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1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 2 1 3 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 44 "Errors in computer floating-point arithmetic" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B .C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 24.3.2007" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "Absolute and relative error" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "If " }{XPPEDIT 18 0 "x[approx];" "6#&%\"xG6#%'approxG" }{TEXT -1 29 " \+ approximates a real number " }{TEXT 263 1 "x" }{TEXT -1 6 ", the " } {TEXT 260 14 "absolute error" }{TEXT -1 4 " in " }{XPPEDIT 18 0 "x[app rox];" "6#&%\"xG6#%'approxG" }{TEXT -1 14 " is given by " }{XPPEDIT 18 0 "abs(x[approx]-x);" "6#-%$absG6#,&&%\"xG6#%'approxG\"\"\"F(!\"\" " }{TEXT -1 14 " , while the " }{TEXT 260 14 "relative error" }{TEXT 259 2 " " }{TEXT -1 14 "is given by " }{XPPEDIT 18 0 "abs(x[approx] -x)/abs(x);" "6#*&-%$absG6#,&&%\"xG6#%'approxG\"\"\"F)!\"\"F,-F%6#F)F- " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "For example, consider the rational approximation " } {XPPEDIT 18 0 "22/7" "6#*&\"#A\"\"\"\"\"(!\"\"" }{TEXT -1 31 " for the mathematical constant " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 93 "Converting both numbers to decimals ( using 10 digit floating-point arithmetic) we have . . . " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Pi_ac curate := evalf(Pi,10);\nPi_approx := evalf(22/7,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,Pi_accurateG$\"+aEfTJ!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*Pi_approxG$\"+Vr&G9$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "The more accurate value can be \+ used to estimate the absolute error in the approximation " }{XPPEDIT 18 0 "22/7" "6#*&\"#A\"\"\"\"\"(!\"\"" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "abserr := abs(Pi_approx-Pi_ accurate);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"(*[k7!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "An est imate for the absolute error in the approximation is 0.001264489. " }} {PARA 0 "" 0 "" {TEXT -1 52 "We can also estimate the relative error a s follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 44 "relerr := evalf[7](abserr/abs(Pi_accurate));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"($*\\-%!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 157 ": Since the value \+ given for the absolute error only has 7 significant digits, it is appr opriate to express the relative error with the same number of digits. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "The \+ percentage error is obtained by multiplying the relative error by 100% , and is therefore about 0.04%. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 248 "In perfo rmimg calculations using floating-point arithmetic on a computer, it i s usually more approriate to consider the relative error as an indicat ion of the accuracy of a particular result rather than the absolute er ror. Of course, if the number " }{TEXT 264 1 "x" }{TEXT -1 133 " is no t too far away from 1 (between say 0.5 and 5), then using absolute err or would not be much different from using relative error." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The following exam ple which examines the " }{TEXT 260 22 "Stirling approximation" } {TEXT -1 1 " " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "sqrt(2*Pi*n)*(n/exp( 1))^n;" "6#*&-%%sqrtG6#*(\"\"#\"\"\"%#PiGF)%\"nGF)F))*&F+F)-%$expG6#F) !\"\"F+F)" }}{PARA 0 "" 0 "" {TEXT -1 5 " to " }{XPPEDIT 18 0 "n! = n *`.`*(n-1)*` . . . `*3*`.`*2*`.`*1;" "6#/-%*factorialG6#%\"nG*4F'\"\" \"%\".GF),&F'F)F)!\"\"F)%(~.~.~.~GF)\"\"$F)F*F)\"\"#F)F*F)F)F)" } {TEXT -1 67 " demonstrates the difference between relative and absolu te error.\n" }}{PARA 0 "" 0 "" {TEXT -1 145 "This example has been ada pted from \"An Introduction to Scientific Computing\", by Charles F . Van Loan, Prentice Hall. (Matlab Curriculum Series)" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 448 "st := n - > sqrt(2*Pi*n)*n^n/exp(n);\nprintf(\"\\n\\n\"):\nprintf(\" n \+ n! stirling absolute rela tive\\n\");\nprintf(\" approximati on error error\\n\\n\");\nfor n from 1 to 20 do\n s := evalf(st(n),30);\n fact:= n!;\n abserr := evalf(abs(fact-s),3 0);\n relerr := abserr/abs(fact); \n printf(\"%4d %20d%25.2f%25.2f \+ %4.2e\\n\",n,n!,s,abserr,relerr);\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#stGf*6#%\"nG6\"6$%)operatorG%&arrowGF(*(-%%sqrtG6#,$ *(\"\"#\"\"\"%#PiGF39$F3F3F3)F5F5F3-%$expG6#F5!\"\"F(F(F(" }}{PARA 6 " " 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 " " {TEXT -1 86 " n n! stirling \+ absolute relative" }}{PARA 6 "" 1 "" {TEXT -1 83 " \+ approximation error \+ error" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 86 " 1 1 .92 \+ .08 7.79e-02" }}{PARA 6 "" 1 "" {TEXT -1 86 " 2 \+ 2 1.92 .08 4.05e-02 " }}{PARA 6 "" 1 "" {TEXT -1 86 " 3 6 \+ 5.84 .16 2.73e-02" }}{PARA 6 "" 1 "" {TEXT -1 86 " 4 24 23.51 \+ .49 2.06e-02" }}{PARA 6 "" 1 "" {TEXT -1 86 " 5 \+ 120 118.02 1.98 \+ 1.65e-02" }}{PARA 6 "" 1 "" {TEXT -1 86 " 6 720 \+ 710.08 9.92 1.38e-02" }}{PARA 6 " " 1 "" {TEXT -1 86 " 7 5040 4980.40 59.60 1.18e-02" }}{PARA 6 "" 1 "" {TEXT -1 86 " \+ 8 40320 39902.40 41 7.60 1.04e-02" }}{PARA 6 "" 1 "" {TEXT -1 86 " 9 362 880 359536.87 3343.13 9.21e-03" }} {PARA 6 "" 1 "" {TEXT -1 86 " 10 3628800 3 598695.62 30104.38 8.30e-03" }}{PARA 6 "" 1 "" {TEXT -1 86 " 11 39916800 39615625.05 \+ 301174.95 7.55e-03" }}{PARA 6 "" 1 "" {TEXT -1 86 " 12 \+ 479001600 475687486.47 3314113.53 \+ 6.92e-03" }}{PARA 6 "" 1 "" {TEXT -1 86 " 13 6227020800 \+ 6187239475.19 39781324.81 6.39e-03" }}{PARA 6 " " 1 "" {TEXT -1 86 " 14 87178291200 86661001740.60 517289459.40 5.93e-03" }}{PARA 6 "" 1 "" {TEXT -1 86 " \+ 15 1307674368000 1300430722199.47 724364580 0.53 5.54e-03" }}{PARA 6 "" 1 "" {TEXT -1 86 " 16 20922789888 000 20814114415223.13 108675472776.87 5.19e-03" }} {PARA 6 "" 1 "" {TEXT -1 86 " 17 355687428096000 353948328 666100.52 1739099429899.48 4.89e-03" }}{PARA 6 "" 1 "" {TEXT -1 86 " 18 6402373705728000 6372804626194309.19 \+ 29569079533690.81 4.62e-03" }}{PARA 6 "" 1 "" {TEXT -1 86 " 19 1 21645100408832000 121112786592293963.42 532313816538036.58 \+ 4.38e-03" }}{PARA 6 "" 1 "" {TEXT -1 86 " 20 2432902008176640000 2 422786846761133393.68 10115161415506606.32 4.16e-03" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "In terms of relat ive error, the Stirling approximation for " }{XPPEDIT 18 0 "n!" "6#-%* factorialG6#%\"nG" }{TEXT -1 27 " improves progressively as " }{TEXT 267 1 "n" }{TEXT -1 12 " increases. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 69 ": The following Map le code constructs the Stirling approximation for " }{XPPEDIT 18 0 "n! " "6#-%*factorialG6#%\"nG" }{TEXT -1 25 " as a single term in the " } {TEXT 260 19 "asympotic expansion" }{TEXT -1 4 " of " }{XPPEDIT 18 0 " n!" "6#-%*factorialG6#%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "k := 1;\nexpand(si mplify(convert(asympt(n!,n,k),polynom),symbolic));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"kG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*,\" \"##\"\"\"F$%#PiGF%%\"nGF%)F(F(F&-%$expG6#F(!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "More accurate approximati ons for " }{XPPEDIT 18 0 "n!" "6#-%*factorialG6#%\"nG" }{TEXT -1 78 " \+ can be obtained by increasing the number of terms in the asympotic exp ansion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "k := 3;\nfactor(expand(simplify(convert(asympt(n!,n,k ),polynom),symbolic)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"\"$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&#\"\"\"\"$)GF&*.\"\"##F&F)%#Pi GF*)%\"nGF-F&,(*&F'F&)F-F)F&F&*&\"#CF&F-F&F&F&F&F&F-#!\"$F)-%$expG6#F- !\"\"F&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 482 "st3 := n -> sqrt(2*Pi)*n^n*(288*n^2+24*n+1)/n^(3/2 )/exp(n)/288;\n\nprintf(\"\\n\\n\"):\nprintf(\" n \+ n! asymptotic absolute relative\\n\" );\nprintf(\" approximation \+ error error\\n\\n\");\nfor n from 1 to 20 do\n s := \+ evalf(st3(n),30);\n fact:= n!;\n abserr := evalf(abs(fact-s),30);\n \+ relerr := abserr/abs(fact); \n printf(\"%4d %20d%27.4f%25.4f %4.2e \\n\",n,n!,s,abserr,relerr);\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$st3Gf*6#%\"nG6\"6$%)operatorG%&arrowGF(,$*&#\"\"\"\"$)GF/*,-%%sq rtG6#,$*&\"\"#F/%#PiGF/F/F/)9$F:F/,(*&F0F/)F:F7F/F/*&\"#CF/F:F/F/F/F/F /F:#!\"$F7-%$expG6#F:!\"\"F/F/F(F(F(" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 88 " n \+ n! asymptotic absolute relative" }}{PARA 6 "" 1 "" {TEXT -1 85 " \+ approximation error error" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 88 " 1 \+ 1 1.0022 .0022 2.18e -03" }}{PARA 6 "" 1 "" {TEXT -1 88 " 2 2 \+ 2.0006 .0006 3.14e-04" }}{PARA 6 "" 1 "" {TEXT -1 88 " 3 6 6.0006 .0006 9.64e-05" }}{PARA 6 "" 1 "" {TEXT -1 88 " \+ 4 24 24.0010 \+ .0010 4.12e-05" }}{PARA 6 "" 1 "" {TEXT -1 88 " 5 \+ 120 120.0025 .0025 2.12e-05" }}{PARA 6 "" 1 "" {TEXT -1 88 " 6 720 \+ 720.0089 .0089 1.23e-05" }}{PARA 6 "" 1 "" {TEXT -1 88 " 7 5040 5040.0392 \+ .0392 7.77e-06" }}{PARA 6 "" 1 "" {TEXT -1 88 " 8 \+ 40320 40320.2102 .210 2 5.21e-06" }}{PARA 6 "" 1 "" {TEXT -1 88 " 9 362880 362881.3302 1.3302 3.67e-06" }} {PARA 6 "" 1 "" {TEXT -1 88 " 10 3628800 3 628809.7036 9.7036 2.67e-06" }}{PARA 6 "" 1 "" {TEXT -1 88 " 11 39916800 39916880.2342 \+ 80.2342 2.01e-06" }}{PARA 6 "" 1 "" {TEXT -1 88 " 12 \+ 479001600 479002341.8834 741.883 4 1.55e-06" }}{PARA 6 "" 1 "" {TEXT -1 88 " 13 6227020800 6227028387.8270 7587.8270 1.22e-06" }} {PARA 6 "" 1 "" {TEXT -1 88 " 14 87178291200 87178 376272.6740 85072.6740 9.76e-07" }}{PARA 6 "" 1 "" {TEXT -1 88 " 15 1307674368000 1307675405698.1388 \+ 1037698.1388 7.94e-07" }}{PARA 6 "" 1 "" {TEXT -1 88 " 16 \+ 20922789888000 20922803570630.3151 13682630.315 1 6.54e-07" }}{PARA 6 "" 1 "" {TEXT -1 88 " 17 355687428096000 355687622043941.2001 193947941.2001 5.45e-07" }} {PARA 6 "" 1 "" {TEXT -1 88 " 18 6402373705728000 6402376646 975582.0824 2941247582.0824 4.59e-07" }}{PARA 6 "" 1 "" {TEXT -1 88 " 19 121645100408832000 121645147929173371.9033 \+ 47520341371.9033 3.91e-07" }}{PARA 6 "" 1 "" {TEXT -1 88 " 20 \+ 2432902008176640000 2432902823091794028.7739 814915154028.773 9 3.35e-07" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "Again, the approximation improves (in terms of relative e rror) as " }{TEXT 279 1 "n" }{TEXT -1 12 " increases. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "Rounding errors" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 101 "When p erforming floating-point computer arithmetic (hardware or software ari thmetic) the results are " }{TEXT 260 35 "approximations to the exact \+ answers" }{TEXT -1 193 " whether or not the original numbers are exact representations of the intended numbers. This is simply because the e xact answer usually would not fit into the floating-point format being used.\n" }}{PARA 0 "" 0 "" {TEXT -1 45 "Let's start with a 10-digit a pproximation of " }{XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Digits := 10:\nr := evalf(sqrt(2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG$\"+iN@99!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "If we increase the precision to 20 digits, we can obtain the exact square of " }{TEXT 265 1 "r" } {TEXT -1 38 ". (Actually 19 digits are sufficient.)" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Digits := \+ 20:\ns := r*r;\nDigits:=10:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG$ \"4WysW*)*******>!#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "We can check that this is the exact product by using i nteger arithmetic." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "1414213562*1414213562;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"4WysW*)*******>" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 22 "A 10 digit version of " }{TEXT 268 3 "r^2 " }{TEXT -1 53 " can be obtained either by rounding our exact answer \+ " }{TEXT 268 1 "s" }{TEXT -1 38 ", or by performing the multiplication " }{TEXT 268 5 "r * r" }{TEXT -1 26 " with 10 digit precision." }} {PARA 0 "" 0 "" {TEXT -1 267 "In both cases we fail to get the exact a nswer 2.\nWhen performing a single floating-point arithmetic operation with Maple, any resulting error will be due to the intial rounding of the floating-point numbers involved so the resulting error is called \+ a rounding error. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 21 "evalf(s);\nevalf(r*r);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+********>!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+********>!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 107 "When fl oating-point arithmetic is being performed, one should generally assum e that an (absolute) error of " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\" #!\"\"" }{TEXT 261 23 " unit in the last place" }{TEXT 260 45 " is int roduced with each arithmetic operation" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 260 65 "This corresponds roughly to a relative error of the order of the " }{TEXT 261 15 "machine epsilon" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 7 "Warning" }{TEXT -1 77 ": In the case of addition (or subtraction) the error can be much more severe." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 319 "In a complex computation rounding errors can accumula te to produce significant problems.\n\nThe following remarks come from \"Numerical Recipies in C', Cambridge University press, page 29.\n\n \+ \"Roundoff errors accumulate with increasing amounts of calculation. I f, in the course of obtaining a calculated value, you perform " } {TEXT 269 1 "N" }{TEXT -1 98 " such arithmetic operations, you might b e lucky as to have a total roundoff error on the order of " }{XPPEDIT 18 0 "sqrt(N);" "6#-%%sqrtG6#%\"NG" }{TEXT -1 188 " times the machine \+ epsilon, if the roundoff errors come in randomly up or down. (The squa re root comes from a random walk.) However this estimate can be badly \+ off the mark for two reasons:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 15 "" 0 "" {TEXT -1 135 "It very frequently happens that the reg ularities of your calculation or the peculiarities of you computer, ca use the roudoff errors to " }{TEXT 260 42 "accumulate preferentially i n one direction" }{TEXT -1 49 ", In this case the total will be of the order of " }{TEXT 270 1 "N" }{TEXT -1 27 " times the machine epsilon. " }}{PARA 15 "" 0 "" {TEXT -1 135 "Some especially unfavourable circum stances can vastly increase the roundoff of single operations. General ly these can be traced to the " }{TEXT 260 38 "subtraction of nearly e qual quantities" }{TEXT -1 3 ".\"\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 73 "Absolute a nd relative error in relation to ulp's and the machine epsilon " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 142 "If the r esult of a numerical calculation is the floating-point number nearest \+ to the correct result, it still might be in error by as much as " } {XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 6 " ulp. " }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "largest absolute erro r" }{TEXT -1 91 " which can result from rounding an \"exact\" number t o \"fit\" a floating-point representation " }{TEXT 271 1 "x" }{TEXT -1 53 ", is half a unit in the last decimal place ( written " } {XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 8 " ulp ). " }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 22 "largest relative erro r" }{TEXT -1 34 " involved in such rounding is the " }{TEXT 260 15 "ma chine epsilon" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 165 "This i s suggested by the following table in which the 10 digit floating-poi nt representations of the values in the first column all have an absol ute error close to " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" } {TEXT -1 5 " ulp." }}{PARA 0 "" 0 "" {TEXT -1 77 "Note that the machin e epsilon for 10 digit floating-point arithmetic is 0.5 " }{TEXT 280 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9)" "6#)\"#5,$\"\"*!\"\"" } {TEXT -1 3 " = " }{XPPEDIT 18 0 "5*`.`*10^(-10)" "6#*(\"\"&\"\"\"%\".G F%)\"#5,$F(!\"\"F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "matrix([[`true valu e`, `representing value`, `absolute error`, `relative error`], [.99999 999995, 1.000000000, 5*`.`*10^(-11), 5*`.`*10^(-11)], [1.000000000499, 1.000000000, ``(4.99)*`.`*10^(-10), ``(4.99)*`.`*10^(-10)], [1.999999 9995, 2.000000000, 5*`.`*10^(-10), ``(2.5)*`.`*10^(-10)], [2.000000000 499, 2.000000000, ``(4.99)*`.`*10^(-10), ``(2.495)*`.`*10^(-10)], [4.9 999999995, 5.000000000, 5*`.`*10^(-10), 10^(-10)], [5.000000000499, 5. 000000000, ``(4.99)*`.`*10^(-10), ``(9.98)*`.`*10^(-11)], [9.999999999 5, 10.00000000, 5*`.`*10^(-10), 5*`.`*10^(-11)]]);" "6#-%'matrixG6#7*7 &%+true~valueG%3representing~valueG%/absolute~errorG%/relative~errorG7 &-%&FloatG6$\",&**********!#6-F.6$\"+++++5!\"**(\"\"&\"\"\"%\".GF8)\"# 5,$\"#6!\"\"F8*(F7F8F9F8)F;,$F=F>F87&-F.6$\".*\\++++5!#7-F.6$F4F5*(-%! G6#-F.6$\"$*\\!\"#F8F9F8)F;,$F;F>F8*(-FK6#-F.6$FOFPF8F9F8)F;,$F;F>F87& -F.6$\",&********>!#5-F.6$\"+++++?F5*(F7F8F9F8)F;,$F;F>F8*(-FK6#-F.6$ \"#DF>F8F9F8)F;,$F;F>F87&-F.6$\".*\\++++?FF-F.6$\"+++++?F5*(-FK6#-F.6$ FOFPF8F9F8)F;,$F;F>F8*(-FK6#-F.6$\"%&\\#!\"$F8F9F8)F;,$F;F>F87&-F.6$\" ,&********\\Fhn-F.6$\"+++++]F5*(F7F8F9F8)F;,$F;F>F8)F;,$F;F>7&-F.6$\". *\\++++]FF-F.6$\"+++++]F5*(-FK6#-F.6$FOFPF8F9F8)F;,$F;F>F8*(-FK6#-F.6$ \"$)**FPF8F9F8)F;,$F=F>F87&-F.6$\",&**********Fhn-F.6$F4!\")*(F7F8F9F8 )F;,$F;F>F8*(F7F8F9F8)F;,$F=F>F8" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 133 "The following graph plot s the (signed) relative error involved in rounding numbers to fit the \+ 10 digit floating-point representation." }}{PARA 0 "" 0 "" {TEXT -1 52 "The graph indicates that for many sampled values of " }{TEXT 303 1 "x" }{TEXT -1 82 " between 1 and 10 this relative rounding error has a magnitude which is less than " }{XPPEDIT 18 0 "epsilon/x" "6#*&%(ep silonG\"\"\"%\"xG!\"\"" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "epsilon = 5*`.`*10^(-10);" "6#/%(epsilonG*(\"\"&\"\"\"%\".GF')\"#5,$F*!\"\"F' " }{TEXT -1 24 " is the machine epsilon." }}{PARA 0 "" 0 "" {TEXT -1 58 "This holds because the absolute error is always less than " } {XPPEDIT 18 0 "epsilon" "6#%(epsilonG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 386 "Digits : = 15:\nmach_eps := 5e-10:\nplot([('evalf[10](x)'-x)/x,mach_eps/x,-mach _eps/x],x=0..10,\n color=[COLOR(RGB,.8,.6,.85),COLOR(RGB,.4,.4,.8)$ 2],linestyle=[1,1$2],\n numpoints=400,font=[HELVETICA,9],labels=[`x `,``],axes=framed,\n view=[0..10,-5.2e-10..5.2e-10],xtickmarks=11,y tickmarks=11,\n title=`relative error in 10 digit floating-point re presentation of x`);\nDigits := 10:" }}{PARA 13 "" 1 "" {GLPLOT2D 610 347 347 {PLOTDATA 2 "6+-%'CURVESG6%7ehy7$$\"05\\'f)=W>)!#=$\"0[49'G=#G %!#D7$$\"0#)H>x$))Q;!#x3Nbb'F1F:7$$F)F1F+7$$\"0&* y:j-L$)*F1F:7$$\"0RaxL;Z)=FJ$\"0r^:z[$*[\"FM7$$\"0z:j_gm'>FJ$!0y& zxor0;FM7$$\"0G7\\r/'[?FJ$\"0$3&GzQ>G%F-7$$\"0y3N!*[08#FJFK7$$\"0F0@4$ \\7AFJ$!0i@r&4)zv%F-7$$\"0w,2GPWH#FJ$\"0>k%4%R)*H\"FM7$$\"0D)Hp9QwBFJ$ !0&zkT%f]D\"FM7$$F4FJF57$$\"0B\"\\Y)p-a#FJ$!0D&\\O4xL>FM7$$\"0s(3NS@AE FJ$!0r9>ok_M$F-7$$\"0LKaB8Np#FJ$!0v9=Vy]g\"FM7$$\"0%pxNC\"[w#FJ$\"0)f2 @n\"y1)F-7$$\"0b@hj6h$GFJ$!0]^OW(z&G%F-7$$\"0;mk$3T2HFJ$!0Y:_(4N.;FM7$ $\"0x5o.5(yHFJ$\"0\\Ru$*\\FN'F-7$$\"0QbrB4+0$FJ$!0G+2!3T%4&F-7$$\"0++v V387$FJ$\"0<#R:B*=g\"FM7$$\"0hWyj2E>$FJ$\"0Z*\\R7=n[F-7$$\"0A*=Qo!RE$F J$!0M1<%oM(z&F-7$$\"0%Q`Qg?NLFJ$\"0(o#*eZp(R\"FM7$$\"0Xy)Q_]1MFJ$\"0$o z%*HL^**e5FM7$$\"06Y4W+V$QFJ$\"0hP?]raS\"F-7$$\"0s!HT' *f0RFJ$!0Cb/JrOW(F-7$$\"0MN;%))*o(RFJ$\"0yD`Ec%p\"*F-7$$\"0&*z>/)>[SFJ $\"0p(>K4#G&\\!#E7$$\"0cCBC(\\>TFJ$!0Dj4KJ'yyF-7$$\"0=pEW'z!>%FJ$\"0fv [*['R*yF-7$$\"0z8Ik&4iUFJ$!0Sdua(\\NKFay7$$\"0SeL%[RLVFJ$!0l**>$=lq#)F -7$$\"0iZSC$*fZ%FJ$!0&G$=%z*Q1\"F-7$$\"0%otW;f=YFJ$\"0gq!>3%yp&F-7$$\" 0X\"3X3*)*o%FJ$!0)4A%>9nt\"F-7$$\"01Ea/!>hZFJ$!0Ng**4.'[*)F-7$$\"0oqdC *[K[FJ$\"0N9ul(Fay7$$\"0? 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1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 " As an example, consider the rounding of the rational number: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "104/99 = 1.0505050505 0505050*` . . . `" "6#/*&\"$/\"\"\"\"\"#**!\"\"*&-%&FloatG6$\"3]]]]]]] ]5!# " 0 "" {MPLTEXT 1 0 35 "104/99;\neva lf(%,15);\nx := evalf(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 "Th e absolute error involved in this rounding is:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1050505051/10^9 - 104/99 = 49/99000000 000" "6#/,&*&\"+^]]]5\"\"\"*$\"#5\"\"*!\"\"F'*&\"$/\"F'\"#**F+F+*&\"# \\F'\",++++!**F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 " ``= 0" "6#/%!G\"\"!" }{TEXT -1 10 ".000000000" } {XPPEDIT 18 0 "494949494949494949*` . . . `" "6#*&\"3\\\\\\\\\\\\\\\\ \\\"\"\"%(~.~.~.~GF%" }{TEXT -1 3 ". " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 273 1 "~" }{TEXT -1 7 " 0.495 " }{TEXT 274 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9)" "6#)\"#5,$\"\"*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 22 " ulp with r espect to " }{XPPEDIT 18 0 "x = 1.050505051" "6#/%\"xG-%&FloatG6$\"+^ ]]]5!\"*" }{TEXT -1 5 " is: " }}{PARA 256 "" 0 "" {TEXT -1 5 " 0.5 " } {TEXT 278 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9)" "6#)\"#5,$\" \"*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "1050505051/10^9 - 104/99;\nevalf(%, 13);\nevalf(%,3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#\\\",++++!** " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\".&\\\\\\\\\\\\!#A" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"$&\\!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "The relative error in this rounding is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``(49/99000000000)/``(104/99)=49/1 04000000000" "6#/*&-%!G6#*&\"#\\\"\"\"\",++++!**!\"\"F*-F&6#*&\"$/\"F* \"#**F,F,*&F)F*\"-++++S5F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{TEXT 275 1 "~" }{TEXT -1 7 " 0.491 " }{TEXT 276 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9 )" "6#)\"#5,$\"\"*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "This is only slightly less than t he machine epsilon for Maple's 10 digit software floating-point arithm etic, namely: " }}{PARA 256 "" 0 "" {TEXT -1 5 " 0.5 " }{TEXT 277 1 "x " }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9)" "6#)\"#5,$\"\"*!\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "(49/99000000000)/(104/99);\nevalf(%,3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6##\"#\\\"-++++S5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$r%!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 36 "The accumulation of rounding errors " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 92 "In this section we invest igate how rounding errors can accumulate in a lengthy calculation. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "The code below constructs two arrays " }{TEXT 291 1 "A" }{TEXT -1 5 " and " } {TEXT 292 1 "B" }{TEXT -1 21 " with 10000 entries. " }}{PARA 0 "" 0 " " {TEXT -1 10 "The array " }{TEXT 293 1 "B" }{TEXT -1 39 " is filled w ith 10000 rational numbers " }{XPPEDIT 18 0 "p/q" "6#*&%\"pG\"\"\"%\"q G!\"\"" }{TEXT -1 8 ", where " }{TEXT 289 1 "q" }{TEXT -1 46 " is a ra ndom integer between 1 and 999, while " }{TEXT 290 1 "p" }{TEXT -1 35 " is a random integer between 1 and " }{TEXT 295 1 "q" }{TEXT -1 45 ". The resulting rational numbers lie between " }{XPPEDIT 18 0 "1/999" " 6#*&\"\"\"F$\"$***!\"\"" }{TEXT -1 8 " and 1. " }}{PARA 0 "" 0 "" {TEXT -1 10 "The array " }{TEXT 294 1 "A" }{TEXT -1 119 " is filled wi th the corresponding 10 digit floating-point approximations for each o f the rational numbers in the array " }{TEXT 296 1 "B" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "n := 10000:\nA := array(1.. n);\nB := array(1..n);\nrandomize():\nfor i to n do\n q := rand(1..9 99)();\n p := rand(1..q)();\n r := p/q;\n A[i] := evalf(r);\n \+ B[i] := r;\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%&arrayG 6$;\"\"\"\"&++\"7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG-%&arrayG 6$;\"\"\"\"&++\"7\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 52 "The first and numbers in each array are as follows. " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "A[1],A[n];\nB[1],B[n];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"+(* Rb&3$!#5$\"+Q. " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "First we calculate the exact sum of the rational numbers in the array " }{TEXT 297 1 "A " }{TEXT -1 87 " using exact rational arithmetic and also the sum of t he floating-numbers in the array " }{TEXT 298 1 "B" }{TEXT -1 58 " usi ng Maple's 10 digit decimal floating-point arithmetic." }}{PARA 0 "" 0 "" {TEXT -1 76 "The absolute and relative error in the floating-poin t sum are then computed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 327 "a := add(A[i],i=1..n):\nprint(`app rox sum = `,a);\nb := add(B[i],i=1..n):\nprint(`exact sum = `,b);\nDig its := 15:\nbb := evalf(b):\nprint(`accurate sum = `,bb);\nabserr := a bs(a-bb):\nrelerr := abserr/abs(bb):\nrelerr_per_op := relerr/(n-1):\n Digits := 10:\nprint(`absolute error = `,abserr);\nprint(`relative err or = `,evalf[5](relerr));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%.approx~ sum~=~G$\"+XqXI]!\"'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$%-exact~sum~=~ G#\"_flf.#etlf.+gtIjG-rj&R2$o]v+,U#)4E#4\"H&o$[^Jp]QBwg\"\\/'oY> HR)Q\">_K*R?\"R#=It5N#z=!pqu75=*=A#Gc_o,#3Th\\/ymiw,XIf9ZRY^J+a&ynGJud I`myY+X\">&4tH;1A\"o))*oRb.WMKu/Ol[DHMssG:cM*ol:kmi\\*HupM**3lhYeo/(Hf(p='G0#)fawW%\\$))4B(=(z>dBf\"yqX*322O*3O[2Jheq8A4$>'QZ@ nGF(>ZVch')o?BvDS@JucW5$pe-%)=eZ8L3%Q#HV9;WIp\"R\\r,]u$f(of1'e4'))y^NL \"QUZ)HI7o6QL_1%z-,!pIc^AW,&G]<-IJNYV(GT:5R#QH06YBJ(4g.1z)om%\\/q(Gat) R?&ylj$z]z`^o$ya%H;9'4#z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%0accurate ~sum~=~G$\"0-;k/d/.&!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2absolute~ error~=~G$\"'-;9!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~erro r~=~G$\"&\\\"G!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 244 "Evaluati ng the exact sum as a floating-point number with a few more digits tha n were used to calculate the previous floating-point number enables th e accuracy of the floating-point sum to be checked. The approximate su m is correct to 9 digits. " }}{PARA 0 "" 0 "" {TEXT -1 96 "The absolut e error is about 1.4 ulp and the relative error is less than the machi ne epsilon 0.5 " }{TEXT 299 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^( -9);" "6#)\"#5,$\"\"*!\"\"" }{TEXT -1 50 " for 10 digit decimal floati ng-point arithmetic. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 267 "Instead of adding all the numbers in an array we no w consider constructing a single value from the array of numbers by ma king use of all four basic arithmetic operations in a random way. This is achieved by choosing randomly at each stage to do one of the follo wing: " }{TEXT 268 3 "add" }{TEXT -1 66 " the next number in the array to the current \"accumulated\" value, " }{TEXT 268 8 "subtract" } {TEXT -1 55 " the next number from the current \"accumulated\" value, \+ " }{TEXT 268 8 "multiply" }{TEXT -1 55 " the current \"accumulated\" v alue by the next number or " }{TEXT 268 6 "divide" }{TEXT -1 52 " the \+ current \"accumulated\" value by the next number." }}{PARA 0 "" 0 "" {TEXT -1 139 "The next code segment gives an indication of how this wo rks by constructing an algebraic expression in which each of the index ed variables " }{XPPEDIT 18 0 "a[1],a[2],` . . . `,a[20]" "6&&%\"aG6# \"\"\"&F$6#\"\"#%(~.~.~.~G&F$6#\"#?" }{TEXT -1 22 " occurs exactly onc e. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 212 "a := 'a':\nrandomize():\nr := a[1]:\nm := 20:\nfor i from 2 to m do\n k := rand(1..4)();\n if k=1 then r := r+a[i]\n \+ elif k=2 then r := r-a[i]\n elif k=3 then r := r*a[i]\n else r := r/a[i]\n end if;\nend do:\nr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#** ,**(,**(,**(,&&%\"aG6#\"\"\"F.&F,6#\"\"#F.F.&F,6#\"\"$F.&F,6#\"\"%F.F. &F,6#\"\"&!\"\"&F,6#\"\"'F;&F,6#\"\"(F;F.&F,6#\"\")F;&F,6#\"\"*F;F.&F, 6#\"#5F;&F,6#\"#6F;&F,6#\"#7F;F.&F,6#\"#8F;&F,6#\"#9F;F.&F,6#\"#:F;&F, 6#\"#;F;&F,6#\"#F;&F,6#\"#?F;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "20 numerical trials \+ using the arrays " }{TEXT 300 1 "A" }{TEXT -1 5 " and " }{TEXT 301 1 " B" }{TEXT -1 38 " are performed by the following code. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 583 "#rando mize():\na := A[1]: b := B[1]:\nfor j to 20 do\n for i from 2 to n d o\n k := rand(1..4)();\n if k=1 then a := a+A[i]; b := b+B[i ];\n elif k=2 then a := a-A[i]; b := b-B[i];\n elif k=3 then a := a*A[i]; b := b*B[i];\n else a := a/A[i]; b := b/B[i];\n \+ end if;\n end do:\n a;\n b;\n Digits := 15:\n bb := evalf( b);\n abserr := abs(a-bb);\n relerr := abserr/abs(bb);\n relerr_ per_op := relerr/(n-1);\n Digits := 10:\n #print(`absolute error = `,abserr);\n relerr := evalf[5](abserr/abs(bb));\n print(`relativ e error = `,evalf[5](relerr));\nend do:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~error~=~G$\"&9C\"!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~error~=~G$\"&]_#!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% 2relative~error~=~G$\"&%*G%!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2re lative~error~=~G$\"&hx$!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2relati ve~error~=~G$\"&ZP\"!#8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~ error~=~G$\"&i\"G!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~err or~=~G$\"&!4Q!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~error~= ~G$\"&Nh%!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~error~=~G$ \"&ai$!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~error~=~G$\"&o 4%!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~error~=~G$\"&3h$!# 7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~error~=~G$\"&sj&!#7" } }{PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~error~=~G$\"&?2'!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~error~=~G$\"&)>f!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~error~=~G$\"&]M)!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~error~=~G$\"&wm)!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~error~=~G$\"&(>q!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~error~=~G$\"&x'y!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~error~=~G$\"&#yn!#7" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%2relative~error~=~G$\"&TF(!#7" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The total relative errors may be compared with the s" }{XPPEDIT 18 0 "qrt(N)*`. `*epsilon" "6#*(-%$qrtG6#%\"NG\"\"\"%\".GF(%(epsilonGF(" }{TEXT -1 8 " , where " }{TEXT 302 1 "N" }{TEXT -1 54 " is the number of arithmetic \+ operations performed and " }{XPPEDIT 18 0 "epsilon = 5*`.`*10^(-10);" "6#/%(epsilonG*(\"\"&\"\"\"%\".GF')\"#5,$F*!\"\"F'" }{TEXT -1 82 " is \+ the machine epsilon for Maple's 10 digit software floating-point arit hmetic. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "sqrt(n-1)*5*10^(-10);\nevalf[5](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"$\"\"\"\"+++++?!\"\"\"%66#F&\"\"#F&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"&)**\\!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "In the trials above the r elative error is not vastly different from this value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 101 ": You can, of course, perform new trials by re-executing all the code from the start of this section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 53 "Reducing rounding errors in a sum of positive terms " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "Consider the sum " }{XPPEDIT 18 0 "Sum(1/(k*(k+1)),k = 1 \+ .. n) = 1/(1*`.`*2)+1/(2*`.`*3)+1/(3*`.`*4)+1/(4*`.`*5)+` . . . `+1/(n *(n+1));" "6#/-%$SumG6$*&\"\"\"F(*&%\"kGF(,&F*F(F(F(F(!\"\"/F*;F(%\"nG ,.*&F(F(*(F(F(%\".GF(\"\"#F(F,F(*&F(F(*(F4F(F3F(\"\"$F(F,F(*&F(F(*(F7F (F3F(\"\"%F(F,F(*&F(F(*(F:F(F3F(\"\"&F(F,F(%(~.~.~.~GF(*&F(F(*&F/F(,&F /F(F(F(F(F,F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "1/(k*(k+1))=1/k-1/(k +1)" "6#/*&\"\"\"F%*&%\"kGF%,&F'F%F%F%F%!\"\",&*&F%F%F'F)F%*&F%F%,&F'F %F%F%F)F)" }{TEXT -1 10 " for each " }{TEXT 304 1 "k" }{TEXT -1 5 " so : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/(k*(k+1)) ,k = 1 .. n)=Sum(``(1/k-1/(k+1)),k = 1 .. n)" "6#/-%$SumG6$*&\"\"\"F(* &%\"kGF(,&F*F(F(F(F(!\"\"/F*;F(%\"nG-F%6$-%!G6#,&*&F(F(F*F,F(*&F(F(,&F *F(F(F(F,F,/F*;F(F/" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = ``(1-1/2)+``(1/ 2-1/3)+``(1/3-1/4)+``(1/4-1/5)+` . . . `+``(1/n-1/(n+1));" "6#/%!G,.-F $6#,&\"\"\"F)*&F)F)\"\"#!\"\"F,F)-F$6#,&*&F)F)F+F,F)*&F)F)\"\"$F,F,F)- F$6#,&*&F)F)F2F,F)*&F)F)\"\"%F,F,F)-F$6#,&*&F)F)F8F,F)*&F)F)\"\"&F,F,F )%(~.~.~.~GF)-F$6#,&*&F)F)%\"nGF,F)*&F)F),&FDF)F)F)F,F,F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1+``(-1/2+1/2)+``(-1/3+1/3)+``(-1/4+1/4)+``(-1/5 +1/5)+` . . . `+``(-1/n+1/n)-1/(n+1)" "6#/%!G,2\"\"\"F&-F$6#,&*&F&F&\" \"#!\"\"F,*&F&F&F+F,F&F&-F$6#,&*&F&F&\"\"$F,F,*&F&F&F2F,F&F&-F$6#,&*&F &F&\"\"%F,F,*&F&F&F8F,F&F&-F$6#,&*&F&F&\"\"&F,F,*&F&F&F>F,F&F&%(~.~.~. ~GF&-F$6#,&*&F&F&%\"nGF,F,*&F&F&FEF,F&F&*&F&F&,&FEF&F&F&F,F," }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1-1/(n+ 1)" "6#/%!G,&\"\"\"F&*&F&F&,&%\"nGF&F&F&!\"\"F*" }{XPPEDIT 18 0 "``=n/ (n+1)" "6#/%!G*&%\"nG\"\"\",&F&F'F'F'!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 28 ": It follows from this that " }{XPPEDIT 18 0 "Sum(1/(k*(k+1)),k = 1 .. infinity) = Limit(n/(n+1), n = infinity);" "6#/-%$SumG6$*&\"\"\"F(*&%\"kGF(,&F*F(F(F(F(!\"\"/F*;F (%)infinityG-%&LimitG6$*&%\"nGF(,&F4F(F(F(F,/F4F/" }{XPPEDIT 18 0 "``= 1" "6#/%!G\"\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 109 "Because of the cancellation that occurs \+ here, this finite series is sometimes referred to as an example of a \+ " }{TEXT 260 18 "telescoping series" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "For example" }{XPPEDIT 18 0 "Sum(1/(k*(k+1)),k = 1 .. 99999)=99999/100000" "6#/-%$SumG6$*&\" \"\"F(*&%\"kGF(,&F*F(F(F(F(!\"\"/F*;F(\"&******&F/F(\"'++5F," }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "k := 'k':\nSum(1/(k*(k+1)),k=1..99999);\nexact_sum := value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'*&%\" kGF',&F)F'F'F'F'!\"\"/F);F'\"&*****" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%*exact_sumG#\"&*****\"'++5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The following " }{TEXT 268 8 "for-loop" }{TEXT -1 17 " sums the se ries " }{XPPEDIT 18 0 "Sum(1/(k*(k+1)),k = 1 .. 99999)" "6#-%$SumG6$*& \"\"\"F'*&%\"kGF',&F)F'F'F'F'!\"\"/F);F'\"&*****" }{TEXT -1 3 " \"" } {TEXT 260 8 "forwards" }{TEXT -1 33 "\" beginning at the term given by " }{XPPEDIT 18 0 "k=1" "6#/%\"kG\"\"\"" }{TEXT -1 81 " using Maple's \+ software floating-point arithmetic with a precision of 10 digits. " }} {PARA 0 "" 0 "" {TEXT -1 60 "The absolute and relative errors in the s um are calculated. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 258 "Digits := 10:\npartial_sum := 0.0:\nfor k \+ from 1 to 99999 do\n term := evalf(1/(k*(k+1)));\n partial_sum := \+ partial_sum + term;\nend do:\napprox_sum := partial_sum;\nabserr := ev alf[11](abs(exact_sum-approx_sum));\nrelerr := evalf[5](abserr/exact_s um);\nk := 'k':" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+a pprox_sumG$\"+H!)*)****!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abser rG$\"&5(>!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"&5(>!#6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The fo llowing " }{TEXT 268 8 "for-loop" }{TEXT -1 17 " sums the series " } {XPPEDIT 18 0 "Sum(1/(k*(k+1)),k = 1 .. 99999)" "6#-%$SumG6$*&\"\"\"F' *&%\"kGF',&F)F'F'F'F'!\"\"/F);F'\"&*****" }{TEXT -1 3 " \"" }{TEXT 260 9 "backwards" }{TEXT -1 33 "\" beginning at the term given by " } {XPPEDIT 18 0 "k = 99999;" "6#/%\"kG\"&*****" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 60 "The absolute and relative errors in the s um are calculated. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 264 "Digits := 10:\npartial_sum := 0.0:\nfor k \+ from 99999 to 1 by -1 do\n term := evalf(1/(k*(k+1)));\n partial_s um := partial_sum + term;\nend do:\napprox_sum := partial_sum;\nabserr := evalf[20](abs(exact_sum-approx_sum));\nrelerr := evalf[5](abserr/e xact_sum);\nk := 'k':" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%+approx_sumG$\"+++!*****!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%' abserrG$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"\"!F& " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "The re appears to be a significant advantage in adding \"backwards\" start ing with the smaller numbers." }}{PARA 0 "" 0 "" {TEXT -1 46 "In gener al, when summing a finite sequence of " }{TEXT 260 16 "positive number s" }{TEXT -1 32 " with floating-point arithmetic " }{TEXT 260 20 "erro rs are minimised" }{TEXT -1 29 " if the numbers are added in " }{TEXT 260 26 "reverse order of magnitude" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 80 "Explanation for difference in the errors when summing \"forwards\" and \"backwards\"" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "When summing " }{TEXT 260 8 "forwards" }{TEXT -1 97 " the terms eventually become much small er than the partial sum which the term is being added to. " }}{PARA 0 "" 0 "" {TEXT -1 122 "For example when summing forwards we can obtain \+ an estimate for the (absolute) error involved in adding the term given by " }{XPPEDIT 18 0 "k=5000" "6#/%\"kG\"%+]" }{TEXT -1 13 " as follow s. " }}{PARA 0 "" 0 "" {TEXT -1 16 "The partial sum " }{XPPEDIT 18 0 " Sum(1/(k*(k+1)),k = 1 .. 4999)" "6#-%$SumG6$*&\"\"\"F'*&%\"kGF',&F)F'F 'F'F'!\"\"/F);F'\"%**\\" }{TEXT -1 77 " obtained using 10 digit softwa re floating-point arithmetic is 0.9997999949. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "partial_sum := 0.0;\nfor k from 1 to 4999 do\n term := evalf(1/(k*(k+1)));\n \+ partial_sum := partial_sum + term;\nend do:\npartial_sum;\nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,partial_sumG$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+\\***z***!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "The next term for " }{XPPEDIT 18 0 "k=5000" "6#/%\"kG\"%+ ]" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "1/(5000*`.`*5001)" "6#*&\"\"\"F$ *(\"%+]F$%\".GF$\"%,]F$!\"\"" }{TEXT -1 1 " " }{TEXT 305 1 "~" }{TEXT -1 21 " 0.00000003999200160." }}{PARA 0 "" 0 "" {TEXT -1 52 "Adding th is term to the previous partial sum gives: " }}{PARA 256 "" 0 "" {TEXT -1 28 " 0.9997999949 +" }}{PARA 256 "" 0 "" {TEXT -1 21 " 0.00000003999200160 " }}{PARA 256 "" 0 "" {TEXT -1 20 " ______ ____________ " }}{PARA 256 "" 0 "" {TEXT -1 21 " 0.99980003489200160 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Howev er, using 10 digit floating-point arithmetic, the value obtained is 0. 9998000349. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 28 " 0.9998000349 -" }}{PARA 256 "" 0 "" {TEXT -1 21 " 0.99980003489200160 " }}{PARA 256 "" 0 "" {TEXT -1 20 " ______ ____________ " }}{PARA 256 "" 0 "" {TEXT -1 21 " 0.00000000000799840 \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The a bsolute error introduced here is about 0.799840 " }{TEXT 306 1 "x" } {TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-11)" "6#)\"#5,$\"#6!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "k := 5000;\nterm := evalf(1/(k*(k+1)));\nsum1 := par tial_sum + term;\nsum2 := evalf[20](partial_sum + term);\nabserr := ev alf[20](abs(sum1-sum2));\nk := 'k': " }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"kG\"%+]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%termG$\"+g,?**R!#< " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sum1G$\"+\\.+)***!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sum2G$\"2g,?*[.+)***!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"'S)*z!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "We can also obtain an estimate for the (absolute) error involved in adding the term given by " } {XPPEDIT 18 0 "k=5000" "6#/%\"kG\"%+]" }{TEXT -1 14 " when summing " } {TEXT 260 9 "backwards" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "The partial sum " }{XPPEDIT 18 0 "Sum(1/(k*(k+1)),k = 5001 .. 9999 9);" "6#-%$SumG6$*&\"\"\"F'*&%\"kGF',&F)F'F'F'F'!\"\"/F);\"%,]\"&***** " }{TEXT -1 98 " obtained summing backwards using 10 digit software fl oating-point arithmetic is 0.0001899600067. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 153 "partial_sum := 0.0;\nfor k from 99999 to 5001 by -1 \+ do\n term := evalf(1/(k*(k+1)));\n partial_sum := partial_sum + te rm;\nend do:\npartial_sum;\nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,partial_sumG$\"\"!F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+n+g* *=!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The term for " } {XPPEDIT 18 0 "k=5000" "6#/%\"kG\"%+]" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "1/(5000*`.`*5001)" "6#*&\"\"\"F$*(\"%+]F$%\".GF$\"%,]F$!\"\"" } {TEXT -1 1 " " }{TEXT 307 1 "~" }{TEXT -1 21 " 0.00000003999200160." } }{PARA 0 "" 0 "" {TEXT -1 52 "Adding this term to the previous partial sum gives: " }}{PARA 256 "" 0 "" {TEXT -1 27 " 0.0001899600067 \+ +" }}{PARA 256 "" 0 "" {TEXT -1 21 " 0.00000003999200160 " }}{PARA 256 "" 0 "" {TEXT -1 20 " __________________ " }}{PARA 256 "" 0 "" {TEXT -1 21 " 0.00018999999870160 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 90 "However, using 10 digit floating-point ar ithmetic, the value obtained is 0.0001899999987. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 23 " 0.00018999999870160 - " }}{PARA 256 "" 0 "" {TEXT -1 26 " 0.0001899999987 " }} {PARA 256 "" 0 "" {TEXT -1 20 " __________________ " }}{PARA 256 "" 0 "" {TEXT -1 21 " 0.00000000000000160 " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 50 "The absolute error introduced here is about 0.160 " }{TEXT 308 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-1 4);" "6#)\"#5,$\"#9!\"\"" }{TEXT -1 112 ", which is considerable small er than the absolute error involved when adding this term in the forwa rd direction." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 152 "k := 5000;\nterm := evalf(1/(k*(k+1)));\nsum1 : = partial_sum + term;\nsum2 := evalf[20](partial_sum + term);\nabserr \+ := evalf[20](abs(sum1-sum2));\nk := 'k':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"%+]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%termG$\"+g,?** R!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sum1G$\"+()******=!#8" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sum2G$\"/g,()******=!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"$g\"!#<" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 67 "The largest possible floating-point val ue for the \"forward\" sum as " }{TEXT 313 1 "n" }{TEXT -1 12 " increa ses " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 " " {TEXT -1 22 "Consider further sums " }{XPPEDIT 18 0 "Sum(1/(k*(k+1)) ,k = 1 .. n)" "6#-%$SumG6$*&\"\"\"F'*&%\"kGF',&F)F'F'F'F'!\"\"/F);F'% \"nG" }{TEXT -1 6 " with " }{TEXT 312 1 "n" }{TEXT -1 32 " possibly la rger than the value " }{XPPEDIT 18 0 "n=99999" "6#/%\"nG\"&*****" } {TEXT -1 24 " considered previously. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 44 "The partial sums are all between 0 an d 1 so " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 32 " \+ ulp in each of their values is " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\" \"#!\"\"" }{TEXT -1 1 " " }{TEXT 309 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10);" "6#)\"#5,$F$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "1/(k*(k+1) )=5*10^(-11);\nsolve(%);\nop(select(type,[evalf(%)],positive));\nfloor (%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&\"\"\"F%*&%\"kGF%,&F'F%F%F %F%!\"\"#F%\",+++++#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&#\"\"\"\"\"# !\"\"*(\"$f%F%F&F'\"'@(z$#F%F&F',&#F%F&F'*(F)F%F&F'F*F+F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+i&3UT\"!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"'?99" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 311 1 "k" }{TEXT -1 89 " is greater than or equal to 141421 the magnit ude of the corresponding term is less than " }{XPPEDIT 18 0 "1/2" "6#* &\"\"\"F$\"\"#!\"\"" }{TEXT -1 42 " ulp for the current partial sum, t hat is," }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "1/(k*(k+1 )) <1/2" "6#2*&\"\"\"F%*&%\"kGF%,&F'F%F%F%F%!\"\"*&F%F%\"\"#F)" } {TEXT -1 1 " " }{TEXT 310 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-1 0);" "6#)\"#5,$F$!\"\"" }{TEXT -1 7 ", when " }{XPPEDIT 18 0 "141421<= k" "6#1\"'@99%\"kG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 148 "so that there is no change in the 10 di git floating-point value for the partial sum when the corresponding te rm is added to the current partial sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Eval(1/(k*(k+1)),k=141420);\n``=value(%);\n``=evalf(r hs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$*&\"\"\"F'*&%\"kG F',&F)F'F'F'F'!\"\"/F)\"'?99" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G# \"\"\"\",?yv***>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+Y01+]!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Eval(1/(k*(k+1)),k=141421);\n``=value(%);\n``=evalf(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$*&\"\"\"F'*&%\"kGF',&F)F'F 'F'F'!\"\"/F)\"'@99" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"\"\",i 1/++#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+N)*)***\\!#?" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 11 "There is a " }{TEXT 260 22 "lar gest possible value" }{TEXT -1 54 " for the sum when summing in the fo rward direction as " }{TEXT 328 1 "n" }{TEXT -1 20 " increases given b y " }{XPPEDIT 18 0 "n=141420" "6#/%\"nG\"'?99" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 332 "n := 141420:\nSum(1/(k*(k+1)),k=1..n);\nexact_sum := value(%);\nD igits := 10:\nevalf(exact_sum);\npartial_sum := 0.0:\nfor k from 1 to \+ n do\n term := evalf(1/(k*(k+1)));\n partial_sum := partial_sum + \+ term;\nend do:\napprox_sum := partial_sum;\nabserr := abs(exact_sum-ap prox_sum);\nrelerr := evalf[5](abserr/exact_sum);\nk := 'k': n := 'n': " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'*&%\"kGF',&F)F' F'F'F'!\"\"/F);F'\"'?99" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*exact_su mG#\"'?99\"'@99" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+*GH*****!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%+approx_sumG$\"+]%R*****!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"&h,\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"&h,\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "Taking the larger value " } {XPPEDIT 18 0 "n=200000" "6#/%\"nG\"'++?" }{TEXT -1 21 " gives the sam e sum. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 332 "n := 200000:\nSum(1/(k*(k+1)),k=1..n);\nexact_sum := value(%);\nDigits := 10:\nevalf(exact_sum);\npartial_sum := 0.0:\nfor k from 1 to n do\n term := evalf(1/(k*(k+1)));\n partial_sum := p artial_sum + term;\nend do:\napprox_sum := partial_sum;\nabserr := abs (exact_sum-approx_sum);\nrelerr := evalf[5](abserr/exact_sum);\nk := ' k': n := 'n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'*& %\"kGF',&F)F'F'F'F'!\"\"/F);F'\"'++?" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%*exact_sumG#\"'++?\"',+?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++ &*****!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+approx_sumG$\"+]%R**** *!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"&]0\"!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"&]0\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 60 "Summing \"backwards \" reduces the error in the final sum when " }{XPPEDIT 18 0 "n=200000 " "6#/%\"nG\"'++?" }{TEXT -1 11 " to 1 ulp. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 338 "n := 200000:\nSum (1/(k*(k+1)),k=1..n);\nexact_sum := value(%);\nDigits := 10:\nevalf(ex act_sum);\npartial_sum := 0.0:\nfor k from n to 1 by -1 do\n term := evalf(1/(k*(k+1)));\n partial_sum := partial_sum + term;\nend do:\n approx_sum := partial_sum;\nabserr := abs(exact_sum-approx_sum);\nrele rr := evalf[5](abserr/exact_sum);\nk := 'k': n := 'n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'*&%\"kGF',&F)F'F'F'F'!\"\"/F);F '\"'++?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*exact_sumG#\"'++?\"',+? " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+++&*****!#5" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%+approx_sumG$\"+,+&*****!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"\"\"!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'relerrG$\"&++\"!#9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 62 "Another ex ample of rounding errors in a sum of positive terms " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 337 8 "Questio n" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 17 "Consider the sum " }{XPPEDIT 18 0 "Sum(k/``(k^4+k^2+1),k = 1 .. n)" "6#-%$SumG6$*&%\"kG\" \"\"-%!G6#,(*$F'\"\"%F(*$F'\"\"#F(F(F(!\"\"/F';F(%\"nG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "(k^2-k+1)*(k^ 2+k+1)=(k^2+1-k)*(k^2+1+k)" "6#/*&,(*$%\"kG\"\"#\"\"\"F'!\"\"F)F)F),(* $F'F(F)F'F)F)F)F)*&,(*$F'F(F)F)F)F'F*F),(*$F'F(F)F)F)F'F)F)" } {XPPEDIT 18 0 "``=(k^2+1)^2-k^2" "6#/%!G,&*$,&*$%\"kG\"\"#\"\"\"F+F+F* F+*$F)F*!\"\"" }{XPPEDIT 18 0 "``=k^4+k^2+1" "6#/%!G,(*$%\"kG\"\"%\"\" \"*$F'\"\"#F)F)F)" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "Sum(k/``(k^4+k^2+1),k = 1 .. n) = Sum(k/((k^2-k+1)*( k^2+k+1)),k = 1 .. n);" "6#/-%$SumG6$*&%\"kG\"\"\"-%!G6#,(*$F(\"\"%F)* $F(\"\"#F)F)F)!\"\"/F(;F)%\"nG-F%6$*&F(F)*&,(*$F(F1F)F(F2F)F)F),(*$F(F 1F)F(F)F)F)F)F2/F(;F)F5" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/(1*`.`*3)+2 /(3*`.`*7)+ 3/(7*`.`*13)+4/(13*`.`*21)+5/(21*`.`*31)+` . . . `+n/((n^2 -n+1)*(n^2+n+1)" "6#/%!G,0*&\"\"\"F'*(F'F'%\".GF'\"\"$F'!\"\"F'*&\"\"# F'*(F*F'F)F'\"\"(F'F+F'*&F*F'*(F/F'F)F'\"#8F'F+F'*&\"\"%F'*(F2F'F)F'\" #@F'F+F'*&\"\"&F'*(F6F'F)F'\"#JF'F+F'%(~.~.~.~GF'*&%\"nGF'*&,(*$F=F-F' F=F+F'F'F',(*$F=F-F'F=F'F'F'F'F+F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 21 "By the phonomena of \"" }{TEXT 260 11 "telescoping" } {TEXT -1 43 "\", this series can be seen to have the sum " }{XPPEDIT 18 0 "n*(n+1)/(2*(n^2+n+1))" "6#*(%\"nG\"\"\",&F$F%F%F%F%*&\"\"#F%,(*$ F$F(F%F$F%F%F%F%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 42 "In detail, the partial fraction expan sion " }{XPPEDIT 18 0 "k/((k^2-k+1)*(k^2+k+1))=1/2" "6#/*&%\"kG\"\"\"* &,(*$F%\"\"#F&F%!\"\"F&F&F&,(*$F%F*F&F%F&F&F&F&F+*&F&F&F*F+" } {XPPEDIT 18 0 "``(1/((k^2-k+1))-1/((k^2+k+1)))" "6#-%!G6#,&*&\"\"\"F(, (*$%\"kG\"\"#F(F+!\"\"F(F(F-F(*&F(F(,(*$F+F,F(F+F(F(F(F-F-" }{TEXT -1 13 " shows that: " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "Sum(k/``(k^4+k^2 +1),k = 1 .. n)=1/2" "6#/-%$SumG6$*&%\"kG\"\"\"-%!G6#,(*$F(\"\"%F)*$F( \"\"#F)F)F)!\"\"/F(;F)%\"nG*&F)F)F1F2" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``(1/(k^2-k+1)-1/(k^2+k+1)),k = 1 .. n)" "6#-%$SumG6$-%!G6#,&*&\" \"\"F+,(*$%\"kG\"\"#F+F.!\"\"F+F+F0F+*&F+F+,(*$F.F/F+F.F+F+F+F0F0/F.;F +%\"nG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#! \"\"" }{XPPEDIT 18 0 "``(``(1-1/3)+``(1/3-1/7)+``(1/7-1/13)+``(1/13-1/ 21)+` . . . `+``(1/(n^2-n+1)-1/(n^2+n+1)));" "6#-%!G6#,.-F$6#,&\"\"\"F **&F*F*\"\"$!\"\"F-F*-F$6#,&*&F*F*F,F-F**&F*F*\"\"(F-F-F*-F$6#,&*&F*F* F3F-F**&F*F*\"#8F-F-F*-F$6#,&*&F*F*F9F-F**&F*F*\"#@F-F-F*%(~.~.~.~GF*- F$6#,&*&F*F*,(*$%\"nG\"\"#F*FGF-F*F*F-F**&F*F*,(*$FGFHF*FGF*F*F*F-F-F* " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" } {XPPEDIT 18 0 "``(1-``(-1/3+1/3)+``(-1/7+1/7)+``(-1/13+1/13)+``(-1/21+ 1/21)+` . . . `+``(-1/(n^2-n+1)+1/(n^2-n+1))-1/(n^2+n+1));" "6#-%!G6#, 2\"\"\"F'-F$6#,&*&F'F'\"\"$!\"\"F-*&F'F'F,F-F'F--F$6#,&*&F'F'\"\"(F-F- *&F'F'F3F-F'F'-F$6#,&*&F'F'\"#8F-F-*&F'F'F9F-F'F'-F$6#,&*&F'F'\"#@F-F- *&F'F'F?F-F'F'%(~.~.~.~GF'-F$6#,&*&F'F',(*$%\"nG\"\"#F'FHF-F'F'F-F-*&F 'F',(*$FHFIF'FHF-F'F'F-F'F'*&F'F',(*$FHFIF'FHF'F'F'F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{XPPEDIT 18 0 "` `(1-1/(n^2+n+1)) = n*(n+1)/(2*(n^2+n+1));" "6#/-%!G6#,&\"\"\"F(*&F(F(, (*$%\"nG\"\"#F(F,F(F(F(!\"\"F.*(F,F(,&F,F(F(F(F(*&F-F(,(*$F,F-F(F,F(F( F(F(F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 28 ": It follows from this that " } {XPPEDIT 18 0 "Sum(k/``(k^4+k^2+1),k = 1 .. infinity) = Limit(n*(n+1)/ (2*(n^2+n+1)),n = infinity);" "6#/-%$SumG6$*&%\"kG\"\"\"-%!G6#,(*$F(\" \"%F)*$F(\"\"#F)F)F)!\"\"/F(;F)%)infinityG-%&LimitG6$*(%\"nGF),&F:F)F) F)F)*&F1F),(*$F:F1F)F:F)F)F)F)F2/F:F5" }{XPPEDIT 18 0 "``=1/2" "6#/%!G *&\"\"\"F&\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "(a) Check that " }{XPPEDIT 18 0 "Sum( k/``(k^4+k^2+1),k = 1 .. n)=n*(n+1)/(2*(n^2+n+1))" "6#/-%$SumG6$*&%\"k G\"\"\"-%!G6#,(*$F(\"\"%F)*$F(\"\"#F)F)F)!\"\"/F(;F)%\"nG*(F5F),&F5F)F )F)F)*&F1F),(*$F5F1F)F5F)F)F)F)F2" }{TEXT -1 14 " using Maple. " }} {PARA 0 "" 0 "" {TEXT -1 19 "(b) Sum the series " }{XPPEDIT 18 0 "Sum( k/``(k^4+k^2+1),k = 1 .. n);" "6#-%$SumG6$*&%\"kG\"\"\"-%!G6#,(*$F'\" \"%F(*$F'\"\"#F(F(F(!\"\"/F';F(%\"nG" }{TEXT -1 3 " \"" }{TEXT 260 8 "forwards" }{TEXT -1 6 "\" for " }{XPPEDIT 18 0 "n = 5000;" "6#/%\"nG \"%+]" }{TEXT -1 35 ", (starting with the term given by " }{XPPEDIT 18 0 "k=1" "6#/%\"kG\"\"\"" }{TEXT -1 67 ") with Maple's 10 digit soft ware floating-point arithmetic using a " }{TEXT 268 8 "for-loop" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 58 " Calculate the absol ute and relative errors in the sum." }}{PARA 0 "" 0 "" {TEXT -1 19 "(c ) Sum the series " }{XPPEDIT 18 0 "Sum(k/``(k^4+k^2+1),k = 1 .. n);" " 6#-%$SumG6$*&%\"kG\"\"\"-%!G6#,(*$F'\"\"%F(*$F'\"\"#F(F(F(!\"\"/F';F(% \"nG" }{TEXT -1 3 " \"" }{TEXT 260 9 "backwards" }{TEXT -1 6 "\" for \+ " }{XPPEDIT 18 0 "n = 5000;" "6#/%\"nG\"%+]" }{TEXT -1 35 ", (starting with the term given by " }{XPPEDIT 18 0 "k = 5000;" "6#/%\"kG\"%+]" } {TEXT -1 67 ") with Maple's 10 digit software floating-point arithmeti c using a " }{TEXT 268 8 "for-loop" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 58 " Calculate the absolute and relative errors in the s um." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "(d ) Find the " }{TEXT 260 13 "largest value" }{TEXT -1 35 " which can be obtained for the sum " }{XPPEDIT 18 0 "Sum(k/``(k^4+k^2+1),k = 1 .. n );" "6#-%$SumG6$*&%\"kG\"\"\"-%!G6#,(*$F'\"\"%F(*$F'\"\"#F(F(F(!\"\"/F ';F(%\"nG" }{TEXT -1 15 " when summing \"" }{TEXT 260 8 "forwards" } {TEXT -1 85 "\" with Maple's 10 digit software floating-point arithmet ic, and the minimum value of " }{TEXT 339 1 "n" }{TEXT -1 23 " which g ives this sum. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 338 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "k := 'k':\nSum(k/(k^4+k^2+1),k=1..n);\nnormal(value(% ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&%\"kG\"\"\",(*$)F'\" \"%F(F(*$)F'\"\"#F(F(F(F(!\"\"/F';F(%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\"#!\"\"%\"nG\"\"\",&F'F(F(F(F(,(*$)F'F%F(F(F'F(F (F(F&F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Sum(k/``(k ^4+k^2+1),k = 1 .. 5000) = 12502500/25005001;" "6#/-%$SumG6$*&%\"kG\" \"\"-%!G6#,(*$F(\"\"%F)*$F(\"\"#F)F)F)!\"\"/F(;F)\"%+]*&\")+D]7F)\"),] +DF2" }{TEXT -1 2 " " }{TEXT 343 1 "~" }{TEXT -1 25 " 0.4999999800039 9999984. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Sum(k/(k^4+k^2+1),k=1..5000);\nexact_sum := value(%); \nevalf[20](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&%\"kG\" \"\",(*$)F'\"\"%F(F(*$)F'\"\"#F(F(F(F(!\"\"/F';F(\"%+]" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*exact_sumG#\")+D]7\"),]+D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5%)*****R+!)*****\\!#?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The following " }{TEXT 268 8 "for-loop" }{TEXT -1 17 " su ms the series " }{XPPEDIT 18 0 "Sum(k/``(k^4+k^2+1),k = 1 .. 5000) " " 6#-%$SumG6$*&%\"kG\"\"\"-%!G6#,(*$F'\"\"%F(*$F'\"\"#F(F(F(!\"\"/F';F( \"%+]" }{TEXT -1 2 " \"" }{TEXT 260 8 "forwards" }{TEXT -1 33 "\" begi nning at the term given by " }{XPPEDIT 18 0 "k=1" "6#/%\"kG\"\"\"" } {TEXT -1 81 " using Maple's software floating-point arithmetic with a \+ precision of 10 digits. " }}{PARA 0 "" 0 "" {TEXT -1 60 "The absolute \+ and relative errors in the sum are calculated. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 259 "Digits := 1 0:\npartial_sum := 0.0:\nfor k from 1 to 5000 do\n term := evalf(k/( k^4+k^2+1));\n partial_sum := partial_sum + term;\nend do:\napprox_s um := partial_sum;\nabserr := evalf[12](abs(exact_sum-approx_sum));\nr elerr := evalf[5](abserr/exact_sum);\nk := 'k':" }{TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%+approx_sumG$\"+Z%*****\\!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"&/`$!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"&31(!#7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 14 "absolute error" }{TEXT -1 97 " in the approximate sum (obtained by summing \"forwards\") due to rounding er rors is about 0.35304 " }{TEXT 340 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-7);" "6#)\"#5,$\"\"(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "The corresponding " }{TEXT 260 14 "relative error" } {TEXT -1 18 " is about 0.70608 " }{TEXT 344 1 "x" }{TEXT -1 1 " " } {XPPEDIT 18 0 "10^(-7);" "6#)\"#5,$\"\"(!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }} {PARA 0 "" 0 "" {TEXT -1 14 "The following " }{TEXT 268 8 "for-loop" } {TEXT -1 18 " sums the series " }{XPPEDIT 18 0 "Sum(k/``(k^4+k^2+1),k = 1 .. 5000) " "6#-%$SumG6$*&%\"kG\"\"\"-%!G6#,(*$F'\"\"%F(*$F'\"\"#F (F(F(!\"\"/F';F(\"%+]" }{TEXT -1 3 " \"" }{TEXT 260 9 "backwards" } {TEXT -1 33 "\" beginning at the term given by " }{XPPEDIT 18 0 "k = 5 000;" "6#/%\"kG\"%+]" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 " The absolute and relative errors in the sum are calculated. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 265 "D igits := 10:\npartial_sum := 0.0:\nfor k from 5000 to 1 by -1 do\n t erm := evalf(k/(k^4+k^2+1));\n partial_sum := partial_sum + term;\ne nd do:\napprox_sum := partial_sum;\nabserr := evalf[15](abs(exact_sum- approx_sum));\nrelerr := evalf[5](abserr/exact_sum);\nk := 'k':" } {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+approx_sumG$\"++)** ***\\!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"%+S!#:" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"&++)!#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 126 "The rounding error in the final sum is r educed when the finite series is summed \"backwards\" starting with th e smaller numbers." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 14 "absolute error" }{TEXT -1 50 " in the approximate sum in this case is about 0.4 " }{TEXT 345 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-11) ;" "6#)\"#5,$\"#6!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 80 "This means that the value obtained by summing backwards is correct to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 18 "The corresponding " } {TEXT 260 14 "relative error" }{TEXT -1 14 " is about 0.8 " }{TEXT 346 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-11);" "6#)\"#5,$\"#6!\" \"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }}{PARA 0 "" 0 "" {TEXT -1 22 "Consider further sums " }{XPPEDIT 18 0 "Sum(k/``(k^4+k^2+ 1),k = 1 .. n);" "6#-%$SumG6$*&%\"kG\"\"\"-%!G6#,(*$F'\"\"%F(*$F'\"\"# F(F(F(!\"\"/F';F(%\"nG" }{TEXT -1 7 " with " }{TEXT 336 1 "n" }{TEXT -1 32 " possibly larger than the value " }{XPPEDIT 18 0 "n=99999" "6#/ %\"nG\"&*****" }{TEXT -1 24 " considered previously. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "For any positive integ er " }{TEXT 341 1 "n" }{TEXT -1 32 " the partial sums of the series " }{XPPEDIT 18 0 "Sum(k/``(k^4+k^2+1),k = 1 .. n);" "6#-%$SumG6$*&%\"kG \"\"\"-%!G6#,(*$F'\"\"%F(*$F'\"\"#F(F(F(!\"\"/F';F(%\"nG" }{TEXT -1 29 " are all between 0 and 1 so " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$ \"\"#!\"\"" }{TEXT -1 32 " ulp in each of their values is " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 333 1 "x" } {TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10);" "6#)\"#5,$F$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "k/(k^4+k^2+1)=5*10^(-11);\nfsolve(%,k=1..10000);\nflo or(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%\"kG\"\"\",(*$)F%\"\"%F &F&*$)F%\"\"#F&F&F&F&!\"\"#F&\",+++++#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+%\\ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 335 1 "k" }{TEXT -1 88 " is greater than or equal to 2714, the magnitude of the corresp onding term is less than " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\" \"" }{TEXT -1 42 " ulp for the current partial sum, that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "k/``(k^4+k^2+1)<1/2" "6#2*& %\"kG\"\"\"-%!G6#,(*$F%\"\"%F&*$F%\"\"#F&F&F&!\"\"*&F&F&F.F/" }{TEXT -1 1 " " }{TEXT 334 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10);" " 6#)\"#5,$F$!\"\"" }{TEXT -1 7 ", when " }{XPPEDIT 18 0 "2714 <= k;" "6 #1\"%9F%\"kG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 148 "so that there is no change in the 10 dig it floating-point value for the partial sum when the corresponding ter m is added to the current partial sum." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Eval(k/(k^4+k^2+1),k=2714);\n``=value(%);\n``=evalf(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$*&%\"kG\"\"\",(*$)F'\"\"%F (F(*$)F'\"\"#F(F(F(F(!\"\"/F'\"%9F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%!G#\"%9F\"/8%z!e\\Da" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"++yI- ]!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "Eval(k/(k^4+k^2+1),k=2715);\n``=value(%);\n``=evalf(r hs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$*&%\"kG\"\"\",(*$ )F'\"\"%F(F(*$)F'\"\"#F(F(F(F(!\"\"/F'\"%:F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"%:F\"/^=Pl\\La" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%!G$\"+CCy'*\\!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "There is a " }{TEXT 260 22 "largest possible value" } {TEXT -1 54 " for the sum when summing in the forward direction as " } {TEXT 342 1 "n" }{TEXT -1 36 " increases that is first given when " } {XPPEDIT 18 0 "n = 2714;" "6#/%\"nG\"%9F" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 345 "n := 2714:\nSum(k/(k^4+k^2+1),k=1..n);\nexact_sum := value(%);\nDigits := \+ 10:\nevalf(exact_sum);\npartial_sum := 0.0:\nfor k from 1 to n do\n \+ term := evalf(k/(k^4+k^2+1));\n partial_sum := partial_sum + term;\n end do:\napprox_sum := partial_sum;\nabserr := evalf[11](abs(exact_sum -approx_sum));\nrelerr := evalf[5](abserr/exact_sum);\nk := 'k': n := \+ 'n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&%\"kG\"\"\",(*$)F' \"\"%F(F(*$)F'\"\"#F(F(F(F(!\"\"/F';F(\"%9F" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*exact_sumG#\"(bUo$\"(6&ot" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+@$*****\\!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+ approx_sumG$\"+Z%*****\\!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abse rrG$\"%c7!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"&?^#!#7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 119 "Note \+ that the sum of 5000 terms obtained previously by summing \"forwards\" is the same as the last sum with 2714 terms. " }}{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 5 "Tasks" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 15 "(a) The series " }{XPPEDIT 18 0 "1+1/1!+1 /2!+1/3!+1/3!+` . . . `+1/n!+` . . . `;" "6#,2\"\"\"F$*&F$F$-%*factori alG6#F$!\"\"F$*&F$F$-F'6#\"\"#F)F$*&F$F$-F'6#\"\"$F)F$*&F$F$-F'6#F1F)F $%(~.~.~.~GF$*&F$F$-F'6#%\"nGF)F$F5F$" }{TEXT -1 14 " converges to " } {XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 10 ", that is " } {XPPEDIT 18 0 "Limit(Sum(1/i!,i = 0 .. n),n=infinity)=``" "6#/-%&Limit G6$-%$SumG6$*&\"\"\"F+-%*factorialG6#%\"iG!\"\"/F/;\"\"!%\"nG/F4%)infi nityG%!G" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/n!,n=0..infinity)=exp( 1)" "6#/-%$SumG6$*&\"\"\"F(-%*factorialG6#%\"nG!\"\"/F,;\"\"!%)infinit yG-%$expG6#F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 " C heck this fact using Maple's " }{TEXT 0 8 "Sum..sum" }{TEXT -1 8 " and /or " }{TEXT 0 12 "Limit..limit" }{TEXT -1 27 " commands. (Consult Ma ple " }{TEXT 268 4 "help" }{TEXT -1 16 " if necessary.) " }}{PARA 0 " " 0 "" {TEXT -1 30 "(b) Check that the finite sum " }{XPPEDIT 18 0 "Su m(1/n!,n = 0 .. 8) = 1+1/1!+1/2!+1/3!+1/4!+` . . . `+1/8!;" "6#/-%$Sum G6$*&\"\"\"F(-%*factorialG6#%\"nG!\"\"/F,;\"\"!\"\"),0F(F(*&F(F(-F*6#F (F-F(*&F(F(-F*6#\"\"#F-F(*&F(F(-F*6#\"\"$F-F(*&F(F(-F*6#\"\"%F-F(%(~.~ .~.~GF(*&F(F(-F*6#F1F-F(" }{TEXT -1 24 " is the rational number " } {XPPEDIT 18 0 "109601/40320" "6#*&\"','4\"\"\"\"\"&?.%!\"\"" }{TEXT -1 57 ", and estimate the absolute and relative errors in using " } {XPPEDIT 18 0 "109601/40320" "6#*&\"','4\"\"\"\"\"&?.%!\"\"" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 41 "________________________ _________________" }}{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 41 "_________________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 35 "The Macluaurin series expansion of " }{XPPEDIT 18 0 "arctan*x" "6#*&%'arctanG\"\"\"%\"xGF% " }{TEXT -1 6 " is: " }}{PARA 256 "" 0 "" {TEXT -1 8 " " } {XPPEDIT 18 0 "arctan*x = x-x^3/3+x^5/5-x^7/7+x^9/9-x^11/11+` . . . `; " "6#/*&%'arctanG\"\"\"%\"xGF&,0F'F&*&F'\"\"$F*!\"\"F+*&F'\"\"&F-F+F&* &F'\"\"(F/F+F+*&F'\"\"*F1F+F&*&F'\"#6F3F+F+%(~.~.~.~GF&" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Sum((-1)^k*x^(2*k+1)/(2*k+1),k = 0 .. infinity),- 1 <= x;" "6$-%$SumG6$*(),$\"\"\"!\"\"%\"kGF))%\"xG,&*&\"\"#F)F+F)F)F)F )F),&*&F0F)F+F)F)F)F)F*/F+;\"\"!%)infinityG1,$F)F*F-" }{XPPEDIT 18 0 " ``<=1" "6#1%!G\"\"\"" }{TEXT -1 2 ". " }}{PARA 258 "" 0 "" {TEXT -1 7 "Hence " }{XPPEDIT 18 0 "Pi/4 = 1-1/3+1/5-1/7+` . . . `" "6#/*&%#PiG \"\"\"\"\"%!\"\",,F&F&*&F&F&\"\"$F(F(*&F&F&\"\"&F(F&*&F&F&\"\"(F(F(%(~ .~.~.~GF&" }{TEXT -1 2 " " }{XPPEDIT 18 0 "``=Sum((-1)^k/(2*k+1),k = \+ 0 .. infinity)" "6#/%!G-%$SumG6$*&),$\"\"\"!\"\"%\"kGF+,&*&\"\"#F+F-F+ F+F+F+F,/F-;\"\"!%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 82 "Execute the following Maple commands to obtain an approxi mate numerical value for " }{XPPEDIT 18 0 "Pi/4" "6#*&%#PiG\"\"\"\"\"% !\"\"" }{TEXT -1 45 " by evaluating the truncated (finite) series " } {XPPEDIT 18 0 "Sum((-1)^k/(2*k+1),k = 0 .. 30);" "6#-%$SumG6$*&),$\"\" \"!\"\"%\"kGF),&*&\"\"#F)F+F)F)F)F)F*/F+;\"\"!\"#I" }{TEXT -1 80 ", an d then find the absolute and relative error in using this approximatio n for " }{XPPEDIT 18 0 "Pi/4" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Sum((-1)^k/(2*k+1),k=0..30);\nvalue(%);\napprox_val : = evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)!\"\"%\"kG \"\"\",&*&\"\"#F*F)F*F*F*F*F(/F);\"\"!\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\";rIY#H$\\]x%R`dY\"\";D]')Hzf1T1?HZ=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+approx_valG$\"+VegMz!#5" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 46 "____________ __________________________________" }}{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 46 "____________________ __________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 43 "It can be shown using Fourier series that: " }}{PARA 0 " " 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "Pi^2/6 = 1+1/4+1/9+1/16+1/25 +1/36+` . . . `;" "6#/*&%#PiG\"\"#\"\"'!\"\",0\"\"\"F**&F*F*\"\"%F(F** &F*F*\"\"*F(F**&F*F*\"#;F(F**&F*F*\"#DF(F**&F*F*\"#OF(F*%(~.~.~.~GF*" }{TEXT -1 2 " " }{XPPEDIT 18 0 "`` = Sum(1/(k^2),k = 1 .. infinity); " "6#/%!G-%$SumG6$*&\"\"\"F)*$%\"kG\"\"#!\"\"/F+;F)%)infinityG" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 82 "Execute the following M aple commands to obtain an approximate numerical value for " } {XPPEDIT 18 0 "Pi^2/6;" "6#*&%#PiG\"\"#\"\"'!\"\"" }{TEXT -1 45 " by e valuating the truncated (finite) series " }{XPPEDIT 18 0 "Sum(1/(k^2), k = 1 .. 40);" "6#-%$SumG6$*&\"\"\"F'*$%\"kG\"\"#!\"\"/F);F'\"#S" } {TEXT -1 80 ", and then find the absolute and relative error in using \+ this approximation for " }{XPPEDIT 18 0 "Pi^2/6;" "6#*&%#PiG\"\"#\"\"' !\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Sum(1/k^2,k=1..40);\nvalue(%);\napp rox_val := evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\" \"\"F'*$)%\"kG\"\"#F'!\"\"/F*;F'\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6##\"A4je@uU:Wv*\\5#pHDY\"A++C%zo^Q*[v[b;paG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+approx_valG$\"+jRC?;!\"*" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 46 "_________________ _____________________________" }}{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 46 "____________ __________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 54 "Find the absolute and relative errors in the value of " } {XPPEDIT 18 0 "(1+1/t)^t" "6#),&\"\"\"F%*&F%F%%\"tG!\"\"F%F'" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "t=666666666" "6#/%\"tG\"*mmmm'" }{TEXT -1 112 " when performing the calculation using Maple's software floati ng-point arithmetic with a precision of 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 34 "Explain why the error is so large." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 4 "Hint" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 24 "(i) Define the function " }{XPPEDIT 18 0 "e(t)=(1+1/t)^t" "6#/-%\"eG6#%\"tG),&\"\"\"F**&F*F*F'!\"\"F*F'" } {TEXT -1 27 " using the arrow notation: " }{TEXT 261 19 "e := t -> (1+ 1/t)^t" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 14 "(ii) Evaluate \+ " }{XPPEDIT 18 0 "e(666666666.)" "6#-%\"eG6#-%&FloatG6$\"*mmmm'\"\"!" }{TEXT -1 82 " using 10 digit floating-point arithmetic and assign the value to a variable, say " }{TEXT 261 10 "approx_val" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 47 " (Be careful to include the decim al point.)" }}{PARA 0 "" 0 "" {TEXT -1 15 "(iii) Evaluate " }{XPPEDIT 18 0 "e(666666666.)" "6#-%\"eG6#-%&FloatG6$\"*mmmm'\"\"!" }{TEXT -1 199 " using floating-point arithmetic with a precision which is suffic iently high for you to be sure that rounding will give a value which i s correct to 10 digits, and assign the value to a variable, say " } {TEXT 261 12 "accurate_val" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 67 "(iv) Calculate the absolute and relative errors in the first va lue." }}{PARA 0 "" 0 "" {TEXT -1 41 "_________________________________ ________" }}{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 41 "_______________________________________ __" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 81 "Estimate the maximum absolute and rela tive errors involved in using the function " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "f(x) = 4*(210+90*x+15*x^2+x^3)/(840-480*x+1 20*x^2-16*x^3+x^4);" "6#/-%\"fG6#%\"xG*(\"\"%\"\"\",*\"$5#F**&\"#!*F*F 'F*F**&\"#:F**$F'\"\"#F*F**$F'\"\"$F*F*,,\"$S)F**&\"$![F*F'F*!\"\"*&\" $?\"F**$F'F2F*F**&\"#;F**$F'F4F*F9*$F'F)F*F9" }{TEXT -1 17 " to appro ximate " }{XPPEDIT 18 0 "exp(x)" "6#-%$expG6#%\"xG" }{TEXT -1 20 " ove r the interval " }{XPPEDIT 18 0 "-1/2 <= x;" "6#1,$*&\"\"\"F&\"\"#!\" \"F(%\"xG" }{XPPEDIT 18 0 "`` <= 1/2;" "6#1%!G*&\"\"\"F&\"\"#!\"\"" } {TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 42 "_________________________________________\n" }}{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 41 "_______________________________________ __" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 21 "Estimate the maximum " }{TEXT 268 14 " absolute error" }{TEXT -1 32 " involved in using the function " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=(1824818/1824817 -3536/24963*x^2+3133/749926*x^4)/(1+6765/270337*x^2)" "6#/-%\"fG6#%\"x G*&,(*&\"(=[#=\"\"\"\"(<[#=!\"\"F,*(\"%ONF,\"&j\\#F.F'\"\"#F.*(\"%LJF, \"'E*\\(F.F'\"\"%F,F,,&F,F,*(\"%lnF,\"'P.FF.F'F2F,F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "to approximate " }{XPPEDIT 18 0 "sin(x) /x" "6#*&-%$sinG6#%\"xG\"\"\"F'!\"\"" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "-Pi/2 <= x;" "6#1,$*&%#PiG\"\"\"\"\"#!\"\"F)%\"xG " }{XPPEDIT 18 0 "`` <= Pi/2;" "6#1%!G*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 18 ", and the maximum " }{TEXT 268 14 "relative error" }{TEXT -1 23 " in using the function " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "g(x)=x*f(x)" "6#/-%\"gG6#%\"xG*&F'\"\"\"-%\"fG6#F'F)" } {XPPEDIT 18 0 "``=(1824818/1824817*x-3536/24963*x^3+3133/749926*x^5)/( 1+6765/270337*x^2)" "6#/%!G*&,(*(\"(=[#=\"\"\"\"(<[#=!\"\"%\"xGF)F)*( \"%ONF)\"&j\\#F+F,\"\"$F+*(\"%LJF)\"'E*\\(F+F,\"\"&F)F),&F)F)*(\"%lnF) \"'P.FF+F,\"\"#F)F+" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "t o approximate " }{XPPEDIT 18 0 "sin(x)" "6#-%$sinG6#%\"xG" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "-Pi/2 <= x;" "6#1,$*&%#PiG\" \"\"\"\"#!\"\"F)%\"xG" }{XPPEDIT 18 0 "`` <= Pi/2;" "6#1%!G*&%#PiG\"\" \"\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 42 "________ _________________________________\n" }}{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 41 "_________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 17 "Consider the sum " }{XPPEDIT 18 0 "Sum(1/ ((2*k-1)*(2*k+1)),k = 1 .. n) = 1/(1*`.`*3)+1/(3*`.`*5)+1/(5*`.`*7)+1/ (7*`.`*9)+` . . . `+1/((2*n-1)*(2*n+1));" "6#/-%$SumG6$*&\"\"\"F(*&,&* &\"\"#F(%\"kGF(F(F(!\"\"F(,&*&F,F(F-F(F(F(F(F(F./F-;F(%\"nG,.*&F(F(*(F (F(%\".GF(\"\"$F(F.F(*&F(F(*(F8F(F7F(\"\"&F(F.F(*&F(F(*(F;F(F7F(\"\"(F (F.F(*&F(F(*(F>F(F7F(\"\"*F(F.F(%(~.~.~.~GF(*&F(F(*&,&*&F,F(F3F(F(F(F. F(,&*&F,F(F3F(F(F(F(F(F.F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "By the phonomena of \"" }{TEXT 260 11 "telescoping" }{TEXT -1 43 "\", this series can be seen to have the sum " }{XPPEDIT 18 0 "n/(2 *n+1)" "6#*&%\"nG\"\"\",&*&\"\"#F%F$F%F%F%F%!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "In detail , the partial fraction expansion " }{XPPEDIT 18 0 "1/((2*k-1)*(2*k+1)) =1/2" "6#/*&\"\"\"F%*&,&*&\"\"#F%%\"kGF%F%F%!\"\"F%,&*&F)F%F*F%F%F%F%F %F+*&F%F%F)F+" }{XPPEDIT 18 0 "``(1/(2*k-1)-1/(2*k+1))" "6#-%!G6#,&*& \"\"\"F(,&*&\"\"#F(%\"kGF(F(F(!\"\"F-F(*&F(F(,&*&F+F(F,F(F(F(F(F-F-" } {TEXT -1 13 " shows that: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(1/((2*k-1)*(2*k+1)),k = 1 .. n) = 1/2;" "6#/-%$SumG 6$*&\"\"\"F(*&,&*&\"\"#F(%\"kGF(F(F(!\"\"F(,&*&F,F(F-F(F(F(F(F(F./F-;F (%\"nG*&F(F(F,F." }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(``(1/(2*k-1)-1/( 2*k+1)),k = 1 .. n);" "6#-%$SumG6$-%!G6#,&*&\"\"\"F+,&*&\"\"#F+%\"kGF+ F+F+!\"\"F0F+*&F+F+,&*&F.F+F/F+F+F+F+F0F0/F/;F+%\"nG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{XPPEDIT 18 0 "`` (``(1-1/3)+``(1/3-1/5)+``(1/5-1/7)+``(1/7-1/9)+` . . . `+``(1/(2*n-1)- 1/(2*n+1))" "6#-%!G6#,.-F$6#,&\"\"\"F**&F*F*\"\"$!\"\"F-F*-F$6#,&*&F*F *F,F-F**&F*F*\"\"&F-F-F*-F$6#,&*&F*F*F3F-F**&F*F*\"\"(F-F-F*-F$6#,&*&F *F*F9F-F**&F*F*\"\"*F-F-F*%(~.~.~.~GF*-F$6#,&*&F*F*,&*&\"\"#F*%\"nGF*F *F*F-F-F**&F*F*,&*&FGF*FHF*F*F*F*F-F-F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{XPPEDIT 18 0 "``(1-``(-1/3+1/3) +``(-1/5+1/5)+``(-1/7+1/7)+``(-1/9+1/9)+` . . . `+``(-1/(2*n-1)+1/(2*n -1))-1/(2*n+1))" "6#-%!G6#,2\"\"\"F'-F$6#,&*&F'F'\"\"$!\"\"F-*&F'F'F,F -F'F--F$6#,&*&F'F'\"\"&F-F-*&F'F'F3F-F'F'-F$6#,&*&F'F'\"\"(F-F-*&F'F'F 9F-F'F'-F$6#,&*&F'F'\"\"*F-F-*&F'F'F?F-F'F'%(~.~.~.~GF'-F$6#,&*&F'F',& *&\"\"#F'%\"nGF'F'F'F-F-F-*&F'F',&*&FHF'FIF'F'F'F-F-F'F'*&F'F',&*&FHF' FIF'F'F'F'F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\" \"F&\"\"#!\"\"" }{XPPEDIT 18 0 "``(1-1/(2*n+1)) =n/(2*n+1)" "6#/-%!G6# ,&\"\"\"F(*&F(F(,&*&\"\"#F(%\"nGF(F(F(F(!\"\"F.*&F-F(,&*&F,F(F-F(F(F(F (F." }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 15 "(a) Check that " }{XPPEDIT 18 0 "Sum(1/((2*k-1)*(2*k+1)),k = 1 .. n)=n/(2*n+1)" "6#/-%$ SumG6$*&\"\"\"F(*&,&*&\"\"#F(%\"kGF(F(F(!\"\"F(,&*&F,F(F-F(F(F(F(F(F./ F-;F(%\"nG*&F3F(,&*&F,F(F3F(F(F(F(F." }{TEXT -1 14 " using Maple. " }} {PARA 0 "" 0 "" {TEXT -1 19 "(b) Sum the series " }{XPPEDIT 18 0 "Sum( 1/((2*k-1)*(2*k+1)),k = 1 .. n)" "6#-%$SumG6$*&\"\"\"F'*&,&*&\"\"#F'% \"kGF'F'F'!\"\"F',&*&F+F'F,F'F'F'F'F'F-/F,;F'%\"nG" }{TEXT -1 3 " \" " }{TEXT 260 8 "forwards" }{TEXT -1 6 "\" for " }{XPPEDIT 18 0 "n = 10 0000;" "6#/%\"nG\"'++5" }{TEXT -1 35 ", (starting with the term given \+ by " }{XPPEDIT 18 0 "k=1" "6#/%\"kG\"\"\"" }{TEXT -1 67 ") with Maple' s 10 digit software floating-point arithmetic using a " }{TEXT 268 8 " for-loop" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 58 " Calculat e the absolute and relative errors in the sum." }}{PARA 0 "" 0 "" {TEXT -1 19 "(c) Sum the series " }{XPPEDIT 18 0 "Sum(1/((2*k-1)*(2*k+ 1)),k = 1 .. n)" "6#-%$SumG6$*&\"\"\"F'*&,&*&\"\"#F'%\"kGF'F'F'!\"\"F' ,&*&F+F'F,F'F'F'F'F'F-/F,;F'%\"nG" }{TEXT -1 3 " \"" }{TEXT 260 9 "ba ckwards" }{TEXT -1 6 "\" for " }{XPPEDIT 18 0 "n = 100000;" "6#/%\"nG \"'++5" }{TEXT -1 35 ", (starting with the term given by " }{XPPEDIT 18 0 "k = 100000;" "6#/%\"kG\"'++5" }{TEXT -1 67 ") with Maple's 10 di git software floating-point arithmetic using a " }{TEXT 268 8 "for-loo p" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 58 " Calculate the a bsolute and relative errors in the sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "(d) Find the " }{TEXT 260 13 "larges t value" }{TEXT -1 35 " which can be obtained for the sum " }{XPPEDIT 18 0 "Sum(1/((2*k-1)*(2*k+1)),k = 1 .. n)" "6#-%$SumG6$*&\"\"\"F'*&,&* &\"\"#F'%\"kGF'F'F'!\"\"F',&*&F+F'F,F'F'F'F'F'F-/F,;F'%\"nG" }{TEXT -1 15 " when summing \"" }{TEXT 260 8 "forwards" }{TEXT -1 85 "\" with Maple's 10 digit software floating-point arithmetic, and the minimum \+ value of " }{TEXT 347 1 "n" }{TEXT -1 23 " which gives this sum. " }} {PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }} {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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 32 "________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q8" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 15 "(a) Check that " }{XPPEDIT 18 0 "Sum((k+1 )/(k^2*(k+2)^2),k = 1 .. n) = n*(5*n^3+30*n^2+57*n+36)/(16*(n+1)^2*(n+ 2)^2);" "6#/-%$SumG6$*&,&%\"kG\"\"\"F*F*F**&F)\"\"#,&F)F*F,F*F,!\"\"/F );F*%\"nG*(F1F*,**&\"\"&F**$F1\"\"$F*F**&\"#IF**$F1F,F*F**&\"#dF*F1F*F *\"#OF*F**(\"#;F**$,&F1F*F*F*F,F*,&F1F*F,F*F,F." }{TEXT -1 15 " using Maple. " }}{PARA 0 "" 0 "" {TEXT -1 19 "(b) Sum the series " } {XPPEDIT 18 0 "Sum((k+1)/(k^2*(k+2)^2),k = 1 .. n)" "6#-%$SumG6$*&,&% \"kG\"\"\"F)F)F)*&F(\"\"#,&F(F)F+F)F+!\"\"/F(;F)%\"nG" }{TEXT -1 3 " \+ \"" }{TEXT 260 8 "forwards" }{TEXT -1 6 "\" for " }{XPPEDIT 18 0 "n = \+ 5000;" "6#/%\"nG\"%+]" }{TEXT -1 35 ", (starting with the term given b y " }{XPPEDIT 18 0 "k=1" "6#/%\"kG\"\"\"" }{TEXT -1 67 ") with Maple's 10 digit software floating-point arithmetic using a " }{TEXT 268 8 "f or-loop" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 58 " Calculate the absolute and relative errors in the sum." }}{PARA 0 "" 0 "" {TEXT -1 19 "(c) Sum the series " }{XPPEDIT 18 0 "Sum((k+1)/(k^2*(k+2) ^2),k = 1 .. n)" "6#-%$SumG6$*&,&%\"kG\"\"\"F)F)F)*&F(\"\"#,&F(F)F+F)F +!\"\"/F(;F)%\"nG" }{TEXT -1 3 " \"" }{TEXT 260 9 "backwards" }{TEXT -1 6 "\" for " }{XPPEDIT 18 0 "n = 5000;" "6#/%\"nG\"%+]" }{TEXT -1 35 ", (starting with the term given by " }{XPPEDIT 18 0 "k = 5000;" "6 #/%\"kG\"%+]" }{TEXT -1 67 ") with Maple's 10 digit software floating- point arithmetic using a " }{TEXT 268 8 "for-loop" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 58 " Calculate the absolute and relative e rrors in the sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "(d) Find the " }{TEXT 260 13 "largest value" }{TEXT -1 36 " which can be obtained for the sum " }{XPPEDIT 18 0 "Sum((k+1)/(k ^2*(k+2)^2),k = 1 .. n)" "6#-%$SumG6$*&,&%\"kG\"\"\"F)F)F)*&F(\"\"#,&F (F)F+F)F+!\"\"/F(;F)%\"nG" }{TEXT -1 15 " when summing \"" }{TEXT 260 8 "forwards" }{TEXT -1 85 "\" with Maple's 10 digit software floating- point arithmetic, and the minimum value of " }{TEXT 348 1 "n" }{TEXT -1 23 " which gives this sum. " }}{PARA 0 "" 0 "" {TEXT -1 32 "_______ _________________________" }}{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 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 32 "__ ______________________________" }}{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 s olutions for some of the task questions " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "solution fo r Q1" }}{PARA 0 "" 0 "" {TEXT 283 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 15 "(a) The series " }{XPPEDIT 18 0 "1+1/1!+1 /2!+1/3!+1/4!+` . . . `+1/n!+` . . . `;" "6#,2\"\"\"F$*&F$F$-%*factori alG6#F$!\"\"F$*&F$F$-F'6#\"\"#F)F$*&F$F$-F'6#\"\"$F)F$*&F$F$-F'6#\"\"% F)F$%(~.~.~.~GF$*&F$F$-F'6#%\"nGF)F$F6F$" }{TEXT -1 14 " converges to \+ " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" }{TEXT -1 10 ", that is \+ " }{XPPEDIT 18 0 "Limit(Sum(1/i!,i = 0 .. n),n=infinity)=``" "6#/-%&Li mitG6$-%$SumG6$*&\"\"\"F+-%*factorialG6#%\"iG!\"\"/F/;\"\"!%\"nG/F4%)i nfinityG%!G" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/n!,n=0..infinity)=e xp(1)" "6#/-%$SumG6$*&\"\"\"F(-%*factorialG6#%\"nG!\"\"/F,;\"\"!%)infi nityG-%$expG6#F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 " \+ Check this fact using Maple's " }{TEXT 0 8 "Sum..sum" }{TEXT -1 8 " \+ and/or " }{TEXT 0 12 "Limit..limit" }{TEXT -1 27 " commands. (Consult Maple " }{TEXT 268 4 "help" }{TEXT -1 16 " if necessary.) " }}{PARA 0 "" 0 "" {TEXT -1 30 "(b) Check that the finite sum " }{XPPEDIT 18 0 "Sum(1/n!,n = 0 .. 8) = 1+1/1!+1/2!+1/3!+1/4!+` . . . `+1/8!;" "6#/-%$ SumG6$*&\"\"\"F(-%*factorialG6#%\"nG!\"\"/F,;\"\"!\"\"),0F(F(*&F(F(-F* 6#F(F-F(*&F(F(-F*6#\"\"#F-F(*&F(F(-F*6#\"\"$F-F(*&F(F(-F*6#\"\"%F-F(%( ~.~.~.~GF(*&F(F(-F*6#F1F-F(" }{TEXT -1 24 " is the rational number " } {XPPEDIT 18 0 "109601/40320" "6#*&\"','4\"\"\"\"\"&?.%!\"\"" }{TEXT -1 57 ", and estimate the absolute and relative errors in using " } {XPPEDIT 18 0 "109601/40320" "6#*&\"','4\"\"\"\"\"&?.%!\"\"" }{TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\"\"\"" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 286 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 " (a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Sum(1/n!,n=0..infini ty);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F' -%*factorialG6#%\"nG!\"\"/F+;\"\"!%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6#\"\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Sum(1/n!,n=0..8);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%$SumG6$*&\"\"\"F'-%*factorialG6#%\"nG!\"\"/F+;\"\"!\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"','4\"\"&?.%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "accurate_val := ev alf(exp(1));\napprox_val := evalf(109601/40320);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-accurate_valG$\"+G=G=F!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+approx_valG$\"+q(y#=F!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The absolute eror involved in u sing " }{XPPEDIT 18 0 "109601/40320" "6#*&\"','4\"\"\"\"\"&?.%!\"\"" } {TEXT -1 16 " to approximate " }{XPPEDIT 18 0 "exp(1)" "6#-%$expG6#\" \"\"" }{TEXT -1 10 " is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "abserr := abs(approx_val-acc urate_val);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"%eI!\"*" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " . . . \+ and the relative error is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "relerr := evalf[4](abserr/ab s(accurate_val));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"%D6! \"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "N ote" }{TEXT -1 158 ": Since the value given for the absolute error onl y has 4 significant digits, it is appropriate to express the relative \+ error with the same number of digits. " }}{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 284 8 "Qu estion" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 35 "The Macluaurin series expansion of " }{XPPEDIT 18 0 "arctan*x" "6#*&%'arctanG\"\"\"% \"xGF%" }{TEXT -1 6 " is: " }}{PARA 256 "" 0 "" {TEXT -1 8 " \+ " }{XPPEDIT 18 0 "arctan*x = x-x^3/3+x^5/5-x^7/7+x^9/9-x^11/11+` . . . `;" "6#/*&%'arctanG\"\"\"%\"xGF&,0F'F&*&F'\"\"$F*!\"\"F+*&F'\"\"&F-F+ F&*&F'\"\"(F/F+F+*&F'\"\"*F1F+F&*&F'\"#6F3F+F+%(~.~.~.~GF&" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Sum((-1)^k*x^(2*k+1)/(2*k+1),k = 0 .. infinity ),-1 <= x;" "6$-%$SumG6$*(),$\"\"\"!\"\"%\"kGF))%\"xG,&*&\"\"#F)F+F)F) F)F)F),&*&F0F)F+F)F)F)F)F*/F+;\"\"!%)infinityG1,$F)F*F-" }{XPPEDIT 18 0 "``<=1" "6#1%!G\"\"\"" }{TEXT -1 2 ". " }}{PARA 258 "" 0 "" {TEXT -1 7 "Hence " }{XPPEDIT 18 0 "Pi/4 = 1-1/3+1/5-1/7+` . . . `" "6#/*&% #PiG\"\"\"\"\"%!\"\",,F&F&*&F&F&\"\"$F(F(*&F&F&\"\"&F(F&*&F&F&\"\"(F(F (%(~.~.~.~GF&" }{TEXT -1 2 " " }{XPPEDIT 18 0 "``=Sum((-1)^k/(2*k+1), k = 0 .. infinity)" "6#/%!G-%$SumG6$*&),$\"\"\"!\"\"%\"kGF+,&*&\"\"#F+ F-F+F+F+F+F,/F-;\"\"!%)infinityG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 82 "Execute the following Maple commands to obtain an approxi mate numerical value for " }{XPPEDIT 18 0 "Pi/4" "6#*&%#PiG\"\"\"\"\"% !\"\"" }{TEXT -1 45 " by evaluating the truncated (finite) series " } {XPPEDIT 18 0 "Sum((-1)^k/(2*k+1),k = 0 .. 30);" "6#-%$SumG6$*&),$\"\" \"!\"\"%\"kGF),&*&\"\"#F)F+F)F)F)F)F*/F+;\"\"!\"#I" }{TEXT -1 80 ", an d then find the absolute and relative error in using this approximatio n for " }{XPPEDIT 18 0 "Pi/4" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Sum((-1)^k/(2*k+1),k=0..30);\nvalue(%);\napprox_val : = evalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&)!\"\"%\"kG \"\"\",&*&\"\"#F*F)F*F*F*F*F(/F);\"\"!\"#I" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\";rIY#H$\\]x%R`dY\"\";D]')Hzf1T1?HZ=" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+approx_valG$\"+VegMz!#5" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 287 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "accurate_val := evalf(Pi/4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-accurate_valG$ \"+N;)R&y!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The absolute eror involved in using " }{XPPEDIT 18 0 "Sum ((-1)^k/(2*k+1),k = 0 .. 30)=14657533947750493292463071/18472920064106 597929865025" "6#/-%$SumG6$*&),$\"\"\"!\"\"%\"kGF*,&*&\"\"#F*F,F*F*F*F *F+/F,;\"\"!\"#I*&\";rIY#H$\\]x%R`dY\"F*\";D]')Hzf1T1?HZ=F+" }{TEXT -1 17 " to approximate " }{XPPEDIT 18 0 "Pi/4;" "6#*&%#PiG\"\"\"\"\"% !\"\"" }{TEXT -1 10 " is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "abserr := abs(approx_val-acc urate_val);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\")3Ui!)!#5 " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " . . . and the relative error is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "relerr := evalf[8](abserr/a bs(accurate_val));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\")$R l-\"!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 4 "Note" }{TEXT -1 152 ": Since the value given for the absolute e rror has 8 significant digits, it is appropriate to express the relati ve error with the same number of digits." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "solution for Q4" }}{PARA 0 "" 0 "" {TEXT 285 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 54 "Find the ab solute and relative errors in the value of " }{XPPEDIT 18 0 "(1+1/t)^t " "6#),&\"\"\"F%*&F%F%%\"tG!\"\"F%F'" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "t=666666666" "6#/%\"tG\"*mmmm'" }{TEXT -1 112 " when performing \+ the calculation using Maple's software floating-point arithmetic with \+ a precision of 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 34 "Explain why \+ the error is so large." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 4 "Hint" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 24 "(i) Define the function " }{XPPEDIT 18 0 "e(t)=(1+1/t)^t" "6#/-%\" eG6#%\"tG),&\"\"\"F**&F*F*F'!\"\"F*F'" }{TEXT -1 27 " using the arrow \+ notation: " }{TEXT 261 19 "e := t -> (1+1/t)^t" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 14 "(ii) Evaluate " }{XPPEDIT 18 0 "e(6666666 66.)" "6#-%\"eG6#-%&FloatG6$\"*mmmm'\"\"!" }{TEXT -1 82 " using 10 dig it floating-point arithmetic and assign the value to a variable, say \+ " }{TEXT 261 10 "approx_val" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 " (Be careful to include the decimal point.)" }}{PARA 0 "" 0 "" {TEXT -1 15 "(iii) Evaluate " }{XPPEDIT 18 0 "e(666666666.)" "6#- %\"eG6#-%&FloatG6$\"*mmmm'\"\"!" }{TEXT -1 199 " using floating-point \+ arithmetic with a precision which is sufficiently high for you to be s ure that rounding will give a value which is correct to 10 digits, and assign the value to a variable, say " }{TEXT 261 12 "accurate_val" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 67 "(iv) Calculate the absol ute and relative errors in the first value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 288 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "e := t -> (1+1/t)^t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"eGf*6# %\"tG6\"6$%)operatorG%&arrowGF(),&*&\"\"\"F/9$!\"\"F/F/F/F0F(F(F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "Digits := 10:\napprox_val := e(666666666.);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+approx_valG$\"+&)ym$z$!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 199 "for digits \+ in [seq(2*i,i=4..10),seq(5*i,i=5..10)] do\n print(`numerical value`= evalf[10](evalf[digits](e(666666666.))),` when calculated using `||dig its||` digit floating-point arithmetic`);\nend do:" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$/%0numerical~valueG$\"\"\"\"\"!%Y~when~calculated~usi ng~8~digit~floating-point~arithmeticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG$\"+&)ym$z$!\"*%Z~when~calculated~using~10~digit ~floating-point~arithmeticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0nume rical~valueG$\"+C=G=F!\"*%Z~when~calculated~using~12~digit~floating-po int~arithmeticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG $\"+C=G=F!\"*%Z~when~calculated~using~14~digit~floating-point~arithmet icG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG$\"+C=G=F!\" *%Z~when~calculated~using~16~digit~floating-point~arithmeticG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG$\"+C=G=F!\"*%Z~whe n~calculated~using~18~digit~floating-point~arithmeticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG$\"+E=G=F!\"*%Z~when~calculated ~using~20~digit~floating-point~arithmeticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG$\"+E=G=F!\"*%Z~when~calculated~usin g~25~digit~floating-point~arithmeticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG$\"+E=G=F!\"*%Z~when~calculated~using~30~digit~f loating-point~arithmeticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numeri cal~valueG$\"+E=G=F!\"*%Z~when~calculated~using~35~digit~floating-poin t~arithmeticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG$ \"+E=G=F!\"*%Z~when~calculated~using~40~digit~floating-point~arithmeti cG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG$\"+E=G=F!\"* %Z~when~calculated~using~45~digit~floating-point~arithmeticG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%0numerical~valueG$\"+E=G=F!\"*%Z~when~calc ulated~using~50~digit~floating-point~arithmeticG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 159 "We can be reasonably confident that the value ob tained by using 50 digit floating-point arithmetic and then rounding t o 10 digits will be correct to 10 digits." }}{PARA 0 "" 0 "" {TEXT -1 32 "You could always try evaluating " }{XPPEDIT 18 0 "e(666666666)=eva l( (1+1/t)^t,t=666666666)" "6#/-%\"eG6#\"*mmmm'-%%evalG6$),&\"\"\"F-*& F-F-%\"tG!\"\"F-F//F/F'" }{TEXT -1 26 " using exact arithmetic! " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "accurate_val := evalf[10](evalf[50](e(666666666.)));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%-accurate_valG$\"+E=G=F!\"*" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The absolute error is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "abserr := abs(approx_val-accurate_val);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%'abserrG$\"+fgQv5!\"*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 " . . . and the relative e rror is . . . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "relerr := abserr/abs(accurate_val);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"+GU7cR!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "The error is about 40%. T his is " }{TEXT 282 4 "HUGE" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "solution for Q5" }}{PARA 0 "" 0 "" {TEXT 316 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 80 "Estim ate the maximum absolute and relative errors involved in using the fun ction" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "f(x) = 4*(21 0+90*x+15*x^2+x^3)/(840-480*x+120*x^2-16*x^3+x^4);" "6#/-%\"fG6#%\"xG* (\"\"%\"\"\",*\"$5#F**&\"#!*F*F'F*F**&\"#:F**$F'\"\"#F*F**$F'\"\"$F*F* ,,\"$S)F**&\"$![F*F'F*!\"\"*&\"$?\"F**$F'F2F*F**&\"#;F**$F'F4F*F9*$F'F )F*F9" }{TEXT -1 17 " to approximate " }{XPPEDIT 18 0 "exp(x)" "6#-%$ expG6#%\"xG" }{TEXT -1 20 " over the interval " }{XPPEDIT 18 0 "-1/2 \+ <= x;" "6#1,$*&\"\"\"F&\"\"#!\"\"F(%\"xG" }{XPPEDIT 18 0 "`` <= 1/2;" "6#1%!G*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 317 8 "Solution" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "f := x -> 4*(210+90*x+15*x^2+x^3)/(840-480*x+120*x^ 2-16*x^3+x^4);\nplot(exp(x)-f(x),x=-1/2..1/2,color=blue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*(\" \"%\"\"\",*\"$5#F/*&\"#!*F/9$F/F/*&\"#:F/)F4\"\"#F/F/*$)F4\"\"$F/F/F/, ,\"$S)F/*&\"$![F/F4F/!\"\"*&\"$?\"F/F7F/F/*&\"#;F/F:F/F@*$)F4F.F/F/F@F /F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 429 218 218 {PLOTDATA 2 "6&-%'CU RVESG6#7fn7$$!3++++++++]!#=$\"3-!pg()4_')e\"!#E7$$!3MLLLe%G?y%F*$\"3') =O%4b7!R6F-7$$!3OmmT&esBf%F*$\"3hrCqM848%)!#F7$$!3KLL$3s%3zVF*$\"3c&p! H1([p)eF87$$!33LL$e/$QkTF*$\"3K+nmA&p:.%F87$$!3!pm;/\"=q]RF*$\"3g*zX3[ ^\"3FF87$$!3SLL3_>f_PF*$\"3(>b_d.MS$=F87$$!3))***\\(o1YZNF*$\"3q\"F87$$!3]LL3-OJNLF*$\"31P$HKA1\"zu!#G7$$!3C++v$*o%Q7$F*$\"3'H8.w=HM `%FW7$$!3ammm\"RFj!HF*$\"3LwDN'yDpg#FW7$$!3JLL$e4OZr#F*$\"3%)G#QvpwJa \"FW7$$!3=+++v'\\!*\\#F*$\"3u'e&GNBq]\")!#H7$$!33+++DwZ#G#F*$\"3b2DN@>Ffo7$$!3OLL3-TC%)=F*$\"3Ngr>b7/; \"*!#I7$$!3!omm;4z)e;F*$\"3s]e$=xnRP$Ffp7$$!3+nmmm`'zY\"F*$\"3odvtp-j& H\"Ffp7$$!3E++v=t)eC\"F*$\"31/IHR\"=\\d$!#J7$$!39nmm;1J\\5F*$\"3H*z)Q/ 6&[@*!#K7$$!3&y***\\(=[jL)!#>$\"3&y=]&HwEK8F\\r7$$!3M****\\iXg#G'F`r$ \"\"!Fgr7$$!3WlmmT&Q(RTF`r$\"3ac^iCIA56!#L7$$!3;nm;/'=><#F`rFfr7$$!3vD MLLe*e$\\!#@F[s7$$\"3[em;zRQb@F`r$\"338.D\\gW?AF]s7$$\"3'[***\\(=>Y2%F `r$!338.D\\gW?AF]s7$$\"3Qhmm\"zXu9'F`rFhs7$$\"3]'******\\y))G)F`r$\"3Y ]-SRoNwKBjf%Ffq7$$\"35****\\P![hY\"F*$\"3M_dp:&z'*y\"Ffp7$$\"3kKL L$Qx$o;F*$\"3yuW,/ELS^Ffp7$$\"3!)*****\\P+V)=F*$\"3i'>H*=,W%R\"Ffo7$$ \"3?mm\"zpe*z?F*$\"33V!**)[_PUJFfo7$$\"3%)*****\\#\\'QH#F*$\"3eoeHPbeY qFfo7$$\"3GKLe9S8&\\#F*$\"3D**Gul=489FW7$$\"3R***\\i?=bq#F*$\"3-&3$Rh% )4mFFW7$$\"3\"HLL$3s?6HF*$\"390\"*y6af*3&FW7$$\"3a***\\7`Wl7$F*$\"3L/# p2L.HB*FW7$$\"3#pmmm'*RRL$F*$\"3i!\\Oq(*31e\"F87$$\"3Qmm;a<.YNF*$\"3/q *\\Ds/Fl#F87$$\"3=LLe9tOcPF*$\"3N:#R!=2F4VF87$$\"3u******\\Qk\\RF*$\"3 9Z4dw+`#e'F87$$\"3]mmT5ASgSF*$\"3NX+Kw$o\"=$)F87$$\"3CLL$3dg6<%F*$\"3J UZ%)yB\"\\/\"F-7$$\"3y***\\(oTAqUF*$\"3k#)*)H9IDv7F-7$$\"3ImmmmxGpVF*$ \"3MC@?k7p\\:F-7$$\"3EL$eRA5\\Z%F*$\"3uhxr>p/**=F-7$$\"3A++D\"oK0e%F*$ \"3xKs5a\\#oJ#F-7$$\"3B+]il(z5j%F*$\"3O1=^H\"oUa#F-7$$\"3C+++]oi\"o%F* $\"3[)omkB#Q\"z#F-7$$\"3B+]PMR " 0 "" {MPLTEXT 1 0 44 "numappr ox[infnorm](exp(x)-f(x),x=-1/2..1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+xj`-\\!#=" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "f := x -> 4*(210+90*x+15*x^2+x^3)/(840-480* x+120*x^2-16*x^3+x^4);\nplot((exp(x)-f(x))/exp(x),x=-1/2..1/2,color=bl ue);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operator G%&arrowGF(,$*(\"\"%\"\"\",*\"$5#F/*&\"#!*F/9$F/F/*&\"#:F/)F4\"\"#F/F/ *$)F4\"\"$F/F/F/,,\"$S)F/*&\"$![F/F4F/!\"\"*&\"$?\"F/F7F/F/*&\"#;F/F:F /F@*$)F4F.F/F/F@F/F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 453 265 265 {PLOTDATA 2 "6&-%'CURVESG6#7]o7$$!3++++++++]!#=$\"3X]\"pp]W#>E!#E7$$!3 [LLe9r]X\\F*$\"3/.UI\"pI1S#F-7$$!3%pmm\"HU,\"*[F*$\"3-,gW@=8)>#F-7$$!3 ()***\\PM@l$[F*$\"3))[giV-s5?F-7$$!3MLLLe%G?y%F*$\"3(4i%HCRUP=F-7$$!39 +](=_+so%F*$\"3U&Q#>^K!oc\"F-7$$!3OmmT&esBf%F*$\"3Y95\"ozz;L\"F-7$$!3e **\\7`'Gd[%F*$\"3)o4[U6GY5\"F-7$$!3KLL$3s%3zVF*$\"32?_&[o4;7*!#F7$$!3# HLLL)QtrUF*$\"3)QMKUD9o[(FV7$$!33LL$e/$QkTF*$\"3u3)3-^yS6'FV7$$!3++]7G CadSF*$\"3O!z(>]:dr\\FV7$$!3!pm;/\"=q]RF*$\"31H*p!y$>--%FV7$$!3SLL3_>f _PF*$\"3K#Rv\"Hb>pEFV7$$!3))***\\(o1YZNF*$\"3))y%Gd%3.1'!#G7$$!3ammm\"RFj !HF*$\"3W6KC?C<'[$Fip7$$!3JLL$e4OZr#F*$\"3;@/XAx[C?Fip7$$!3=+++v'\\!* \\#F*$\"3#*=LrU9ZY5Fip7$$!33+++DwZ#G#F*$\"3mHMR'*pXz]!#H7$$!3-+++D.xt? F*$\"3](oT_h?TO#F^r7$$!3OLL3-TC%)=F*$\"3MXsEF9i+6F^r7$$!3!omm;4z)e;F*$ \"3U\"Gx09pF)R!#I7$$!3+nmmm`'zY\"F*$\"3e$\"3A>+xQ24[9!#K7$$!3M****\\iXg#G'Fbt$\"\"!Fjt7$$!3WlmmT&Q(RTFbt$\"3 u[G>Qz9d6!#L7$$!3;nm;/'=><#FbtFit7$$!3vDMLLe*e$\\!#@$\"3!ft4W:r26\"F`u 7$$\"3[em;zRQb@Fbt$\"3CUT=***)4t@F`u7$$\"3'[***\\(=>Y2%Fbt$!3Uz9Y'z*yJ @F`u7$$\"3Qhmm\"zXu9'Fbt$\"3c<`*eJc!)3#F`u7$$\"3]'******\\y))G)Fbt$\"3 u\"*4U()Q0N;Fet7$$\"3'*)***\\i_QQ5F*$\"3xWC1**>32)*Fet7$$\"3@***\\7y%3 T7F*$\"3;J^,N*e)fSFis7$$\"35****\\P![hY\"F*$\"35\"3yxz9ca\"F^s7$$\"3kK LL$Qx$o;F*$\"3%*Gq>1LX]VF^s7$$\"3!)*****\\P+V)=F*$\"3UF9N4o&\\:\"F^r7$ $\"3?mm\"zpe*z?F*$\"3)zeH0$)pAb#F^r7$$\"3%)*****\\#\\'QH#F*$\"3!36p1# \\=-cF^r7$$\"3GKLe9S8&\\#F*$\"3g7^*=p_55\"Fip7$$\"3R***\\i?=bq#F*$\"3[ sZ@^zT5@Fip7$$\"3\"HLL$3s?6HF*$\"3))o52/J4/QFip7$$\"3a***\\7`Wl7$F*$\" 37[&zo8$*Qv'Fip7$$\"3#pmmm'*RRL$F*$\"3])>$H>r[K6FV7$$\"3Qmm;a<.YNF*$\" 3?#**RnMW2'=FV7$$\"3=LLe9tOcPF*$\"3K%H)pY.$)fHFV7$$\"3u******\\Qk\\RF* $\"3V1@beqnMWFV7$$\"3]mmT5ASgSF*$\"3mw;lVsDUbFV7$$\"3CLL$3dg6<%F*$\"3A mSyE?R&)oFV7$$\"3y***\\(oTAqUF*$\"3y@15)fu.K)FV7$$\"3ImmmmxGpVF*$\"39@ dG&eF6+\"F-7$$\"3EL$eRA5\\Z%F*$\"3&4W#*pnFR@\"F-7$$\"3A++D\"oK0e%F*$\" 3kN?3:PUl9F-7$$\"3B+]il(z5j%F*$\"3d#o&f<2<,;F-7$$\"3C+++]oi\"o%F*$\"3O >r$GfGyu\"F-7$$\"3B+]PMRF-7$$\"3A++v=5s#y%F*$\"3w4F 55l!p2#F-7$$\"3;+D1k2/P[F*$\"3mn,K/;CvAF-7$$\"35+]P40O\"*[F*$\"3y*RZ,8 3+\\#F-7$$\"31+voa-oX\\F*$\"3>=-hJZOAFF-7$$\"3++++++++]F*$\"3!Q[AHiQN( HF--%'COLOURG6&%$RGBGFitFit$\"*++++\"!\")-%+AXESLABELSG6$Q\"x6\"Q!Fj`l -%%VIEWG6$;$!+++++]!#5$\"+++++]Fbal%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The previous graph shows that t he maximum relative error occurs when " }{XPPEDIT 18 0 "x=1/2" "6#/%\" xG*&\"\"\"F&\"\"#!\"\"" }{TEXT -1 24 " and is approximately 3 " } {TEXT 315 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-9)" "6#)\"#5,$\" \"*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedur e " }{TEXT 0 7 "infnorm" }{TEXT -1 8 " in the " }{TEXT 0 9 "numapprox " }{TEXT -1 75 " package may be used to obtain an estimate for the max imum relative error. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "numapprox[infnorm]((exp(x)-f(x))/exp(x),x =-1/2..1/2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+B'QN(H!#=" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "solution for Q7 " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT 322 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 17 "Consi der the sum " }{XPPEDIT 18 0 "Sum(1/((2*k-1)*(2*k+1)),k = 1 .. n) = 1/ (1*`.`*3)+1/(3*`.`*5)+1/(5*`.`*7)+1/(7*`.`*9)+` . . . `+1/((2*n-1)*(2* n+1));" "6#/-%$SumG6$*&\"\"\"F(*&,&*&\"\"#F(%\"kGF(F(F(!\"\"F(,&*&F,F( F-F(F(F(F(F(F./F-;F(%\"nG,.*&F(F(*(F(F(%\".GF(\"\"$F(F.F(*&F(F(*(F8F(F 7F(\"\"&F(F.F(*&F(F(*(F;F(F7F(\"\"(F(F.F(*&F(F(*(F>F(F7F(\"\"*F(F.F(%( ~.~.~.~GF(*&F(F(*&,&*&F,F(F3F(F(F(F.F(,&*&F,F(F3F(F(F(F(F(F.F(" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "By the phonomena of \" " }{TEXT 260 11 "telescoping" }{TEXT -1 43 "\", this series can be see n to have the sum " }{XPPEDIT 18 0 "n/(2*n+1)" "6#*&%\"nG\"\"\",&*&\" \"#F%F$F%F%F%F%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 42 "In detail, the partial fraction expan sion " }{XPPEDIT 18 0 "1/((2*k-1)*(2*k+1))=1/2" "6#/*&\"\"\"F%*&,&*&\" \"#F%%\"kGF%F%F%!\"\"F%,&*&F)F%F*F%F%F%F%F%F+*&F%F%F)F+" }{XPPEDIT 18 0 "``(1/(2*k-1)-1/(2*k+1))" "6#-%!G6#,&*&\"\"\"F(,&*&\"\"#F(%\"kGF(F(F (!\"\"F-F(*&F(F(,&*&F+F(F,F(F(F(F(F-F-" }{TEXT -1 13 " shows that: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/((2*k-1)*(2*k+ 1)),k = 1 .. n) = 1/2;" "6#/-%$SumG6$*&\"\"\"F(*&,&*&\"\"#F(%\"kGF(F(F (!\"\"F(,&*&F,F(F-F(F(F(F(F(F./F-;F(%\"nG*&F(F(F,F." }{TEXT -1 1 " " } {XPPEDIT 18 0 "Sum(``(1/(2*k-1)-1/(2*k+1)),k = 1 .. n);" "6#-%$SumG6$- %!G6#,&*&\"\"\"F+,&*&\"\"#F+%\"kGF+F+F+!\"\"F0F+*&F+F+,&*&F.F+F/F+F+F+ F+F0F0/F/;F+%\"nG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\" \"F&\"\"#!\"\"" }{XPPEDIT 18 0 "``(``(1-1/3)+``(1/3-1/5)+``(1/5-1/7)+` `(1/7-1/9)+` . . . `+``(1/(2*n-1)-1/(2*n+1))" "6#-%!G6#,.-F$6#,&\"\"\" F**&F*F*\"\"$!\"\"F-F*-F$6#,&*&F*F*F,F-F**&F*F*\"\"&F-F-F*-F$6#,&*&F*F *F3F-F**&F*F*\"\"(F-F-F*-F$6#,&*&F*F*F9F-F**&F*F*\"\"*F-F-F*%(~.~.~.~G F*-F$6#,&*&F*F*,&*&\"\"#F*%\"nGF*F*F*F-F-F**&F*F*,&*&FGF*FHF*F*F*F*F-F -F*" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\" " }{XPPEDIT 18 0 "``(1-``(-1/3+1/3)+``(-1/5+1/5)+``(-1/7+1/7)+``(-1/9+ 1/9)+` . . . `+``(-1/(2*n-1)+1/(2*n-1))-1/(2*n+1))" "6#-%!G6#,2\"\"\"F '-F$6#,&*&F'F'\"\"$!\"\"F-*&F'F'F,F-F'F--F$6#,&*&F'F'\"\"&F-F-*&F'F'F3 F-F'F'-F$6#,&*&F'F'\"\"(F-F-*&F'F'F9F-F'F'-F$6#,&*&F'F'\"\"*F-F-*&F'F' F?F-F'F'%(~.~.~.~GF'-F$6#,&*&F'F',&*&\"\"#F'%\"nGF'F'F'F-F-F-*&F'F',&* &FHF'FIF'F'F'F-F-F'F'*&F'F',&*&FHF'FIF'F'F'F'F-F-" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=1/2" "6#/%!G*&\"\"\"F&\"\"#!\"\"" }{XPPEDIT 18 0 "`` (1-1/(2*n+1)) =n/(2*n+1)" "6#/-%!G6#,&\"\"\"F(*&F(F(,&*&\"\"#F(%\"nGF( F(F(F(!\"\"F.*&F-F(,&*&F,F(F-F(F(F(F(F." }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 15 "(a) Check that " }{XPPEDIT 18 0 "Sum(1/((2*k-1)*(2 *k+1)),k = 1 .. n)=n/(2*n+1)" "6#/-%$SumG6$*&\"\"\"F(*&,&*&\"\"#F(%\"k GF(F(F(!\"\"F(,&*&F,F(F-F(F(F(F(F(F./F-;F(%\"nG*&F3F(,&*&F,F(F3F(F(F(F (F." }{TEXT -1 14 " using Maple. " }}{PARA 0 "" 0 "" {TEXT -1 19 "(b) \+ Sum the series " }{XPPEDIT 18 0 "Sum(1/((2*k-1)*(2*k+1)),k = 1 .. n)" "6#-%$SumG6$*&\"\"\"F'*&,&*&\"\"#F'%\"kGF'F'F'!\"\"F',&*&F+F'F,F'F'F'F 'F'F-/F,;F'%\"nG" }{TEXT -1 3 " \"" }{TEXT 260 8 "forwards" }{TEXT -1 6 "\" for " }{XPPEDIT 18 0 "n = 100000;" "6#/%\"nG\"'++5" }{TEXT -1 35 ", (starting with the term given by " }{XPPEDIT 18 0 "k=1" "6#/% \"kG\"\"\"" }{TEXT -1 67 ") with Maple's 10 digit software floating-po int arithmetic using a " }{TEXT 268 8 "for-loop" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 58 " Calculate the absolute and relative e rrors in the sum." }}{PARA 0 "" 0 "" {TEXT -1 19 "(c) Sum the series \+ " }{XPPEDIT 18 0 "Sum(1/((2*k-1)*(2*k+1)),k = 1 .. n)" "6#-%$SumG6$*& \"\"\"F'*&,&*&\"\"#F'%\"kGF'F'F'!\"\"F',&*&F+F'F,F'F'F'F'F'F-/F,;F'%\" nG" }{TEXT -1 3 " \"" }{TEXT 260 9 "backwards" }{TEXT -1 6 "\" for " }{XPPEDIT 18 0 "n = 100000;" "6#/%\"nG\"'++5" }{TEXT -1 35 ", (startin g with the term given by " }{XPPEDIT 18 0 "k = 100000;" "6#/%\"kG\"'++ 5" }{TEXT -1 67 ") with Maple's 10 digit software floating-point arith metic using a " }{TEXT 268 8 "for-loop" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 58 " Calculate the absolute and relative errors in th e sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "(d) Find the " }{TEXT 260 13 "largest value" }{TEXT -1 35 " which can be obtained for the sum " }{XPPEDIT 18 0 "Sum(1/((2*k-1)*(2*k+1)),k = 1 .. n)" "6#-%$SumG6$*&\"\"\"F'*&,&*&\"\"#F'%\"kGF'F'F'!\"\"F',&*&F+F 'F,F'F'F'F'F'F-/F,;F'%\"nG" }{TEXT -1 15 " when summing \"" }{TEXT 260 8 "forwards" }{TEXT -1 85 "\" with Maple's 10 digit software float ing-point arithmetic, and the minimum value of " }{TEXT 324 1 "n" } {TEXT -1 23 " which gives this sum. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 323 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "k := 'k':\nSum(1/((2*k-1)*(2*k+1)), k=1..n);\nnormal(value(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG 6$*&\"\"\"F'*&,&*&\"\"#F'%\"kGF'F'F'!\"\"F',&*&F+F'F,F'F'F'F'F'F-/F,;F '%\"nG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"nG\"\"\",&*&\"\"#F%F$F% F%F%F%!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Sum(1/((2 *k-1)*(2*k+1)),k = 1 .. 100000) = 100000/200001;" "6#/-%$SumG6$*&\"\" \"F(*&,&*&\"\"#F(%\"kGF(F(F(!\"\"F(,&*&F,F(F-F(F(F(F(F(F./F-;F(\"'++5* &F3F(\"',+?F." }{TEXT -1 1 " " }{TEXT 325 1 "~" }{TEXT -1 25 " 0.49999 750001249993750. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "Sum(1/((2*k-1)*(2*k+1)),k=1..10^5);\nexact_su m := value(%);\nevalf[20](%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$Su mG6$*&\"\"\"F'*&,&*&\"\"#F'%\"kGF'F'F'!\"\"F',&*&F+F'F,F'F'F'F'F'F-/F, ;F'\"'++5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*exact_sumG#\"'++5\"',+ ?" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"5]P**\\7+](***\\!#?" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "The following " }{TEXT 268 8 "f or-loop" }{TEXT -1 17 " sums the series " }{XPPEDIT 18 0 "Sum(1/((2*k- 1)*(2*k+1)),k = 1 .. 100000);" "6#-%$SumG6$*&\"\"\"F'*&,&*&\"\"#F'%\"k GF'F'F'!\"\"F',&*&F+F'F,F'F'F'F'F'F-/F,;F'\"'++5" }{TEXT -1 3 " \"" } {TEXT 260 8 "forwards" }{TEXT -1 33 "\" beginning at the term given by " }{XPPEDIT 18 0 "k=1" "6#/%\"kG\"\"\"" }{TEXT -1 81 " using Maple's \+ software floating-point arithmetic with a precision of 10 digits. " }} {PARA 0 "" 0 "" {TEXT -1 60 "The absolute and relative errors in the s um are calculated. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 267 "Digits := 10:\npartial_sum := 0.0:\nfor k \+ from 1 to 100000 do\n term := evalf(1/((2*k-1)*(2*k+1)));\n partia l_sum := partial_sum + term;\nend do:\napprox_sum := partial_sum;\nabs err := evalf[11](abs(exact_sum-approx_sum));\nrelerr := evalf[5](abser r/exact_sum);\nk := 'k':" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+approx_sumG$\"+c(p***\\!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%'abserrG$\"&TC&!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\" &)[5!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 14 " absolute error" }{TEXT -1 97 " in the approximate sum (obtained by sum ming \"forwards\") due to rounding errors is about 0.52441 " }{TEXT 329 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-6)" "6#)\"#5,$\"\"'!\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "The corresponding \+ " }{TEXT 260 14 "relative error" }{TEXT -1 18 " is about 0.10488 " } {TEXT 330 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-5);" "6#)\"#5,$\" \"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 4 "(c) " }}{PARA 0 "" 0 "" {TEXT -1 14 "The following \+ " }{TEXT 268 8 "for-loop" }{TEXT -1 17 " sums the series " }{XPPEDIT 18 0 "Sum(1/((2*k-1)*(2*k+1)),k = 1 .. 100000);" "6#-%$SumG6$*&\"\"\"F '*&,&*&\"\"#F'%\"kGF'F'F'!\"\"F',&*&F+F'F,F'F'F'F'F'F-/F,;F'\"'++5" } {TEXT -1 3 " \"" }{TEXT 260 9 "backwards" }{TEXT -1 33 "\" beginning \+ at the term given by " }{XPPEDIT 18 0 "k = 100000;" "6#/%\"kG\"'++5" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 60 "The absolute and relati ve errors in the sum are calculated. " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 273 "Digits := 10:\npartial_ sum := 0.0:\nfor k from 100000 to 1 by -1 do\n term := evalf(1/((2*k -1)*(2*k+1)));\n partial_sum := partial_sum + term;\nend do:\napprox _sum := partial_sum;\nabserr := evalf[15](abs(exact_sum-approx_sum)); \nrelerr := evalf[5](abserr/exact_sum);\nk := 'k':" }{TEXT -1 0 "" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%+approx_sumG$\"++](***\\!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"&+D\"!#:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"&+]#!#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "The rounding error in the final sum is reduced when the \+ finite series is summed \"backwards\" starting with the smaller number s." }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 260 14 "absolute error " }{TEXT -1 52 " in the approximate sum in this case is about 0.125 " }{TEXT 331 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10);" "6#)\"#5,$ F$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 80 "This means th at the value obtained by summing backwards is correct to 10 digits." } }{PARA 0 "" 0 "" {TEXT -1 18 "The corresponding " }{TEXT 260 14 "relat ive error" }{TEXT -1 15 " is about 0.25 " }{TEXT 332 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10);" "6#)\"#5,$F$!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }} {PARA 0 "" 0 "" {TEXT -1 22 "Consider further sums " }{XPPEDIT 18 0 "S um(1/((2*k-1)*(2*k+1)),k = 1 .. n)" "6#-%$SumG6$*&\"\"\"F'*&,&*&\"\"#F '%\"kGF'F'F'!\"\"F',&*&F+F'F,F'F'F'F'F'F-/F,;F'%\"nG" }{TEXT -1 6 " wi th " }{TEXT 321 1 "n" }{TEXT -1 32 " possibly larger than the value " }{XPPEDIT 18 0 "n = 100000;" "6#/%\"nG\"'++5" }{TEXT -1 24 " considere d previously. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "For any positive integer " }{TEXT 326 1 "n" }{TEXT -1 32 " the partial sums of the series " }{XPPEDIT 18 0 "Sum(1/((2*k-1)*(2*k +1)),k = 1 .. n)" "6#-%$SumG6$*&\"\"\"F'*&,&*&\"\"#F'%\"kGF'F'F'!\"\"F ',&*&F+F'F,F'F'F'F'F'F-/F,;F'%\"nG" }{TEXT -1 28 " are all between 0 a nd 1 so " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 32 " ulp in each of their values is " }{XPPEDIT 18 0 "1/2" "6#*&\"\"\"F$\" \"#!\"\"" }{TEXT -1 1 " " }{TEXT 318 1 "x" }{TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10);" "6#)\"#5,$F$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "1/((2*k-1) *(2*k+1))=5*10^(-11);\nsolve(%);\nop(select(type,[evalf(%)],positive)) ;\nfloor(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&\"\"\"F%*&,&*&\"\" #F%%\"kGF%F%F%!\"\"F%,&*&F)F%F*F%F%F%F%F%F+#F%\",+++++#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$,$*&\"\"#!\"\"\",,++++##\"\"\"F%F&,$*&F%F&F'F(F) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+5y1rq!\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&52(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 320 1 "k" }{TEXT -1 89 " is greater than or equal to 70710, the magnitude of the corresponding term is less than " }{XPPEDIT 18 0 "1/ 2" "6#*&\"\"\"F$\"\"#!\"\"" }{TEXT -1 42 " ulp for the current partial sum, that is," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/( (2*k-1)*(2*k+1))<1/2" "6#2*&\"\"\"F%*&,&*&\"\"#F%%\"kGF%F%F%!\"\"F%,&* &F)F%F*F%F%F%F%F%F+*&F%F%F)F+" }{TEXT -1 1 " " }{TEXT 319 1 "x" } {TEXT -1 1 " " }{XPPEDIT 18 0 "10^(-10);" "6#)\"#5,$F$!\"\"" }{TEXT -1 7 ", when " }{XPPEDIT 18 0 "70710 <= k;" "6#1\"&52(%\"kG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "so that there is no change in the 10 digit floating-point value f or the partial sum when the corresponding term is added to the current partial sum." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Eval(1/((2*k -1)*(2*k+1)),k=70710);\n``=value(%);\n``=evalf(rhs(%));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%%EvalG6$*&\"\"\"F'*&,&*&\"\"#F'%\"kGF'F'F'!\" \"F',&*&F+F'F,F'F'F'F'F'F-/F,\"&52(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G#\"\"\"\",*R;'***>" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+-f4 +]!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "Eval(1/((2*k-1)*(2*k+1)),k=70711);\n``=value(%);\n``= evalf(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%EvalG6$*&\"\"\"F' *&,&*&\"\"#F'%\"kGF'F'F'!\"\"F',&*&F+F'F,F'F'F'F'F'F-/F,\"&62(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"\"\",$3#=++#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"+![a***\\!#?" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 "There is a " }{TEXT 260 22 "larges t possible value" }{TEXT -1 54 " for the sum when summing in the forwa rd direction as " }{TEXT 327 1 "n" }{TEXT -1 36 " increases that is fi rst given when " }{XPPEDIT 18 0 "n = 70710;" "6#/%\"nG\"&52(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 358 "n := 70710:\nSum(1/((2*k-1)*(2*k+1)),k=1..n);\nexact _sum := value(%);\nDigits := 10:\nevalf(exact_sum);\npartial_sum := 0. 0:\nfor k from 1 to n do\n term := evalf(1/((2*k-1)*(2*k+1)));\n p artial_sum := partial_sum + term;\nend do:\napprox_sum := partial_sum; \nabserr := evalf[11](abs(exact_sum-approx_sum));\nrelerr := evalf[5]( abserr/exact_sum);\nk := 'k': n := 'n':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$SumG6$*&\"\"\"F'*&,&*&\"\"#F'%\"kGF'F'F'!\"\"F',&*&F+F'F,F'F' F'F'F'F-/F,;F'\"&52(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*exact_sumG# \"&52(\"'@99" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+XY'***\\!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%+approx_sumG$\"+c(p***\\!#5" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'abserrG$\"&96&!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'relerrG$\"&B-\"!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "Note that the sum of 100000 terms obtained previously by summing \"forwards\" is the same as the last s um with 70710 terms. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "4 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }