{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 35 "" 0 1 104 64 92 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE " Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Maple Input" -1 260 "Courier" 0 0 255 0 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 262 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 272 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 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 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Ti mes" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 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 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "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" -1 258 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 40 "A procedure for calculating squar e roots" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 25.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 68 "load interpolation and function approxima tion procedures including: " }{TEXT 0 5 "remez" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 272 10 "fcnapprx.m" }{TEXT -1 37 " contains the code for the procedure " }{TEXT 0 5 "remez" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It c an be read into a Maple session by a command similar to the one that f ollows, where the file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procdrs/fcnapprx.m\";" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 45 "Newton's method for calculating square roots " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 43 "Newton's method for calculating a solution " }{XPPEDIT 18 0 "phi(x) = 0;" "6#/-%$phiG6#% \"xG\"\"!" }{TEXT -1 31 " uses on the iterative formula " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x[n+1] = x[n]-phi*` '`(x[n])/( phi*``(x[n]));" "6#/&%\"xG6#,&%\"nG\"\"\"F)F),&&F%6#F(F)*(%$phiGF)-%#~ 'G6#&F%6#F(F)*&F.F)-%!G6#&F%6#F(F)!\"\"F:" }{TEXT -1 13 " ------- (i) \+ " }}{PARA 0 "" 0 "" {TEXT -1 40 "to provide a sequence of approximatio ns " }{XPPEDIT 18 0 "x[0],x[1],` . . . `,x[n],x[n+1], ` . . . `" "6(&% \"xG6#\"\"!&F$6#\"\"\"%(~.~.~.~G&F$6#%\"nG&F$6#,&F-F)F)F)F*" }{TEXT -1 68 " which, under suitable circumstances, converges to the desired \+ root." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "T he " }{TEXT 259 11 "square root" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt( a)" "6#-%%sqrtG6#%\"aG" }{TEXT -1 58 " of a non-negative number a is a solution of the equation " }{XPPEDIT 18 0 "x^2=a" "6#/*$%\"xG\"\"#%\" aG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 59 "We can apply Newton 's method, as outlined above, if we let " }{XPPEDIT 18 0 "phi(x) = x^2 -a;" "6#/-%$phiG6#%\"xG,&*$F'\"\"#\"\"\"%\"aG!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 13 "In this case " }{XPPEDIT 18 0 "phi*`'`(x) = 2*x;" "6#/*&%$phiG\"\"\"-%\"'G6#%\"xGF&*&\"\"#F&F*F&" }{TEXT -1 39 ", so the iterative formula (i) becomes " }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x[n+1] = x[n]-(x[n]^2-a)/(2*x[n]);" "6#/&%\" xG6#,&%\"nG\"\"\"F)F),&&F%6#F(F)*&,&*$&F%6#F(\"\"#F)%\"aG!\"\"F)*&F2F) &F%6#F(F)F4F4" }{TEXT -1 1 " " }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "`` = (2*x[n]^2-x[n]^2+a)/(2*x[n]);" "6#/%!G*&,(*&\"\"# \"\"\"*$&%\"xG6#%\"nGF(F)F)*$&F,6#F.F(!\"\"%\"aGF)F)*&F(F)&F,6#F.F)F2 " }{TEXT -1 2 " " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = (x[n]^2+a)/(2*x[n]);" "6#/%!G*&,&*$&%\"xG6#%\"nG\"\"#\"\"\"%\"aGF -F-*&F,F-&F)6#F+F-!\"\"" }{TEXT -1 1 "." }}{PARA 258 "" 0 "" {TEXT -1 42 "Dividing the numerator and denominator by " }{XPPEDIT 18 0 "x[n]" "6#&%\"xG6#%\"nG" }{TEXT -1 33 " we obtain the iterative formula " }} {PARA 257 "" 0 "" {XPPEDIT 18 0 "x[n+1] = (x[n]+a/x[n])/2" "6#/&%\"xG6 #,&%\"nG\"\"\"F)F)*&,&&F%6#F(F)*&%\"aGF)&F%6#F(!\"\"F)F)\"\"#F2" } {TEXT -1 14 " ------- (ii)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "It is possible to give a heuristic derivation o f this formula that does not involve any calculus." }}{PARA 0 "" 0 "" {TEXT -1 55 "This is suggested in the example of the calculation of " }{XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 32 " given in t he following section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Hero's formula for calculating square roots" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 39 "We consider the problem of calculating " } {XPPEDIT 18 0 "sqrt(2);" "6#-%%sqrtG6#\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 27 " must lie between 1 and 2." }}{PARA 15 "" 0 "" {TEXT -1 16 "Try the number " }{XPPEDIT 18 0 "(3/2)" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 35 " which is mid-way b etween 1 and 2." }}{PARA 15 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "(3/2)^2;" "6#*$*&\"\"$\"\"\"\"\"#!\"\"F'" }{TEXT -1 4 " = " } {XPPEDIT 18 0 "9/4;" "6#*&\"\"*\"\"\"\"\"%!\"\"" }{TEXT -1 33 " is gr eater than 2, we see that " }{XPPEDIT 18 0 "(3/2)" "6#*&\"\"$\"\"\"\" \"#!\"\"" }{TEXT -1 13 " is too big." }}{PARA 15 "" 0 "" {TEXT -1 14 "Also dividing " }{XPPEDIT 18 0 "(3/2)" "6#*&\"\"$\"\"\"\"\"#!\"\"" } {TEXT -1 15 " into 2 gives " }{XPPEDIT 18 0 "4/3;" "6#*&\"\"%\"\"\"\" \"$!\"\"" }{TEXT -1 25 " , which is smaller than " }{XPPEDIT 18 0 "(3/ 2)" "6#*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 9 " itself." }}{PARA 0 "" 0 "" {TEXT -1 8 " Since " }{XPPEDIT 18 0 "4/3;" "6#*&\"\"%\"\"\"\"\"$ !\"\"" }{TEXT -1 5 " < " }{XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\"\"# " }{TEXT -1 5 " < " }{XPPEDIT 18 0 "(3/2)" "6#*&\"\"$\"\"\"\"\"#!\" \"" }{TEXT -1 61 ", we could now try the number mid-way between these, namely " }{XPPEDIT 18 0 "(3/2+4/3)/2;" "6#*&,&*&\"\"$\"\"\"\"\"#!\" \"F'*&\"\"%F'F&F)F'F'F(F)" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "17/12; " "6#*&\"#<\"\"\"\"#7!\"\"" }{TEXT -1 2 ". " }}{PARA 15 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "17/12" "6#*&\"#<\"\"\"\"#7!\"\"" }{TEXT -1 27 " is a bit too large since " }{XPPEDIT 18 0 "(17/12)^2;" "6#*$*&\"#< \"\"\"\"#7!\"\"\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "289/144;" "6#* &\"$*G\"\"\"\"$W\"!\"\"" }{TEXT -1 17 ", which is 2 + " }{XPPEDIT 18 0 "1/144;" "6#*&\"\"\"F$\"$W\"!\"\"" }{TEXT -1 2 " ." }}{PARA 15 " " 0 "" {TEXT -1 9 "Dividing " }{XPPEDIT 18 0 "17/12" "6#*&\"#<\"\"\"\" #7!\"\"" }{TEXT -1 14 " into 2 gives " }{XPPEDIT 18 0 "24/17;" "6#*&\" #C\"\"\"\"# " 0 "" {MPLTEXT 1 0 230 "x[0] := 3/2;\nx[1] := (x[0]+2/x[0])/2;\n``=eval f(x[1],2);\nx[2] := (x[1]+2/x[1])/2;\n``=evalf(x[2],5);\nx[3] := (x[2] +2/x[2])/2;\n``=evalf(x[3],12);\nx[4] := (x[3]+2/x[3])/2;\n``=evalf(x[ 4],24);\nx[5] := (x[4]+2/x[4])/2;\n``=evalf(x[5],49);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"!#\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"\"#\"#<\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"#9!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"# #\"$x&\"$3%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"&UT\"!\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"$#\"'dem\"'K3Z" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"-PiN@99!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"%#\"-(*))3Jn))\"-[gc8qi" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/%!G$\"9p,)[]4tBc8UT\"!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"&#\":6\"" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G$\"Rx`(=np&y!)p4Us)o,)[]4tBc8UT\" !#[" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 116 " The decimal values are correct to the number of digits given, which sh ows that the number of correct decimal places " }{TEXT 259 41 "approxi mately doubles with each iteration" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 58 "Starting approximation for a general sq uare root procedure" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 16 "We first make a " }{TEXT 259 15 "range re duction" }{TEXT -1 17 " to the interval " }{XPPEDIT 18 0 "[1/100, 1]; " "6#7$*&\"\"\"F%\"$+\"!\"\"F%" }{TEXT -1 94 " by multiplying or divid ing a by a suitable power of 100 to obtain a number z in the interval \+ " }{XPPEDIT 18 0 "[1/100, 1];" "6#7$*&\"\"\"F%\"$+\"!\"\"F%" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "a = z*100^t;" "6#/%\"aG*&%\"zG\"\" \")\"$+\"%\"tGF'" }{TEXT -1 15 ". Then we have " }{XPPEDIT 18 0 "sqrt( a) = sqrt(z)*10^t;" "6#/-%%sqrtG6#%\"aG*&-F%6#%\"zG\"\"\")\"#5%\"tGF, " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 22 "In order to calculate " }{XPPEDIT 18 0 "sqrt(z);" "6#-%%sqrtG6#%\"zG" }{TEXT -1 26 " by the iterative formula " }{XPPEDIT 18 0 "x[n+1] = (x[n]+z/x[n])/2;" "6#/&% \"xG6#,&%\"nG\"\"\"F)F)*&,&&F%6#F(F)*&%\"zGF)&F%6#F(!\"\"F)F)\"\"#F2" }{TEXT -1 22 " , we need a suitable " }{TEXT 259 22 "starting approxim ation" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " } {XPPEDIT 18 0 "y=sqrt(x)" "6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT -1 17 " on \+ the interval " }{XPPEDIT 18 0 "[1/100, 1];" "6#7$*&\"\"\"F%\"$+\"!\"\" F%" }{TEXT -1 23 " extends from the point" }{XPPEDIT 18 0 "``(1/100,1/ 10);" "6#-%!G6$*&\"\"\"F'\"$+\"!\"\"*&F'F'\"#5F)" }{TEXT -1 13 " to th e point" }{XPPEDIT 18 0 "``(1,1)" "6#-%!G6$\"\"\"F&" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "We could simply replace the curve " }{XPPEDIT 18 0 "y=sqrt(x)" "6#/%\"yG-%%sqr tG6#%\"xG" }{TEXT -1 62 " with the straight line which passes through \+ these two points." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "This straight line has gradient " }{XPPEDIT 18 0 "``(1-1 /10)/``(1-1/100) = ``(9/10)/``(99/100);" "6#/*&-%!G6#,&\"\"\"F)*&F)F) \"#5!\"\"F,F)-F&6#,&F)F)*&F)F)\"$+\"F,F,F,*&-F&6#*&\"\"*F)F+F,F)-F&6#* &\"#**F)F1F,F," }{TEXT -1 3 " = " }{XPPEDIT 18 0 "90/99 = 10/11;" "6#/ *&\"#!*\"\"\"\"#**!\"\"*&\"#5F&\"#6F(" }{TEXT -1 26 ", and so has the \+ equation " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-1 = 10 /11;" "6#/,&%\"yG\"\"\"F&!\"\"*&\"#5F&\"#6F'" }{XPPEDIT 18 0 " ``(x-1) " "6#-%!G6#,&%\"xG\"\"\"F(!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y \+ = 10*x/11+1/11;" "6#/%\"yG,&*(\"#5\"\"\"%\"xGF(\"#6!\"\"F(*&F(F(F*F+F( " }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "The linear function " } {XPPEDIT 18 0 "y = (10*x+1)/11;" "6#/%\"yG*&,&*&\"#5\"\"\"%\"xGF)F)F)F )F)\"#6!\"\"" }{TEXT -1 33 " gives a rough approximation to " } {XPPEDIT 18 0 "y = sqrt(x);" "6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT -1 17 " \+ on the interval " }{XPPEDIT 18 0 "[1/100, 1];" "6#7$*&\"\"\"F%\"$+\"! \"\"F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "plot([(10*x+1)/11,sqrt(x)],x=1/100. .1,color=[red,brown],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 403 314 314 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"3-+++++++5!#>$\"3/+++++++5!# =7$$\"31++]i#=z:$F*$\"3)*****\\(Quh>\"F-7$$\"3J+]P/9^N]F*$\"3/+]7tY'oO \"F-7$$\"3+++vj71ZrF*$\"3+++D^P#)e:F-7$$\"3-++DYygs#*F*$\"3!****\\(e_0 _F-7$$\"3#**\\Pu'R$\\L\"F-$\"3 ))**\\7VsmA@F-7$$\"3>+v$z$R,Q:F-$\"3A+]7)R&G2BF-7$$\"3;+v$R`R![F-$\"31+]i&zP&)o#F-7$$\"3O++D#)ets@F-$\"3A ++]Z`I%)GF-7$$\"35+]7l7TiBF-$\"3o***\\P^Pn0$F-7$$\"3;++v@3%fd#F-$\"31+ +]#Hb3D$F-7$$\"3)****\\7:Z.z#F-$\"3Q****\\P,xXMF-7$$\"3e++Dyt'p*HF-$\" 3.++]2ngLOF-7$$\"3W+v$*Q$)f%=$F-$\"3I+]73.=/QF-7$$\"3c++D*p4xS$F-$\"3r ++]<)3q+%F-7$$\"3;+++(G9nf$F-$\"3E+++q6$)yTF-7$$\"32+vVa:d;QF-$\"3;+]7 89qyVF-7$$\"3y****\\*[#=6SF-$\"3I*****\\W?cb%F-7$$\"3m+vVH:qCUF-$\"3?+ ]7j'G(\\ZF-7$$\"3T+DJ[@-GWF-$\"3n+]P*elX$\\F-7$$\"3W+]P%)e;SYF-$\"3]++ DJNUF^F-7$$\"3w+v=e+)\\$[F-$\"3H+]iDt_/`F-7$$\"3U**\\7jM6X]F-$\"3Y*** \\Ppdb\\&F-7$$\"3S+v$R,$Qj_F-$\"3Q+]7eX)Rp&F-7$$\"3!**\\i&*H(Q`aF-$\"3 K+](os:n'eF-7$$\"3q+]PLrfecF-$\"3k++D@,F`gF-7$$\"3a****\\r*)fqeF-$\"3o *****\\1**fC'F-7$$\"39+]()49+ygF-$\"3t***\\itYXV'F-7$$\"3y+vVLRnyiF-$ \"3i+]7.j(ph'F-7$$\"3B+]7dl[,lF-$\"3J++vLK`>oF-7$$\"3:++]4Op,nF-$\"3.+ ++X'R:+(F-7$$\"3/++DrtX:pF-$\"3%*****\\P.(e>(F-7$$\"3k*\\P46f\"4rF-$\" 3Q**\\7GG'>P(F-7$$\"3a,+vvi#4K(F-$\"3S,+]K%yWc(F-7$$\"3***\\PWn#=?vF-$ \"3]+]781iXxF-7$$\"3v+v=CIYGxF-$\"3y+]i&Qm\\$zF-7$$\"3H++DO^4KzF-$\"31 ++](['3?\")F-7$$\"3;+v$f3z_9)F-$\"3D+]7y+*QJ)F-7$$\"3k,++n0g]$)F-$\"3* =+++(fa+&)F-7$$\"3u+]iO9dg&)F-$\"3=,+vy&G9p)F-7$$\"3Y,vVTO!)o()F-$\"3k ,]7$eI2)))F-7$$\"3E,+]6u9g*)F-$\"3m+++l%zY0*F-7$$\"33,]7l*[%z\"*F-$\"3 [++v8X/a#*F-7$$\"3q+++*)[fv$*F-$\"3W+++!**eBV*F-7$$\"37,vVats%e*F-$\"3 K,]78%zCi*F-7$$\"3O+Dc3Q*[y*F-$\"3U+](o\"*[W!)*F-7$$\"\"\"\"\"!Fdz-%'C OLOURG6&%$RGBG$\"*++++\"!\")$FfzFfzF^[l-F$6$7SF'7$F/$\"3-\\&yDD`qx\"F- 7$F4$\"3?2Zo>X*RC#F-7$F9$\"3%*QiUA))RtEF-7$F>$\"3a)ybRq&4XIF-7$FC$\"3! [JwY!)=YP$F-7$FH$\"3E4/#y\"[n`OF-7$FM$\"3s'z/.3_<#RF-7$FR$\"3QbN#4Fc4= %F-7$FW$\"3bk_\"z#=CCWF-7$Ffn$\"3!HD$[0:EhYF-7$F[o$\"3#R&*eBCk/'[F-7$F `o$\"3gr;5XmYVZOAvF-7$F`t$\"3IX q21M)>m(F-7$Fet$\"3EP^j,Q:'z(F-7$Fjt$\"3B)[odQ3Q#zF-7$F_u$\"3c(f&=>'zJ 1)F-7$Fdu$\"3Wi;;VsQ'=)F-7$Fiu$\"3^v&H?!H#fJ)F-7$F^v$\"3fi?&R&HeJ%)F-7 $Fcv$\"3E4J@e6Cc&)F-7$Fhv$\"3'=KD;o)*=n)F-7$F]w$\"3K$>:0un6z)F-7$Fbw$ \"36ILQT5B1*)F-7$Fgw$\"39s9T\"**>^-*F-7$F\\x$\"3_(e\")40i\"Q\"*F-7$Fax $\"3%))f=ohNBD*F-7$Ffx$\"3-=L+U!*=k$*F-7$F[y$\"37'Q\\&=b!eY*F-7$F`y$\" 39fuw*3W4e*F-7$Fey$\"3!>-/+clFo*F-7$Fjy$\"3Ukjdz?;!z*F-7$F_z$\"3W\"zc_ @i=*)*F-Fcz-Fhz6&Fjz$\")#)eqkF][l$\"))eqk\"F][lFcdl-%*THICKNESSG6#\"\" #-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;$\"+++++5!#6Fdz%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "T he absolute error curve has equation " }{XPPEDIT 18 0 "y = sqrt(x)-(10 *x+1)/11;" "6#/%\"yG,&-%%sqrtG6#%\"xG\"\"\"*&,&*&\"#5F*F)F*F*F*F*F*\"# 6!\"\"F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot(sqrt(x)-(10*x+1)/11,x=1/100..1 ,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6 &-%'CURVESG6#7en7$$\"3-+++++++5!#>$\"\"!F,7$$\"3=+D\"GyR(p7F*$\"3%pI)) *)4bI-\"F*7$$\"3;+]il&z%R:F*$\"3#Ql*>519<>F*7$$\"39+vV[$>#4=F*$\"3i&=p B^o]r#F*7$$\"37++DJ\"f*y?F*$\"3!\\fr'\\TsPMF*7$$\"33+](opQ%=EF*$\"3E+` !)\\DG5ZF*7$$\"31++]i#=z:$F*$\"3B!\\&y]')y3eF*7$$\"3=+vVL[r'4%F*$\"3]f s%>mZ^U(F*7$$\"3J+]P/9^N]F*$\"3irqfl%)Hr()F*7$$\"3#)*\\iSL'G\"4'F*$\"3 &\\eNV$*3_+\"!#=7$$\"3+++vj71ZrF*$\"3%*Qi+v$z$R,Q: FZ$\"3A'zz@omWh\"FZ7$$\"3;+v$R`R![ FZ$\"3@k-HKSqN!*[N,i&)F*7$$\"3H++DO^4KzFZ$\"3A.L$)QbWhyF*7$$\"3;+v$f3z_9 )FZ$\"3r@Z'G8*H7rF*7$$\"3k,++n0g]$)FZ$\"3)*ee\")43;wjF*7$$\"3u+]iO9dg& )FZ$\"3Izfo!Qq!4cF*7$$\"3Y,vVTO!)o()FZ$\"3xmJy)e%eM[F*7$$\"3E,+]6u9g*) FZ$\"3^dQ\\N0E6TF*7$$\"33,]7l*[%z\"*FZ$\"3[*eu,w&**oKF*7$$\"3q+++*)[fv $*FZ$\"3A<-/+d1/DF*7$$\"37,vVats%e*FZ$\"3sLO^km#on\"F*7$$\"3O+Dc3Q*[y* FZ$\"3NH\"z\"Q)H8u)!#?7$$\"\"\"F,$\"3O\"*GchvbvF!#M-%+AXESLABELSG6$Q\" x6\"Q!6\"-%'COLOURG6&%$RGBGF+F+$\"*++++\"!\")-%%VIEWG6$;$\"+++++5!#6F] ]l%(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 80 "We can find the coordinates of the maximum point on this error \+ curve as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "f := x -> sqrt(x)-(10*x+1)/11;\ndiff(f(x),x); \nxmax := solve(%,x);\nf(xmax);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,(-%%sqrtG6#9$\"\"\"*&#\"#5\"#6 F1F0F1!\"\"#F1F5F6F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\" F%*&\"\"#F%%\"xG#F%F'!\"\"F%#\"#5\"#6F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xmaxG#\"$@\"\"$+%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#\") \"$S%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "The maximum point has \+ coordinates" }{XPPEDIT 18 0 "``(121/400,81/440);" "6#-%!G6$*&\"$@\"\" \"\"\"$+%!\"\"*&\"#\")F(\"$S%F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 72 "This means that the maximum absolute error in using the l inear function " }{XPPEDIT 18 0 "g(x) = (10*x+1)/11;" "6#/-%\"gG6#%\"x G*&,&*&\"#5\"\"\"F'F,F,F,F,F,\"#6!\"\"" }{TEXT -1 16 " to approximate \+ " }{XPPEDIT 18 0 "f(x)=sqrt(x)" "6#/-%\"fG6#%\"xG-%%sqrtG6#F'" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[1/100, 1];" "6#7$*&\"\"\"F% \"$+\"!\"\"F%" }{TEXT -1 4 " is " }{XPPEDIT 18 0 "81/440;" "6#*&\"#\") \"\"\"\"$S%!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 14 "If we replace " }{XPPEDIT 18 0 "y = (10*x +1)/11;" "6#/%\"yG*&,&*&\"#5\"\"\"%\"xGF)F)F)F)F)\"#6!\"\"" }{TEXT -1 46 " by a parallel line moved up by a distance of " }{XPPEDIT 18 0 "81 /880;" "6#*&\"#\")\"\"\"\"$!))!\"\"" }{TEXT -1 48 ", the maximum absol ute error will be reduced to " }{XPPEDIT 18 0 "81/880;" "6#*&\"#\")\" \"\"\"$!))!\"\"" }{TEXT -1 51 ", and this error will now occur at thre e values of " }{TEXT 270 1 "x" }{TEXT -1 3 ": " }{XPPEDIT 18 0 "x = 1 /100,x = 121/400;" "6$/%\"xG*&\"\"\"F&\"$+\"!\"\"/F$*&\"$@\"F&\"$+%F( " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "1/11+81/880;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"$h \"\"$!))" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "plot(sqrt(x)-(10/11*x+161/880),x=1/100..1,color=blue) ;" }}{PARA 13 "" 1 "" {GLPLOT2D 494 243 243 {PLOTDATA 2 "6&-%'CURVESG6 #7en7$$\"3-+++++++5!#>$!39aaaaaa/#*F*7$$\"3=+D\"GyR(p7F*$!3AZrbb.\\\"= )F*7$$\"3;+]il&z%R:F*$!3M+eMW[S(G(F*7$$\"39+vV[$>#4=F*$!3_oi)f#z(RzF*7$$\"3q+++bX$)4#)F*$\"3SdKFa$oQ*GF*7$$ \"3-++DYygs#*F*$\"3bHC^(**ees$F*7$$\"3$**\\(o2_!)Q6!#=$\"3K$pnf))ez4&F *7$$\"3#**\\Pu'R$\\L\"Fgo$\"3lR'3CHIb5'F*7$$\"3>+v$z$R,Q:Fgo$\"313DDn8 7SpF*7$$\"3;+v$R`R![Fgo$\"3'z=d$o[ \\_\")F*7$$\"3O++D#)ets@Fgo$\"3YvqGDh,l&)F*7$$\"35+]7l7TiBFgo$\"3I&3W: $=sK))F*7$$\"3;++v@3%fd#Fgo$\"34k7Z\"zG1/*F*7$$\"3)****\\7:Z.z#Fgo$\"3 wp=c(z*[h\"*F*7$$\"3e++Dyt'p*HFgo$\"3k:k%))H_R?*F*7$$\"3W+v$*Q$)f%=$Fg o$\"3S9d9Cq*e=*F*7$$\"3c++D*p4xS$Fgo$\"3kcY&)4!e45*F*7$$\"3;+++(G9nf$F go$\"3O$4-Llb(z*)F*7$$\"32+vVa:d;QFgo$\"3%)\\HXws%oy)F*7$$\"3y****\\*[ #=6SFgo$\"3#zVx)Rp9t&)F*7$$\"3m+vVH:qCUFgo$\"3gS4DY>(eH)F*7$$\"3T+DJ[@ -GWFgo$\"3q_*f/6%=$*zF*7$$\"3W+]P%)e;SYFgo$\"3/Gi,>B)*RwF*7$$\"3w+v=e+ )\\$[Fgo$\"3Aj*Q!e@?%G(F*7$$\"3U**\\7jM6X]Fgo$\"3]')z/^(R)ooF*7$$\"3S+ v$R,$Qj_Fgo$\"3C$zLc[&y/kF*7$$\"3!**\\i&*H(Q`aFgo$\"3'fkcD:T%F*7$$\"3y+vVLRnyiFgo$\"3mC%*)=PvP'QF* 7$$\"3B+]7dl[,lFgo$\"34@0\")*R=>B$F*7$$\"3:++]4Op,nFgo$\"3rs62F0$Rk#F* 7$$\"3/++DrtX:pFgo$\"3U/.v!>!)f*>F*7$$\"3k*\\P46f\"4rFgo$\"3#3=DP!el\" R\"F*7$$\"3a,+vvi#4K(Fgo$\"3csi&e-=38(FU7$$\"3***\\PWn#=?vFgo$\"3F8ixX I_Be!#@7$$\"3v+v=CIYGxFgo$!3.e_Vc!>`U'FU7$$\"3H++DO^4KzFgo$!3$4:7d\"** 4V8F*7$$\"3;+v$f3z_9)Fgo$!3WK2o@jC#4#F*7$$\"3k,++n0g]$)Fgo$!3<&fHZk%QG GF*7$$\"3u+]iO9dg&)Fgo$!3'[ZfQ2vaf$F*7$$\"3Y,vVTO!)o()Fgo$!3Q(Gid'3'*p VF*7$$\"3E,+]6u9g*)Fgo$!3k'f^!>\\G$4&F*7$$\"33,]7l*[%z\"*Fgo$!3mk3P%p \\b$fF*7$$\"3q+++*)[fv$*Fgo$!3#pB0Xvz/q'F*7$$\"37,vVats%e*Fgo$!3V?=.!z =x_(F*7$$\"3O+Dc3Q*[y*Fgo$!3@TvsqCTI$)F*7$$\"\"\"\"\"!$!3P^aaaaa/#*F*- %+AXESLABELSG6$Q\"x6\"Q!6\"-%'COLOURG6&%$RGBG$F`]lF`]lF^^l$\"*++++\"! \")-%%VIEWG6$;$\"+++++5!#6F^]l%(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 33 "Here is a list of sample values . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 276 "printf(\"\\n\"):\nprintf(\" x square r oot linear approx absolute error\\n\\n\");\nfor i from 0 to \+ 18 do\n xx := 0.1+i*0.05;\n r1 := sqrt(xx);\n r2 := 10/11*xx+161 /880;\nprintf(\" %.3f %12.10f %12.10f %2.2e\\n\" ,xx,r1,r2,abs(r1-r2));\nend do:" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }} {PARA 6 "" 1 "" {TEXT -1 67 " x square root linear approx absolute error" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 62 " .100 .3162277660 .2738636364 \+ 4.24e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .150 .3872983 346 .3193181819 6.80e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .200 .4472135955 .3647727273 8.24e-02" }} {PARA 6 "" 1 "" {TEXT -1 62 " .250 .5000000000 .41022 72728 8.98e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .300 \+ .5477225575 .4556818182 9.20e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .350 .5916079783 .5011363637 9.0 5e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .400 .6324555320 \+ .5465909091 8.59e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .450 .6708203932 .5920454546 7.88e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .500 .7071067812 .6375000000 \+ 6.96e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .550 .7416198487 .6829545455 5.87e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " \+ .600 .7745966692 .7284090910 4.62e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .650 .8062257748 .7738636364 \+ 3.24e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .700 .83666 00265 .8193181819 1.73e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .750 .8660254038 .8647727273 1.25e-03" }} {PARA 6 "" 1 "" {TEXT -1 62 " .800 .8944271910 .91022 72728 1.58e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .850 \+ .9219544457 .9556818182 3.37e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .900 .9486832981 1.0011363640 5.2 5e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .950 .9746794345 \+ 1.0465909090 7.19e-02" }}{PARA 6 "" 1 "" {TEXT -1 63 " 1.00 0 1.0000000000 1.0920454550 9.20e-02" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "evalf(161/880);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+baaH=!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 30 ": There is a procedure call ed " }{TEXT 0 7 "minimax" }{TEXT -1 16 " in the package " }{TEXT 260 9 "numapprox" }{TEXT -1 59 " which can be used to achieve essentially \+ the same result. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "numapprox[minimax](sqrt(x),x=1/100..1,1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"+`aaH=!#5\"\"\"*&$\"+!4444*F&F'% \"xGF'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The special procedure " }{TEXT 0 5 "remez" }{TEXT -1 19 " may also be used. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 28 "remez(sqrt(x),x=1/100..1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"+baaH=!#5\"\"\"*&$\"+\"4444*F&F'%\"xGF'F'" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "A procedure for calcula ting " }{XPPEDIT 18 0 "sqrt(a)" "6#-%%sqrtG6#%\"aG" }{TEXT -1 2 " :" }{TEXT 260 11 "squareroot " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 17 "squareroot: usage" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 20 "Calling S equence: " }{TEXT -1 16 "squareroot( a ) " }{TEXT 265 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 15 "Parameters: " }{TEXT 266 19 "a - a real number" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT 269 12 "Description:" }{TEXT -1 1 " " }{TEXT 268 14 "T he procedure " }{TEXT 0 10 "squareroot" }{TEXT 267 45 " calculates the square root of a real number." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 16 "How to a ctivate:" }{TEXT 256 1 "\n" }{TEXT -1 154 "To make the procedure activ e open the subsection, place the cursor anywhere after the prompt [ > \+ and press [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "squareroot: implementation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 21 " : The computation of " }{TEXT 35 9 "ilog10(x)" }{TEXT -1 74 " is very \+ efficient and does not require the computation of any logarithms\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1190 "squareroot := proc(a::real cons)\n local z,s,x,h,saveDigits,i,eps,maxit,isneg;\n \n\011 # H andle a special case\n\011 if a = 0 then\n x := 0\n else \n \+ eps := Float(1,-Digits); # desired relative error\n maxit := \+ length(Digits)*3+4; # maximum no. of iterations\n\n\011\011 # Incr ease precision for the computation by a few digits\n\011\011 saveDi gits := Digits;\n Digits := Digits+length(Digits)+1;\n z := \+ evalf(a);\n\n # negative argument gives an imaginary result\n \+ if z<0 then\n\011\011 z := -z;\n isneg := true;\n \+ else isneg := false end if;\n\n\011\011 # Apply range reduction.\n s := -ilog10(z)-1;\n if s mod 2 =1 then s := s-1 end if;\n \+ z := z*Float(1,s);\n s := -s/2; \n\011 \n\011\011 # O btain an initial approximation.\n\011\011 x := .1829545455+.9090909 090*z;\n\011\011 h := 1:\n\n for i from 1 to maxit do\n\011 \+ h := (z/x-x)/2; # Newton iteration formula\n\011\011\011 x : = x+h;\n if abs(h) " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 39 "Examples are given in the next section." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 260 10 "squareroot" }{TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 15 "Calculation of " }{XPPEDIT 18 0 "sqrt(2)" "6#-%%sqrtG6#\" \"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``[4]*sqrt(2)" "6#*&&%!G6#\" \"%\"\"\"-%%sqrtG6#\"\"#F(" }{TEXT -1 21 " using the procedure " } {TEXT 0 10 "squareroot" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "squareroot(.20);\nsquar eroot(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bf8sW!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]IS(o'!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "Calculation of " }{XPPEDIT 18 0 "sqrt (2)" "6#-%%sqrtG6#\"\"#" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``[4]*sqr t(2)" "6#*&&%!G6#\"\"%\"\"\"-%%sqrtG6#\"\"#F(" }{TEXT -1 21 " using th e procedure " }{TEXT 0 10 "evalf/sqrt" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalf(s qrt(.20));\nevalf(sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+bf8 sW!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+]IS(o'!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Calculation of " } {XPPEDIT 18 0 "sqrt(700000);" "6#-%%sqrtG6#\"'++q" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "``[4]*sqrt(700000);" "6#*&&%!G6#\"\"%\"\"\"-%%sqrtG6 #\"'++qF(" }{TEXT -1 21 " using the procedure " }{TEXT 260 10 "squarer oot" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "evalf[50](squareroot(700000));\nevalf[50] (squareroot(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"Sn)Hp`\"GR*[(=& yD?&3w]#*G!#[" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Calculation of " }{XPPEDIT 18 0 "sqrt(700000);" "6#- %%sqrtG6#\"'++q" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``[4]*sqrt(700000 );" "6#*&&%!G6#\"\"%\"\"\"-%%sqrtG6#\"'++qF(" }{TEXT -1 21 " using the procedure " }{TEXT 0 10 "evalf/sqrt" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "evalf[50] (sqrt(700000));\nevalf[50](sqrt(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#$\"Sn)Hp`\"GR*[(=&yD?&3w]#*G!#[" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 15 " Calculation of " }{XPPEDIT 18 0 "sqrt(1/70000000);" "6#-%%sqrtG6#*&\" \"\"F'\")+++q!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``[4]*sqrt(1/7 0000000)" "6#*&&%!G6#\"\"%\"\"\"-%%sqrtG6#*&F(F(\")+++q!\"\"F(" } {TEXT -1 21 " using the procedure " }{TEXT 0 10 "squareroot" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "evalf[500](squareroot(1/70000000));\nevalf[500](squar eroot(%));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"_jl()QLWV\"4\"3p!)=&= G2Uc<<_DgG?-Vcvl[B:mv?OjzN#)pK9X#)p=@#eq*HX.O#eX(fVufrZ@4_y\"*>4ojKHlL 0*y(3Lm&)[3)G-W)ogjQYET7)[p]78cCQ4C3A(fem!)ztG8e`v&))*35?!3z\"['za3u&f%=q`*Gk$e-C0:d4)pv*>_ %>-&y8t1$e(*HC!\\3'R)[6!\\;ueUJ5)**)4zo%[)\\7$pzr\")o*RORM$4'G_>\"!$.& " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"_jlr@\"pBIa*)o9J%R-&)zB/yV$o$Gm .$yE164Ni<83Q7esX8Ht^iN)H#>zD-mqtzvnP)*=<)\\#\\EU4ktcAekf&p%H(p>V*Rnq& *odQ9%G&zArf+mb6;Qn=fG`%3)\\h7gM8@Uez9YN*>%H!oG)[*>gd!y.aklQ7y@TvA@Jm! Q4hp&=Hy;NV,V'[.?LT.^rTb#yKr,,HUk<`#o<`'*R&>B$G`o\\uma%*QH8:Z'*Ru*HP'3 -8j$zRPv " 0 "" {MPLTEXT 1 0 50 "evalf[500 ](sqrt(1/70000000));\nevalf[500](sqrt(%));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"_jl()QLWV\"4\"3p!)=&=G2Uc<<_DgG?-Vcvl[B:mv?OjzN#)pK9 X#)p=@#eq*HX.O#eX(fVufrZ@4_y\"*>4ojKHlL0*y(3Lm&)[3)G-W)ogjQYET7)[p]78c CQ4C3A(fem!)ztG8e`v&) )*35?!3z\"['za3u&f%=q`*Gk$e-C0:d4)pv*>_%>-&y8t1$e(*HC!\\3'R)[6!\\;ueUJ 5)**)4zo%[)\\7$pzr\")o*RORM$4'G_>\"!$.&" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#$\"_jlr@\"pBIa*)o9J%R-&)zB/yV$o$Gm.$yE164Ni<83Q7esX8Ht^iN)H#>zD- mqtzvnP)*=<)\\#\\EU4ktcAekf&p%H(p>V*Rnq&*odQ9%G&zArf+mb6;Qn=fG`%3)\\h7 gM8@Uez9YN*>%H!oG)[*>gd!y.aklQ7y@TvA@Jm!Q4hp&=Hy;NV,V'[.?LT.^rTb#yKr,, HUk<`#o<`'*R&>B$G`o\\uma%*QH8:Z'*Ru*HP'3-8j$zRPv " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "E xample 4 " }}{PARA 0 "" 0 "" {TEXT -1 22 "We can plot the graph " } {XPPEDIT 18 0 "y=sqrt(x)" "6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT -1 31 " usi ng the numerical procedure " }{TEXT 0 10 "squareroot" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot('squareroot(x)',x=0..2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7W7$$\"\"!F)F(7$$\"+3x&)*3\"!#6$\"+ .D'R/\"!#57$$\"+3=F07$$\"+M3VfV F-$\"+2]#z3#F07$$\"+j&*)fD'F-$\"+Ew>,DF07$$\"+#H[D:)F-$\"+knEbGF07$$\" +e0$=C\"F0$\"+J;'R_$F07$$\"+3RBr;F0$\"+$ys!)3%F07$$\"+zjf)4#F0$\"+jR/ \"e%F07$$\"+'4;[\\#F0$\"+1M\"[*\\F07$$\"+j'y]!HF0$\"+l\"y)*Q&F07$$\"+' zs$HLF0$\"+Z<2qdF07$$\"+8iI_PF0$\"+$32c7'F07$$\"+<_M(=%F0$\"+c*p4Z'F07 $$\"+4y_qXF0$\"+#)ycgnF07$$\"+]1!>+&F0$\"+Z;TsqF07$$\"+]Z/NaF0$\"+)fvA P(F07$$\"+]$fC&eF0$\"+Ho8]wF07$$\"+'z6:B'F0$\"+BD*R*yF07$$\"+<=C#o'F0$ \"+Q!)\\u\")F07$$\"+n#pS1(F0$\"+jTT7Fet7$$\"+U9C#e\"Fet$\"+\"zryD\"Fet7$ $\"+1*3`i\"Fet$\"+Og([F\"Fet7$$\"+$*zym;Fet$\"+P9/\"H\"Fet7$$\"+^j?4#)yL \"Fet7$$\"+9@BM=Fet$\"+d#QVN\"Fet7$$\"+`v&Q(=Fet$\"+o\"*))o8Fet7$$\"+O l5;>Fet$\"++]B%Q\"Fet7$$\"+/Uac>Fet$\"+V_w)R\"Fet7$$\"\"#F)$\"+iN@99Fe t-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q\"x6\"Q!Ff\\l-%%VIEW G6$;F(Fg[l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{PARA 0 "" 0 "" {TEXT -1 35 "Numerical solu tion of the equation " }{XPPEDIT 18 0 "sqrt(x) = Pi;" "6#/-%%sqrtG6#% \"xG%#PiG" }{TEXT -1 11 " using the " }{TEXT 260 6 "fsolve" }{TEXT -1 11 " procedure " }{TEXT 0 10 "squareroot" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "st := time():\nevalf[30](fsolve('squareroot(x)'=Pi,x=10));\ntime()-st;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?))**4\\M)='e$*3,Wgp)*!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"$1%!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "st := time():\nevalf[30] (fsolve(sqrt(x)=Pi,x=50));\ntime()-st;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"?))**4\\M)='e$*3,Wgp)*!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# $\"$_#!\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 6 "Tasks " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 174 "It is possible to improve on the startin g approximation for the square root procedure discussed above by using a polynomial of degree 2 or more, or using a rational function." }} {PARA 0 "" 0 "" {TEXT -1 70 "The following commands construct a ration al minimax approximation for " }{XPPEDIT 18 0 "sqrt(x)" "6#-%%sqrtG6#% \"xG" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[1/100,1]" "6#7 $*&\"\"\"F%\"$+\"!\"\"F%" }{TEXT -1 53 " with numerator and denominato r each having degree 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "numapprox[minimax](sqrt(x),x=1/100. .1,[2,2]);\ng := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,& $\")nfuU!#5\"\"\"*&,&$\"*I65)\\!\"*F(*&$\"+;`Qh:F-F(%\"xGF(F(F(F1F(F(F (,&$\"+s\"*fBz!#6F(*&,&$\"+!echL\"F-F(*&$\"+](Rg^'F'F(F1F(F(F(F1F(F(! \"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operator G%&arrowGF(*&,&$\")nfuU!#5\"\"\"*&,&$\"*I65)\\!\"*F1*&$\"+;`Qh:F6F19$F 1F1F1F:F1F1F1,&$\"+s\"*fBz!#6F1*&,&$\"+!echL\"F6F1*&$\"+](Rg^'F0F1F:F1 F1F1F:F1F1!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "(a) Check the absolute error of this rational approx imation graphically." }}{PARA 0 "" 0 "" {TEXT -1 25 "(b) Modify the pr ocedure " }{TEXT 0 10 "squareroot" }{TEXT -1 48 " to incorporate this \+ new starting approximation." }}{PARA 0 "" 0 "" {TEXT -1 39 "__________ _____________________________" }}{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 39 "____________ ___________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 18 "Write a procedure " }{TEXT 0 8 "cuberoot" }{TEXT -1 65 " which will compute a numerical approximation for the cube root " }{XPPEDIT 18 0 "a^(1/3);" "6#)%\"a G*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 89 " of a general real number a by us ing Newton's method. The following suggestions may help." }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 67 "If a is negative, record this fact, and calculate the cube root of " }{XPPEDIT 18 0 "-a " "6#,$%\"aG!\"\"" }{TEXT -1 9 " instead." }}{PARA 15 "" 0 "" {TEXT -1 7 "Make a " }{TEXT 259 15 "range reduction" }{TEXT -1 17 " to the i nterval " }{XPPEDIT 18 0 "[1/1000, 1];" "6#7$*&\"\"\"F%\"%+5!\"\"F%" } {TEXT -1 95 " by multiplying or dividing a by a suitable power of 1000 to obtain a number z in the interval " }{XPPEDIT 18 0 "[1/1000, 1];" "6#7$*&\"\"\"F%\"%+5!\"\"F%" }{TEXT -1 11 " such that " }{XPPEDIT 18 0 "a = z*1000^t;" "6#/%\"aG*&%\"zG\"\"\")\"%+5%\"tGF'" }{TEXT -1 15 ". Then we have " }{XPPEDIT 18 0 "sqrt(a) = sqrt(z)*10^t;" "6#/-%%sqrtG6 #%\"aG*&-F%6#%\"zG\"\"\")\"#5%\"tGF," }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 37 " Find the maximum point on the curve " }{XPPEDIT 18 0 "y = x^(1/3)-(100*x+11)/111;" "6#/%\"yG,&)%\"xG*&\"\"\"F)\"\"$!\"\"F)* &,&*&\"$+\"F)F'F)F)\"#6F)F)\"$6\"F+F+" }{TEXT -1 150 ", and use the re sult to help in constructing a linear function to provide a starting a pproximation by a similar method to that used for the procedure " } {TEXT 0 10 "squareroot" }{TEXT -1 93 ".\nAlternatively, use the method of question 1 to obtain a rational minimax approximation for " } {XPPEDIT 18 0 "x^(1/3)" "6#)%\"xG*&\"\"\"F&\"\"$!\"\"" }{TEXT -1 17 " \+ on the interval " }{XPPEDIT 18 0 "[1/1000,1]" "6#7$*&\"\"\"F%\"%+5!\" \"F%" }{TEXT -1 93 ".\nIt seems to be hard to get a good approximation , but you could try the following commands.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "numapprox[minimax](surd(x,3),x=1/1000..1,[1,2]); " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "The \+ special procedure " }{TEXT 0 5 "remez" }{TEXT -1 19 " may also be used . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "remez(surd(x,3),x=1/1000..1,[1,2]);" }}}{PARA 15 "" 0 "" {TEXT -1 79 "Check that the Newton iteration formula for calculat ing the cube root of z is " }{XPPEDIT 18 0 "x[n+1]=x[n]+h" "6#/&%\"xG 6#,&%\"nG\"\"\"F)F),&&F%6#F(F)%\"hGF)" }{TEXT -1 8 ", where " } {XPPEDIT 18 0 "h = (z/(x[n]^2)-x[n])/3;" "6#/%\"hG*&,&*&%\"zG\"\"\"*$& %\"xG6#%\"nG\"\"#!\"\"F)&F,6#F.F0F)\"\"$F0" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{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 39 "_______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "You may consult the \+ following subsections for partial answers if necessary." }}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 2 "1." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 13 "First make a " }{TEXT 259 15 " range reduction" }{TEXT -1 17 " to the interval " }{XPPEDIT 18 0 "[1/1 000, 1];" "6#7$*&\"\"\"F%\"%+5!\"\"F%" }{TEXT -1 94 " by multiplying o r dividing a by a suitable power of 1000 to obtain a number z in the i nterval" }{XPPEDIT 18 0 "[1/1000, 1];" "6#7$*&\"\"\"F%\"%+5!\"\"F%" } {TEXT -1 11 " such that " }{XPPEDIT 18 0 "a = z*1000^t;" "6#/%\"aG*&% \"zG\"\"\")\"%+5%\"tGF'" }{TEXT -1 15 ". Then we have " }{XPPEDIT 18 0 "sqrt(a) = sqrt(z)*10^t;" "6#/-%%sqrtG6#%\"aG*&-F%6#%\"zG\"\"\")\"#5 %\"tGF," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "In order to calculate " }{XPPEDIT 18 0 "sqrt(z);" "6 #-%%sqrtG6#%\"zG" }{TEXT -1 39 " by Newton's method we need a suitable " }{TEXT 259 22 "starting approximation" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 14 "The graph of " }{XPPEDIT 18 0 "y = x^(1/3);" "6#/ %\"yG)%\"xG*&\"\"\"F(\"\"$!\"\"" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[1/1000, 1];" "6#7$*&\"\"\"F%\"%+5!\"\"F%" }{TEXT -1 23 " extends from the point" }{XPPEDIT 18 0 "``(1/1000,1/10);" "6#-%!G 6$*&\"\"\"F'\"%+5!\"\"*&F'F'\"#5F)" }{TEXT -1 13 " to the point" } {XPPEDIT 18 0 "``(1,1);" "6#-%!G6$\"\"\"F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 10 "The curve " }{XPPEDIT 18 0 "y = x^(1/3);" "6#/%\" yG)%\"xG*&\"\"\"F(\"\"$!\"\"" }{TEXT -1 78 " could be replaced by the \+ straight line which passes through these two points." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "This straight line has \+ gradient " }{XPPEDIT 18 0 "``(1-1/10)/``(1-1/1000) = ``(9/10)/``(999/1 000);" "6#/*&-%!G6#,&\"\"\"F)*&F)F)\"#5!\"\"F,F)-F&6#,&F)F)*&F)F)\"%+5 F,F,F,*&-F&6#*&\"\"*F)F+F,F)-F&6#*&\"$***F)F1F,F," }{XPPEDIT 18 0 "`` \+ = 900/999;" "6#/%!G*&\"$+*\"\"\"\"$***!\"\"" }{XPPEDIT 18 0 " ``= 100/ 111" "6#/%!G*&\"$+\"\"\"\"\"$6\"!\"\"" }{TEXT -1 27 ", and so has the \+ equation " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-1 = 1 00/111;" "6#/,&%\"yG\"\"\"F&!\"\"*&\"$+\"F&\"$6\"F'" }{XPPEDIT 18 0 "` `(x-1)" "6#-%!G6#,&%\"xG\"\"\"F(!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = (100*x+11)/111;" "6#/%\"yG*&,&*&\"$+\"\"\"\"%\"xGF)F)\"#6F)F)\" $6\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 20 "The linear function " }{XPPEDIT 18 0 "y = (100*x+1 1)/111;" "6#/%\"yG*&,&*&\"$+\"\"\"\"%\"xGF)F)\"#6F)F)\"$6\"!\"\"" } {TEXT -1 33 " gives a rough approximation to " }{XPPEDIT 18 0 "y = x^ (1/3);" "6#/%\"yG)%\"xG*&\"\"\"F(\"\"$!\"\"" }{TEXT -1 18 " on the int erval " }{XPPEDIT 18 0 "[1/1000, 1];" "6#7$*&\"\"\"F%\"%+5!\"\"F%" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot([(100*x+11)/111,x^(1/3)],x=1/1000..1,color= [red,brown],thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6'-%'CURVESG6$7S7$$\"3-+++++++5!#?$\"3/+++++++5!#=7$$\"31 ++D,d`xA!#>$\"3)*****\\(Quh>\"F-7$$\"3Q+vory>sTF1$\"3/+]7tY'oO\"F-7$$ \"3()**\\()QO%HI'F1$\"3+++D^P#)e:F-7$$\"3],]7sL\"yW)F1$\"3=++ve_0_F-7$$\"3)*\\(o)R1;c7F-$\"3))**\\7VsmA @F-7$$\"31](=>z'3h9F-$\"3A+]7)R&G2BF-7$$\"37](=:r@In\"F-$\"3M+]7ex@)\\ #F-7$$\"3/]P9`pF%)=F-$\"31+]i&zP&)o#F-7$$\"3K+]sN*y:5#F-$\"3A++]Z`I%)G F-7$$\"3?+DES'yHH#F-$\"3C++v8vtcIF-7$$\"3H+]nu$\\%3DF-$\"31++]#Hb3D$F- 7$$\"3;+]i_[![s#F-$\"3%*****\\P,xXMF-7$$\"34+]KXMILHF-$\"3.++]2ngLOF-7 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\\\")F-7$F\\t$\"3eYT2)eX>D)F-7$Fat$\"3;mIzbbOb$)F-7$Fft$\"3CRAT'yATX)F -7$F[u$\"3-%H_^%y_Z&)F-7$F`u$\"3#)Qf)4*>!*[')F-7$Feu$\"3x1)po`1!Q()F-7 $Fju$\"3MMyFU2=J))F-7$F_v$\"3s60)3qWR\"*)F-7$Fdv$\"303$pyK.F+*F-7$Fiv$ \"3C,-EL%\\Y3*F-7$F^w$\"3!=vbyLb(o\"*F-7$Fcw$\"3%**RF#[(=&\\#*F-7$Fhw$ \"3k)*R%z='eK$*F-7$F]x$\"3=n!*e&o:7T*F-7$Fbx$\"3'QQRN^*G!\\*F-7$Fgx$\" 3#)*\\SYHGuc*F-7$F\\y$\"3+'*p'yNIsj*F-7$Fay$\"35'3VHl2gr*F-7$Ffy$\"3sR nxt/S&y*F-7$F[z$\"3O2&Gs_>$e)*F-7$F`z$\"36cEt%=;r#**F-Fdz-Fiz6&F[[l$\" )#)eqkF^[l$\"))eqk\"F^[lF[fl-%*THICKNESSG6#\"\"#-%+AXESLABELSG6$Q\"x6 \"Q!6\"-%%VIEWG6$;$\"+++++5!#7Fez%(DEFAULTG" 1 2 0 1 10 2 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The absolute error cur ve has equation " }{XPPEDIT 18 0 "y = x^(1/3)-(100*x+11)/111;" "6#/%\" yG,&)%\"xG*&\"\"\"F)\"\"$!\"\"F)*&,&*&\"$+\"F)F'F)F)\"#6F)F)\"$6\"F+F+ " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(x^(1/3)-(100*x+11)/111,x=1/1000..1,color =blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CU RVESG6#7in7$$\"3-+++++++5!#?$\"3qX9y!yyxQ\"!#M7$$\"3#\\iSm!*z/o\"F*$\" 3Xt'oqDrw#=!#>7$$\"3!)\\7G8)f4O#F*$\"3w(HdX9cI>$F37$$\"3nu=#*>(R9/$F*$ \"3Q?[O0fo/VF37$$\"3)**\\ili>>s$F*$\"3#*eT?q&o?D&F37$$\"3?]P%)R%zG3&F* $\"3\"\\A%pSk*e#oF37$$\"3&***\\7`#RQW'F*$\"3`!y1PR:#=\")F37$$\"3[*\\(o z)ed;*F*$\"3fs&3%44@>5!#=7$$\"33+]i]yw)=\"F3$\"3y:71wd<%=\"FR7$$\"3;+v $fx^Jt\"F3$\"3QgLYRvvS9FR7$$\"31++D,d`xAF3$\"3>$)*[\">NSQ;FR7$$\"3R](o kym[A$F3$\"3%*HsSA][,>FR7$$\"3Q+vory>sTF3$\"3>6=mp]Z,@FR7$$\"3x\\7Gb2d P_F3$\"3!R]nJpM'yAFR7$$\"3()**\\()QO%HI'F3$\"3$pC/24`3U#FR7$$\"3o++]0& y`P(F3$\"3D@]\\*\\N#QDFR7$$\"3],]7sL\"yW)F3$\"3Z$\\!zrwvNEFR7$$\"3-]PR \"*)[#e5FR$\"33i95.wk&y#FR7$$\"3)*\\(o)R1;c7FR$\"3#\\tAc[Lb)GFR7$$\"31 ](=>z'3h9FR$\"3YrDfZ#e'fHFR7$$\"37](=:r@In\"FR$\"3y$pKQ3!)>,$FR7$$\"3/ ]P9`pF%)=FR$\"3XBI97uXWIFR7$$\"3K+]sN*y:5#FR$\"37f'He#f5hIFR7$$\"3?+DE S'yHH#FR$\"3%34`!=r%R1$FR7$$\"3H+]nu$\\%3DFR$\"3OF`c1!Re0$FR7$$\"3;+]i _[![s#FR$\"3FY2(=Qms.$FR7$$\"34+]KXMILHFR$\"3ZLX!=oS2,$FR7$$\"3&*\\(=? 9SE7$FR$\"3;rp^K^:!)HFR7$$\"3Q+]U(yzxM$FR$\"3W(o8@51m$HFR7$$\"3<++q)f- &QNFR$\"3w4\\p;_@%*GFR7$$\"3=](o&o&e.w$FR$\"3`)ySXsTFR$\"33n&R$fLeAFFR7$$\"3E]i ?/yOxVFR$\"3g3\")HEhFeEFR7$$\"3U+vo>,W\"f%FR$\"3C+D****yA(e#FR7$$\"3U] PWJ`-)y%FR$\"3t&[;*G:p=DFR7$$\"3I+D1S!p++&FR$\"3XqaM(4%[TCFR7$$\"33](= &fGK?_FR$\"37k#\\6KHzN#FR7$$\"33]7$oXa?T&FR$\"3Q['HE.#f#G#FR7$$\"3r+ve M)H\">cFR$\"3%f9CoYv')>#FR7$$\"3`***\\@'*eI$eFR$\"3[mIz!\\m$4@FR7$$\"3 (**\\Ps(oMUgFR$\"3^RA;]gd>?FR7$$\"3P](okpV[C'FR$\"3S$HF?a^0$>FR7$$\"3m *\\i%*)=opkFR$\"3iRfBd(o$H=FR7$$\"3]***\\f+4a\"FR7$$\" 37,]2g5d'H(FR$\"3k1$p`*[AQ9FR7$$\"3%*\\(o0))Qw\\(FR$\"3'=?N,#)G!R8FR7$ $\"3I]P/)o7yq(FR$\"3*3vIA&*)yL7FR7$$\"3w**\\7,gH8zFR$\"3u*RF2EK%H6FR7$ $\"3/](=n)zTG\")FR$\"3D)**=)4hp=5FR7$$\"3&4++n-1cL)FR$\"3fi1*e:(p1\"*F 37$$\"3/+DTAd[Z&)FR$\"3kOQ*yM4'))zF37$$\"3M^(os%4hd()FR$\"3]!)\\::r(p' oF37$$\"3c++:1Up]*)FR$\"38_*p'G*3b#eF37$$\"3/,DE5%*)><*FR$\"3([&3$>RJ' >YF37$$\"31++!*y%=*p$*FR$\"3N\"Rnx$[TINF37$$\"3)4vo&[@&4e*FR$\"3))e].T 6SeBF37$$\"3o]7t(HQHy*FR$\"3bbldyEnE7F37$$\"\"\"\"\"!$!3qX9y!yyxQ\"F-- %+AXESLABELSG6$Q\"x6\"Q!6\"-%'COLOURG6&%$RGBG$Fd^lFd^lFb_l$\"*++++\"! \")-%%VIEWG6$;$\"+++++5!#7Fb^l%(DEFAULTG" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 44.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "We can find the coordinates of \+ the maximum point on this error curve as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "f := x -> x^ (1/3)-(100*x+11)/111;\ndiff(f(x),x);\nxmax := fsolve(%,x=0.23);\nf(xma x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG %&arrowGF(,(*$)9$#\"\"\"\"\"$F1F1*&#\"$+\"\"$6\"F1F/F1!\"\"#\"#6F6F7F( F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%*$)%\"xG#\"\"#\"\" $F%!\"\"#F%F+#\"$+\"\"$6\"F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%xma xG$\"+O@i]A!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+HS=kI!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The maximum point has coordinat es (0.2250622136, 0.3064184029), approximately. " }}{PARA 0 "" 0 "" {TEXT -1 72 "This means that the maximum absolute error in using the l inear function " }{XPPEDIT 18 0 "g(x) = (100*x+11)/111;" "6#/-%\"gG6#% \"xG*&,&*&\"$+\"\"\"\"F'F,F,\"#6F,F,\"$6\"!\"\"" }{TEXT -1 16 " to app roximate " }{XPPEDIT 18 0 "f(x) = ``[3]*sqrt(x);" "6#/-%\"fG6#%\"xG*&& %!G6#\"\"$\"\"\"-%%sqrtG6#F'F-" }{TEXT -1 17 " on the interval " } {XPPEDIT 18 0 "[1/1000, 1];" "6#7$*&\"\"\"F%\"%+5!\"\"F%" }{TEXT -1 18 " is 0.3064184029. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "If we replace " }{XPPEDIT 18 0 "y = (100*x+11)/111; " "6#/%\"yG*&,&*&\"$+\"\"\"\"%\"xGF)F)\"#6F)F)\"$6\"!\"\"" }{TEXT -1 44 " by a parallel line moved up a distance of " }{XPPEDIT 18 0 ".306 4184029/2;" "6#*&-%&FloatG6$\"+HS=kI!#5\"\"\"\"\"#!\"\"" }{TEXT -1 48 ", the maximum absolute error will be reduced to " }{XPPEDIT 18 0 ".30 64184029/2;" "6#*&-%&FloatG6$\"+HS=kI!#5\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{TEXT 263 1 "~" }{TEXT -1 64 " 0.1532092014, and this error will \+ now occur at three values of " }{TEXT 271 1 "x" }{TEXT -1 2 ": " } {XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 5 ".01, " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 16 ".2250622136 and " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "0.3064184029/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"+9?4K:!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "evalf(100/111);\nevalf(11/111+0.1532092014);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+4!4!4!*!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+0I3BD!#5" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "plot(x^(1/3)-(.9009009009*x+ 0.2523083005),x=1/1000..1,color=blue);" }}{PARA 13 "" 1 "" {GLPLOT2D 494 243 243 {PLOTDATA 2 "6&-%'CURVESG6#7in7$$\"3-+++++++5!#?$!3(****3S ,#4K:!#=7$$\"3#\\iSm!*z/o\"F*$!3%p7-$))[K\\8F-7$$\"3!)\\7G8)f4O#F*$!37 eKb*R'y77F-7$$\"3nu=#*>(R9/$F*$!3v*\\sMUB;5\"F-7$$\"3)**\\ili>>s$F*$!3 zf&))p:&)o+\"F-7$$\"3?]P%)R%zG3&F*$!3!)4aR*pB]\\)!#>7$$\"3&***\\7`#RQW 'F*$!31KFQYZq-sFG7$$\"3[*\\(oz)ed;*F*$!3mRN+Y5\")G^FG7$$\"33+]i]yw)=\" FG$!3OjoZzB;zMFG7$$\"3;+v$fx^Jt\"FG$!34w#\\XXZM8*F*7$$\"31++D,d`xAFG$ \"3?'y,90:J1\"FG7$$\"3R](okym[A$FG$\"3e3^)R3IRp$FG7$$\"3Q+vory>sTFG$\" 3%fxJlbIQp&FG7$$\"3x\\7Gb2dP_FG$\"3Gj'*e\"zEaY(FG7$$\"3()**\\()QO%HI'F G$\"3)R0epw5w)))FG7$$\"3o++]0&y`P(FG$\"3Jwm[&[Vh+\"F-7$$\"3],]7sL\"yW) FG$\"37XAydcm.6F-7$$\"3-]PR\"*)[#e5F-$\"3#eS$4*ebND\"F-7$$\"3)*\\(o)R1 ;c7F-$\"3Sd[hr9W`8F-7$$\"31](=>z'3h9F-$\"3Ay[eLicF9F-7$$\"37](=:r@In\" F-$\"3]\">D)p!)))z9F-7$$\"3/]P9`pF%)=F-$\"3e6d8)RlB^\"F-7$$\"3K+]sN*y: 5#F-$\"3?VD#=\"R,H:F-7$$\"3?+DES'yHH#F-$\"3+Zh//^&=`\"F-7$$\"3H+]nu$\\ %3DF-$\"3#ydeD*puB:F-7$$\"3;+]i_[![s#F-$\"3c\">kyOu^]\"F-7$$\"34+]KXMI LHF-$\"3'p;)zn'['y9F-7$$\"3&*\\(=?9SE7$F-$\"31v2^=J1[9F-7$$\"3Q+]U(yzx M$F-$\"3'Rp2\")39XS\"F-7$$\"3<++q)f-&QNF-$\"3Q)3*o-K7i8F-7$$\"3=](o&o& e.w$F-$\"3(p;N0;)328F-7$$\"3))***\\RpQn&RF-$\"3YQ1PG:p`7F-7$$\"3k](og \"))>sTF-$\"3)oJM`M\"\\!>\"F-7$$\"3E]i?/yOxVF-$\"3sUIH7T=E6F-7$$\"3U+v o>,W\"f%F-$\"3_Fw)f)e8b5F-7$$\"3U]PWJ`-)y%F-$\"3+,z6\\^*f')*FG7$$\"3I+ D1S!p++&F-$\"3ud'4M$3#R4*FG7$$\"33](=&fGK?_F-$\"3Aw&\\92t$e#)FG7$$\"33 ]7$oXa?T&F-$\"3?X^D'=+]](FG7$$\"3r+veM)H\">cF-$\"3_\"*>?GX$em'FG7$$\"3 `***\\@'*eI$eF-$\"3uGJ*ywWFx&FG7$$\"3(**\\Ps(oMUgF-$\"3GNneh.%[([FG7$$ \"3P](okpV[C'F-$\"3/1\"R-G&f%)RFG7$$\"3m*\\i%*)=opkF-$\"3y)eFBVnF(HFG7 $$\"3]***\\f+44o'y([P/#FG7$$\"3;+]iugT()oF-$\"3M+.v2R=K5FG 7$$\"3w\\(=#R(yG3(F-$\"3!fR$)Gve)*))*!#@7$$\"37,]2g5d'H(F-$!3=-GJj=r'Q *F*7$$\"3%*\\(o0))Qw\\(F-$!3^H0nQ>jI>FG7$$\"3I]P/)o7yq(F-$!3QRJr<1.$)H FG7$$\"3w**\\7,gH8zF-$!3q'zWF`(fESFG7$$\"3/](=n)zTG\")F-$!3!*zo#=/fR8& FG7$$\"3&4++n-1cL)F-$!3hMV7%)HA9iFG7$$\"3/+DTAd[Z&)F-$!3(4D>@z5BL(FG7$ $\"3M^(os%4hd()F-$!3k3i&[-VRX)FG7$$\"3c++:1Up]*)F-$!3'R\\R8@6a\\*FG7$$ \"3/,DE5%*)><*F-$!3Qgw![()G,2\"F-7$$\"31++!*y%=*p$*F-$!3+GQAI00z6F-7$$ \"3)4vo&[@&4e*F-$!3Qro*)**=D'H\"F-7$$\"3o]7t(HQHy*F-$!3lRD9YZU49F-7$$ \"\"\"\"\"!$!3&******R,#4K:F--%+AXESLABELSG6$Q\"x6\"Q!6\"-%'COLOURG6&% $RGBG$Fd^lFd^lFb_l$\"*++++\"!\")-%%VIEWG6$;$\"+++++5!#7Fb^l%(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 33 "Here is a list of sample values. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "printf(\"\\n\"):\nprintf(\" x \+ cube root linear approx absolute error\\n\\n\"); \nfor i from 0 to 19 do\n xx := 0.001+i*0.05;\n r1 := xx^(1/3);\n \+ r2 := .9009009009*xx+0.2523083005;\nprintf(\" %.3f %12.10f \+ %12.10f %2.2e\\n\",xx,r1,r2,abs(r1-r2));\nend do:" }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 66 " x \+ cube root linear approx absolute error" }}{PARA 6 " " 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 62 " .001 .1 000000000 .2532092014 1.53e-01" }}{PARA 6 "" 1 "" {TEXT -1 62 " .051 .3708429769 .2982542464 7.26e-02 " }}{PARA 6 "" 1 "" {TEXT -1 62 " .101 .4657009508 .3 432992915 1.22e-01" }}{PARA 6 "" 1 "" {TEXT -1 62 " .151 \+ .5325074022 .3883443365 1.44e-01" }}{PARA 6 "" 1 "" {TEXT -1 62 " .201 .5857766003 .4333893816 1.5 2e-01" }}{PARA 6 "" 1 "" {TEXT -1 62 " .251 .6307993549 \+ .4784344266 1.52e-01" }}{PARA 6 "" 1 "" {TEXT -1 62 " .301 .6701759395 .5234794717 1.47e-01" }}{PARA 6 "" 1 "" {TEXT -1 62 " .351 .7054004063 .5685245167 \+ 1.37e-01" }}{PARA 6 "" 1 "" {TEXT -1 62 " .401 .7374197940 .6135695618 1.24e-01" }}{PARA 6 "" 1 "" {TEXT -1 62 " \+ .451 .7668766491 .6586146068 1.08e-01" }}{PARA 6 "" 1 "" {TEXT -1 62 " .501 .7942293073 .7036596519 \+ 9.06e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .551 .81981 75283 .7487046969 7.11e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .601 .8439009789 .7937497419 5.02e-02" }} {PARA 6 "" 1 "" {TEXT -1 62 " .651 .8666831029 .83879 47870 2.79e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .701 \+ .8883266120 .8838398320 4.49e-03" }}{PARA 6 "" 1 "" {TEXT -1 62 " .751 .9089639217 .9288848771 1.9 9e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .801 .9287044047 \+ .9739299221 4.52e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .851 .9476395693 1.0189749670 7.13e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .901 .9658468409 1.0640200120 \+ 9.82e-02" }}{PARA 6 "" 1 "" {TEXT -1 62 " .951 .9833923805 1.1090650570 1.26e-01" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "numapprox[minimax](x^(1/3) ,x=1/1000..1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&$\"+)*H3BD!#5\" \"\"*&$\"+4!4!4!*F&F'%\"xGF'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 258 "" 0 "" {TEXT -1 2 "2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1217 "cuberoot := proc(a::realcons)\n local z ,s,x,h,saveDigits,i,eps,maxit,isneg;\n \n\011 # Handle a special c ase\n\011 if a=0 then\n x := 0\n else \n eps := Float(1, -Digits); # desired relative error\n maxit := Digits*5; # maxim um no. of iterations\n\n\011\011 # Increase precision for the compu tation by a few digits\n\011\011 saveDigits := Digits;\n Digit s := Digits+length(Digits)+1;\n z := evalf(a);\n\n # negativ e argument gives a negative result\n if z<0 then\n\011\011 \+ z := -z;\n isneg := true;\n else isneg := false end if;\n \n\011\011 # Apply range reduction.\n s := -ilog10(z)-1;\n \+ if s mod 3=1 then s := s-1 end if;\n if s mod 3=2 then s := s-2 end if;\n z := z*Float(1,s);\n s := -s/3; \n\011 \n\011 \011 # Obtain an initial approximation.\n\011\011 x := .90090090 09*z+0.2523083005;\n\011\011 h := 1:\n\n for i from 1 to maxit do\n\011 h := (z/(x*x)-x)/3; # Newton iteration formula\n\011 \011\011 x := x+h;\n if abs(h) " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }