{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 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 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 1 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 "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 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 "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Author" -1 19 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 8 8 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 35 "Best approximation to a r eal number" }}{PARA 19 "" 0 "" {TEXT -1 31 "by Alex Quandt, Uni Greifs wald " }}{PARA 0 "" 0 "" {TEXT -1 25 "We want to approximate a " } {TEXT 263 13 "real number x" }{TEXT -1 6 " as a " }{TEXT 264 9 "fracti on " }{TEXT -1 7 "of two " }{TEXT 265 16 "integers p and q" }{TEXT -1 8 ". \n\nThe " }{TEXT 267 24 "mathematical derivation " }{TEXT -1 9 "o f this " }{TEXT 272 10 "classical " }{TEXT -1 30 "algorithm can be fo und in : " }{TEXT 266 1 "\n" }{TEXT 270 0 "" }{TEXT 271 30 "Fundamen tals of Number Theory " }{TEXT -1 11 "by William " }{TEXT -1 47 "J. Le Veque, Addison-Wesley 1997, \nChapter 9.2. " }}}{SECT 0 {PARA 3 "" 0 " " {TEXT -1 17 "General settings " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Let us specify a number of things : " }{MPLTEXT 1 0 1 " " }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 36 "Clear everything + general settin gs " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "restart; \nDigits:=15 ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#:" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "Initial values" }}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 12 "Real number " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Spe cify the " }{TEXT 258 11 "real number" }{TEXT -1 1 " " }{TEXT 262 6 "x start" }{TEXT -1 32 " which should be approximated : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "xstart:=12.3682671;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'xstartG$\"*rEoB\"!\"(" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 "Initial values " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Specify " }{TEXT 259 14 "initial values" }{TEXT -1 22 " to start re cursion : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "p[-2]:= 0; p[ -1]:=1; q[-2]:=1; q[-1]:=0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"p G6#!\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"pG6#!\"\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"qG6#!\"#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"qG6#!\"\"\"\"!" }}}}{EXCHG }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 9 "Recursion" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "k max : maximum number of " }{TEXT 261 16 "recursion steps " } {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "kmax := 10 ; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%kmaxG\"#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 18 "Best approximation" }{TEXT -1 21 " through recursion : " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 396 " printlevel:=0; \nx[0]:=evalf(xstart); \n\nfor k from 0 to kmax do\n\n \+ lambda[k]:=trunc(x[k]); \n\n p[k]:=lambda[k]*p[k-1]+p[k-2]; \n q[k ]:=lambda[k]*q[k-1]+q[k-2]; \n fracpq:=evalf(p[k]/q[k]);\n\n printf( \"step = %d\\n p = %d q = %d \\n\",k,p[k],q[k]);\n printf(\" p/q = \+ %20.15f x= %20.15f \\n |p/q-x| = %20.15f \\n\\n\",fracpq,xstart,abs( fracpq-xstart)); \n\n x[k+1]:=1/(x[k]-trunc(x[k])); \n\nend do; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+printlevelG\"\"!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%\"xG6#\"\"!$\"*rEoB\"!\"(" }}{PARA 6 "" 1 "" {TEXT -1 8 "step = 0" }}{PARA 6 "" 1 "" {TEXT -1 16 " p = 12 q = 1 \+ " }}{PARA 6 "" 1 "" {TEXT -1 54 " p/q = 12.000000000000000 x= 12 .368267100000000 " }}{PARA 6 "" 1 "" {TEXT -1 32 " |p/q-x| = .3682 67100000000 " }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 8 "step = 1" }}{PARA 6 "" 1 "" {TEXT -1 16 " p = 25 q = 2 " }} {PARA 6 "" 1 "" {TEXT -1 54 " p/q = 12.500000000000000 x= 12.368 267100000000 " }}{PARA 6 "" 1 "" {TEXT -1 32 " |p/q-x| = .13173290 0000000 " }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 8 "step = 2" }}{PARA 6 "" 1 "" {TEXT -1 16 " p = 37 q = 3 " }}{PARA 6 "" 1 "" {TEXT -1 54 " p/q = 12.333333333333300 x= 12.368267100 000000 " }}{PARA 6 "" 1 "" {TEXT -1 32 " |p/q-x| = .03493376666670 0 " }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 8 "step = 3" }}{PARA 6 "" 1 "" {TEXT -1 16 " p = 99 q = 8 " }}{PARA 6 "" 1 "" {TEXT -1 54 " p/q = 12.375000000000000 x= 12.368267100000000 \+ " }}{PARA 6 "" 1 "" {TEXT -1 32 " |p/q-x| = .006732900000000 " }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 8 "step = 4" }}{PARA 6 "" 1 "" {TEXT -1 18 " p = 136 q = 11 " }}{PARA 6 "" 1 "" {TEXT -1 54 " p/q = 12.363636363636400 x= 12.368267100000000 " } }{PARA 6 "" 1 "" {TEXT -1 32 " |p/q-x| = .004630736363600 " }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 8 "step = 5" }}{PARA 6 "" 1 "" {TEXT -1 18 " p = 235 q = 19 " }}{PARA 6 "" 1 "" {TEXT -1 54 " p/q = 12.368421052631600 x= 12.368267100000000 " } }{PARA 6 "" 1 "" {TEXT -1 32 " |p/q-x| = .000153952631600 " }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 8 "step = 6" }}{PARA 6 "" 1 "" {TEXT -1 20 " p = 4131 q = 334 " }}{PARA 6 "" 1 " " {TEXT -1 54 " p/q = 12.368263473053900 x= 12.368267100000000 \+ " }}{PARA 6 "" 1 "" {TEXT -1 32 " |p/q-x| = .000003626946100 " }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 8 "step = 7" }}{PARA 6 "" 1 "" {TEXT -1 20 " p = 8497 q = 687 " }}{PARA 6 "" 1 " " {TEXT -1 54 " p/q = 12.368267831149900 x= 12.368267100000000 \+ " }}{PARA 6 "" 1 "" {TEXT -1 32 " |p/q-x| = .000000731149900 " }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 8 "step = 8" }}{PARA 6 "" 1 "" {TEXT -1 22 " p = 21125 q = 1708 " }}{PARA 6 "" 1 "" {TEXT -1 54 " p/q = 12.368266978922700 x= 12.368267100000000 \+ " }}{PARA 6 "" 1 "" {TEXT -1 32 " |p/q-x| = .000000121077300 " }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 8 "step = 9" }}{PARA 6 "" 1 "" {TEXT -1 22 " p = 50747 q = 4103 " }}{PARA 6 "" 1 "" {TEXT -1 54 " p/q = 12.368267121618300 x= 12.368267100000000 \+ " }}{PARA 6 "" 1 "" {TEXT -1 32 " |p/q-x| = .000000021618300 " }} {PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 9 "step = 10 " }}{PARA 6 "" 1 "" {TEXT -1 23 " p = 122619 q = 9914 " }}{PARA 6 " " 1 "" {TEXT -1 54 " p/q = 12.368267097034500 x= 12.368267100000 000 " }}{PARA 6 "" 1 "" {TEXT -1 32 " |p/q-x| = .000000002965500 \+ " }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 2 8" 9 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }