi`l$\"3oC++++++GF-Fg]m7$7$Fd`lF\\^m7$Fi`l$\" 3'[-++++++$F-7$7$Fi`l$\"3gC++++++_F-F`^m7$7$Fd`lFe^m7$Fi`l$\"3wC++++++ aF-7$7$Fi`l$\"3]C++++++wF-Fi^m7$7$Fd`lF^_m7$Fi`l$\"3oC++++++yF-7$7$Fi` l$\"3W-++++++5F*Fb_m-Fh_l6&Fj_lF_`lF_`lF_`l-%*LINESTYLEG6#\"\"#-%+AXES LABELSG6%Q\"x6\"Q\"yFc`m-%%FONTG6#%(DEFAULTG-Ff`m6$%*HELVETICAG\"\"*-% *AXESTICKSG6$\"\"%\"\"$-%%VIEWG6$;$!#7F]`l$\"#7F]`l;Ff`lFd`l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }} }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 42 ": The re should be a \"missing point\" where " }{XPPEDIT 18 0 "x=0" "6#/%\"x G\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = -1;" "6#/%\"yG,$\"\"\" !\"\"" }{TEXT -1 13 ", but, since " }{XPPEDIT 18 0 "Limit(f(x),x=0)=-1 " "6#/-%&LimitG6$-%\"fG6#%\"xG/F*\"\"!,$\"\"\"!\"\"" }{TEXT -1 120 " ( see below), we can fill it in and obtain a continuous function, indeed one which has derivatives of all orders at 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "The Maclaurin series of \+ " }{XPPEDIT 18 0 "f(x)=ln(1-x)/x" "6#/-%\"fG6#%\"xG*&-%#lnG6#,&\"\"\"F -F'!\"\"F-F'F." }{TEXT -1 4 " is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "-1-1/2" "6#,&\"\"\"!\"\"*&F$F$\"\"#F%F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x-1/3" "6#,&%\"xG\"\"\"*&F%F%\"\"$!\"\"F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^2-1/4" "6#,&*$%\"xG\"\"#\"\"\"*&F'F'\"\"%! \"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^3-1/5" "6#,&*$%\"xG\"\"$\"\" \"*&F'F'\"\"&!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^4-` . . . . `- 1/(n+1);" "6#,(*$%\"xG\"\"%\"\"\"%*~.~.~.~.~G!\"\"*&F'F',&%\"nGF'F'F'F )F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^n-` . . . `" "6#,&)%\"xG%\"nG\" \"\"%(~.~.~.~G!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "taylor(f(x),x=0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+/%\"xG!\"\"\"\"!#F%\"\"#\"\"\"#F%\"\"$F(#F% \"\"%F+#F%\"\"&F--%\"OG6#F)F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 " It follo ws that " }{XPPEDIT 18 0 "Limit(f(x),x=0)=-1" "6#/-%&LimitG6$-%\"fG6#% \"xG/F*\"\"!,$\"\"\"!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Limit(f(x),x=0);\n ``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&-%#lnG6#, &\"\"\"F+%\"xG!\"\"F+F,F-/F,\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %!G!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Using the expressi on " }{TEXT 260 17 "'evalf[10](f(x))'" }{TEXT -1 52 " in the following plot command will prevent Maple's " }{TEXT 0 4 "plot" }{TEXT -1 27 " \+ procedure from evaluating " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 43 " using hardware floating point arithmetic. " }{TEXT 259 4 "Note" }{TEXT -1 37 ": It is essential to use the quotes. " }}{PARA 0 "" 0 "" {TEXT -1 22 "Plotting the graph of " }{XPPEDIT 18 0 "f(x)" " 6#-%\"fG6#%\"xG" }{TEXT -1 186 " in a narrow interval containing 0 sho ws up some strange behaviour involving oscillations in value. This giv es an indication that there are some accuracy problems involved in eva luating " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 6 " when \+ " }{TEXT 448 1 "x" }{TEXT -1 11 " is near 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 141 "f := x -> ln(1-x) /x:\n'f(x)'=f(x);\nplot('evalf[10](f(x))',x=-4e-8..1e-8,-2.3..0.1,\n \+ color=red,numpoints=200,font=[HELVETICA,9],axes=framed);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&-%#lnG6#,&\"\"\"F-F'!\"\"F-F'F. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "x[0] = -7" "6#/&%\"xG6#\"\"!,$\"\"(!\"\"" }{TEXT -1 1 " " }{TEXT 403 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10)" "6 #)\"#5,$F$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 16 "We wou ld expect " }{XPPEDIT 18 0 "f(x[0])" "6#-%\"fG6#&%\"xG6#\"\"!" }{TEXT -1 19 " to be close to -1." }}{PARA 0 "" 0 "" {TEXT -1 5 "When " } {XPPEDIT 18 0 "x=x[0]" "6#/%\"xG&F$6#\"\"!" }{TEXT -1 47 " the first 4 terms of the Maclaurin series for " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6# %\"xG" }{TEXT -1 7 " give: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "f(x[0])" "6#-%\"fG6#&%\"xG6#\"\"!" }{TEXT -1 1 " " } {TEXT 404 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-1+3.5;" "6#,&\"\"\"! \"\"-%&FloatG6$\"#NF%F$" }{TEXT -1 1 " " }{TEXT 405 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10)-49/3;" "6#,&)\"#5,$F%!\"\"\"\"\"*&\"#\\F( \"\"$F'F'" }{TEXT -1 1 " " }{TEXT 406 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-20)+343/4;" "6#,&)\"#5,$\"#?!\"\"\"\"\"*&\"$V$F)\"\"%F(F)" }{TEXT -1 1 " " }{TEXT 407 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(- 30)" "6#)\"#5,$\"#I!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 408 1 "~" }{TEXT -1 98 " -1 + 0.00000000035 - 0.00000000000000000016333333333333333 + 0.00000 000000000000000000000008575. " }}{PARA 256 "" 0 "" {TEXT -1 45 " = -0 .99999999965000000016333333324758333. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 131 "taylor(f(x),x=0,5):\np := unapply(convert(%,polynom),x):\n'p(x)'=p(x);\nx0 := -7*10^(-10);\n Eval(p(x),x=x0);\n``=value(%);\nevalf[35](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,,\"\"\"!\"\"*&#F)\"\"#F)F'F)F**&#F)\"\"$ F)*$)F'F-F)F)F**&#F)\"\"%F)*$)F'F0F)F)F**&#F)\"\"&F)*$)F'F5F)F)F*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G#!\"(\",+++++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$,,\"\"\"!\"\"*&#F'\"\"#F'%\"xGF'F(*&#F' \"\"$F'*$)F,F+F'F'F(*&#F'\"\"%F'*$)F,F/F'F'F(*&#F'\"\"&F'*$)F,F4F'F'F( /F,#!\"(\",+++++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#!K.s++v8()* ****\\C+++v%*******\\\"\"K++++++++++++++++++++:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$!DL$eZKLLL;+++]'*********!#N" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "This last value agrees to 35 digits with the value of " }{XPPEDIT 18 0 "f(x)=ln(1-x)/x" "6#/-% \"fG6#%\"xG*&-%#lnG6#,&\"\"\"F-F'!\"\"F-F'F." }{TEXT -1 104 ", when ca lculated using Maple's software floating point arithmetic with 35 deci mal digits of precision. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "f := x -> ln(1-x)/x:\nx0 := -7*10^( -10):\nEval(f(x),x=x0);\n``=value(%);\nevalf[35](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$*&-%#lnG6#,&\"\"\"F+%\"xG!\"\"F+F,F-/F,#! \"(\",+++++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\",+++++\"\" \"(\"\"\"-%#lnG6##\",2++++\"\",+++++\"F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$!DN$eZKLLL;+++]'*********!#N" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "If we try evaluating " } {XPPEDIT 18 0 "f(x)=ln(1-x)/x" "6#/-%\"fG6#%\"xG*&-%#lnG6#,&\"\"\"F-F' !\"\"F-F'F." }{TEXT -1 4 " at " }{XPPEDIT 18 0 "x=x[0]" "6#/%\"xG&F$6# \"\"!" }{XPPEDIT 18 0 "`` = -7;" "6#/%!G,$\"\"(!\"\"" }{TEXT -1 1 " " }{TEXT 409 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10)" "6#)\"#5,$F $!\"\"" }{TEXT -1 111 " using Maple's software floating point arithmet ic with 10 decimal digits of precision, there is a large error. " }} {PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 16 ": The procedure " } {TEXT 0 6 "forget" }{TEXT -1 74 " is used to remove the value obtained by the previous evaluation from the " }{TEXT 259 14 "remember table" }{TEXT -1 4 " of " }{TEXT 0 5 "evalf" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "forget(ev alf);\nf := x -> ln(1-x)/x:\nx0 := -7*10^(-10):\nEval(f(x),x=x0);\n``= value(%);\nevalf[10](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$ *&-%#lnG6#,&\"\"\"F+%\"xG!\"\"F+F,F-/F,#!\"(\",+++++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*&#\",+++++\"\"\"(\"\"\"-%#lnG6##\",2++++\" \",+++++\"F*!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$!+G9dG9!\"* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "This value clearly exhibi ts a large error." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "When calculating with a precision of 10 decimal digits gi ven " }{XPPEDIT 18 0 "x = -7;" "6#/%\"xG,$\"\"(!\"\"" }{TEXT -1 1 " " }{TEXT 413 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10)" "6#)\"#5,$F $!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "1-x" "6#,&\"\"\"F$%\"xG!\"\" " }{TEXT -1 26 " evaluates to 1.000000001." }}{PARA 0 "" 0 "" {TEXT -1 17 "Now suppose that " }{XPPEDIT 18 0 "ln*u" "6#*&%#lnG\"\"\"%\"uGF %" }{TEXT -1 42 " is evaluated by the Maclaurin series for " } {XPPEDIT 18 0 "ln(1+v);" "6#-%#lnG6#,&\"\"\"F'%\"vGF'" }{TEXT -1 8 ", \+ where " }{XPPEDIT 18 0 "v = u-1;" "6#/%\"vG,&%\"uG\"\"\"F'!\"\"" } {TEXT -1 11 ", namely: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "ln(1+v) = v-v^2/2+v^3/3-v^4/4+v^5/5-v^6/6+` . . . `;" " 6#/-%#lnG6#,&\"\"\"F(%\"vGF(,0F)F(*&F)\"\"#F,!\"\"F-*&F)\"\"$F/F-F(*&F )\"\"%F1F-F-*&F)\"\"&F3F-F(*&F)\"\"'F5F-F-%(~.~.~.~GF(" }{TEXT -1 1 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ln*u = ``(u-1)-(u-1)^2/2+(u-1)^3/3-(u-1)^4/4+ (u-1)^5/5-(u-1)^6/6+` . . . `;" "6#/*&%#lnG\"\"\"%\"uGF&,0-%!G6#,&F'F& F&!\"\"F&*&,&F'F&F&F-\"\"#F0F-F-*&,&F'F&F&F-\"\"$F3F-F&*&,&F'F&F&F-\" \"%F6F-F-*&,&F'F&F&F-\"\"&F9F-F&*&,&F'F&F&F-\"\"'F " 0 "" {MPLTEXT 1 0 71 "taylor(ln(1+v),v=0,7) :\nq := unapply(convert(%,polynom),v):\n'q(v)'=q(v);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"vG,.F'\"\"\"*&#F)\"\"#F)*$)F'F,F)F)!\"\" *&#F)\"\"$F)*$)F'F2F)F)F)*&#F)\"\"%F)*$)F'F7F)F)F/*&#F)\"\"&F)*$)F'F " 0 "" {MPLTEXT 1 0 129 "Digits := 10:\nf := x -> ln (1-x)/x:\nuu := 1.000000001;\nvv := uu-1:\nEval(v-v^2/2,v=vv);\n``=val ue(%);\nEval(ln(u),u=uu);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#uuG$\"+,+++5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6 $,&%\"vG\"\"\"*&#F(\"\"#F(*$)F'F+F(F(!\"\"/F'$F(!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+&*********!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$-%#lnG6#%\"uG/F)$\"+,+++5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+&*********!#>" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 22 "Dividing 0.9999999995 " }{TEXT 424 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9)" "6#)\"#5,$\"\"*!\"\"" } {TEXT -1 4 " by " }{XPPEDIT 18 0 "x[0] = -7" "6#/&%\"xG6#\"\"!,$\"\"(! \"\"" }{TEXT -1 1 " " }{TEXT 420 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10)" "6#)\"#5,$F$!\"\"" }{TEXT -1 43 " gives the 10 digit value \+ -1.428571428 for " }{XPPEDIT 18 0 "f(x)=ln(1-x)/x" "6#/-%\"fG6#%\"xG*& -%#lnG6#,&\"\"\"F-F'!\"\"F-F'F." }{TEXT -1 6 " when " }{XPPEDIT 18 0 " x=x[0]" "6#/%\"xG&F$6#\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "Digits := 10:\nx0 \+ := -7*10^(-10):\nvv := 0.9999999995*10^(-9);\nvv/x0;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#vvG$\"+&*********!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+G9dG9!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "A brief summary of the error problem here is th at " }{XPPEDIT 18 0 "ln(1-x)" "6#-%#lnG6#,&\"\"\"F'%\"xG!\"\"" }{TEXT -1 1 " " }{TEXT 441 1 "~" }{XPPEDIT 18 0 " ``(1-x)-1" "6#,&-%!G6#,&\" \"\"F(%\"xG!\"\"F(F(F*" }{TEXT -1 6 " when " }{TEXT 442 1 "x" }{TEXT -1 66 " is near 0. A large relative error may occur in the evaluation \+ of " }{XPPEDIT 18 0 "ln(1-x)" "6#-%#lnG6#,&\"\"\"F'%\"xG!\"\"" }{TEXT -1 97 " since the second (outer) subtraction will involve the subtract ion of \"nearly equal quantities\". " }}{PARA 0 "" 0 "" {TEXT -1 85 "A s a check on the method of calculation, note that, if the natural loga rthm function " }{TEXT 260 2 "ln" }{TEXT -1 40 " is replaced by the fo llowing procedure " }{TEXT 260 3 "ln2" }{TEXT -1 38 ", we obtain the s ame curve as before. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "ln2 := proc(x)\n local p;\n p := x-1; \n (p-p^2/2+p^3/3-p^4/4)/x;\nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ln2Gf*6#%\"xG6#%\"pG6\"F*C$>8$,&9$\"\"\"F0!\"\"*&,*F-F0*&#F0 \"\"#F0*$)F-F6F0F0F1*&#F0\"\"$F0*$)F-F;F0F0F0*&#F0\"\"%F0*$)F-F@F0F0F1 F0F/F1F*F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 258 "f := x -> ln(1-x)/x:\n'f(x)'=f(x);\nf2 := x -> \+ 'ln2'(1-x)/x:\n'f2(x)'=f2(x);\nplot(['evalf[10](f(x))','evalf[10](f2(x ))','evalf[20](f(x))'],\n x=-3e-9..3e-9,-2.5..0.5,color=[red,green,b rown],\n thickness=[1,2,1],numpoints=100,font=[HELVETICA,9],axes=f ramed);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&-%#lnG6#,& \"\"\"F-F'!\"\"F-F'F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#f2G6#%\"x G*&-%$ln2G6#,&\"\"\"F-F'!\"\"F-F'F." }}{PARA 13 "" 1 "" {GLPLOT2D 533 309 309 {PLOTDATA 2 "6)-%'CURVESG6%7hel7$$!\"$!\"*$!+()********!#57$$! +(4!fOH!#=$!+PIf@5F*7$$!+zuT\")GF1$!+IU:T5F*7$$!+,,P>GF1$!+lu1k5F*7$$! +K9\"pv#F1$!+QU<)3\"F*7$$!+='\\Zp#F1$!+qfF86F*7$$!+fw6PEF1$!+ocgP6F*7$ $!+/JWxDF1$!+\" F*7$$!+%*y.3DF1$!+uT:'>\"F*7$$!+'>[.]#F1$!+))G$)*>\"F*7$$!+sdU)\\#F1$! +'pS]+)F-7$$!+ZL]'\\#F1$!+a\\?6!)F-7$$!+A4e%\\#F1$!+7(yt,)F-7$$!+)\\eE \\#F1$!+\"*>cB!)F-7$$!+[O\"))[#F1$!+5s&f.)F-7$$!+*zo\\[#F1$!+%y!R[!)F- 7$$!+-%*epCF1$!+KA^)4)F-7$$!+0+@aCF1$!+c=E\\\")F-7$$!+03dACF1$!+YEpb#) F-7$$!+0;$4R#F1$!+M-%\\O)F-7$$!+bf>NBF1$!+PF1$!+.A%R,\"F*7$$!+$\\%*y!>F 1$!+&*fF[5F*7$$!+V&32&=F1$!+*=n13\"F*7$$!+$\\mzy\"F1$!+%>*e=6F*7$$!+C7 AGmO \"F1$!+)F1$!+u76?7F*7$$!*.ZRe(F1$!+5Yd=8F*7$$!*.zb)pF1$!+f0_J9F*7$$!*= V\"fjF1$!+'*)QDd\"F*7$$!+!4xu0'!#>$!+,B&3l\"F*7$$!*+6ev&F1$!+GYPPF*7$$!+!)Qd+^F]_l$!+*pj0'> F*7$$!+]5Li]F]_l$!+lXPv>F*7$$!+N'4K/&F]_l$!+JU'G)>F*7$$!+?#)3C]F]_l$!+ 34T!*>F*7$$!+5o'\\+&F]_l$!+Z_,)*>F*7$$!+&RXe)\\F]_l$!\"!\"\"!7$$!+Q(f$ 4\\F]_lF[bl7$$!+!3uG$[F]_lF[bl7$$!+lF!*zYF]_lF[bl7$$!*XJp_%F1F[bl7$$!+ S>!eC%F]_lF[bl7$$!*VsY'RF1F[bl7$$!*;g-K$F1F[bl7$$!*8!*Qu#F1F[bl7$$!*Zf $H@F1F[bl7$$!*Av6a\"F1F[bl7$$!)>s!)))F1F[bl7$$!)'H`A$F1F[bl7$$!+++#pe \"!#AF[bl7$$\")7f$>$F1$F]blF]bl7$$\"+)))f[#R!#?Fidl7$$\"+v&Ghl%F]elFid l7$$\"+[d%*Q[F]elFidl7$$\"+?Hw@]F]el$!+eDL\"*>F*7$$\"+!4!e/_F]el$!+$f% Q@>F*7$$\"+gsR(Q&F]el$!+*)Q=c=F*7$$\"+0;.`dF]el$!+')R@Q8F* 7$$\"+5?Z7$)F]el$!+q:,.7F*7$$\")2uV!*F1$!+Vrt06F*7$$\"+vENk5F]_l$!+L?Q &R*F-7$$\"+!GJVA\"F]_l$!+PnF-7$$\"+RQH/:F]_l$!+#[F&H8F*7$$\"+k6HC:F ]_l$!+\"G&378F*7$$\"*\\)GW:F1$!+/[4&H\"F*7$$\"+qHK#o\"F]_l$!+oB$))=\"F *7$$\"+]uN?=F]_l$!+;`o)4\"F*7$$\"+I>Re>F]_l$!+QgC@5F*7$$\"*TEk4#F1$!+2 R/S&*F-7$$\"+s:j`AF]_l$!+d]cu))F-7$$\"+Nn$3T#F]_l$!+<\\(eH)F-7$$\"+E!Q ,X#F]_l$!+Pa!G;)F-7$$\"+;$R%*[#F]_l$!+]v$R.)F-7$$\"+i**34DF]_l$!+DEl&> \"F*7$$\"+21uGDF]_l$!+-8O'=\"F*7$$\"+_7R[DF]_l$!+qK@x6F*7$$\"+)*=/oDF] _l$!+R_?o6F*7$$\"+zWkYEF]_l$!+i2^L6F*7$$\"*1Z_s#F1$!+Ew\"35\"F*7$$\"+0 >&*GIF]_l$!+&4;W!**F-7$$\"*vcEL$F1$!+Yr#=+*F-7$$\"+hL`3MF]_l$!+`%R9!)) 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" }}{PARA 0 "" 0 "" {TEXT -1 5 "For " }{XPPEDIT 18 0 "-3" "6#,$\"\"$!\"\"" }{TEXT -1 1 " \+ " }{TEXT 428 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9) <= x;" "6#1 )\"#5,$\"\"*!\"\"%\"xG" }{XPPEDIT 18 0 "``<=3" "6#1%!G\"\"$" }{TEXT -1 1 " " }{TEXT 427 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9)" "6# )\"#5,$\"\"*!\"\"" }{TEXT -1 19 " this happens when " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x=-3" "6#/%\"xG,$\"\"$!\"\"" } {TEXT -1 1 " " }{TEXT 429 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9 )" "6#)\"#5,$\"\"*!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "-2;" "6#,$ \"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 430 1 "x" }{TEXT -1 1 " " } {XPPEDIT 18 0 "10^(-9)" "6#)\"#5,$\"\"*!\"\"" }{TEXT -1 3 ", " } {XPPEDIT 18 0 "-1;" "6#,$\"\"\"!\"\"" }{TEXT -1 1 " " }{TEXT 431 1 "x " }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9)" "6#)\"#5,$\"\"*!\"\"" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "1;" "6#\"\"\"" }{TEXT -1 1 " " }{TEXT 432 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10);" "6#)\"#5,$F$!\"\" " }{TEXT -1 2 ", " }{XPPEDIT 18 0 "2;" "6#\"\"#" }{TEXT -1 1 " " } {TEXT 433 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10);" "6#)\"#5,$F $!\"\"" }{TEXT -1 2 ", " }{TEXT 434 7 ". . . ." }{TEXT -1 3 " , " } {XPPEDIT 18 0 "2.9;" "6#-%&FloatG6$\"#H!\"\"" }{TEXT -1 1 " " }{TEXT 436 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9);" "6#)\"#5,$\"\"*!\" \"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "3;" "6#\"\"$" }{TEXT -1 1 " " } {TEXT 435 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9);" "6#)\"#5,$\" \"*!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[x, 1-x], [________, __ ______], [-3*`.`*10^(-9), 1.000000003], [-2*`.`*10^(-9), 1.000000002], [-1*`.`*10^(-9), 1.000000001], [1*`.`*10^(-10), .9999999999], [2*`.`* 10^(-10), .9999999998], [`.`, `.`], [`.`, `.`], [29*`.`*10^(-10), .999 9999971], [3*`.`*10^(-9), .9999999970]]);" "6#-%'matrixG6#7-7$%\"xG,& \"\"\"F*F(!\"\"7$%)________GF-7$,$*(\"\"$F*%\".GF*)\"#5,$\"\"*F+F*F+-% &FloatG6$\"+.+++5!\"*7$,$*(\"\"#F*F2F*)F4,$F6F+F*F+-F86$\"+-+++5F;7$,$ *(F*F*F2F*)F4,$F6F+F*F+-F86$\"+,+++5F;7$*(F*F*F2F*)F4,$F4F+F*-F86$\"+* *********!#57$*(F?F*F2F*)F4,$F4F+F*-F86$\"+)*********FT7$F2F27$F2F27$* (\"#HF*F2F*)F4,$F4F+F*-F86$\"+r********FT7$*(F1F*F2F*)F4,$F6F+F*-F86$ \"+q********FT" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 47 "The discontinuities or \"jumps\" in the g raph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 34 " plott ed over the narrow interval " }{XPPEDIT 18 0 "-3" "6#,$\"\"$!\"\"" } {TEXT -1 1 " " }{TEXT 426 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9 ) <= x;" "6#1)\"#5,$\"\"*!\"\"%\"xG" }{XPPEDIT 18 0 "``<=3" "6#1%!G\" \"$" }{TEXT -1 1 " " }{TEXT 425 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 " 10^(-9)" "6#)\"#5,$\"\"*!\"\"" }{TEXT -1 130 " occur exactly half way \+ in between the previous values, and are the thresholds where the 10 di git floating point representation of" }{XPPEDIT 18 0 "``(1-x);" "6#-%! G6#,&\"\"\"F'%\"xG!\"\"" }{TEXT -1 60 " suddenly changes from one floa ting point value to the next." }}{PARA 0 "" 0 "" {TEXT -1 20 "In parti cular, when " }{XPPEDIT 18 0 "-5*`.`*10^(-10) " 0 "" {MPLTEXT 1 0 45 "fn := proc(x)\n evalhf(ln( 1-x)/x);\nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#fnGf*6#%\"xG 6\"F(F(-%'evalhfG6#*&-%#lnG6#,&\"\"\"F19$!\"\"F1F2F3F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "f(x)" "6 #-%\"fG6#%\"xG" }{TEXT -1 31 " is plotted over the interval " } {XPPEDIT 18 0 "-6;" "6#,$\"\"'!\"\"" }{TEXT -1 1 " " }{TEXT 438 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-16) <= x;" "6#1)\"#5,$\"#;!\"\"% \"xG" }{XPPEDIT 18 0 "`` <= 6;" "6#1%!G\"\"'" }{TEXT -1 1 " " }{TEXT 437 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-16);" "6#)\"#5,$\"#;!\" \"" }{TEXT -1 45 " with hardware floating point arithmetic (in " } {TEXT 259 6 "purple" }{TEXT -1 34 ") and with 25 digit precision (in \+ " }{TEXT 261 5 "brown" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 48 ": The same graph is obtained if we simply plot \+ " }{XPPEDIT 18 0 "f(x)=ln(1-x)/x" "6#/-%\"fG6#%\"xG*&-%#lnG6#,&\"\"\" F-F'!\"\"F-F'F." }{TEXT -1 30 " with the default setting of " }{TEXT 260 6 "Digits" }{TEXT -1 25 ", because, in this case, " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 61 " will be evaluated using hard ware floating point arithmetic. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 782 "y0 := -2.5: y1 := 0.5:\np1 \+ := plot(['fn(x)','evalf[25](f(x))'],x=-6e-16..6e-16,y0..y1,\n \+ color=[COLOR(RGB,.5,0,1),brown]):\nx0 := -2^(-51): x1 := -2^(-52) : x2 := 2^(-53):\nx3 := -x1: x4 := x2+x3: x5 := -x0: x6 := x5+x2:\np2 \+ := plot([[[x0,y0],[x0,y1]],[[x1,y0],[x1,y1]],[[x2,y0],[x2,y1]],\n \+ [[x3,y0],[x3,y1]],[[x4,y0],[x4,y1]],[[x5,y0],[x5,y1]],\n [[x6,y0] ,[x6,y1]]],color=COLOR(RGB,0,.7,0),linestyle=3):\nt1 := plots[textplot ]([[x0,y1+.12,`-2^(-51)`],[x1,y1+.12,`-2^(-52)`],\n [x2,y1+.12,`2^ (-53)`],[x3,y1+.12,`2^(-52)`],[x4,y1+.12,`2^(-53)`],\n [x4,y1+.24, `2^(-52) +`],[x5,y1+.12,`2^(-51)`],\n [x6,y1+.12,`2^(-53)`],[x6,y1 +.24,`2^(-51) +`]],font=[HELVETICA,8]):\nplots[display]([p1,p2,t1],axe s=framed,view=[-6e-16..6e-16,y0..y1+.24],\n font=[HELVETICA,9]);" }}{PARA 13 "" 1 "" {GLPLOT2D 731 387 387 {PLOTDATA 2 "68-%'CURVESG6$7j s7$$!3')**************f!#L$!3Kc^iCIA56!#<7$$!3o*****\\2<#peF*$!3<5=cl> '\\8\"F-7$$!3\\******\\TVQdF*$!3#zc2?aG3;\"F-7$$!3g**\\7)QP:o&F*$!3W90 HcMXs6F-7$$!3r***\\iiSYi&F*$!3M\"eZrb8V=\"F-7$$!3G*\\7`C#>'f&F*$!35)yZ U/M.>\"F-7$$!3#)**\\PkQunbF*$!3)f'>pagT'>\"F-7$$!3YD19p\"F-7$$!35]i!Rn>Nb&F*$!3CQ#HmU![*>\"F-7$$!3uu=nyvSYbF*$!3h^YuW+z1!) !#=7$$!3P+vV$[&HRbF*$!3?Q'eDCqq,)Fen7$$!3m\\(oHHr]_&F*$!3=?^qO+rP!)Fen 7$$!3$*****\\-r%3^&F*$!3'G>o'yjXe!)Fen7$$!3m***\\PQuGQ&F*$!33Mcc9&Q+D) Fen7$$!3M+++l;!\\D&F*$!3qXk:d=&4X)Fen7$$!3K+++b'fs*\\F*$!3uaV&ppam)))F en7$$!3b****\\s@%3u%F*$!3i`,i\\jIn$*Fen7$$!3')****\\U.6.XF*$!3WR]+NF$= ')*Fen7$$!3))****\\-G&pD%F*$!3)H'y>Y\"4K/\"F-7$$!3B++]AjP-SF*$!3``rBaQ c46F-7$$!39++]sih[PF*$!3Kc=\"4.vY=\"F-7$$!3.++DrX5=OF*$!32%GDIW3uA\"F- 7$$!3#*******pGf([$F*$!3ed-:q,Mt7F-7$$!3%****\\7[:,V$F*$!37S@BRrn%H\"F -7$$!3'*****\\#4QEP$F*$!3hVT$)o8u;8F-7$$!3A+DJX(o#eLF*$!3K@*emVvBK\"F- 7$$!3)***\\7)R**QM$F*$!3W/JsDz0G8F-7$$!3')\\7`CZrOLF*$!3`R-rBv\"4L\"F- 7$$!3t*\\P40I&HLF*$!3g\">FZJZ*omFen7$$!3h\\PMx`MALF*$!3+/%ou6pLo'Fen7$ $!3*****\\Pqg^J$F*$!3'45q0V`yp'Fen7$$!3++]P4?U'G$F*$!3a6T2')RUcnFen7$$ !3-+++:LodKF*$!3o/?KRz-;oFen7$$!3=++]i9FGJF*$!3MS++#p'*z4(Fen7$$!3%)** ****4'f))*HF*$!3Sxh$zU,VS(Fen7$$!39+++]J(*QFF*$!3wunxXe&o5)Fen7$$!3w** *****QC&)[#F*$!3]SK#)eF*$!3!\\O)fW\\V:6F-7$$!3%)*****\\n1h(=F*$!3!\\2ckLRN= \"F-7$$!3w******R%e:w\"F*$!3#yA8&G4]g7F-7$$!3)****\\7h6$G;F*$!3w&)e3P$ \\OO\"F-7$$!3?++]#yk]\\\"F*$!3A!4Sj$Q=&[\"F-7$$!3S++Dh(=rP\"F*$!3ZS'Q; I&Q7;F-7$$!3g+++SF\"F*$!35\\!QTJQ*e =F-7$$!3@++DhswH6F*$!3MfUo/8Sl>F-7$$!3;D\"G)=%z;7\"F*$!3%*zSV$ys&z>F-7 $$!35]iSw:f86F*$!3GBgn7,&R*>F-7$$!30vV)Rt.b5\"F*$\"\"!F][l7$$!37+Dc\"* eT(4\"F*F\\[l7$$!39](=n?S73\"F*F\\[l7$$!3-+](=_k]1\"F*F\\[l7$$!3#**\\( =_JrK5F*F\\[l7$$!3$)****\\#yh.+\"F*F\\[l7$$!3;&*****\\;Pr()!#MF\\[l7$$ !3/#****\\ZD\"RvF`\\lF\\[l7$$!3Y#*****\\ion\\F`\\lF\\[l7$$!3?'****\\K- jg#F`\\lF\\[l7$$!3aO0++]2Bf!#OF\\[l7$$\"3'G++]xgke#F`\\lF\\[l7$$\"3X++ ++>+QPF`\\lF\\[l7$$\"31)****\\-V&*)[F`\\lF\\[l7$$\"3=**\\i:qY+_F`\\lF \\[l7$$\"3K++D15R6bF`\\lF\\[l7$$\"3I]i!R+A\"*e&F`\\l$!3KD[U&p*R')>F-7$ $\"3G+Dc,I&om&F`\\l$!3m%e*=!o_\"f>F-7$$\"3D](=#**ReWdF`\\l$!3\"z?mM.VE $>F-7$$\"3Y,](o*\\JAeF`\\l$!3&G'3ZA7%o!>F-7$$\"3m-v=#*pxxfF`\\l$!3s4(= ql]s&=F-7$$\"3g-+]()*QK8'F`\\l$!3p#\\b23u,\"=F-7$$\"3)[+](op3bnF`\\l$! 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+-$[m>$Fe\\n$!+%*********Fjgm7$$!*K-jg#Fdgm$!+B********Fjgm7$$!+GW`p>F e\\n$!+e)*******Fjgm7$$!+NlwK8Fe\\n$!+P(*******Fjgm7$$!+[0<`7Fe\\n$!+< '*******Fjgm7$$!+iXdt6Fe\\n$!+K+++5F`gm7$$!+w&yR4\"Fe\\n$!+A+++5F`gm7$ $!+*e#Q95Fe\\n$!+6+++5F`gm7$$!+gh!>b)!#F$!+8)*******Fjgm7$$!+Dk)*fpFdf n$!+*Q*******Fjgm7$$!+gl-khFdfn$!+r+++5F`gm7$$!+&pm!o`Fdfn$!+d+++5F`gm 7$$!+Go5sXFdfn$!+Q+++5F`gm7$$!+ip9wPFdfnFcbn7$$!+'4(=!)HFdfn$!+y'***** **Fjgm7$$!+JsA%=#Fdfn$!+U*)******Fjgm7$$!+mtE)Q\"Fdfn$!+kt******Fjgm7$ $!(vI#fFdgm$!+W3++5F`gm7$$\"+I5rWB!#G$!+s)*******Fjgm7$$\"+1(\\71\"Fdf nF[]n7$$\"+4$G!))=Fdfn$!+_********Fjgm7$$\"+7p![r#Fdfn$!+c********Fjgm 7$$\"+=TOoVFdfn$!+f********Fjgm7$$\"+D8#>-'Fdfn$!+e********Fjgm7$$\"+I &yan(Fdfn$!+h********Fjgm7$$\"+Sd.H$*Fdfn$!+d********Fjgm7$$\"+M9e:5Fe \\nFb[o7$$\"+%Hf#)4\"Fe\\n$!+k********Fjgm7$$\"+ar$4=\"Fe\\n$!+m****** **Fjgm7$$\"+:]hj7Fe\\nFe]n7$$\"+ckK%f\"Fe\\nFa_n7$$\"+)*y.D>Fe\\nFchm7 $$\"+R$\\dD#Fe\\nF^gm7$$\"*ygke#FdgmF^gm7$$\"++>+QPFe\\nF^gm7$$\"*-V&* )[FdgmF^gm7$$\"*&\\$pP(FdgmF^gm7$$\"*?am%**FdgmF^gm7$$\"+:B1Y7FdgmF^gm 7$$\"+Q " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The values of " }{TEXT 439 1 "x" }{TEXT -1 85 " for which the error in the lower precision values is effectively 0 are shown by the " }{TEXT 257 5 "green" }{TEXT -1 51 " vertical dashed lines. This happens for values of " }{TEXT 440 1 "x" }{TEXT -1 11 " such that " } {XPPEDIT 18 0 "``(1-x);" "6#-%!G6#,&\"\"\"F'%\"xG!\"\"" }{TEXT -1 8 " \+ has an " }{TEXT 259 5 "exact" }{TEXT -1 84 " 53 binary digit floating \+ point representation as indicated by the following table. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[x, `binary representat ion of `*(1-x)], [_______________________________, __________________ ______________________________________], [-2^(-51) = -.4440892098*`.`* 10^(-15), 1.0000000000000000000000000000000000000000000000000010*``[2] ], [-2^(-52) = -.2220446049*`.`*10^(-15), 1.00000000000000000000000000 00000000000000000000000001*``[2]], [2^(-53) = .1110223025*`.`*10^(-15) , .11111111111111111111111111111111111111111111111111111*``[2]], [2^(- 52) = .2220446049*`.`*10^(-15), .1111111111111111111111111111111111111 1111111111111110*``[2]], [2^(-52)+2^(-53) = .3330669074*`.`*10^(-15), \+ .11111111111111111111111111111111111111111111111111101*``[2]], [2^(-51 ) = .4440892098*`.`*10^(-15), .111111111111111111111111111111111111111 11111111111100*``[2]], [2^(-51)+2^(-53) = .5551115123*`.`*10^(-15), .1 1111111111111111111111111111111111111111111111111011*``[2]]]);" "6#-%' matrixG6#7+7$%\"xG*&%;binary~representation~of~~G\"\"\",&F+F+F(!\"\"F+ 7$%@_______________________________G%Y________________________________ ________________________G7$/,$)\"\"#,$\"#^F-F-,$*(-%&FloatG6$\"+)4#*3W %!#5F+%\".GF+)\"#5,$\"#:F-F+F-*&-F;6$\"V5+++++++++++++++++++++++++\"!# _F+&%!G6#F5F+7$/,$)F5,$\"#_F-F-,$*(-F;6$\"+\\gW?AF>F+F?F+)FA,$FCF-F+F- *&-F;6$\"V,+++++++++++++++++++++++++\"FHF+&FJ6#F5F+7$/)F5,$\"#`F-*(-F; 6$\"+DIA56F>F+F?F+)FA,$FCF-F+*&-F;6$\"V66666666666666666666666666\"!#` F+&FJ6#F5F+7$/)F5,$FQF-*(-F;6$\"+\\gW?AF>F+F?F+)FA,$FCF-F+*&-F;6$\"V56 666666666666666666666666\"FhoF+&FJ6#F5F+7$/,&)F5,$FQF-F+)F5,$F]oF-F+*( -F;6$\"+u!p1L$F>F+F?F+)FA,$FCF-F+*&-F;6$\"V,6666666666666666666666666 \"FhoF+&FJ6#F5F+7$/)F5,$F7F-*(-F;6$\"+)4#*3W%F>F+F?F+)FA,$FCF-F+*&-F;6 $\"V+6666666666666666666666666\"FhoF+&FJ6#F5F+7$/,&)F5,$F7F-F+)F5,$F]o F-F+*(-F;6$\"+B^6^bF>F+F?F+)FA,$FCF-F+*&-F;6$\"V6566666666666666666666 6666\"FhoF+&FJ6#F5F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "As before the discontinuities or \"jump s\" in the graph of " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 113 " occur exactly half way in between the previous values, and ar e the thresholds where the binary representation of" }{XPPEDIT 18 0 "` `(1-x);" "6#-%!G6#,&\"\"\"F'%\"xG!\"\"" }{TEXT -1 45 " suddenly change s from one value to the next." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 259 "" 0 "" {TEXT 398 50 "Subtraction round-off error in the evaluation of \+ " }{XPPEDIT 18 0 "g(x) = (1-cos(x))/(x^2);" "6#/-%\"gG6#%\"xG*&,&\"\" \"F*-%$cosG6#F'!\"\"F**$F'\"\"#F." }{TEXT 397 7 " when " }{TEXT 395 1 "x" }{TEXT 396 10 " is near 0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 375 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 46 "This question is concerned with the function " }{XPPEDIT 18 0 "g(x) \+ = (1-cos(x))/(x^2);" "6#/-%\"gG6#%\"xG*&,&\"\"\"F*-%$cosG6#F'!\"\"F**$ F'\"\"#F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "(a) Plot a \+ graph of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 19 " ove r the interval " }{XPPEDIT 18 0 "-5*Pi <= x;" "6#1,$*&\"\"&\"\"\"%#PiG F'!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 5*Pi;" "6#1%!G*&\"\"&\"\"\"%#PiGF' " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "(b) Use the expressions " }{TEXT 260 17 "'evalf[10](g(x)) '" }{TEXT -1 5 " and " }{TEXT 260 17 "'evalf[20](g(x))'" }{TEXT -1 46 " in a suitable plot command to plot graphs of " }{XPPEDIT 18 0 "g(x); " "6#-%\"gG6#%\"xG" }{TEXT -1 18 " over the interval" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "-4;" "6#,$\"\"%!\"\"" }{TEXT -1 1 " " }{TEXT 374 1 "x" }{TEXT -1 2 " 1" }{XPPEDIT 18 0 "0^(-5) <= x;" "6#1)\"\"!,$\"\"&!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 4;" "6#1%!G\"\"%" } {TEXT -1 1 " " }{TEXT 373 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-5 );" "6#)\"#5,$\"\"&!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 53 "with 10 digit precision and with 20 digit precision. " }{TEXT 259 4 "Note" }{TEXT -1 37 ": It is essential to use the quotes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "(c) Find the fi rst 4 terms of the Maclaurin series for " }{XPPEDIT 18 0 "g(x);" "6#-% \"gG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 15 "(d) Check \+ that " }{XPPEDIT 18 0 "Limit(g(x),x = 0) = 1;" "6#/-%&LimitG6$-%\"gG6# %\"xG/F*\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "(e ) Let " }{XPPEDIT 18 0 "x[0] = 11;" "6#/&%\"xG6#\"\"!\"#6" }{TEXT -1 1 " " }{TEXT 379 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-6);" "6#) \"#5,$\"\"'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 " ( i) Find the sum of the first " }{TEXT 262 7 "4 terms" }{TEXT -1 29 " o f the Maclaurin series for " }{XPPEDIT 18 0 "g(x) = (1-cos(x))/(x^2); " "6#/-%\"gG6#%\"xG*&,&\"\"\"F*-%$cosG6#F'!\"\"F**$F'\"\"#F." }{TEXT -1 14 " evaluated at " }{XPPEDIT 18 0 "x=x[0]" "6#/%\"xG&F$6#\"\"!" } {TEXT -1 82 ", performing the calculation with Maple's base 10 floatin g point arithmetic using " }{TEXT 262 9 "35 digits" }{TEXT -1 15 " of \+ precision. " }}{PARA 0 "" 0 "" {TEXT -1 26 " (ii) Find the value of \+ " }{XPPEDIT 18 0 "g(x[0]);" "6#-%\"gG6#&%\"xG6#\"\"!" }{TEXT -1 34 ", \+ performing the calculation with " }{TEXT 262 9 "45 digits" }{TEXT -1 14 " of precision." }}{PARA 0 "" 0 "" {TEXT -1 27 " (iii) Find the v alue of " }{XPPEDIT 18 0 "g(x[0]);" "6#-%\"gG6#&%\"xG6#\"\"!" }{TEXT -1 34 ", performing the calculation with " }{TEXT 262 9 "10 digits" } {TEXT -1 15 " of precision. " }}{PARA 0 "" 0 "" {TEXT -1 3 " " } {TEXT 259 4 "Note" }{TEXT -1 54 " for (iii): In order to prevent Maple from evaluating " }{XPPEDIT 18 0 "g(x[0]);" "6#-%\"gG6#&%\"xG6#\"\"! " }{TEXT -1 4 " as " }{XPPEDIT 18 0 "1/(x[0]^2)-cos(x)/(x[0]^2);" "6#, &*&\"\"\"F%*$&%\"xG6#\"\"!\"\"#!\"\"F%*&-%$cosG6#F(F%*$&F(6#F*F+F,F," }{TEXT -1 10 ", compute " }{XPPEDIT 18 0 "g(x[0]);" "6#-%\"gG6#&%\"xG6 #\"\"!" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" } {TEXT -1 45 " first evaluated as a floating point number. " }}{PARA 0 "" 0 "" {TEXT -1 28 " Also execute the command " }{TEXT 260 13 "forg et(evalf)" }{TEXT -1 34 " before performing the evaluation." }}{PARA 0 "" 0 "" {TEXT -1 18 " (iv) Calculate " }{XPPEDIT 18 0 "cos(x[0]); " "6#-%$cosG6#&%\"xG6#\"\"!" }{TEXT -1 34 ", performing the calculatio n with " }{TEXT 262 9 "10 digits" }{TEXT -1 54 " of precision, and che ck that using the approximation " }{XPPEDIT 18 0 "cos(x);" "6#-%$cosG6 #%\"xG" }{TEXT -1 1 " " }{TEXT 380 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1-x^2/2;" "6#,&\"\"\"F$*&%\"xG\"\"#F'!\"\"F(" }{TEXT -1 64 ", obtai ned from the first two terms of the Maclaurin series for " }{XPPEDIT 18 0 "cos(x);" "6#-%$cosG6#%\"xG" }{TEXT -1 24 ", gives the same value . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 " \+ (v) Use the answer from (iv) to explain the result of (iii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "(f) Give one po sitive value of " }{TEXT 378 1 "x" }{TEXT -1 37 " for which the error \+ in the value of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 65 " calculated using 10 digit precision is effectively equal to 0. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 376 8 "Solut ion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "g := x -> (1-cos(x))/x^2:\n'g(x)'=g (x);\nplot(g(x),x=-5*Pi..5*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"gG6#%\"xG*&,&\"\"\"F*-%$cosGF&!\"\"F*F'!\"#" }}{PARA 13 "" 1 "" {GLPLOT2D 525 362 362 {PLOTDATA 2 "6%-%'CURVESG6$7ep7$$!3'****\\OK'zq: !#;$\"36)*Q%Qs%p0\")!#?7$$!3WNzQW&=B]\"F*$\"3agt_NjgiyF-7$$!3742![ROFW \"F*$\"3aDUPi*[*yhF-7$$!3Ext3O+tv8F*$\"3CB*f<1([CLF-7$$!3'y2Ld^z#38F*$ 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\"3oVTkbLVk=F-7$$\"3cY\"p0*QPtgFfn$\"3CkhdK#4`%fFB7$$\"3NeOtxN/FiFfn$ \"3M<%*G[y:jSFH7$$\"39q\")*[E82Q'Ffn$\"3U;K-+2?n6FB7$$\"3%>oi?&HQMlFfn $\"31K?ZYUL]tFB7$$\"3'f-rr1'QqoFfn$\"3Y(4'\\@In[NF-7$$\"3)*p$zA=*Q1sFf n$\"3gbU\"p7W%RwF-7$$\"3:HJCuYpQyFfn$\"3nyG#p4!f-;FX7$$\"3C3s1*Ql\"p\" )Ffn$\"3s-v\"*zp'H'>FX7$$\"3K(G\"*Q5O'*\\)Ffn$\"3\"[)e^>^5'zMM#FX7$$\"3qenYgr#e9*Ffn$\"3*HJ.Vv6[M#FX7$$\" 3'3*)[cByS[*Ffn$\"3#y:Jo:f:A#FX7$$\"3,B5$3JHB#)*Ffn$\"3s(GSl1t@*>FX7$$ \"3%3'>eG\")QZ5F*$\"3u7]#y%H$eO\"FX7$$\"3:K#)fG(=S6\"F*$\"37J(e>Lrl*oF -7$$\"3'\\I0Of(4!=\"F*$\"3%orOmm?E+#F-7$$\"3rm8'y?<3C\"F*$\"3\"Rb6I@21 6)FH7$$\"3BnP+Q(3/J\"F*$\"3dmBn(=[\"=#)FB7$$\"3#Ryp@B_EP\"F*$\"3#zer3e i')=$F-7$$\"3hNK@zn,R9F*$\"3RF(R$eH!z.'F-7$$\"3x;#f'=h`-:F*$\"3U@bHCpT myF-7$$\"3'****\\OK'zq:F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F[hlFjgl- %+AXESLABELSG6$Q\"x6\"Q!F`hl-%%VIEWG6$;$!+Fjzq:!\")$\"+Fjzq:Fhhl%(DEFA ULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1 " }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 42 ": The re should be a \"missing point\" where " }{XPPEDIT 18 0 "x=0" "6#/%\"x G\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 1/2;" "6#/%\"yG*&\"\" \"F&\"\"#!\"\"" }{TEXT -1 13 ", but, since " }{XPPEDIT 18 0 "Limit(g(x ),x = 0) = 1/2;" "6#/-%&LimitG6$-%\"gG6#%\"xG/F*\"\"!*&\"\"\"F.\"\"#! \"\"" }{TEXT -1 120 " (see below), we can fill it in and obtain a cont inuous function, indeed one which has derivatives of all orders at 0. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "g := x -> (1-cos(x))/x^ 2:\n'g(x)'=g(x);\nplot(['evalf[10](g(x))','evalf[20](g(x))'],x=-4e-5.. 4e-5,-.1..1,\n color=[red,brown],numpoints=100,axes=framed);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&,&\"\"\"F*-%$cosGF&!\" \"F*F'!\"#" }}{PARA 13 "" 1 "" {GLPLOT2D 805 296 296 {PLOTDATA 2 "6'-% 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eoFh^p7$FahoFh^p7$F[ioFh^p7$FeioFh^p7$F_joFh^p7$Fa]pFh^p7$F[^pFh^p-F^^ p6&F`^p$\")#)eqkFc^p$\"))eqk\"Fc^pFaiv-%*AXESSTYLEG6#%&FRAMEG-%+AXESLA BELSG6$Q\"x6\"Q!F[jv-%%VIEWG6$;F(F[^p;$!\"\"Fcjv$\"\"\"Faam" 1 2 0 1 10 0 2 9 1 3 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "(c)" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "g := x -> (1-cos(x))/x^2:\n 'g(x)'=g(x);\ntaylor(g(x),x=0,9);\np := unapply(convert(%,polynom),x): \n'p(x)'=p(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&,&\" \"\"F*-%$cosGF&!\"\"F*F'!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#+-%\"x G#\"\"\"\"\"#\"\"!#!\"\"\"#CF'#F&\"$?(\"\"%#F*\"&?.%\"\"'-%\"OG6#F&\" \"(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"xG,*#\"\"\"\"\"#F** &#F*\"#CF**$)F'F+F*F*!\"\"*&#F*\"$?(F**$)F'\"\"%F*F*F**&#F*\"&?.%F**$) F'\"\"'F*F*F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "The Macla urin series for " }{XPPEDIT 18 0 "g(x) = (1-cos(x))/(x^2);" "6#/-%\"gG 6#%\"xG*&,&\"\"\"F*-%$cosG6#F'!\"\"F**$F'\"\"#F." }{TEXT -1 4 " is " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/2!-x^2/4!+x^4/6!-x ^6/8!+` . . . `" "6#,,*&\"\"\"F%-%*factorialG6#\"\"#!\"\"F%*&%\"xGF)-F '6#\"\"%F*F**&F,F/-F'6#\"\"'F*F%*&F,F3-F'6#\"\")F*F*%(~.~.~.~GF%" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "(d) " }{XPPEDIT 18 0 "L imit((1-cos(x))/x^2,x = 0)=1/2" "6#/-%&LimitG6$*&,&\"\"\"F)-%$cosG6#% \"xG!\"\"F)*$F-\"\"#F./F-\"\"!*&F)F)F0F." }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "g := \+ x -> (1-cos(x))/x^2:\nLimit(g(x),x=0);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,&\"\"\"F(-%$cosG6#%\"xG!\"\"F(F,!\"#/F ,\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "(e) (i) " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "x0 := 11*10^(-6);\nEval(p(x) ,x=x0);\n``=value(%);\nevalf[35](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%#x0G#\"#6\"(+++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$,*# \"\"\"\"\"#F(*&#F(\"#CF(*$)%\"xGF)F(F(!\"\"*&#F(\"$?(F(*$)F/\"\"%F(F(F (*&#F(\"&?.%F(*$)F/\"\"'F(F(F0/F/#\"#6\"(+++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"JR%G#)****f*)>3+++s'z******f,#\"J+++++++++++++++ +++?.%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"D^bb0oONLLLe\\******* **\\!#N" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "(e) (ii) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "g := x -> (1 -cos(x))/x^2:\nx0 := 11*10^(-6);\nEval(g(x),x=x0);\n``=value(%);\neval f[45](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G#\"#6\"(+++\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$*&,&\"\"\"F(-%$cosG6#%\"xG! \"\"F(F,!\"#/F,#\"#6\"(+++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,& #\".++++++\"\"$@\"\"\"\"*&#\".++++++\"F(F)-%$cosG6##\"#6\"(+++\"F)!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"D^bb0oONLLLe\\*********\\ !#N" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "( e) (iii) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "forget(evalf); \ng := x -> (1-cos(x))/x^2:\nx0 := evalf[10](11*10^(-6));\nEval(g(x),x =x0);\n``=value(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"++++ +6!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$*&,&\"\"\"F(-%$cosG 6#%\"xG!\"\"F(F,!\"#/F,$\"+++++6!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G$\"+5GYk#)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 9 "(e) (iv) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "t aylor(cos(x),x=0,7):\nq := unapply(convert(%,polynom),x):\n'q(x)'=q(x) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG,0\"\"\"F)F'F)*&#F) \"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F'\"\"%F )F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "D igits := 10:\nx0 := evalf(11*10^(-6));\nEval(cos(x),x=x0);\n``=value(% );\nEval(1-x^2/2,x=x0);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"+++++6!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$-%$ cosG6#%\"xG/F)$\"+++++6!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+ **********!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$,&\"\"\"F'* &\"\"#!\"\"%\"xGF)F*/F+$\"+++++6!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G$\"+**********!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "(e) (v) " }}{PARA 0 "" 0 "" {TEXT -1 44 "We have the \+ 10 digit value 0.9999999999 for " }{XPPEDIT 18 0 "cos(x[0]);" "6#-%$co sG6#&%\"xG6#\"\"!" }{TEXT -1 13 " which gives " }{XPPEDIT 18 0 "1-cos( x[0]);" "6#,&\"\"\"F$-%$cosG6#&%\"xG6#\"\"!!\"\"" }{TEXT -1 1 " " } {TEXT 381 1 "~" }{TEXT -1 14 " 0.0000000001." }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(1-co s(x[0]))/(x[0]^2);" "6#*&,&\"\"\"F%-%$cosG6#&%\"xG6#\"\"!!\"\"F%*$&F*6 #F,\"\"#F-" }{TEXT -1 1 " " }{TEXT 382 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "10^(-10)/(121*`.`*10^(-12));" "6#*&)\"#5,$F%!\"\"\"\"\" *(\"$@\"F(%\".GF()F%,$\"#7F'F(F'" }{TEXT -1 1 " " }{TEXT 383 1 "~" } {TEXT -1 1 " " }{XPPEDIT 18 0 "100/121;" "6#*&\"$+\"\"\"\"\"$@\"!\"\" " }{TEXT -1 1 " " }{TEXT 384 1 "~" }{TEXT -1 15 " 0.8264462810. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "100/121;\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$+\"\"$@ \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5GYk#)!#5" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "A brief summary of the \+ error problem here is that " }{XPPEDIT 18 0 "1-cos(x);" "6#,&\"\"\"F$- %$cosG6#%\"xG!\"\"" }{TEXT -1 1 " " }{TEXT 390 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "1-``(1-x^2/2);" "6#,&\"\"\"F$-%!G6#,&F$F$*&%\"xG\"\"#F+ !\"\"F,F," }{TEXT -1 6 " when " }{TEXT 391 1 "x" }{TEXT -1 66 " is nea r 0. A large relative error may occur in the evaluation of " } {XPPEDIT 18 0 "1-cos(x);" "6#,&\"\"\"F$-%$cosG6#%\"xG!\"\"" }{TEXT -1 98 " since the last step in the evaluation will involve the subtractio n of \"nearly equal quantities\". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "cs0 := 0.9999999999;\nnum : = 1-cs0;\nx0 := evalf(11*10^(-6));\nnum/x0^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$cs0G$\"+**********!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$numG$\"\"\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"++ +++6!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5GYk#)!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(f) " }}{PARA 0 "" 0 "" {TEXT -1 50 "The error is effectively 0 when the approximation " }{XPPEDIT 18 0 "cos(x )" "6#-%$cosG6#%\"xG" }{TEXT -1 1 " " }{TEXT 393 1 "~" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "1-x^2/2;" "6#,&\"\"\"F$*&%\"xG\"\"#F'!\"\"F(" } {TEXT -1 45 " has an exact floating point representation. " }}{PARA 0 "" 0 "" {TEXT -1 44 "The following table indicates the values of " } {TEXT 377 1 "x" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "0 <= \+ x;" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "`` <= 4;" "6#1%!G\"\"%" }{TEXT -1 1 " " }{TEXT 385 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-5);" "6#) \"#5,$\"\"&!\"\"" }{TEXT -1 29 " for which this is the case. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "matrix([[x, cos(x) = 1-x^2/2+O(x^4)], [________________ ____________, ____________________], [sqrt(2)*`.`*10^(-5) = `0.0000141 4213562`, `0.9999999999`], [2*`.`*10^(-5) = `0.00002000000000`, `0 .99 99999998`], [sqrt(6)*`.`*10^(-5) = `0.00002449489743`, `0.9999999997`] , [sqrt(8)*`.`*10^(-5) = `0.00002828427124`, `0.9999999996`], [sqrt(10 )*`.`*10^(-5) = `0.00003162277660`, `0.9999999995`], [sqrt(12)*`.`*10^ (-5) = `0.00003464101615`, `0 .9999999994`], [sqrt(14)*`.`*10^(-5) = ` 0.00003741657387`, `0 .9999999993`], [4*`.`*10^(-5) = `0.0000400000000 0`, `0.9999999992`]]);" "6#-%'matrixG6#7,7$%\"xG/-%$cosG6#F(,(\"\"\"F. *&F(\"\"#F0!\"\"F1-%\"OG6#*$F(\"\"%F.7$%=____________________________G %5____________________G7$/*(-%%sqrtG6#F0F.%\".GF.)\"#5,$\"\"&F1F.%10.0 0001414213562G%-0.9999999999G7$/*(F0F.F@F.)FB,$FDF1F.%10.0000200000000 0G%.0~.9999999998G7$/*(-F>6#\"\"'F.F@F.)FB,$FDF1F.%10.00002449489743G% -0.9999999997G7$/*(-F>6#\"\")F.F@F.)FB,$FDF1F.%10.00002828427124G%-0.9 999999996G7$/*(-F>6#FBF.F@F.)FB,$FDF1F.%10.00003162277660G%-0.99999999 95G7$/*(-F>6#\"#7F.F@F.)FB,$FDF1F.%10.00003464101615G%.0~.9999999994G7 $/*(-F>6#\"#9F.F@F.)FB,$FDF1F.%10.00003741657387G%.0~.9999999993G7$/*( F6F.F@F.)FB,$FDF1F.%10.00004000000000G%-0.9999999992G" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 67 "The thresholds where the 10 digit floating point representation of " } {XPPEDIT 18 0 "cos(x);" "6#-%$cosG6#%\"xG" }{TEXT -1 1 " " }{TEXT 394 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1-x^2/2" "6#,&\"\"\"F$*&%\"xG\" \"#F'!\"\"F(" }{TEXT -1 119 " suddenly changes from one floating point value to the next correspond to the discontinuities or jumps in the g raph of " }{XPPEDIT 18 0 "g(x) = (1-cos(x))/(x^2);" "6#/-%\"gG6#%\"xG* &,&\"\"\"F*-%$cosG6#F'!\"\"F**$F'\"\"#F." }{TEXT -1 27 " and are tabul ated below. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matr ix([[x, cos(x) = 1-x^2/2+O(x^4)], [____________________________, _____ ______________], [10^(-5) = `0.00001000000000`, `0.99999999995`], [sqr t(3)*`.`*10^(-5) = `0.00001732050808`, `0 .99999999985`], [sqrt(5)*`.` *10^(-5) = `0.00002236067977`, `0.99999999975`], [sqrt(7)*`.`*10^(-5) \+ = `0.00002645751311`, `0.99999999965`], [3*`.`*10^(-5) = `0.0000300000 0000`, `0.99999999955`], [sqrt(11)*`.`*10^(-5) = `0.00003316624791`, ` 0 .99999999945`], [sqrt(13)*`.`*10^(-5) = `0.00003605551276`, `0 .9999 9999935`], [sqrt(15)*`.`*10^(-5) = `0.00003872983346`, `0.99999999925` ]]);" "6#-%'matrixG6#7,7$%\"xG/-%$cosG6#F(,(\"\"\"F.*&F(\"\"#F0!\"\"F1 -%\"OG6#*$F(\"\"%F.7$%=____________________________G%4________________ ___G7$/)\"#5,$\"\"&F1%10.00001000000000G%.0.99999999995G7$/*(-%%sqrtG6 #\"\"$F.%\".GF.)F=,$F?F1F.%10.00001732050808G%/0~.99999999985G7$/*(-FF 6#F?F.FIF.)F=,$F?F1F.%10.00002236067977G%.0.99999999975G7$/*(-FF6#\"\" (F.FIF.)F=,$F?F1F.%10.00002645751311G%.0.99999999965G7$/*(FHF.FIF.)F=, $F?F1F.%10.00003000000000G%.0.99999999955G7$/*(-FF6#\"#6F.FIF.)F=,$F?F 1F.%10.00003316624791G%/0~.99999999945G7$/*(-FF6#\"#8F.FIF.)F=,$F?F1F. %10.00003605551276G%/0~.99999999935G7$/*(-FF6#\"#:F.FIF.)F=,$F?F1F.%10 .00003872983346G%.0.99999999925G" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 21 ": In \+ the graph below " }{XPPEDIT 18 0 "g(x) = (1-cos(x))/(x^2);" "6#/-%\"gG 6#%\"xG*&,&\"\"\"F*-%$cosG6#F'!\"\"F**$F'\"\"#F." }{TEXT -1 31 " is pl otted over the interval " }{XPPEDIT 18 0 "-4.1;" "6#,$-%&FloatG6$\"#T !\"\"F(" }{TEXT -1 1 " " }{TEXT 387 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-5) <= x;" "6#1)\"#5,$\"\"&!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 4.1;" "6#1%!G-%&FloatG6$\"#T!\"\"" }{TEXT -1 1 " " }{TEXT 386 1 "x" } {TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-5);" "6#)\"#5,$\"\"&!\"\"" }{TEXT -1 54 " with 10 digit software floating point arithmetic (in " }{TEXT 260 3 "red" }{TEXT -1 34 ") and with 20 digit precision (in " }{TEXT 261 5 "brown" }{TEXT -1 17 "). The values of " }{TEXT 389 1 "x" } {TEXT -1 85 " for which the error in the lower precision values is eff ectively 0 are shown by the " }{TEXT 257 5 "green" }{TEXT -1 43 " vert ical dashed lines. They are values of " }{TEXT 388 1 "x" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "cos(x);" "6#-%$cosG6#%\"xG" }{TEXT -1 1 " " }{TEXT 392 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1-x^2/2" "6#,&\" \"\"F$*&%\"xG\"\"#F'!\"\"F(" }{TEXT -1 8 " has an " }{TEXT 259 5 "exac t" }{TEXT -1 125 " 10 digit floating point representation. The thresho lds where the last digit in the10 digit floating point representation \+ of " }{XPPEDIT 18 0 "cos(x)" "6#-%$cosG6#%\"xG" }{TEXT -1 26 " changes are shown by the " }{TEXT 256 4 "blue" }{TEXT -1 17 " vertical lines. " }}{PARA 0 "" 0 "" {TEXT -1 20 "In particular, when " }{XPPEDIT 18 0 "-10^(-5) < x;" "6#2,$)\"#5,$\"\"&!\"\"F)%\"xG" }{XPPEDIT 18 0 "`` < 10^(-5);" "6#2%!G)\"#5,$\"\"&!\"\"" }{TEXT -1 47 " the 10 digit float ing point representation of " }{XPPEDIT 18 0 "cos(x);" "6#-%$cosG6#%\" xG" }{TEXT -1 24 " is 1.000000000 so that " }{XPPEDIT 18 0 "1-cos(x); " "6#,&\"\"\"F$-%$cosG6#%\"xG!\"\"" }{TEXT -1 49 " evaluates to 0 rath er than to a number close to " }{XPPEDIT 18 0 "x^2/2;" "6#*&%\"xG\"\"# F%!\"\"" }{TEXT -1 11 ", and then " }{XPPEDIT 18 0 "(1-cos(x))/(x^2); " "6#*&,&\"\"\"F%-%$cosG6#%\"xG!\"\"F%*$F)\"\"#F*" }{TEXT -1 53 " also evaluates to 0 instead of to a number close to " }{XPPEDIT 18 0 "1/2; " "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 622 "g := x -> (1-cos(x)) /x^2:\n'g(x)'=g(x);\ny0 := -.1: y1 := 1.1:\np1 := plot(['evalf[10](g(x ))','evalf[20](g(x))'],x=-4e-5-1e-6..4e-5+1e-6,y0..y1,\n \+ color=[red,brown]):\nfor i to 16 do\n x||i := evalf(sqrt(i)*10^(-5)) ;\nend do: \np2 := plot([seq([[x||(2*i),y0],[x||(2*i),y1]],i=1..8),\n \+ seq([[-x||(2*i),y0],[-x||(2*i),y1]],i=1..8)],\n color=COLOR(RG B,0,.7,0),linestyle=3):\np3 := plot([seq([[x||(2*i-1),y0],[x||(2*i-1), y1]],i=1..8),\n seq([[-x||(2*i-1),y0],[-x||(2*i-1),y1]],i=1..8)], \n color=blue,linestyle=3):\nplots[display]([p1,p2,p3],axes=framed ,font=[HELVETICA,9],view=[-4e-5-1e-6..4e-5+1e-6,y0..y1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&,&\"\"\"F*-%$cosGF&!\"\"F*F'!\" #" }}{PARA 13 "" 1 "" {GLPLOT2D 767 353 353 {PLOTDATA 2 "6H-%'CURVESG6 $7ix7$$!#T!\"'$\"+\")>2fZ!#57$$!+o;j5S!#9$\"+Ak_t\\F-7$$!+OLE@RF1$\"+R +\"G?&F-7$$!+4O#=!RF1$\"+GAya_F-7$$!+#)QQ#)QF1$\"+sq`2`F-7$$!+]R_xQF1$ \"+<*\\3K&F-7$$!+=SmsQF1$\"+'HOum%F-7$$!+'3/y'QF1$\"+?IaF-7$$!+5#4lWF-7$$!+i'p4 &HF1$\"+*=gLf%F-7$$!+\\x\"*3HF1$\"+O_7FZF-7$$!+_u$>#GF1$\"+oO-B]F-7$$! 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" }}{PARA 0 "" 0 "" {TEXT -1 20 "(a) Plot a graph of " }{XPPEDIT 18 0 "h(x);" "6 #-%\"hG6#%\"xG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "-5* Pi <= x;" "6#1,$*&\"\"&\"\"\"%#PiGF'!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= \+ 5*Pi;" "6#1%!G*&\"\"&\"\"\"%#PiGF'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "(b) Use the expressions " }{TEXT 260 17 "'evalf[10](h(x))'" }{TEXT -1 5 " and " }{TEXT 260 17 "'evalf[20](h(x))'" }{TEXT -1 46 " in a suitable plot command to pl ot graphs of " }{XPPEDIT 18 0 "h(x);" "6#-%\"hG6#%\"xG" }{TEXT -1 18 " over the interval" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "-7;" "6#,$\"\"(!\"\"" }{TEXT -1 1 " " }{TEXT 347 1 "x" }{TEXT -1 2 " 1" }{XPPEDIT 18 0 "0^(-5) <= x;" "6#1)\"\"!,$\"\"&!\"\"%\"xG" } {XPPEDIT 18 0 "`` <= 7;" "6#1%!G\"\"(" }{TEXT -1 1 " " }{TEXT 346 1 "x " }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-5);" "6#)\"#5,$\"\"&!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 53 "with 10 digit precision \+ and with 20 digit precision. " }{TEXT 259 4 "Note" }{TEXT -1 37 ": It \+ is essential to use the quotes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 55 "(c) Find the first 4 terms of the Maclaur in series for " }{XPPEDIT 18 0 "h(x);" "6#-%\"hG6#%\"xG" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 15 "(d) Check that " }{XPPEDIT 18 0 "Lim it(h(x),x = 0) = 1/6;" "6#/-%&LimitG6$-%\"hG6#%\"xG/F*\"\"!*&\"\"\"F. \"\"'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "(e) Let " } {XPPEDIT 18 0 "x[0] = 32;" "6#/&%\"xG6#\"\"!\"#K" }{TEXT -1 1 " " } {TEXT 352 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-6);" "6#)\"#5,$\" \"'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 33 " (i) Find \+ the sum of the first " }{TEXT 262 7 "4 terms" }{TEXT -1 29 " of the Ma claurin series for " }{XPPEDIT 18 0 "h(x) = (x-sin(x))/(x^3);" "6#/-% \"hG6#%\"xG*&,&F'\"\"\"-%$sinG6#F'!\"\"F**$F'\"\"$F." }{TEXT -1 14 " e valuated at " }{XPPEDIT 18 0 "x=x[0]" "6#/%\"xG&F$6#\"\"!" }{TEXT -1 82 ", performing the calculation with Maple's base 10 floating point a rithmetic using " }{TEXT 262 9 "35 digits" }{TEXT -1 15 " of precision . " }}{PARA 0 "" 0 "" {TEXT -1 26 " (ii) Find the value of " } {XPPEDIT 18 0 "h(x[0]);" "6#-%\"hG6#&%\"xG6#\"\"!" }{TEXT -1 34 ", per forming the calculation with " }{TEXT 262 9 "44 digits" }{TEXT -1 14 " of precision." }}{PARA 0 "" 0 "" {TEXT -1 27 " (iii) Find the value of " }{XPPEDIT 18 0 "h(x[0]);" "6#-%\"hG6#&%\"xG6#\"\"!" }{TEXT -1 34 ", performing the calculation with " }{TEXT 262 9 "10 digits" } {TEXT -1 15 " of precision. " }}{PARA 0 "" 0 "" {TEXT -1 3 " " } {TEXT 259 4 "Note" }{TEXT -1 54 " for (iii): In order to prevent Maple from evaluating " }{XPPEDIT 18 0 "h(x[0]);" "6#-%\"hG6#&%\"xG6#\"\"! " }{TEXT -1 4 " as " }{XPPEDIT 18 0 "1/(x[0]^2)-sin(x)/(x[0]^3);" "6#, &*&\"\"\"F%*$&%\"xG6#\"\"!\"\"#!\"\"F%*&-%$sinG6#F(F%*$&F(6#F*\"\"$F,F ," }{TEXT -1 10 ", compute " }{XPPEDIT 18 0 "h(x[0]);" "6#-%\"hG6#&%\" xG6#\"\"!" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"! " }{TEXT -1 45 " first evaluated as a floating point number. " }} {PARA 0 "" 0 "" {TEXT -1 28 " Also execute the command " }{TEXT 260 13 "forget(evalf)" }{TEXT -1 34 " before performing the evaluation." } }{PARA 0 "" 0 "" {TEXT -1 18 " (iv) Calculate " }{XPPEDIT 18 0 "sin( x[0]);" "6#-%$sinG6#&%\"xG6#\"\"!" }{TEXT -1 34 ", performing the calc ulation with " }{TEXT 262 9 "10 digits" }{TEXT -1 54 " of precision, a nd check that using the approximation " }{XPPEDIT 18 0 "sin(x);" "6#-% $sinG6#%\"xG" }{TEXT -1 1 " " }{TEXT 353 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x-x^3/6;" "6#,&%\"xG\"\"\"*&F$\"\"$\"\"'!\"\"F)" } {TEXT -1 64 ", obtained from the first two terms of the Maclaurin seri es for " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 24 ", g ives the same value. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 " (v) Use the answer from (iv) to explain the result \+ of (iii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "(e) Give one positive and one negative value of " }{TEXT 351 1 "x " }{TEXT -1 37 " for which the error in the value of " }{XPPEDIT 18 0 "h(x);" "6#-%\"hG6#%\"xG" }{TEXT -1 65 " calculated using 10 digit pre cision is effectively equal to 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT 349 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "h : = x -> (x-sin(x))/x^3:\n'h(x)'=h(x);\nplot(h(x),x=-5*Pi..5*Pi);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"xG*&,&F'\"\"\"-%$sinGF&!\" \"F*F'!\"$" }}{PARA 13 "" 1 "" {GLPLOT2D 525 362 362 {PLOTDATA 2 "6%-% 'CURVESG6$7eo7$$!3'****\\OK'zq:!#;$\"3cIx!QNZG0%!#?7$$!3WNzQW&=B]\"F*$ \"3+V7Ft`>WUF-7$$!3742![ROFW\"F*$\"3_d\"=i3#=&[%F-7$$!3Ext3O+tv8F*$\"3 &oF\\.=hp#\\F-7$$!3'y2Ld^z#38F*$\"3.9o^VM*>i&F-7$$!3')zz#fd\\6C\"F*$\" 3o%>?O(4GslF-7$$!3U:#)\\G:\"*y6F*$\"3&>S/9)R9BwF-7$$!3,%4``jnW6\"F*$\" 3k2M&oJ+dw)F-7$$!3ynW()o'>y/\"F*$\"3'oD8\"4ebj)*F-7$$!3/WfpKW&Q\")*!#< $\"3ieARh\"G%y5!#>7$$!33&e1Vw'\\I\"*FY$\"3CtVWfUUh6Ffn7$$!331*H!e\\fG& )FY$\"3s.D(*yx#*[7Ffn7$$!3w`u@%3'*4&yFY$\"3puhcnws:9Ffn7$$!3smnKC\\hqr FY$\"3eJKZ`D`MDIdHF#Ffn7$$!3X!RBoTF&>f FY$\"3?c&Hl6(HDIFfn7$$!3O,(RrNA:@&FY$\"3sn'\\Hn'=-VFfn7$$!3Zu`xzdj6\\F Y$\"3yZ42NM[s\\Ffn7$$!3eZ5T-#\\FY$\"3WIS'e)zcq8F]s7$$!3$=R%=By8P;FY$ \"3mYRkY-0d9F]s7$$!3JTWU;s`+8FY$\"3khwoM=EJ:F]s7$$!3NMU\"3%yK9**F]s$\" 3GEq$)\\hk'e\"F]s7$$!3fbSQEYyM2 rFfn$\"3cqN,kdCm;F]s7$$\"3%)o2$)4``w:F]s$\"3!*y7Hsmfk;F]s7$$\"3-V`.NeL UCF]s$\"3k[bl$)Gqh;F]s7$$\"3><*R-OO\"3LF]s$\"3'3+M&)eqvl\"F]s7$$\"3bl! \\1TP(R]F]s$\"3;S#e#>%Gck\"F]s7$$\"3!R@e5YQ8x'F]s$\"3ujc^g>()G;F]s7$$ \"3/fy_qg1'y*F]s$\"3Gss&RZc')e\"F]s7$$\"3U](**zOz+G\"FY$\"3\"GL7iOC`` \"F]s7$$\"3mT'Q*3#ycg\"FY$\"3-?puCQak9F]s7$$\"3!H`x)\\qFJ>FY$\"3eru)>K n?Q\"F]s7$$\"3c>pV7DlnAFY$\"3Of#oc8MqG\"F]s7$$\"3B1j*\\(z-/EFY$\"3yrs# HfMZ=\"F]s7$$\"3w#\\l.u0J$HFY$\"3e+If8dM!3\"F]s7$$\"3IzYt0N=iKFY$\"3]C lx]jRV(*Ffn7$$\"37/GfkJe!e$FY$\"3/A=yA/%es)Ffn7$$\"3%*G4XBG)*)*QFY$\"3 .L$)oY1:PxFfn7$$\"3GFr&*e8^_UFY$\"3?Yx1$Qc]p'Ffn7$$\"3iDLY%*)Rgg%FY$\" 3'GN\")\\?j5t&Ffn7$$\"3IoCV25qB\\FY$\"3sJ_'39mS%\\Ffn7$$\"3)4h,/7i8C&F Y$\"3za9@n$f'RUFfn7$$\"3xMYS.Uq>fFY$\"3t1Q-_;.DIFfn7$$\"3%>oi?&HQMlFY$ \"3uwRi8@$HD#Ffn7$$\"3)*p$zA=*Q1sFY$\"3P8SaUf[79Ffn7$$\"3K(G\"*Q5O'*\\)FY$\"3AOL/-*GTD\"Ffn7$$\"3qenYgr#e 9*FY$\"3k&H(Ha(>&f6Ffn7$$\"3,B5$3JHB#)*FY$\"3S+\"z\"[%ft2\"Ffn7$$\"3%3 '>eG\")QZ5F*$\"3iGr4PA8q)*F-7$$\"3:K#)fG(=S6\"F*$\"3MPk)fYHNx)F-7$$\"3 '\\I0Of(4!=\"F*$\"3'\\c'>=XA-wF-7$$\"3rm8'y?<3C\"F*$\"3en;08DaxlF-7$$ \"3BnP+Q(3/J\"F*$\"3#f3U'\\v\"ff&F-7$$\"3#Ryp@B_EP\"F*$\"3#H`xPMfG&\\F -7$$\"3hNK@zn,R9F*$\"3&3*H939A/XF-7$$\"3x;#f'=h`-:F*$\"34SGeb0\\VUF-7$ $\"3'****\\OK'zq:F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!FiblFhbl-%+AXES LABELSG6$Q\"x6\"Q!F^cl-%%VIEWG6$;$!+Fjzq:!\")$\"+Fjzq:Ffcl%(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 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 42 ": There should be a \"missi ng point\" where " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 5 " \+ and " }{XPPEDIT 18 0 "y = 1/6;" "6#/%\"yG*&\"\"\"F&\"\"'!\"\"" }{TEXT -1 13 ", but, since " }{XPPEDIT 18 0 "Limit(h(x),x = 0) = 1/6;" "6#/-% &LimitG6$-%\"hG6#%\"xG/F*\"\"!*&\"\"\"F.\"\"'!\"\"" }{TEXT -1 120 " (s ee below), we can fill it in and obtain a continuous function, indeed \+ one which has derivatives of all orders at 0. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }{XPPEDIT 18 0 "Limit((x-sin(x))/x^3,x = 0)=1/6 " "6#/-%&LimitG6$*&,&%\"xG\"\"\"-%$sinG6#F)!\"\"F**$F)\"\"$F./F)\"\"!* &F*F*\"\"'F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "h := x -> (x-sin(x))/x^3:\nL imit(h(x),x=0);\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&Li mitG6$*&,&%\"xG\"\"\"-%$sinG6#F(!\"\"F)F(!\"$/F(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"\"\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "(e) (i) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "x0 := 32*10^(-6);\nEval(p(x),x=x0);\n``=value(%);\nev alf[35](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G#\"\"\"\"&]7$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$,*#\"\"\"\"\"'F(*&#F(\"$?\"F (*$)%\"xG\"\"#F(F(!\"\"*&#F(\"%S]F(*$)F/\"\"%F(F(F(*&#F(\"'!)GOF(*$)F/ F)F(F(F1/F/#F(\"&]7$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"@dGkWDK qqulU/Fm/)\"A++++++]iSm!oAwz#[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G $\"D<\")p7%QTNLLL8emmmm;!#N" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "(e) (ii) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "h := x -> (x-sin(x))/x^3:\nx0 := 32*10^(-6);\nEval(h(x),x=x0); \n``=value(%);\nevalf[44](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0 G#\"\"\"\"&]7$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$*&,&%\"xG \"\"\"-%$sinG6#F(!\"\"F)F(!\"$/F(#F)\"&]7$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&\"*+Dcw*\"\"\"*&\"/+]7yv^IF'-%$sinG6##F'\"&]7$F'! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"D<\")p7%QTNLLL8emmmm;!# N" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "(e) (iii) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "forget(evalf);\nh := x -> (x-sin(x))/x^3:\nx0 := evalf(32*10^(-6));\nEval(h(x),x=x0);\n ``=value(%);\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"+++++K!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$*&,&%\"xG\"\"\"-%$sinG6#F( !\"\"F)F(!\"$/F($\"+++++K!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$ \"+7yv^I!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "(e) (iv) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "taylor(si n(x),x=0,10):\nq := unapply(convert(%,polynom),x):\n'q(x)'=q(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG,,F'\"\"\"*&#F)\"\"'F)*$ )F'\"\"$F)F)!\"\"*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"%S]F)*$)F'\"\"(F) F)F0*&#F)\"'!)GOF)*$)F'\"\"*F)F)F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "Digits := 10:\nx0 := evalf (32*10^(-6));\nEval(sin(x),x=x0);\ny0 := value(%);\nEval(x-x^3/6,x=x0) ;\n``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"+++++K!#9 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$-%$sinG6#%\"xG/F)$\"++++ +K!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G$\"+*******>$!#9" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$,&%\"xG\"\"\"*&#F(\"\"'F(*$) F'\"\"$F(F(!\"\"/F'$\"+++++K!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%! G$\"+*******>$!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "(e) (v) " }}{PARA 0 "" 0 "" {TEXT -1 40 "We have the 10 di git value 0.3199999999 " }{TEXT 368 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-6)" "6#)\"#5,$\"\"'!\"\"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "sin(x[0]);" "6#-%$sinG6#&%\"xG6#\"\"!" }{TEXT -1 13 " which gives \+ " }{XPPEDIT 18 0 "x-sin(x[0]);" "6#,&%\"xG\"\"\"-%$sinG6#&F$6#\"\"!!\" \"" }{TEXT -1 1 " " }{TEXT 354 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "1 0^(-14)" "6#)\"#5,$\"#9!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x[0] -sin(x[0]))/(x[0]^3);" "6#*&,&&%\"xG6#\"\"!\"\"\"-%$sinG6#&F&6#F(!\"\" F)*$&F&6#F(\"\"$F/" }{TEXT -1 1 " " }{TEXT 355 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "10^(-14)/(32^3*`.`*10^(-18));" "6#*&)\"#5,$\"#9!\"\"\" \"\"*(\"#K\"\"$%\".GF))F%,$\"#=F(F)F(" }{TEXT -1 1 " " }{TEXT 356 1 "~ " }{TEXT -1 1 " " }{XPPEDIT 18 0 "10000/32768;" "6#*&\"&++\"\"\"\"\"&o F$!\"\"" }{TEXT -1 1 " " }{TEXT 357 1 "~" }{TEXT -1 15 " 0.3051757812. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "10000/32^3;\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$D'\"%[?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7yv^I!#5" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "A brief s ummary of the error problem here is that " }{XPPEDIT 18 0 "x-sin(x);" "6#,&%\"xG\"\"\"-%$sinG6#F$!\"\"" }{TEXT -1 1 " " }{TEXT 363 1 "~" } {TEXT -1 1 " " }{XPPEDIT 18 0 "x-``(x-x^3/6);" "6#,&%\"xG\"\"\"-%!G6#, &F$F%*&F$\"\"$\"\"'!\"\"F-F-" }{TEXT -1 6 " when " }{TEXT 364 1 "x" } {TEXT -1 66 " is near 0. A large relative error may occur in the evalu ation of " }{XPPEDIT 18 0 "x-sin(x);" "6#,&%\"xG\"\"\"-%$sinG6#F$!\"\" " }{TEXT -1 98 " since the last step in the evaluation will involve th e subtraction of \"nearly equal quantities\". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "sn0 := 0.00 003199999999;\nx0 := evalf(32*10^(-6));\nnum := x0-sn0;\nnum/x0^3;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sn0G$\"+*******>$!#9" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#x0G$\"+++++K!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$numG$\"\"\"!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+7yv^I!# 5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(f) " }}{PARA 0 "" 0 "" {TEXT -1 50 "The error is effectively 0 when the approximation " } {XPPEDIT 18 0 "x-sin(x);" "6#,&%\"xG\"\"\"-%$sinG6#F$!\"\"" }{TEXT -1 1 " " }{TEXT 366 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^3/6-x^5/120; " "6#,&*&%\"xG\"\"$\"\"'!\"\"\"\"\"*&F%\"\"&\"$?\"F(F(" }{TEXT -1 46 " has an exact floating point representation. \n" }}{PARA 0 "" 0 "" {TEXT -1 44 "The following table indicates the values of " }{TEXT 350 1 "x" }{TEXT -1 17 " in the interval " }{XPPEDIT 18 0 "0 <= x;" "6#1\" \"!%\"xG" }{XPPEDIT 18 0 "`` <= 7;" "6#1%!G\"\"(" }{TEXT -1 1 " " } {TEXT 358 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-5);" "6#)\"#5,$\" \"&!\"\"" }{TEXT -1 29 " for which this is the case. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "ma trix([[x, x-sin(x) = x^3/6-x^5/120+O(x^7)], [_________________, ______ ____________________], [3.914867641*`.`*10^(-5), 10^(-14)], [4.9324241 49*`.`*10^(-5), 2*`.`*10^(-14)], [5.646216174*`.`*10^(-5), 3*`.`*10^(- 14)], [6.214465012*`.`*10^(-5), 4*`.`*10^(-14)], [6.694329501*`.`*10^( -5), 5*`.`*10^(-14)]]);" "6#-%'matrixG6#7)7$%\"xG/,&F(\"\"\"-%$sinG6#F (!\"\",(*&F(\"\"$\"\"'F/F+*&F(\"\"&\"$?\"F/F/-%\"OG6#*$F(\"\"(F+7$%2__ _______________G%;__________________________G7$*(-%&FloatG6$\"+Tw'[\"R !\"*F+%\".GF+)\"#5,$F5F/F+)FH,$\"#9F/7$*(-FB6$\"+\\TUK\\FEF+FFF+)FH,$F 5F/F+*(\"\"#F+FFF+)FH,$FLF/F+7$*(-FB6$\"+uh@YcFEF+FFF+)FH,$F5F/F+*(F2F +FFF+)FH,$FLF/F+7$*(-FB6$\"+7]Y9iFEF+FFF+)FH,$F5F/F+*(\"\"%F+FFF+)FH,$ FLF/F+7$*(-FB6$\"+,&HVp'FEF+FFF+)FH,$F5F/F+*(F5F+FFF+)FH,$FLF/F+" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 67 "The thresholds where the 10 digit floating point represen tation of " }{XPPEDIT 18 0 "sin(x);" "6#-%$sinG6#%\"xG" }{TEXT -1 1 " \+ " }{TEXT 367 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x-x^3/6+x^5/120;" " 6#,(%\"xG\"\"\"*&F$\"\"$\"\"'!\"\"F)*&F$\"\"&\"$?\"F)F%" }{TEXT -1 119 " suddenly changes from one floating point value to the next corre spond to the discontinuities or jumps in the graph of " }{XPPEDIT 18 0 "h(x) = (x-sin(x))/(x^3);" "6#/-%\"hG6#%\"xG*&,&F'\"\"\"-%$sinG6#F'! \"\"F**$F'\"\"$F." }{TEXT -1 26 " and are tabulated below. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[x, x-sin(x) = x^3/6-x^5/120+O(x^7)], [_________________ , __________________________], [3.107232506*`.`*10^(-5), 5*`.`*10^(-15 )], [4.481404747*`.`*10^(-5), 15*`.`*10^(-15)], [5.313292846*`.`*10^(- 5), 25*`.`*10^(-15)], [5.943921953*`.`*10^(-5), 35*`.`*10^(-15)], [6.9 10423231*`.`*10^(-5), 45*`.`*10^(-15)]])" "6#-%'matrixG6#7)7$%\"xG/,&F (\"\"\"-%$sinG6#F(!\"\",(*&F(\"\"$\"\"'F/F+*&F(\"\"&\"$?\"F/F/-%\"OG6# *$F(\"\"(F+7$%2_________________G%;__________________________G7$*(-%&F loatG6$\"+1DB2J!\"*F+%\".GF+)\"#5,$F5F/F+*(F5F+FFF+)FH,$\"#:F/F+7$*(-F B6$\"+ZZS\"[%FEF+FFF+)FH,$F5F/F+*(FMF+FFF+)FH,$FMF/F+7$*(-FB6$\"+YGH8` FEF+FFF+)FH,$F5F/F+*(\"#DF+FFF+)FH,$FMF/F+7$*(-FB6$\"+`>#R%fFEF+FFF+)F H,$F5F/F+*(\"#NF+FFF+)FH,$FMF/F+7$*(-FB6$\"+JKU5pFEF+FFF+)FH,$F5F/F+*( \"#XF+FFF+)FH,$FMF/F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 20 "In particular, when " }{XPPEDIT 18 0 "-3.107232506*`.`*10^(-5) " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 13 "Calculations " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "y0 := 10^(-14);\nx-sin(x)=y0;\nevalf(solve(%)); \nx^3/6=y0;\nop(remove(has,[solve(%)],Complex(1)));\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G#\"\"\"\"0+++++++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"xG\"\"\"-%$sinG6#F%!\"\"#F&\"0+++++++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Tw'[\"R!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,$*&\"\"'!\"\"%\"xG\"\"$\"\"\"#F*\"0+++++++\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"'++5!\"\"\"#g#\"\"\"\"\"$F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Tw'[\"R!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 127 "y0 := 5*10^ (-14);\nx-sin(x)=y0;\nx=evalf(solve(%));\nx^3/6-1/120*x^5=y0;\n[fsolve (%,x=0)]:\nx=op(select(_u->evalb(abs(_u)<1e-3),%));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#y0G#\"\"\"\"/++++++?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%\"xG\"\"\"-%$sinG6#F%!\"\"#F&\"/++++++?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG$\"+,&HVp'!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /,&*&#\"\"\"\"\"'F'*$)%\"xG\"\"$F'F'F'*&#F'\"$?\"F'*$)F+\"\"&F'F'!\"\" #F'\"/++++++?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG$\"+,&HVp'!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 128 "for i to 12 do\n x-sin(x)=5*i*10^(-15);\n x||i := evalf(e valf[15](fsolve(%,x=0..1e-4)));\nend do:\nxvals := [seq(x||i,i=1..12)] ;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&xvalsG7.$\"+1DB2J!#9$\"+Tw'[\" RF($\"+ZZS\"[%F($\"+\\TUK\\F($\"+YGH8`F($\"+uh@YcF($\"+`>#R%fF($\"+7]Y 9iF($\"+rSIjkF($\"+,&HVp'F($\"+JKU5pF($\"+5my8rF(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 21 ": In the graph below " }{XPPEDIT 18 0 "h(x);" "6#-%\"hG6#%\"xG " }{TEXT -1 31 " is plotted over the interval " }{XPPEDIT 18 0 "-7.1; " "6#,$-%&FloatG6$\"#r!\"\"F(" }{TEXT -1 1 " " }{TEXT 360 1 "x" } {TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-5) <= x;" "6#1)\"#5,$\"\"&!\"\"%\" xG" }{XPPEDIT 18 0 "`` <= 7.1;" "6#1%!G-%&FloatG6$\"#r!\"\"" }{TEXT -1 1 " " }{TEXT 359 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-5);" "6 #)\"#5,$\"\"&!\"\"" }{TEXT -1 53 " with 10 digit sofware floating poin t arithmetic (in " }{TEXT 260 3 "red" }{TEXT -1 34 ") and with 20 digi t precision (in " }{TEXT 261 5 "brown" }{TEXT -1 17 "). The values of \+ " }{TEXT 362 1 "x" }{TEXT -1 55 " for which the error is effectively 0 are shown by the " }{TEXT 257 5 "green" }{TEXT -1 43 " vertical dashe d lines. They are values of " }{TEXT 361 1 "x" }{TEXT -1 11 " such tha t " }{XPPEDIT 18 0 "x-sin(x);" "6#,&%\"xG\"\"\"-%$sinG6#F$!\"\"" } {TEXT -1 1 " " }{TEXT 365 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^3/6- x^5/120;" "6#,&*&%\"xG\"\"$\"\"'!\"\"\"\"\"*&F%\"\"&\"$?\"F(F(" } {TEXT -1 8 " has an " }{TEXT 259 5 "exact" }{TEXT -1 125 " 10 digit fl oating point representation. The thresholds where the last digit in th e10 digit floating point representation of " }{XPPEDIT 18 0 "x-sin(x); " "6#,&%\"xG\"\"\"-%$sinG6#F$!\"\"" }{TEXT -1 26 " changes are shown b y the " }{TEXT 256 4 "blue" }{TEXT -1 17 " vertical lines. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 663 "h := x -> (x-sin(x))/x^3:\n'h(x)'=h(x);\ny0 := -.1: y1 := 0.4:\np1 := p lot(['evalf[10](h(x))','evalf[20](h(x))'],x=-7e-5-1e-6..7e-5+1e-6,y0.. y1,\n color=[red,brown]):\nfor i to 12 do\n x-sin(x)=5* i*10^(-15);\n x||i := evalf(evalf[15](fsolve(%,x=0..1e-4)));\nend do : \np2 := plot([seq([[x||(2*i),y0],[x||(2*i),y1]],i=1..6),\n seq([ [-x||(2*i),y0],[-x||(2*i),y1]],i=1..6)],\n color=COLOR(RGB,0,.7,0) ,linestyle=3):\np3 := plot([seq([[x||(2*i-1),y0],[x||(2*i-1),y1]],i=1. .6),\n seq([[-x||(2*i-1),y0],[-x||(2*i-1),y1]],i=1..6)],\n col or=blue,linestyle=3):\nplots[display]([p1,p2,p3],axes=framed,font=[HEL VETICA,9],view=[-7e-5-1e-6..7e-5+1e-6,y0..y1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"hG6#%\"xG*&,&F'\"\"\"-%$sinGF&!\"\"F*F'!\"$" }} {PARA 13 "" 1 "" {GLPLOT2D 767 353 353 {PLOTDATA 2 "6@-%'CURVESG6$7_v7 $$!#r!\"'$\"+6WRw;!#57$$!++,iAq!#9$\"+Q@UKR-1=F-7$$!+XFA;pF1$\"+4=h 8=F-7$$!+g-b1pF1$\"+*)=q<:F-7$$!+&G0s)oF1$\"+Nn_I:F-7$$!+5.'y'oF1$\"+& ['\\V:F-7$$!+g.hZ$=F-7$$!+Ie%QZ'F1$\"+B,#G%=F-7$$!+!=\"QkkF1$\"+Fj#4&=F-7 $$!+Il\"\\X'F1$\"+:TE([\"F-7$$!+!)=XXkF1$\"+Rb#Q\\\"F-7$$!+&G$f2kF1$\" +G1Y?:F-7$$!+!pM(pjF1$\"+2FtZ:F-7$$!+&\\F-7$$!+qyd>`F1$\"+O%>H*>F-7$$!+q^Z5`F1$\"+8oXN8F-7$$!+qCP,`F1$\"+`xM U8F-7$$!+q(pAH&F1$\"+&='G\\8F-7$$!+qV1u_F1$\"+mrIj8F-7$$!+l*eeD&F1$\"+ *3BvP\"F-7$$!+g\"[%>_F1$\"+jk:F-7$$!+K(pn)[F1$\"+(p;Qr\"F-7$$!+:` 9OZF1$\"+%\\'e#)=F-7$$!+MY2hYF1$\"+([@](>F-7$$!+_R+'e%F1$\"+'H3O2#F-7$ $!+6'o%[XF1$\"+M#p`7#F-7$$!+qK$4^%F1$\"+Lt')y@F-7$$!+N%\\:]%F1$\"+?=_# >#F-7$$!++c;#\\%F1$\"+R1H1AF-7$$!+lu7F-7$$!+(*[)p7%F1$\"+)ygEU\"F-7$$!+w]&4*RF1$\"+Pw9t:F-7$$!+c_# \\&QF1$\"+o'Hcu\"F-7$$!+)*yy,PF1$\"+wvNr>F-7$$!+R0l[NF1$\"+yMuPAF-7$$! 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His l atest book, Isaac Newton, was a Pulitzer Prize finalist this year and \+ a national bestseller, as were Chaos: Making a New Science (Viking Pen guin, 1987) and Genius: The Life and Science of Richard Feynman (Panth eon, 1992). His other books include Faster: The Acceleration of Just A bout Everything (Pantheon, 1999) and What Just Happened: A Chronicle f rom the Electronic Frontier (Pantheon, 2002)." }}{PARA 256 "" 0 "" {TEXT -1 32 " ______________________________ " }}{PARA 0 "" 0 "" {TEXT -1 9 " " }}{PARA 0 "" 0 "" {TEXT -1 6343 "It took the Eu ropean Space Agency 10 years and $7 billion to produce Ariane 5, a gia nt rocket capable of hurling a pair of three-ton satellites into orbit with each launch and intended to give Europe overwhelming supremacy i n the commercial space business.\n All it took to explode that rocke t less than a minute into its maiden voyage last June, scattering fier y rubble across the mangrove swamps of French Guiana, was a small comp uter program trying to stuff a 64-bit number into a 16-bit space.\n \+ One bug, one crash. Of all the careless lines of code recorded in the \+ annals of computer science, this one may stand as the most devastating ly efficient. From interviews with rocketry experts and an analysis pr epared for the space agency, a clear path from an arithmetic error to \+ total destruction emerges.\n To play the tape backward:\n At 39 se conds after launch, as the rocket reached an altitude of two and a hal f miles, a self-destruct mechanism finished off Ariane 5, along with i ts payload of four expensive and uninsured scientific satellites. Self -destruction was triggered automatically because aerodynamic forces we re ripping the boosters from the rocket.\n This disintegration had b egun an instant before, when the spacecraft swerved off course under t he pressure of the three powerful nozzles in its boosters and main eng ine. The rocket was making an abrupt course correction that was not ne eded, compensating for a wrong turn that had not taken place.\n Stee ring was controlled by the on-board computer, which mistakenly thought the rocket needed a course change because of numbers coming from the \+ inertial guidance system. That device uses gyroscopes and acceleromete rs to track motion. The numbers looked like flight data -- bizarre and impossible flight data -- but were actually a diagnostic error messag e. The guidance system had in fact shut down.\n This shutdown occurr ed 36.7 seconds after launch, when the guidance system's own computer \+ tried to convert one piece of data -- the sideways velocity of the roc ket -- from a 64-bit format to a 16-bit format. The number was too big , and an overflow error resulted. When the guidance system shut down, \+ it passed control to an identical, redundant unit, which was there to \+ provide backup in case of just such a failure. But the second unit had failed in the identical manner a few milliseconds before. And why not ? It was running the same software.\n This bug belongs to a species \+ that has existed since the first computer programmers realized they co uld store numbers as sequences of bits, atoms of data, ones and zeroes : 1001010001101001. . . . A bug like this might crash a spreadsheet or word processor on a bad day. Ordinarily, though, when a program conve rts data from one form to another, the conversions are protected by ex tra lines of code that watch for errors and recover gracefully. Indeed , many of the data conversions in the guidance system's programming in cluded such protection.\n But in this case, the programmers had deci ded that this particular velocity figure would never be large enough t o cause trouble. After all, it never had been before. Unluckily, Arian e 5 was a faster rocket than Ariane 4. One extra absurdity: the calcul ation containing the bug, which shut down the guidance system, which c onfused the on-board computer, which forced the rocket off course, act ually served no purpose once the rocket was in the air. Its only funct ion was to align the system before launch. So it should have been turn ed off. But engineers chose long ago, in an earlier version of the Ari ane, to leave this function running for the first 40 seconds of flight -- a \"special feature\" meant to make it easy to restart the system \+ in the event of a brief hold in the countdown.\n The Europeans hope \+ to launch a new Ariane 5 next spring, this time with a newly designate d \"software architect\" who will oversee a process of more intensive \+ and, they hope, realistic ground simulation. Simulation is the great h ope of software debuggers everywhere, though it can never anticipate e very feature of real life. \"Very tiny details can have terrible conse quences,\" says Jacques Durand, head of the project, in Paris. \"That' s not surprising, especially in a complex software system such as this is.\"\n These days, we have complex software systems everywhere. We have them in our dishwashers and in our wristwatches, though they're \+ not quite so mission-critical. We have computers in our cars -- from 1 5 to 50 microprocessors, depending how you count: in the engine, the t ransmission, the suspensions, the steering, the brakes and every other major subsystem. Each runs its own software, thoroughly tested, simul ated and debugged, no doubt.\n Bill Powers, vice president for resea rch at Ford, says that cars' computing power is increasingly devoted n ot just to actual control but to diagnostics and contingency planning \+ -- \"Should I abort the mission, and if I abort, where would I go?\" h e says. \"We also have what's called a limp-home strategy.\" That is, \+ in the worst case, the car is supposed to behave more or less normally , like a car of the pre-computer era, instead of, say, taking it upon \+ itself to swerve into the nearest tree.\n The European investigators chose not to single out any particular contractor or department for b lame. \"A decision was taken,\" they wrote. \"It was not analyzed or f ully understood.\" And \"the possible implications of allowing it to c ontinue to function during flight were not realized.\" They did not at tempt to calculate how much time or money was saved by omitting the st andard error-protection code.\n \"The board wishes to point out,\" t hey added, with the magnificent blandness of many official accident re ports, \"that software is an expression of a highly detailed design an d does not fail in the same sense as a mechanical system.\" No. It fai ls in a different sense. Software built up over years from millions of lines of code, branching and unfolding and intertwining, comes to beh ave more like an organism than a machine.\n \"There is no life today without software,\" says Frank Lanza, an executive vice president of \+ the American rocket maker Lockheed Martin. \"The world would probably \+ just collapse.\" Fortunately, he points out, really important software has a reliability of 99.9999999 percent. At least, until it doesn't. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "The Patriot Missile Failure" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 "Reference: " }{URLLINK 17 "The Patriot Mi ssile Failure" 4 "http://www.ima.umn.edu/~arnold/disasters/patriot.htm l" "" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 158 "Douglas N. Arnol d, Director of the Institute for Mathematics and its Applications in M inneapolis and Professor of Mathematics at the University of Minnesota . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 287 27 "The Patriot Missile Failure" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {OLE 1 182799 1 "[xm]Br=Wf oRrB:::wk;nyyI;G:;:j;>:=:;:<;?:=:wyyyyyB:J:C:c?yyyyyy::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::uyyyyya?wAwyyyqj>J?JDJ@j@>:W:YJ:><AFANAVA^AfAnA;jYJZB:=EFENEVE^EfEnEvE>F_lqvGcMJfBWMtNHm=;:: :::::nwhGYX>efI`:B:::::::JFNZ;f :vYxI>:<::::::JDJ:j:VBYmp>HYLkNG>::::::::N<^:vYxI>:<::::::=J:fH>:nYN:: wAyA:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::J:B:::U=;B:cl`>Duk[NAsL\\vCAldfCELyF@sl\\>`:f@=Jyyy;dZ :vYxYZ`MD;:V>ZFB::JQH]nIalHH>::::=J:D:<::::::::::yayY:;J<>:=Z:vYxY;J:JP M;]s=>_WG:>::::::::::::::::::::::::::::::::::::Nfr?l]GiKHfqok[whH<::ve o_kZohGHfqok[;JjTq@yotlrjXqAEPv;\\?kTUOVPWu\\otUS>q[E[yvUAfaWeZuVU=Fa[e[wfUCFaSeZovT=V`OEZkVTyu_EuXafRkE^KeYgFTwe_IUYeV 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`i`?lrOfJ;K^Al>S\\NKdrp^kg?eE`gQWkfwDbTgt\\O eOIfTXBB\\QM>S\\FKJ?c\\VkhkqZGLAc]^KiD]]uOUfVkd]aUPYvVqt]auP[FWsD^nkkc ]vklC^>llweKXhZOlnGfLTRVFH@::JDUqlLSuPpCQQlDStNL@ssDPBEqjBWDssVlm;DaNXHeFW;e_eURgVW==R_cERIw]m=EW]ZEDcfNPt@ YALDA_\\PK??Z]\\Cxg\\lJK>u`A<;?nMrULrq:[]i;Vm=T`ZSHJmVZJEuaN`i`olx?gPp IB_FlsWfKHi]Olt_F>LZILAUaGNKVHDA]=TRqZAL<_d@XgPOkg?eFxGR]:vJXBM>>KXBM ^CL?S\\Vkf;OvTgloTo?e`El=s[ZBL>S\\FK:drpNKhB:U>VKgD]]uOS^G\\lOeIXHY>nk kc]vKxB[fV@CZ[kkDC:::cfWLCa>NL@C:U^GTQWFWsd]vKxrtpOCs^VlqwEflpC_eMDQAB WuD_cUQafVHCcvVPCg^PlD?fO`HIovJMub_LMuRvTDIw]mEd[fXEW]iEcS>NK??Z]\\Cxg [\\B<;g[L< Cr]iMAUaGhHb]g=pBU^FL@;O>a]l;x`@pGr[fjdc[To>s\\WUP:U>vJXBKVU\\B:M^C>KgD\\FKdrpZEL@C]YM?gdDxgOWkjoDRVmT ]]UPWfVod]_eP@QhtBr]ZKlBIQjDCb^:::^lpC_VLJCS^:@p@QaGLAIQitrstoAMqi^lmC_auQGg]kecWnMybaxKK_b@L MW^qed_FYEW]iM@;ONK;rbFnBpg[<@hY `Jw^WLFC`;nEyaSlGIbUXJm^R@JcFYGU_umDqaMXI]VXAE^keR]^LLB]Qk:: ZILBS^nKtruZI@IYFX:;:S>njh?e@pGK^BK`RpFKj=epdX BK>FKdrpVKhBMNdngNOkigDVkiC][EPUVVmT]BVqT]nK:fLnHcVWfhZglnS_NLud^VLjDMaOLDC_vnI HiaxKKG^FLMW^dTCOoLNvjgc[RUPBB[BV;W[:]uaCFVw^WlGIbTPJ mVZdS;mPGERpJF=RoPsxXPCeQlHswPPDmQmHsw>LjAYaJLB;r]vKtBW>vKxBW^HlAs]>L@ CjuZI@IYFXlsS^oES\\aILBC^:vKxBWfWtBY>>lr;YvWtRunK:pBU^FL@;OvVVHDA] ]i;I^A`GG^@l>S\\NKj?C]Z@L>C\\:NKdBM^Cl?koNK`RoRTdBI^A<\\BK^C`GM>vjdgdA <\\Bb\\UM?s\\FK`Bb\\rUi>lrOfJlAs]:>L @CW^HxHYFXxRv>lsS^Flt;Y>ZKK\\BM>ZFlK`B M>vjds[>Kj>c\\Vkhs\\FK`BO>[=:FHSvUpBU^HZOLN=Sw\\TBGoKdd_>\\D Iw]TDE?S>DD??Z]LJCbZYpjfwm?hgYLIcaCFU;_ZhJ=or@T=?NhCb`?nEyq nPsxVluS_uuSgfYTSyVlugfKlBaA>lrC^ZIlAc]nKjBaqkJBs]vKJBS^nKtruvkrs]kMBa aKPI_vXEe^qUS[VX>lr;r]iMAUaHq`A\\eodB<`B:EVUiT[Xo=kp>K`RqVKhB:UfUmT]^kg?eF\\ewdFXgQwjds[ZBL >S\\FKJ>i`B<`BO>VKlrqNKdBr\\^kj?EfKtBSvUlBU>ZIHhX?L>NRV=g[:_EbEVV:sF[Zm:PgPpIcF YXsy`PFC`;NEs^Nlt;[FXq`A`Gh`A<`B:E^ ?tOIFUfGK^CK`BM>>keC\\FKdBr\\^KlB O^Dl?krZGLAc]_M@C]fkj;WFVxrrt?Z:v^;vyyYZ:>Z:>::::WTJWTL;B:;B ::::::::::::::::::::::::::::4:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 684 "On February 25, 1991, du ring the Gulf War, an American Patriot Missile battery in Dharan, Saud i Arabia, failed to track and intercept an incoming Iraqi Scud missile . The Scud struck an American Army barracks, killing 28 soldiers and i njuring around 100 other people. A report of the General Accounting o ffice, GAO/IMTEC-92-26, entitled Patriot Missile Defense: Software Pro blem Led to System Failure at Dhahran, Saudi Arabia reported on the ca use of the failure. It turns out that the cause was an inaccurate calc ulation of the time since boot due to computer arithmetic errors. Spec ifically, the time in tenths of second as measured by the system's int ernal clock was multiplied by " }{XPPEDIT 18 0 "1/10" "6#*&\"\"\"F$\"# 5!\"\"" }{TEXT -1 126 " to produce the time in seconds. This calculati on was performed using a 24 bit fixed point register. In particular, t he value " }{XPPEDIT 18 0 "1/10" "6#*&\"\"\"F$\"#5!\"\"" }{TEXT -1 426 ", which has a non-terminating binary expansion, was chopped at 24 bits after the radix point. The small chopping error, when multiplied by the large number giving the time in tenths of a second, lead to a \+ significant error. Indeed, the Patriot battery had been up around 100 \+ hours, and an easy calculation shows that the resulting time error due to the magnified chopping error was about 0.34 seconds. (The number 1 /10 equals " }{XPPEDIT 18 0 "1/2^4+1/2^5+1/2^8+1/2^9+1/2^12+1/2^13+` . . . `" "6#,0*&\"\"\"F%*$\"\"#\"\"%!\"\"F%*&F%F%*$F'\"\"&F)F%*&F%F%*$F '\"\")F)F%*&F%F%*$F'\"\"*F)F%*&F%F%*$F'\"#7F)F%*&F%F%*$F'\"#8F)F%%(~.~ .~.~GF%" }{TEXT -1 42 ". In other words, the binary expansion of " } {XPPEDIT 18 0 "1/10" "6#*&\"\"\"F$\"#5!\"\"" }{TEXT -1 2111 " is 0.000 1100110011001100110011001100.... Now the 24 bit register in the Patrio t stored instead 0.00011001100110011001100 introducing an error of 0.0 000000000000000000000011001100... binary, or about 0.000000095 decimal . Multiplying by the number of tenths of a second in 100 hours gives 0 .000000095\327100\32760\32760\32710=0.34.) A Scud travels at about 1,6 76 meters per second, and so travels more than half a kilometer in thi s time. This was far enough that the incoming Scud was outside the \"r ange gate\" that the Patriot tracked. Ironically, the fact that the ba d time calculation had been improved in some parts of the code, but no t all, contributed to the problem, since it meant that the inaccuracie s did not cancel. \n\nThe following paragraph is excerpted from the GA O report. \n\nThe range gate's prediction of where the Scud will next \+ appear is a function of the Scud's known velocity and the time of the \+ last radar detection. Velocity is a real number that can be expressed \+ as a whole number and a decimal (e.g., 3750.2563...miles per hour). Ti me is kept continuously by the system's internal clock in tenths of se conds but is expressed as an integer or whole number (e.g., 32, 33, 34 ...). The longer the system has been running, the larger the number re presenting time. To predict where the Scud will next appear, both time and velocity must be expressed as real numbers. Because of the way th e Patriot computer performs its calculations and the fact that its reg isters are only 24 bits long, the conversion of time from an integer t o a real number cannot be any more precise than 24 bits. This conversi on results in a loss of precision causing a less accurate time calcula tion. The effect of this inaccuracy on the range gate's calculation is directly proportional to the target's velocity and the length of the \+ the system has been running. Consequently, performing the conversion a fter the Patriot has been running continuously for extended periods ca uses the range gate to shift away from the center of the target, makin g it less likely that the target, in this case a Scud, will be success fully intercepted. \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 104 "(a) Us ing 10 digit floating point arithmetic find numerical values for the s olutions quadratic equation " }{XPPEDIT 18 0 "2*x^2-157*x+1=0" "6#/,(* &\"\"#\"\"\"*$%\"xGF&F'F'*&\"$d\"F'F)F'!\"\"F'F'\"\"!" }{TEXT -1 4 " i n " }{TEXT 259 10 "three ways" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 21 " (i) Use Maple's " }{TEXT 0 5 "solve" }{TEXT -1 139 " to obtain \"exact\" expressions for the solutions (involving radicals ), and then evaluate these expressions as floating point numbers using " }{TEXT 0 5 "evalf" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 " (ii) Use the Maple procedure " }{TEXT 0 6 "fsolve" }{TEXT -1 13 " rather than " }{TEXT 0 5 "solve" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 " (iii) Use the special procedure " }{TEXT 0 15 "solve _quadratic" }{TEXT -1 3 " - " }{HYPERLNK 17 "" 1 "" "" }{TEXT -1 1 " \+ " }{HYPERLNK 17 "solve_quadratic" 1 "" "solve_quadratic" }{TEXT -1 37 " given in one of the sections above. " }}{PARA 0 "" 0 "" {TEXT -1 96 "(b) Explain any loss of significance which occurs in the numerical va lues obtained in part (a). " }}{PARA 0 "" 0 "" {TEXT -1 35 "__________ _________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "____________________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 31 "(i) Set up Maple functions for " }{XPPEDIT 18 0 "f(x) = (x-1)^6;" "6# /-%\"fG6#%\"xG*$,&F'\"\"\"F*!\"\"\"\"'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "p(x)=x^6+15*x^4+15*x^2+1" "6#/-%\"pG6#%\"xG,**$F'\"\"'\"\"\"*&\"#:F +*$F'\"\"%F+F+*&F-F+*$F'\"\"#F+F+F+F+" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q(x)=6*x^5+20*x^3+6*x" "6#/-%\"qG6#%\"xG,(*&\"\"'\"\"\"*$F'\"\"& F+F+*&\"#?F+*$F'\"\"$F+F+*&F*F+F'F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "(ii) Check that " }{XPPEDIT 18 0 "g(x)=p(x)-q(x)" "6 #/-%\"gG6#%\"xG,&-%\"pG6#F'\"\"\"-%\"qG6#F'!\"\"" }{TEXT -1 33 " is ma thematically equivalent to " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "(iii) Given " } {XPPEDIT 18 0 "a = 1+sqrt(85)*`.`*10^(-2);" "6#/%\"aG,&\"\"\"F&*(-%%sq rtG6#\"#&)F&%\".GF&)\"#5,$\"\"#!\"\"F&F&" }{TEXT -1 32 ", check that t he exact value of " }{XPPEDIT 18 0 "f(a)" "6#-%\"fG6#%\"aG" }{TEXT -1 5 " is 0" }{XPPEDIT 18 0 ".614125*`.`*10^(-6)" "6#*(-%&FloatG6$\"'DTh! \"'\"\"\"%\".GF))\"#5,$\"\"'!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 23 "(iv) Set up a variable " }{TEXT 280 1 "b" }{TEXT -1 18 " which represents " }{TEXT 279 1 "a" }{TEXT -1 45 " in floating point form correct to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 13 " Compu te " }{XPPEDIT 18 0 "p(b)" "6#-%\"pG6#%\"bG" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "q(b)" "6#-%\"qG6#%\"bG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(b)=p(b)-q(b)" "6#/-%\"gG6#%\"bG,&-%\"pG6#F'\"\"\"-%\"qG6#F'!\" \"" }{TEXT -1 42 " using 10 digit floating point arithmetic." }}{PARA 0 "" 0 "" {TEXT -1 49 "(v) Calculate the absolute and relative error i n " }{XPPEDIT 18 0 "g(b)" "6#-%\"gG6#%\"bG" }{TEXT -1 25 " as an appro ximation for " }{XPPEDIT 18 0 "f(a)" "6#-%\"fG6#%\"aG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 96 "(vi) Compare the relative error with \+ the machine epsilon for 10 digit floating point arithmetic." }}{PARA 0 "" 0 "" {TEXT -1 53 " Comment on the magnitude of the relative \+ error." }}{PARA 0 "" 0 "" {TEXT -1 35 "_______________________________ ____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 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 35 "____________________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 31 "The exact analytical value of " }{XPPEDIT 18 0 "Int(x^9/(x^2+12),x = 0 .. 1);" "6#-%$IntG6$*&%\"xG\"\"*,&*$F'\"\"#\"\"\"\"#7F,!\"\"/F';\" \"!F," }{TEXT -1 5 " is " }{XPPEDIT 18 0 "10368*ln(13/12)-6639/8" "6# ,&*&\"&o.\"\"\"\"-%#lnG6#*&\"#8F&\"#7!\"\"F&F&*&\"%RmF&\"\")F-F-" } {TEXT -1 69 ", as can be verified using Maple by executing the followi ng commands." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Int(x^9/(x^2+12),x=0..1);\nvalue(%);\ncombine(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"*,&*$)F'\"\"# \"\"\"F-\"#7F-!\"\"/F';\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**& \"&o.\"\"\"\"-%#lnG6#\"#8F&F&*&\"&O2#F&-F(6#\"\"#F&!\"\"*&F%F&-F(6#\" \"$F&F0#\"%Rm\"\")F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"%Rm\"\")! \"\"*&\"&o.\"\"\"\"-%#lnG6##\"#8\"#7F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "(i) Find a numerical value for " }{XPPEDIT 18 0 "Int(x^9/ (x^2+12),x = 0 .. 1);" "6#-%$IntG6$*&%\"xG\"\"*,&*$F'\"\"#\"\"\"\"#7F, !\"\"/F';\"\"!F," }{TEXT -1 15 " by evaluating " }{XPPEDIT 18 0 "10368 *ln(13/12)-6639/8" "6#,&*&\"&o.\"\"\"\"-%#lnG6#*&\"#8F&\"#7!\"\"F&F&*& \"%RmF&\"\")F-F-" }{TEXT -1 36 " using 10 digit decimal arithmetic. " }}{PARA 0 "" 0 "" {TEXT -1 32 "(ii) Find a numerical value for " } {XPPEDIT 18 0 "Int(x^9/(x^2+12),x = 0 .. 1);" "6#-%$IntG6$*&%\"xG\"\"* ,&*$F'\"\"#\"\"\"\"#7F,!\"\"/F';\"\"!F," }{TEXT -1 108 " which is cor rect to 10 figures, and hence calculate the relative error in the valu e obtained in part (i). " }}{PARA 0 "" 0 "" {TEXT -1 113 "(iii) Why is the relative obtained in part (ii) so much larger than the machine ep silon for 10 digit arithmetic? " }}{PARA 0 "" 0 "" {TEXT -1 35 "______ _____________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "____________ _______________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 30 "The exact analytical value of " }{XPPEDIT 18 0 "Int((x^16+1)/(x ^2+5),x=0..1)" "6#-%$IntG6$*&,&*$%\"xG\"#;\"\"\"F+F+F+,&*$F)\"\"#F+\" \"&F+!\"\"/F);\"\"!F+" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "390626/5*sqr t(5)" "6#*(\"'E1R\"\"\"\"\"&!\"\"-%%sqrtG6#F&F%" }{TEXT -1 1 " " } {XPPEDIT 18 0 "arctan(sqrt(5)/5)-3309200072/45045" "6#,&-%'arctanG6#*& -%%sqrtG6#\"\"&\"\"\"F+!\"\"F,*&\"+s+?4LF,\"&X]%F-F-" }{TEXT -1 69 ", \+ as can be verified using Maple by executing the following commands." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Int((x^16+1)/(x^2+5),x=0..1);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&,&*$)%\"xG\"#;\"\"\"F,F,F,F,,&*$)F*\"\"#F,F, \"\"&F,!\"\"/F*;\"\"!F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"+s+?4L \"&X]%!\"\"*&#\"'E1R\"\"&\"\"\"*&F+#F,\"\"#-%'arctanG6#,$*&F+F'F+F.F,F ,F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "(i) Find a numerical v alue for " }{XPPEDIT 18 0 "Int((x^16+1)/(x^2+5),x=0..1)" "6#-%$IntG6$* &,&*$%\"xG\"#;\"\"\"F+F+F+,&*$F)\"\"#F+\"\"&F+!\"\"/F);\"\"!F+" } {TEXT -1 15 " by evaluating " }{XPPEDIT 18 0 "390626/5*sqrt(5)" "6#*( \"'E1R\"\"\"\"\"&!\"\"-%%sqrtG6#F&F%" }{TEXT -1 1 " " }{XPPEDIT 18 0 " arctan(sqrt(5)/5)-3309200072/45045" "6#,&-%'arctanG6#*&-%%sqrtG6#\"\"& \"\"\"F+!\"\"F,*&\"+s+?4LF,\"&X]%F-F-" }{TEXT -1 37 " using 10 digit \+ decimal arithmetic. " }}{PARA 0 "" 0 "" {TEXT -1 32 "(ii) Find a numer ical value for " }{XPPEDIT 18 0 "Int((x^16+1)/(x^2+5),x = 0 .. 1)" "6# -%$IntG6$*&,&*$%\"xG\"#;\"\"\"F+F+F+,&*$F)\"\"#F+\"\"&F+!\"\"/F);\"\"! F+" }{TEXT -1 107 " which is correct to 10 figures, and hence calculat e the relative error in the value obtained in part (i). " }}{PARA 0 " " 0 "" {TEXT -1 113 "(iii) Why is the relative obtained in part (ii) s o much larger than the machine epsilon for 10 digit arithmetic? " }} {PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 17 "Find a value for " } {XPPEDIT 18 0 "Int(x^25/(x+9),x = 0 .. 1);" "6#-%$IntG6$*&%\"xG\"#D,&F '\"\"\"\"\"*F*!\"\"/F';\"\"!F*" }{TEXT -1 115 " which is accurate to a bout 10 digits by evaluating numerically an \"exact\" analytical expre ssion for the integral. " }}{PARA 0 "" 0 "" {TEXT -1 35 "_____________ ______________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "____________________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6" }}{PARA 0 "" 0 "" {TEXT -1 11 "Reference: " }{URLLINK 17 "A Remarkable Example of Catastrophic Cance llation Unravelled" 4 "http://link.springer.de/link/service/journals/0 0607/bibs/1066003/10660309.htm" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 49 "Consider the problem of evaluating the ex pression" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "G(u,v) = \+ 1335/4*v^6+u^2*(11*u^2*v^2-v^6-121*v^4-2)+11/2*v^8+u/(2*v);" "6#/-%\"G G6$%\"uG%\"vG,**(\"%N8\"\"\"\"\"%!\"\"F(\"\"'F,*&F'\"\"#,**(\"#6F,*$F' F1F,F(F1F,*$F(F/F.*&\"$@\"F,*$F(F-F,F.F1F.F,F,*(F4F,F1F.F(\"\")F,*&F'F ,*&F1F,F(F,F.F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "when " }{XPPEDIT 18 0 "u=77617" "6#/%\"uG\"& " 0 "" {MPLTEXT 1 0 73 "G := (u,v) -> 1335/4*v^6+u^2*(11*u^2*v^2-v^6-121*v^4-2)+11/2*v^8+u /(2*v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "(a) Show using exact arithmetic that " }{XPPEDIT 18 0 "G(77617,33096)=-54767/66192" "6#/-% \"GG6$\"&m!\"\"F." }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 63 "(b) Use the result of part (a) to obtain \+ a numerical value for " }{XPPEDIT 18 0 "G(77617,33096)" "6#-%\"GG6$\"& " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "_______________________________ ____" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 259 "" 0 "" {TEXT 338 2 "Q7" }{TEXT -1 41 " .. round-off error in the \+ evaluation of " }{XPPEDIT 18 0 "g(x)=(exp(x)-1)/x" "6#/-%\"gG6#%\"xG*& ,&-%$expG6#F'\"\"\"F-!\"\"F-F'F." }{TEXT -1 6 " when " }{TEXT 337 1 "x " }{TEXT -1 10 " is near 0" }}{PARA 0 "" 0 "" {TEXT -1 46 "This questi on is concerned with the function " }{XPPEDIT 18 0 "g(x) = (exp(x)-1) /x;" "6#/-%\"gG6#%\"xG*&,&-%$expG6#F'\"\"\"F-!\"\"F-F'F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "(a) Plot a graph of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 19 " over the interval " } {XPPEDIT 18 0 "-3<=x" "6#1,$\"\"$!\"\"%\"xG" }{XPPEDIT 18 0 "``<=3" "6 #1%!G\"\"$" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "(b) Use the expressions " }{TEXT 260 17 "'evalf [10](g(x))'" }{TEXT -1 5 " and " }{TEXT 260 17 "'evalf[20](g(x))'" } {TEXT -1 46 " in a suitable plot command to plot graphs of " } {XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 18 " over the interv al" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "-3" "6#,$\"\"$ !\"\"" }{TEXT -1 1 " " }{TEXT 450 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9) <= x;" "6#1)\"#5,$\"\"*!\"\"%\"xG" }{XPPEDIT 18 0 "``<=3" " 6#1%!G\"\"$" }{TEXT -1 1 " " }{TEXT 449 1 "x" }{TEXT -1 1 " " } {XPPEDIT 18 0 "10^(-9);" "6#)\"#5,$\"\"*!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 53 "with 10 digit precision and with 20 digit precision. " }{TEXT 259 4 "Note" }{TEXT -1 37 ": It is essential to u se the quotes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "(c) Find the first 5 terms of the Maclaurin series for " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 15 "(d) Check that " }{XPPEDIT 18 0 "Limit(g(x),x=0)=1" "6#/-%&LimitG6$-%\"gG6#%\"xG/F*\"\"!\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 8 "(e) Let " }{XPPEDIT 18 0 "x[0] = 51;" "6#/&%\"xG 6#\"\"!\"#^" }{TEXT -1 1 " " }{TEXT 451 1 "x" }{TEXT -1 1 " " } {XPPEDIT 18 0 "10^(-11);" "6#)\"#5,$\"#6!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 33 " (i) Find the sum of the first " } {TEXT 262 7 "5 terms" }{TEXT -1 29 " of the Maclaurin series for " } {XPPEDIT 18 0 "g(x)=(exp(x)-1)/x" "6#/-%\"gG6#%\"xG*&,&-%$expG6#F'\"\" \"F-!\"\"F-F'F." }{TEXT -1 14 " evaluated at " }{XPPEDIT 18 0 "x=x[0] " "6#/%\"xG&F$6#\"\"!" }{TEXT -1 82 ", performing the calculation with Maple's base 10 floating point arithmetic using " }{TEXT 262 9 "35 di gits" }{TEXT -1 15 " of precision. " }}{PARA 0 "" 0 "" {TEXT -1 26 " \+ (ii) Find the value of " }{XPPEDIT 18 0 "g(x[0])" "6#-%\"gG6#&%\"xG6# \"\"!" }{TEXT -1 34 ", performing the calculation with " }{TEXT 262 9 "45 digits" }{TEXT -1 14 " of precision." }}{PARA 0 "" 0 "" {TEXT -1 27 " (iii) Find the value of " }{XPPEDIT 18 0 "g(x[0])" "6#-%\"gG6#& %\"xG6#\"\"!" }{TEXT -1 34 ", performing the calculation with " } {TEXT 262 9 "10 digits" }{TEXT -1 15 " of precision. " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 259 4 "Note" }{TEXT -1 54 " for (iii): In o rder to prevent Maple from evaluating " }{XPPEDIT 18 0 "g(x[0])" "6#-% \"gG6#&%\"xG6#\"\"!" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "exp(x[0])/x[0] -1/x[0]" "6#,&*&-%$expG6#&%\"xG6#\"\"!\"\"\"&F)6#F+!\"\"F,*&F,F,&F)6#F +F/F/" }{TEXT -1 10 ", compute " }{XPPEDIT 18 0 "g(x[0])" "6#-%\"gG6#& %\"xG6#\"\"!" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\" \"!" }{TEXT -1 45 " first evaluated as a floating point number. " }} {PARA 0 "" 0 "" {TEXT -1 28 " Also execute the command " }{TEXT 260 13 "forget(evalf)" }{TEXT -1 34 " before performing the evaluation." } }{PARA 0 "" 0 "" {TEXT -1 18 " (iv) Calculate " }{XPPEDIT 18 0 "exp( x[0])" "6#-%$expG6#&%\"xG6#\"\"!" }{TEXT -1 34 ", performing the calcu lation with " }{TEXT 262 9 "10 digits" }{TEXT -1 54 " of precision, an d check that using the approximation " }{XPPEDIT 18 0 "exp(x)" "6#-%$e xpG6#%\"xG" }{TEXT -1 1 " " }{TEXT 334 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "1+x" "6#,&\"\"\"F$%\"xGF$" }{TEXT -1 64 ", obtained fro m the first two terms of the Maclaurin series for " }{XPPEDIT 18 0 "ex p(x)" "6#-%$expG6#%\"xG" }{TEXT -1 24 ", gives the same value. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 " (v) Us e the answer from (iv) to explain the result of (iii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "(e) Give one positiv e and one negative value of " }{TEXT 333 1 "x" }{TEXT -1 37 " for whic h the error in the value of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG " }{TEXT -1 65 " calculated using 10 digit precision is effectively eq ual to 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "(f) Plot a graph of " }{XPPEDIT 18 0 "g(x)=(exp(x)-1)/x" "6#/-% \"gG6#%\"xG*&,&-%$expG6#F'\"\"\"F-!\"\"F-F'F." }{TEXT -1 21 " over the interval " }{XPPEDIT 18 0 "-6;" "6#,$\"\"'!\"\"" }{TEXT -1 1 " " } {TEXT 336 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-16) <= x;" "6#1) \"#5,$\"#;!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 6;" "6#1%!G\"\"'" }{TEXT -1 1 " " }{TEXT 335 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-16);" " 6#)\"#5,$\"#;!\"\"" }{TEXT -1 11 " in which " }{XPPEDIT 18 0 "g(x)" " 6#-%\"gG6#%\"xG" }{TEXT -1 59 " is evaluated using hardware floating p oint arithmetic. " }}{PARA 0 "" 0 "" {TEXT -1 35 "_________________ __________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 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 35 "___________________________________" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q8 " }} {PARA 0 "" 0 "" {TEXT -1 17 "Find a value for " }{XPPEDIT 18 0 "Int(x^ 25/(x+9),x = 0 .. 1);" "6#-%$IntG6$*&%\"xG\"#D,&F'\"\"\"\"\"*F*!\"\"/F ';\"\"!F*" }{TEXT -1 61 " which is accurate to about 10 digits by down ward recursion. " }}{PARA 0 "" 0 "" {TEXT -1 35 "_____________________ ______________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "_______________________________ ____" }}{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 50 "Specimen solutions for some of th e task questions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "solution for Q1 " }}{PARA 0 "" 0 "" {TEXT 300 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 104 "(a) Using 10 digit floating point arithmetic find numerical value s for the solutions quadratic equation " }{XPPEDIT 18 0 "2*x^2-157*x+1 =0" "6#/,(*&\"\"#\"\"\"*$%\"xGF&F'F'*&\"$d\"F'F)F'!\"\"F'F'\"\"!" } {TEXT -1 4 " in " }{TEXT 259 10 "three ways" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 21 " (i) Use Maple's " }{TEXT 0 5 "solve" } {TEXT -1 139 " to obtain \"exact\" expressions for the solutions (invo lving radicals), and then evaluate these expressions as floating point numbers using " }{TEXT 0 5 "evalf" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 34 " (ii) Use the Maple procedure " }{TEXT 0 6 "fsolve " }{TEXT -1 13 " rather than " }{TEXT 0 5 "solve" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 37 " (iii) Use the special procedure " } {TEXT 0 15 "solve_quadratic" }{TEXT -1 3 " - " }{HYPERLNK 17 "" 1 "" " " }{TEXT -1 1 " " }{HYPERLNK 17 "solve_quadratic" 1 "" "solve_quadrati c" }{TEXT -1 37 " given in one of the sections above. " }}{PARA 0 "" 0 "" {TEXT -1 96 "(b) Explain any loss of significance which occurs in the numerical values obtained in part (a). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 303 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 4 "(i) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 10:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "2*x^ 2-157*x+1=0;\nsolve(%);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /,(*&\"\"#\"\"\")%\"xGF&F'F'*&\"$d\"F'F)F'!\"\"F'F'\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$,&#\"$d\"\"\"%\"\"\"*&F&!\"\"\"&TY##F'\"\"#F',&F $F'*&F&F)F*F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+0IO\\y!\")$\"'& *pjF%" }}}{PARA 0 "" 0 "" {TEXT -1 5 "(ii) " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 27 "2*x^2-157*x+1=0;\nfsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&\"\"#\"\"\")%\"xGF&F'F'*&\"$d\"F'F)F'!\"\"F'F'\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+YO%*pj!#7$\"+1IO\\y!\")" }}} {PARA 0 "" 0 "" {TEXT -1 6 "(iii) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "2*x^2-157*x+1=0;\nsolve_quadratic(%);\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,(*&\"\"#\"\"\")%\"xGF&F'F'*&\"$d\"F' F)F'!\"\"F'F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"$d\"\"\"%\" \"\"*&F&!\"\"\"&TY##F'\"\"#F'*&F'F',&#F%F,F'*&F,F)F*F+F'F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+0IO\\y!\")$\"+YO%*pj!#7" }}}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" {TEXT -1 15 "The expression \+ " }{XPPEDIT 18 0 "157/4-sqrt(24641)/4" "6#,&*&\"$d\"\"\"\"\"\"%!\"\"F& *&-%%sqrtG6#\"&TY#F&F'F(F(" }{TEXT -1 33 " obtained by the Maple proce dure " }{TEXT 0 5 "solve" }{TEXT -1 219 " in (i) of part (a) produces \+ the numerical value 0.00636995 when evaluated using 10 digit software \+ floating point arithmetic. The value 0.00636995 only has 6 significant digits, so that 4 digits of precision are \"lost\"." }}{PARA 0 "" 0 " " {TEXT -1 58 "This loss of significance is due to \"subtraction error \". " }}{PARA 0 "" 0 "" {TEXT -1 20 "This arises because " }{XPPEDIT 18 0 "a = 157/4;" "6#/%\"aG*&\"$d\"\"\"\"\"\"%!\"\"" }{TEXT -1 32 " ev aluates to 39.25000000 while " }{XPPEDIT 18 0 "b = sqrt(24641)/4;" "6# /%\"bG*&-%%sqrtG6#\"&TY#\"\"\"\"\"%!\"\"" }{TEXT -1 43 " evaluates to \+ 39.24363005 which shows that " }{TEXT 288 1 "a" }{TEXT -1 5 " and " } {TEXT 289 1 "b" }{TEXT -1 14 " are \"close\". " }}{PARA 0 "" 0 "" {TEXT -1 56 "In particular the first three digits of the two numbers \+ " }{TEXT 290 1 "a" }{TEXT -1 5 " and " }{TEXT 291 1 "b" }{TEXT -1 76 " are the same. Hence the difference could have at most 7 significant d igits." }}{PARA 0 "" 0 "" {TEXT -1 124 "Setting out the subtraction as it would be performed \"by hand\" shows why there are 6 significant d igits in the final value. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 14 " 39.25000000 -" }}{PARA 256 "" 0 "" {TEXT -1 14 " 39.24363005 " }}{PARA 256 "" 0 "" {TEXT -1 12 " __________ " }} {PARA 256 "" 0 "" {TEXT -1 13 " 0.00636995 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "The fact that there are 6 sign ificant digits rather than 7 is a consequence of the fourth digits of \+ " }{TEXT 292 1 "a" }{TEXT -1 5 " and " }{TEXT 293 1 "b" }{TEXT -1 21 " only differing by 1." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "157/4;\na := evalf(%);\nsqrt(24641)/4;\nb := evalf(%);\n'a-b' =a-b; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$d \"\"\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG$\"++++DR!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"\"%!\"\"\"&TY##\"\"\"\"\"#F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"+0IOCR!\")" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/,&%\"aG\"\"\"%\"bG!\"\"$\"'&*pj!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "solution for Q2" }}{PARA 0 "" 0 " " {TEXT 301 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 31 "(i) Set up Maple functions for " }{XPPEDIT 18 0 "f(x) = (x-1)^6;" "6#/-%\"fG6#%\"xG*$,&F'\"\"\"F*!\"\"\"\"'" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "p(x)=x^6+15*x^4+15*x^2+1" "6#/-%\"pG6#%\"xG,**$F'\"\"'\"\"\"*&\" #:F+*$F'\"\"%F+F+*&F-F+*$F'\"\"#F+F+F+F+" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "q(x)=6*x^5+20*x^3+6*x" "6#/-%\"qG6#%\"xG,(*&\"\"'\"\"\" *$F'\"\"&F+F+*&\"#?F+*$F'\"\"$F+F+*&F*F+F'F+F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 16 "(ii) Check that " }{XPPEDIT 18 0 "g(x)=p( x)-q(x)" "6#/-%\"gG6#%\"xG,&-%\"pG6#F'\"\"\"-%\"qG6#F'!\"\"" }{TEXT -1 33 " is mathematically equivalent to " }{XPPEDIT 18 0 "f(x)" "6#-% \"fG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "(iii) Giv en " }{XPPEDIT 18 0 "a = 1+sqrt(85)*`.`*10^(-2);" "6#/%\"aG,&\"\"\"F&* (-%%sqrtG6#\"#&)F&%\".GF&)\"#5,$\"\"#!\"\"F&F&" }{TEXT -1 32 ", check \+ that the exact value of " }{XPPEDIT 18 0 "f(a)" "6#-%\"fG6#%\"aG" } {TEXT -1 5 " is 0" }{XPPEDIT 18 0 ".614125*`.`*10^(-6)" "6#*(-%&FloatG 6$\"'DTh!\"'\"\"\"%\".GF))\"#5,$\"\"'!\"\"F)" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 23 "(iv) Set up a variable " }{TEXT 295 1 "b " }{TEXT -1 18 " which represents " }{TEXT 294 1 "a" }{TEXT -1 45 " in floating point form correct to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 13 " Compute " }{XPPEDIT 18 0 "p(b)" "6#-%\"pG6#%\"bG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "q(b)" "6#-%\"qG6#%\"bG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "g(b)=p(b)-q(b)" "6#/-%\"gG6#%\"bG,&-%\"pG6#F'\"\"\"-% \"qG6#F'!\"\"" }{TEXT -1 42 " using 10 digit floating point arithmetic ." }}{PARA 0 "" 0 "" {TEXT -1 49 "(v) Calculate the absolute and relat ive error in " }{XPPEDIT 18 0 "g(b)" "6#-%\"gG6#%\"bG" }{TEXT -1 25 " \+ as an approximation for " }{XPPEDIT 18 0 "f(a)" "6#-%\"fG6#%\"aG" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 96 "(vi) Compare the relativ e error with the machine epsilon for 10 digit floating point arithmeti c." }}{PARA 0 "" 0 "" {TEXT -1 78 " Comment on the magnitude of t he relative error and give an explanation. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 302 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 4 "(i) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "f := x -> (x-1)^6;\np := x -> x^6+15*x^4+15*x^2+1;\nq := x -> \+ 6*x^5+20*x^3+6*x;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6 \"6$%)operatorG%&arrowGF(*$),&9$\"\"\"F0!\"\"\"\"'F0F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,** $)9$\"\"'\"\"\"F1*&\"#:F1)F/\"\"%F1F1*&F3F1)F/\"\"#F1F1F1F1F(F(F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qGf*6#%\"xG6\"6$%)operatorG%&arrow GF(,(*&\"\"'\"\"\")9$\"\"&F/F/*&\"#?F/)F1\"\"$F/F/*&F.F/F1F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(ii) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "g := p-q;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gG,&%\"pG\"\"\"%\"qG!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "g(x);\nfacto r(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,0*$)%\"xG\"\"'\"\"\"F(*&\"#: F()F&\"\"%F(F(*&F*F()F&\"\"#F(F(F(F(*&F'F()F&\"\"&F(!\"\"*&\"#?F()F&\" \"$F(F3*&F'F(F&F(F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),&%\"xG\"\" \"F'!\"\"\"\"'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "(iii) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "a := 1 +sqrt(85)*10^(-2);\nf(a);\nfa := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG,&\"\"\"F&*&\"$+\"!\"\"\"#&)#F&\"\"#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"%8\\\"+++++!)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#faG$\"+++DTh!#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 19 " Maple simplifies " }{XPPEDIT 18 0 "6141 25/10^12" "6#*&\"'DTh\"\"\"*$\"#5\"#7!\"\"" }{TEXT -1 5 " to " } {XPPEDIT 18 0 "4913/8000000000" "6#*&\"%8\\\"\"\"\"+++++!)!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "614125/10^12;\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"%8\\\"+++++!)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 5 "(iv) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "a := \+ 1+sqrt(85)*10^(-2);\nb := evalf(evalf(%,15));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG,&\"\"\"F&*&\"$+\"!\"\"\"#&)#F&\"\"#F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG$\"+Xa>#4\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "'p(b)'=p(b); \n'q(b)'=q(b);\n'g(b)'=g(b);\ngb := rhs(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"bG$\"+sjc$>%!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"bG$\"+6jc$>%!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"bG$\"#h!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gbG$\"#h!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 167 "The value obtained for g(b) only has 2 significant \+ digits. This happens because of the excessive error resulting from the subtraction of the \"nearly equal\" quantities " }{XPPEDIT 18 0 "p(b) " "6#-%\"pG6#%\"bG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "q(b)" "6#-%\"q G6#%\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 14 " 41.93566372 -" }}{PARA 256 "" 0 "" {TEXT -1 14 " 41.93566311 " }}{PARA 256 "" 0 "" {TEXT -1 15 " __________ " }}{PARA 256 "" 0 "" {TEXT -1 12 " 0.00000061 " }}{PARA 0 "" 0 "" {TEXT -1 4 "(v) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "abserr : = abs(gb-fa);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\")++DT!#; " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "relerr := abserr/abs(fa);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"+,O(or'!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "mach_eps := 5.*10^(-10) ;\nevalf(relerr/mach_eps,3);\nround(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)mach_epsG$\"+++++]!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$ M\"\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\")++S8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 143 "The relative error is about 13400000 tim es the machine epsilon for 10 digit floating point arithmetic. This ca n only be regarded as excessive. " }}{PARA 0 "" 0 "" {TEXT -1 89 "A st ated above, it results from the subtraction error involved in the the \+ subtraction of " }{XPPEDIT 18 0 "q(b)" "6#-%\"qG6#%\"bG" }{TEXT -1 6 " from " }{XPPEDIT 18 0 "p(b)" "6#-%\"pG6#%\"bG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 15 "solution for Q3" }}{PARA 0 "" 0 "" {TEXT 298 8 "Questio n" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 31 "The exact analytica l value of " }{XPPEDIT 18 0 "Int(x^9/(x^2+12),x = 0 .. 1);" "6#-%$Int G6$*&%\"xG\"\"*,&*$F'\"\"#\"\"\"\"#7F,!\"\"/F';\"\"!F," }{TEXT -1 5 " \+ is " }{XPPEDIT 18 0 "10368*ln(13/12)-6639/8" "6#,&*&\"&o.\"\"\"\"-%#l nG6#*&\"#8F&\"#7!\"\"F&F&*&\"%RmF&\"\")F-F-" }{TEXT -1 69 ", as can be verified using Maple by executing the following commands." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Int( x^9/(x^2+12),x=0..1);\nvalue(%);\ncombine(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"*,&*$)F'\"\"#\"\"\"F-\"#7F-!\"\"/F' ;\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"&o.\"\"\"\"-%#lnG6# \"#8F&F&*&\"&O2#F&-F(6#\"\"#F&!\"\"*&F%F&-F(6#\"\"$F&F0#\"%Rm\"\")F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&#\"%Rm\"\")!\"\"*&\"&o.\"\"\"\"-%# lnG6##\"#8\"#7F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 31 "(i) Find \+ a numerical value for " }{XPPEDIT 18 0 "Int(x^9/(x^2+12),x = 0 .. 1); " "6#-%$IntG6$*&%\"xG\"\"*,&*$F'\"\"#\"\"\"\"#7F,!\"\"/F';\"\"!F," } {TEXT -1 15 " by evaluating " }{XPPEDIT 18 0 "10368*ln(13/12)-6639/8" "6#,&*&\"&o.\"\"\"\"-%#lnG6#*&\"#8F&\"#7!\"\"F&F&*&\"%RmF&\"\")F-F-" } {TEXT -1 36 " using 10 digit decimal arithmetic. " }}{PARA 0 "" 0 "" {TEXT -1 32 "(ii) Find a numerical value for " }{XPPEDIT 18 0 "Int(x^9 /(x^2+12),x = 0 .. 1);" "6#-%$IntG6$*&%\"xG\"\"*,&*$F'\"\"#\"\"\"\"#7F ,!\"\"/F';\"\"!F," }{TEXT -1 108 " which is correct to 10 figures, an d hence calculate the relative error in the value obtained in part (i) . " }}{PARA 0 "" 0 "" {TEXT -1 113 "(iii) Why is the relative obtained in part (ii) so much larger than the machine epsilon for 10 digit ari thmetic? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 299 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Digits := 10:" }}}{PARA 0 " " 0 "" {TEXT -1 4 "(i) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "1 0368*ln(13/12)-6639/8;\napprox_val := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"&o.\"\"\"\"-%#lnG6##\"#8\"#7F&F&#\"%Rm\"\")!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+approx_valG$\"&+z(!\"(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" } {TEXT -1 159 ": There is considerable loss of significance in this val ue because it is obtained by a subtraction of the \"nearly equal\" qua ntities 10368*ln(13/12) and 6639/8." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "10368*ln(13/12);\nevalf(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&\"&o.\"\"\"\"-%#lnG6##\"#8\"#7 F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"++z#))H)!\"(" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "6639/ 8;\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"%Rm\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++v)H)!\"(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 153 "The first 4 digits of these two quantities are the same \+ and the 5th digits only differ by 1. As a result their difference only has 5 significant digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 14 " 829.8827900 -" }}{PARA 256 "" 0 "" {TEXT -1 14 " 829.8750000 " }}{PARA 256 "" 0 "" {TEXT -1 11 "__________ " }} {PARA 256 "" 0 "" {TEXT -1 14 " 0.0077900 " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(ii) " }}{PARA 0 "" 0 "" {TEXT -1 13 "An value for " }{XPPEDIT 18 0 "Int(x^9/(x^2+12),x = 0 .. \+ 1);" "6#-%$IntG6$*&%\"xG\"\"*,&*$F'\"\"#\"\"\"\"#7F,!\"\"/F';\"\"!F," }{TEXT -1 61 " which is correct to 10 figures can be obtained in two \+ ways." }}{PARA 0 "" 0 "" {TEXT 296 8 "Method I" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 129 "Compensate for the loss of significance \+ in the value computed in part (i) by increasing the precision used for the evaluation of " }{XPPEDIT 18 0 "10368*ln(13/12)-6639/8" "6#,&*&\" &o.\"\"\"\"-%#lnG6#*&\"#8F&\"#7!\"\"F&F&*&\"%RmF&\"\")F-F-" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "10368*ln(13/12)-6639/8;\nevalf[20](%);\naccurate_val \+ := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"&o.\"\"\"\"-%#ln G6##\"#8\"#7F&F&#\"%Rm\"\")!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"0iicAfJz(!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-accurate_valG$\"+ E#fJz(!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 297 9 "Method II" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 56 "Eval uate the integral by means of numerical integration " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Int(x^9/(x ^2+12),x=0..1);\naccurate_val := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$*&%\"xG\"\"*,&*$)F'\"\"#\"\"\"F-\"#7F-!\"\"/F' ;\"\"!F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-accurate_valG$\"+E#fJz( !#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "abserr := abs(approx_val-accurate_val);\nrelerr := ev alf[5](abserr/abs(accurate_val));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %'abserrG$\"(E#fJ!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"& Q0%!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "mach_eps := 5.*10^(-10);\nevalf[3](relerr/mach_eps); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)mach_epsG$\"+++++]!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$5)\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 101 "The relative error is about 810000 times the machine eps ilon for 10 digit floating point arithmetic. " }}{PARA 0 "" 0 "" {TEXT -1 140 "The large relative error results from the subtraction er ror involved in the the subtraction of 829.875 from 829.8827900 as men tioned above. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "solution for Q6 " }}{PARA 0 "" 0 "" {TEXT -1 11 "Ref erence: " }{URLLINK 17 "A Remarkable Example of Catastrophic Cancelati on Unraveled" 4 "http://link.springer.de/link/service/journals/00607/b ibs/1066003/10660309.htm" "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 304 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 49 "Consider the problem of evaluating the expression" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "G(u,v) = 1335/4*v^6+u ^2*(11*u^2*v^2-v^6-121*v^4-2)+11/2*v^8+u/(2*v);" "6#/-%\"GG6$%\"uG%\"v G,**(\"%N8\"\"\"\"\"%!\"\"F(\"\"'F,*&F'\"\"#,**(\"#6F,*$F'F1F,F(F1F,*$ F(F/F.*&\"$@\"F,*$F(F-F,F.F1F.F,F,*(F4F,F1F.F(\"\")F,*&F'F,*&F1F,F(F,F .F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "when " }{XPPEDIT 18 0 "u=77617" "6#/%\"uG\"& 1335/4*v^6+u^2*(11*u^2*v^2 -v^6-121*v^4-2)+11/2*v^8+u/(2*v);" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "(a) Show using exact arit hmetic that " }{XPPEDIT 18 0 "G(77617,33096)=-54767/66192" "6#/-%\"GG6 $\"&m!\"\"F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 63 "(b) Use the result of part (a) to obtain a nume rical value for " }{XPPEDIT 18 0 "G(77617,33096)" "6#-%\"GG6$\"& " 0 "" {MPLTEXT 1 0 89 "G := (u,v) -> 1335/4*v^6+u^2*(11*u^2*v^2-v^6-121*v^4-2)+11/2*v^8+u/(2*v );\nG(77617,33096);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"GGf*6$%\"uG %\"vG6\"6$%)operatorG%&arrowGF),**&#\"%N8\"\"%\"\"\"*$)9%\"\"'F2F2F2*& )9$\"\"#F2,**(\"#6F2F8F2)F5F:F2F2F3!\"\"*&\"$@\"F2)F5F1F2F?F:F?F2F2*&# F=F:F2*$)F5\"\")F2F2F2*&#F2F:F2*&F9F2F5F?F2F2F)F)F)" }}{PARA 11 "" 1 " " {XPPMATH 20 "6##!&nZ&\"&#>m" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "G(77617,33096);\nevalf(evalf[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##!&nZ&\"&#>m" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+*fgR F)!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " (c) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 193 "for digits in [seq( 5*i,i=2..7),36,37,38] do\n print(`numerical value`=evalf[10](evalf[d igits](G(77617.,33096.))),` when calculated using `||digits||` digit f loating point arithmetic`);\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6$/%0numerical~valueG$\"\"&\"#F%Z~when~calculated~using~10~digit~float ing~point~arithmeticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~ valueG$!\"#\"#A%Z~when~calculated~using~15~digit~floating~point~arithm eticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG$!+++++5\" \")%Z~when~calculated~using~20~digit~floating~point~arithmeticG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG$!+++++I\"\"$%Z~whe n~calculated~using~25~digit~floating~point~arithmeticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG$\"+<,++5!\"#%Z~when~calculated ~using~30~digit~floating~point~arithmeticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG$!+hRF))H!\"(%Z~when~calculated~usin g~35~digit~floating~point~arithmeticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG$\"+%Rgs6#!\")%Z~when~calculated~using~36~digit~ floating~point~arithmeticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numer ical~valueG$!+*fgRF)!#5%Z~when~calculated~using~37~digit~floating~poin t~arithmeticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG$! +*fgRF)!#5%Z~when~calculated~using~38~digit~floating~point~arithmeticG " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 104 "The minimum number of digits of precision needed to obtain a value which \+ is correct to 10 digits is 37. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "(d) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 181 "H := (u,v) -> 1335/4*v^6+u^2*(11*u^2*v^2-v^6-121*v^4-2);\nK : = v -> 11/2*v^8;\nevalf[38](evalf[45](H(77617.,33096.)));\nevalf[38](e valf[45](K(33096.)));\n#SFloatMantissa(%);\n#length(%);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"HGf*6$%\"uG%\"vG6\"6$%)operatorG%&arrowGF),& *&#\"%N8\"\"%\"\"\"*$)9%\"\"'F2F2F2*&)9$\"\"#F2,**(\"#6F2F8F2)F5F:F2F2 F3!\"\"*&\"$@\"F2)F5F1F2F?F:F?F2F2F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"KGf*6#%\"vG6\"6$%)operatorG%&arrowGF(,$*&#\"#6\"\"#\"\"\"*$) 9$\"\")F1F1F1F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!G+&G%\\_,Z8,6 Oh*o1M6r\"z!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"G![G%\\_,Z8,6Oh *o1M6r\"z!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "The first 35 digits in the two decimal expansions are th e same. This leads to incredibly severe subtraction error when these t wo numbers are added. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "solution for Q7 " }}{PARA 0 " " 0 "" {TEXT 319 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 46 "This question is concerned with the function " } {XPPEDIT 18 0 "g(x) = (exp(x)-1)/x;" "6#/-%\"gG6#%\"xG*&,&-%$expG6#F' \"\"\"F-!\"\"F-F'F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 20 "( a) Plot a graph of " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "-3<=x" "6#1,$\"\"$!\"\"%\" xG" }{XPPEDIT 18 0 "``<=3" "6#1%!G\"\"$" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "(b) Use the expres sions " }{TEXT 260 17 "'evalf[10](g(x))'" }{TEXT -1 5 " and " }{TEXT 260 17 "'evalf[20](g(x))'" }{TEXT -1 46 " in a suitable plot command t o plot graphs of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 18 " over the interval" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "-3" "6#,$\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 318 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9) <= x;" "6#1)\"#5,$\"\"*!\"\"%\"xG" } {XPPEDIT 18 0 "``<=3" "6#1%!G\"\"$" }{TEXT -1 1 " " }{TEXT 317 1 "x" } {TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9);" "6#)\"#5,$\"\"*!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 53 "with 10 digit precision and wi th 20 digit precision. " }{TEXT 259 4 "Note" }{TEXT -1 37 ": It is ess ential to use the quotes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "(c) Find the first 5 terms of the Maclaurin serie s for " }{XPPEDIT 18 0 "g(x)" "6#-%\"gG6#%\"xG" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 15 "(d) Check that " }{XPPEDIT 18 0 "Limit(g( x),x=0)=1" "6#/-%&LimitG6$-%\"gG6#%\"xG/F*\"\"!\"\"\"" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 8 "(e) Let " }{XPPEDIT 18 0 "x[0] = 51;" "6#/&%\"xG6#\"\"!\"#^" }{TEXT -1 1 " " }{TEXT 323 1 "x" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "10^(-11);" "6#)\"#5,$\"#6!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 33 " (i) Find the sum of the first " } {TEXT 262 7 "5 terms" }{TEXT -1 29 " of the Maclaurin series for " } {XPPEDIT 18 0 "g(x)=(exp(x)-1)/x" "6#/-%\"gG6#%\"xG*&,&-%$expG6#F'\"\" \"F-!\"\"F-F'F." }{TEXT -1 14 " evaluated at " }{XPPEDIT 18 0 "x=x[0] " "6#/%\"xG&F$6#\"\"!" }{TEXT -1 82 ", performing the calculation with Maple's base 10 floating point arithmetic using " }{TEXT 262 9 "35 di gits" }{TEXT -1 15 " of precision. " }}{PARA 0 "" 0 "" {TEXT -1 26 " \+ (ii) Find the value of " }{XPPEDIT 18 0 "g(x[0])" "6#-%\"gG6#&%\"xG6# \"\"!" }{TEXT -1 34 ", performing the calculation with " }{TEXT 262 9 "45 digits" }{TEXT -1 14 " of precision." }}{PARA 0 "" 0 "" {TEXT -1 27 " (iii) Find the value of " }{XPPEDIT 18 0 "g(x[0])" "6#-%\"gG6#& %\"xG6#\"\"!" }{TEXT -1 34 ", performing the calculation with " } {TEXT 262 9 "10 digits" }{TEXT -1 15 " of precision. " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 259 4 "Note" }{TEXT -1 54 " for (iii): In o rder to prevent Maple from evaluating " }{XPPEDIT 18 0 "g(x[0])" "6#-% \"gG6#&%\"xG6#\"\"!" }{TEXT -1 4 " as " }{XPPEDIT 18 0 "exp(x[0])/x[0] -1/x[0]" "6#,&*&-%$expG6#&%\"xG6#\"\"!\"\"\"&F)6#F+!\"\"F,*&F,F,&F)6#F +F/F/" }{TEXT -1 10 ", compute " }{XPPEDIT 18 0 "g(x[0])" "6#-%\"gG6#& %\"xG6#\"\"!" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\" \"!" }{TEXT -1 45 " first evaluated as a floating point number. " }} {PARA 0 "" 0 "" {TEXT -1 28 " Also execute the command " }{TEXT 260 13 "forget(evalf)" }{TEXT -1 34 " before performing the evaluation." } }{PARA 0 "" 0 "" {TEXT -1 18 " (iv) Calculate " }{XPPEDIT 18 0 "exp( x[0])" "6#-%$expG6#&%\"xG6#\"\"!" }{TEXT -1 34 ", performing the calcu lation with " }{TEXT 262 9 "10 digits" }{TEXT -1 54 " of precision, an d check that using the approximation " }{XPPEDIT 18 0 "exp(x)" "6#-%$e xpG6#%\"xG" }{TEXT -1 1 " " }{TEXT 324 1 "~" }{TEXT -1 1 " " } {XPPEDIT 18 0 "1+x" "6#,&\"\"\"F$%\"xGF$" }{TEXT -1 64 ", obtained fro m the first two terms of the Maclaurin series for " }{XPPEDIT 18 0 "ex p(x)" "6#-%$expG6#%\"xG" }{TEXT -1 24 ", gives the same value. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 " (v) Us e the answer from (iv) to explain the result of (iii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "(e) Give one positiv e and one negative value of " }{TEXT 322 1 "x" }{TEXT -1 37 " for whic h the error in the value of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG " }{TEXT -1 65 " calculated using 10 digit precision is effectively eq ual to 0. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "(f) Plot a graph of " }{XPPEDIT 18 0 "g(x)=(exp(x)-1)/x" "6#/-% \"gG6#%\"xG*&,&-%$expG6#F'\"\"\"F-!\"\"F-F'F." }{TEXT -1 21 " over the interval " }{XPPEDIT 18 0 "-6;" "6#,$\"\"'!\"\"" }{TEXT -1 1 " " } {TEXT 332 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-16) <= x;" "6#1) \"#5,$\"#;!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 6;" "6#1%!G\"\"'" }{TEXT -1 1 " " }{TEXT 331 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-16);" " 6#)\"#5,$\"#;!\"\"" }{TEXT -1 11 " in which " }{XPPEDIT 18 0 "g(x)" " 6#-%\"gG6#%\"xG" }{TEXT -1 56 " is evaluated using hardware floating p oint arithmetic. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 320 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 " (a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "g := x -> (exp(x)-1) /x:\n'g(x)'=g(x);\nplot(g(x),x=-3..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&,&-%$expGF&\"\"\"F,!\"\"F,F'F-" }}{PARA 13 "" 1 "" {GLPLOT2D 427 336 336 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!\"$\"\"!$\"3 #oyt(QkPnJ!#=7$$!3!******\\2<#pG!#<$\"3aLO$yS0vG$F-7$$!3#)***\\7bBav#F 1$\"3gyk]FOX)R$F-7$$!36++]K3XFEF1$\"3Q/b*4nT4`$F-7$$!3%)****\\F)H')\\# F1$\"3??vP%*HAtOF-7$$!3#****\\i3@/P#F1$\"3)H*yJ)QiW#QF-7$$!3;++Dr^b^AF 1$\"3-f@z=Y)R(RF-7$$!3$****\\7Sw%G@F1$\"3R'GT(zS-RTF-7$$!3*****\\7;)=, ?F1$\"34&yU@vf:K%F-7$$!3/++DO\"3V(=F1$\"3YSq\"ooSl^%F-7$$!3#******\\V' zViUC\"F1$\"3+jPc$\\L5s&F-7$$!3-++DhkaI6F1$\"3'*[+(QQ@&*)fF-7 $$!3s******\\XF`**F-$\"3EZ\"\\Z#*pNL'F-7$$!3u*******>#z2))F-$\"3)e=3!H *yzk'F-7$$!3S++]7RKvuF-$\"3I#[EgZBF/(F-7$$!3s,+++P'eH'F-$\"3z*>\"G#G_0 U(F-7$$!3q)***\\7*3=+&F-$\"3Sx1weTtoyF-7$$!3[)***\\PFcpPF-$\"3U>,%GN48 L)F-7$$!3;)****\\7VQ[#F-$\"3GPds^D#[&))F-7$$!32)***\\i6:.8F-$\"3i&=#oJ #HeP*F-7$$!3Wb+++v`hH!#?$\"3geW_>p?&)**F-7$$\"3]****\\(QIKH\"F-$\"3cq) H&G9an5F17$$\"38****\\7:xWCF-$\"3G?^Kb*RG8\"F17$$\"3E,++vuY)o$F-$\"3H$ f]K9`$47F17$$\"3!z******4FL(\\F-$\"3)ecb)[.d&H\"F17$$\"3A)****\\d6.B'F -$\"3cjm/&[&o(Q\"F17$$\"3s****\\(o3lW(F-$\"3oIj**p^'[[\"F17$$\"35***** \\A))oz)F-$\"3l&Q&)[d4Ig\"F17$$\"3e******Hk-,5F1$\"3_&fOpj3$>F17 $$\"3M+++b*=jP\"F1$\"3%45S%o)z3:#F17$$\"3g***\\(3/3(\\\"F1$\"3fpi*[GDp J#F17$$\"33++vB4JB;F1$\"37E8@0\"Gr]#F17$$\"3u*****\\KCnu\"F1$\"3,v^(># oB6FF17$$\"3s***\\(=n#f(=F1$\"3=4%z460i%HF17$$\"3P+++!)RO+?F1$\"3u/%Ru ^\"H&>$F17$$\"30++]_!>w7#F1$\"3C'G/-=[cZ$F17$$\"3O++v)Q?QD#F1$\"3evPzH e/#y$F17$$\"3G+++5jypBF1$\"39Xgfa68\"4%F17$$\"3<++]Ujp-DF1$\"3#e'G%QJ@ 8[%F17$$\"3++++gEd@EF1$\"3aukIzNHm[F17$$\"39++v3'>$[FF1$\"3SN/RPFH=`F1 7$$\"37++D6EjpGF1$\"3%)*Htg,b`z&F17$$\"\"$F*$\"3obD1Tc%=O'F1-%'COLOURG 6&%$RGBG$\"#5!\"\"$F*F*Fa[l-%+AXESLABELSG6$Q\"x6\"Q!Ff[l-%%VIEWG6$;F(F fz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 42 ": There should be a \"missing point\" where " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y = 1;" "6#/%\"yG\"\"\"" }{TEXT -1 13 ", but, since " }{XPPEDIT 18 0 "Limit(g(x),x = 0) = 1;" "6#/-%&LimitG6$-%\"gG6#%\"xG/F*\"\"!\"\" \"" }{TEXT -1 120 " (see below), we can fill it in and obtain a contin uous function, indeed one which has derivatives of all orders at 0. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "g := x -> (exp(x)-1)/x:\n 'g(x)'=g(x);\nplot(['evalf[10](g(x))','evalf[20](g(x))'],x=-3e-9..3e-9 ,-.1..2,\n color=[red,brown],numpoints=100,axes=framed);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#%\"xG*&,&-%$expGF&\"\"\"F,!\"\"F,F' F-" }}{PARA 13 "" 1 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" }}{PARA 0 " " 0 "" {TEXT -1 5 "(d) " }{XPPEDIT 18 0 "Limit((exp(x)-1)/x,x = 0)=1 " "6#/-%&LimitG6$*&,&-%$expG6#%\"xG\"\"\"F-!\"\"F-F,F./F,\"\"!F-" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "g := x -> (exp(x)-1)/x:\nLimit(g(x),x=0);\n``=va lue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%&LimitG6$*&,&-%$expG6#%\" xG\"\"\"F,!\"\"F,F+F-/F+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G \"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "(e) (i) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "x0 := 51*10^(-1 1);\nEval(p(x),x=x0);\n``=value(%);\nevalf[35](%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#x0G#\"#^\"-+++++5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$,*\"\"\"F'*&#F'\"\"#F'%\"xGF'F'*&#F'\"\"'F'*$)F+F*F'F'F'* &#F'\"#CF'*$)F+\"\"$F'F'F'/F+#\"#^\"-+++++5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"C " 0 "" {MPLTEXT 1 0 86 "g := x -> (exp(x)-1)/x: \nx0 := 51*10^(-11);\nEval(g(x),x=x0);\n``=value(%);\nevalf[45](%);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G#\"#^\"-+++++5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$*&,&-%$expG6#%\"xG\"\"\"F,!\"\"F,F+F-/F+ #\"#^\"-+++++5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"-+++++5\" #^\"\"\"-%$expG6##F)F(F*F*#\"-+++++5F)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"E7Fb+++NV+++]D++++5!#N" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "(e) (iii) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "forget(evalf);\ng := x -> (exp(x)-1 )/x:\nx0 := evalf(51*10^(-11));\nEval(g(x),x=x0);\n``=value(%);\n" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"+++++^!#>" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%%EvalG6$*&,&-%$expG6#%\"xG\"\"\"F,!\"\"F,F+F-/F+$\" +++++^!#>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+9Vyg>!\"*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "taylor(exp(x),x=0,7):\nq := unapply(convert(%,polynom),x):\n'q(x)' =q(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"qG6#%\"xG,0\"\"\"F)F'F) *&#F)\"\"#F)*$)F'F,F)F)F)*&#F)\"\"'F)*$)F'\"\"$F)F)F)*&#F)\"#CF)*$)F' \"\"%F)F)F)*&#F)\"$?\"F)*$)F'\"\"&F)F)F)*&#F)\"$?(F)*$)F'F1F)F)F)" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "(e) (iv) \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "Digits := 10:\nx0 := ev alf(51*10^(-11));\nEval(exp(x),x=x0);\n``=value(%);\nEval(1+x,x=x0);\n ``=value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"+++++^!#>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$-%$expG6#%\"xG/F)$\"+++++^!# >" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+,+++5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$,&\"\"\"F'%\"xGF'/F($\"+++++^!#>" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+,+++5!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "(e) (v) " }}{PARA 0 "" 0 " " {TEXT -1 43 "We have the 10 digit value 1.000000001 for " }{XPPEDIT 18 0 "exp(x[0])" "6#-%$expG6#&%\"xG6#\"\"!" }{TEXT -1 13 " which gives " }{XPPEDIT 18 0 "exp(x[0])-1" "6#,&-%$expG6#&%\"xG6#\"\"!\"\"\"F+!\" \"" }{TEXT -1 1 " " }{TEXT 325 1 "~" }{TEXT -1 13 " 0.000000001." }} {PARA 0 "" 0 "" {TEXT -1 6 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "(exp(x[0])-1)/x[0]" "6#*&,&-%$expG6#&%\"xG6#\"\"!\" \"\"F,!\"\"F,&F)6#F+F-" }{TEXT -1 1 " " }{TEXT 326 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9)/(51*`.`*10^(-11))" "6#*&)\"#5,$\"\"*!\"\"\" \"\"*(\"#^F)%\".GF))F%,$\"#6F(F)F(" }{TEXT -1 1 " " }{TEXT 327 1 "~" } {TEXT -1 1 " " }{XPPEDIT 18 0 "100/51" "6#*&\"$+\"\"\"\"\"#^!\"\"" } {TEXT -1 1 " " }{TEXT 328 1 "~" }{TEXT -1 14 " 1.960784314. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "A brief summary of the error problem here is that " }{XPPEDIT 18 0 "exp(x)-1;" "6#,&- %$expG6#%\"xG\"\"\"F(!\"\"" }{TEXT -1 1 " " }{TEXT 343 1 "~" } {XPPEDIT 18 0 "``(1+x)-1;" "6#,&-%!G6#,&\"\"\"F(%\"xGF(F(F(!\"\"" } {TEXT -1 6 " when " }{TEXT 344 1 "x" }{TEXT -1 66 " is near 0. A large relative error may occur in the evaluation of " }{XPPEDIT 18 0 "exp(x )-1;" "6#,&-%$expG6#%\"xG\"\"\"F(!\"\"" }{TEXT -1 98 " since the last \+ step in the evaluation will involve the subtraction of \"nearly equal \+ quantities\". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 67 "ex0 := 1.000000001;\nnum := ex0-1;\nx0 := eval f(51*10^(-11));\nnum/x0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ex0G$\" +,+++5!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$numG$\"\"\"!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$\"+++++^!#>" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#$\"+9Vyg>!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(f) " }}{PARA 0 "" 0 "" {TEXT -1 50 "The error is effectively 0 whe n the approximation " }{XPPEDIT 18 0 "1+x;" "6#,&\"\"\"F$%\"xGF$" } {TEXT -1 5 " for " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 45 " has an exact floating point representation. " }}{PARA 0 "" 0 " " {TEXT -1 44 "The following table indicates the values of " }{TEXT 321 1 "x" }{TEXT -1 18 " in the interval " }{XPPEDIT 18 0 "-3" "6#,$ \"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 330 1 "x" }{TEXT -1 1 " " } {XPPEDIT 18 0 "10^(-9) <= x;" "6#1)\"#5,$\"\"*!\"\"%\"xG" }{XPPEDIT 18 0 "``<=3" "6#1%!G\"\"$" }{TEXT -1 1 " " }{TEXT 329 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9);" "6#)\"#5,$\"\"*!\"\"" }{TEXT -1 29 " f or which this is the case. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[x, 1+x], [________ ___, ___________], [-3*`.`*10^(-9), .9999999970], [-29*`.`*10^(-10), . 9999999971], [`.`, `.`], [`.`, `.`], [-2*`.`*10^(-10), .9999999998], [ -1*`.`*10^(-10), .9999999999], [1*`.`*10^(-9), 1.000000001], [2*`.`*10 ^(-7), 1.000000002], [3*`.`*10^(-9), 1.000000003]]);" "6#-%'matrixG6#7 -7$%\"xG,&\"\"\"F*F(F*7$%,___________GF,7$,$*(\"\"$F*%\".GF*)\"#5,$\" \"*!\"\"F*F6-%&FloatG6$\"+q********!#57$,$*(\"#HF*F1F*)F3,$F3F6F*F6-F8 6$\"+r********F;7$F1F17$F1F17$,$*(\"\"#F*F1F*)F3,$F3F6F*F6-F86$\"+)*** ******F;7$,$*(F*F*F1F*)F3,$F3F6F*F6-F86$\"+**********F;7$*(F*F*F1F*)F3 ,$F5F6F*-F86$\"+,+++5!\"*7$*(FJF*F1F*)F3,$\"\"(F6F*-F86$\"+-+++5Fin7$* (F0F*F1F*)F3,$F5F6F*-F86$\"+.+++5Fin" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 7 ": Wh en " }{XPPEDIT 18 0 "-5*`.`*10^(-11) < x;" "6#2,$*(\"\"&\"\"\"%\".GF') \"#5,$\"#6!\"\"F'F-%\"xG" }{XPPEDIT 18 0 "`` < 5*`.`*10^(-10);" "6#2%! G*(\"\"&\"\"\"%\".GF')\"#5,$F*!\"\"F'" }{TEXT -1 47 " the 10 digit flo ating point representation of " }{XPPEDIT 18 0 "1+x;" "6#,&\"\"\"F$%\" xGF$" }{TEXT -1 24 " is 1.000000000 so that " }{XPPEDIT 18 0 "exp(x)-1 ;" "6#,&-%$expG6#%\"xG\"\"\"F(!\"\"" }{TEXT -1 49 " evaluates to 0 rat her than to a number close to " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 11 ", and then " }{XPPEDIT 18 0 "(exp(x)-1)/x;" "6#*&,&-%$expG6#%\" xG\"\"\"F)!\"\"F)F(F*" }{TEXT -1 53 " also evaluates to 0 instead of t o a number close to " }{XPPEDIT 18 0 "1;" "6#\"\"\"" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "(g) " }}{PARA 0 "" 0 "" {TEXT -1 13 "The g raph of " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT -1 31 " is p lotted over the interval " }{XPPEDIT 18 0 "-6;" "6#,$\"\"'!\"\"" } {TEXT -1 1 " " }{TEXT 340 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-1 6) <= x;" "6#1)\"#5,$\"#;!\"\"%\"xG" }{XPPEDIT 18 0 "`` <= 6;" "6#1%!G \"\"'" }{TEXT -1 1 " " }{TEXT 339 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-16);" "6#)\"#5,$\"#;!\"\"" }{TEXT -1 45 " with hardware floati ng point arithmetic (in " }{TEXT 259 6 "purple" }{TEXT -1 34 ") and wi th 25 digit precision (in " }{TEXT 261 5 "brown" }{TEXT -1 17 "). The \+ values of " }{TEXT 342 1 "x" }{TEXT -1 85 " for which the error in the lower precision values is effectively 0 are shown by the " }{TEXT 257 5 "green" }{TEXT -1 43 " vertical dashed lines. They are values of " }{TEXT 341 1 "x" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "x+1;" " 6#,&%\"xG\"\"\"F%F%" }{TEXT -1 8 " has an " }{TEXT 259 5 "exact" } {TEXT -1 48 " 53 binary digit floating point representation. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 867 "g := x -> (exp(x)-1)/x:\n'g(x)'=g(x);\ngn := proc(x)\n evalhf(( exp(x)-1)/x);\nend proc;\ny0 := -.1: y1 := 2:\np1 := plot(['gn'(x),'ev alf[20](g(x))'],x=-6e-16..6e-16,y0..y1,\n color=[COLOR(RG B,.5,0,1),brown]):\nx1 := -2^(-51): x3 := -2^(-52): x4 := -2^(-53):\nx 0 := x1+x4: x2 := x3+x4: x5 := -x3: x6 := -x1:\np2 := plot([[[x0,y0],[ x0,y1]],[[x1,y0],[x1,y1]],[[x2,y0],[x2,y1]],\n [[x3,y0],[x3,y1]], [[x4,y0],[x4,y1]],[[x5,y0],[x5,y1]],\n [[x6,y0],[x6,y1]]],color=CO LOR(RGB,0,.7,0),linestyle=3):\nt1 := plots[textplot]([[x0,y1+.12,`-2^( -51)`],[x0,y1+.24,`-2^(-53)`],[x1,y1+.12,`-2^(-51)`],\n [x2,y1+.24 ,`-2^(-52)`],[x2,y1+.12,`-2^(-53)`],\n [x3,y1+.12,`-2^(-52)`],[x4, y1+.12,`-2^(-53)`],\n [x5,y1+.12,`2^(-52)`],\n [x6,y1+.12,`2^( -51)`]],font=[HELVETICA,8]):\nplots[display]([p1,p2,t1],axes=framed,fo nt=[HELVETICA,9],view=[-6e-16..6e-16,y0..y1+.24]);" }}{PARA 11 "" 1 " " {XPPMATH 20 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