{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 Input" 2 19 "" 0 1 255 0 0 1 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 "" -1 256 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 0 1 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 "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Ti mes" 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 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 "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 31 "Uniform random numbers in Maple" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "First we define a floating-point random generator based on the intege r random number generator" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart; Digits:=14;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG \"#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "#?random" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "rand();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"-\"3p'>uU" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "Thi s is a 12-digit random number generator." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "randx:=s->evalf(rand()/999999999999);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&randxGj+6#%\"sG6\"6$%)operatorG%&arrowGF(-%&eva lfG6#,$*&#\"\"\"\"-************F2-%%randGF(F2F2F(F(F(6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "randx();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/KqKp56K!#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 " In this form the generator will always produce the same sequence." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "S1:=[seq(randx(),i=1..1000)] :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "S2:=[seq(randx(),i=1.. 1000)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "plot([seq([S1[i] ,S2[i]],i=1..1000)],style=point,symbol=circle,symbolsize=4,scaling=con strained);" }}{PARA 13 "" 1 "" {GLPLOT2D 277 275 275 {PLOTDATA 2 "6(-% 'CURVESG6$7din7$$\"3v*RtptIjV$!#=$\"3Y+jxS,o3jF*7$$\"36+ZjN9cUZF*$\"3I +a!=H!*3X&F*7$$\"3m*fl(*=(e%e&F*$\"3#)*fVz!G_>OF*7$$\"3'**\\(Q0$QvY(F* $\"3c**y0_pKkyF*7$$\"3n*>.&3AA1K!#>$\"3[+qnBGSvpF*7$$\"3-+so<7uHsF*$\" 3\")*\\--)>1+DF*7$$\"3#)**f@Rh0VgF*$\"3i+$3<.l!z#)F?7$$\"3')*\\(4u.!eX (F*$\"3a+))[Yix@))F*7$$\"3C+EbE&>\")f#F*$\"3s*f)**3')\\%e)F*7$$\"3:+Jj r[v+JF*$\"3++76S2jr6F*7$$\"35+!yX!\\zrzF*$\"3A+&QVzr=X)F*7$$\"3!)**Q+; %fp\"RF?$\"3&**H<5\"3v.tF*7$$\"3L*z3u;dI%))F?$\"3c*HOPo0!>jF*7$$\"3S+' f3M))\\g*F*$\"3o*R0K`iGW&F?7$$\"3-+\"o\"zX?H\")F*$\"3&***=+_FwI>F*7$$ \"3>+Xh%>qu`%F*$\"3z*zURlw&QGF*7$$\"3v*RwI&RJSkF*$\"3-+9CkK&HV\"F*7$$ \"3`*>*\\t%\\i?*F*$\"37+nbtuuOnF*7$$\"3R+&p3IN0^*F*$\"3O+(3>W-un)F*7$$ \"31+:)>2j[Y\"F*$\"3K+OSlBm)p.&>F*7 $$\"3;+V4Pn#RH%F*$\"3k*z[b!zs]()F*7$$\"3]+`t4^Ga_F*$\"3z**[y!4#f@\\F*7 $$\"3w*p7)*31gs#F*$\"3C+g4k!\\R*fF*7$$\"3-+Au$*4g(>#F*$\"3))**)z:Q4()* )*F*7$$\"3]*zmwLH)fnF*$\"3i*>;y?%\\zhF*7$$\"3E+&QZ4NZX)F*$\"3Q+#Hhyc+; *F*7$$\"3S+oU$)yqknF*$\"3c+'\\#*zmek*F*7$$\"3v*z7>z(Q8GF*$\"3v**e^Zui; fF*7$$\"3K+z$R+f\\#zF*$\"3!**4G6%)4O5)F*7$$\"3i*\\2IR&47vF*$\"31+6J95X X6F*7$$\"3]+j6IWj$G'F*$\"3)**f-]xnYh#F?7$$\"3()*4$Ql3YPJF*$\"35+g6cHxa fF*7$$\"39+f+8\\mie!#?$\"3%***44&)\\R@5F*7$$\"3g+v+AcO\"[(F?$\"3E+I7uj $H'HF*7$$\"3c*RY%QWUQkF*$\"3w*RoOr-!f$)F*7$$\"37+8lXv0>8F*$\"3Z+!>xDA! 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H`F*$\"3-+RXa7JyQF*7$$\"3)***HW)eFy'HF*$\"3#)*RQKI?8P)F*7$$\"3++W*owZ1 W%F*$\"3_**paB>\">)pF*7$$\"3$**z-k<2l$GF?$\"37+]3!RX7)\\F*7$$\"3f*f4L9 'e5'*F*$\"3#**pU`$\\EqEF*7$$\"3<+*)=ZVL:*)F*$\"3/+U(y_B_<%F*7$$\"3U+l+ kitClF[v$\"31+5^K+3J5F*7$$\"3R+$)Q,1d.$)F*$\"3]*H1Ddk')H'F*7$$\"37+;:< `'**f\"F*$\"3W*zfAFPAy*F*7$$\"3y**[b\\rj_[F*$\"3D+HF)4J9\"HF*7$$\"3&** f.ZFYLk$F*$\"3g*ReAnBUQ)F*7$$\"3q**op+!Rx*oF*$\"3?+O3&)e+,OF*7$$\"33+X 5)>`zX%F*$\"3s*fcHRjwb'F*7$$\"3))*RJTjBsW\"F*$\"3/+6S-(eG3\"F?7$$\"3%) **GA\\w5^GF*$\"3#)*fk%oY-QYF*7$$\"3y*HxU8U]K(F*$\"3A+XEW0:%\\%F*7$$\"3 `*\\Q:W2\\])F*$\"3&)**)3zu:]$*)F*7$$\"3%**H$z+$oPF$F*$\"3w**Ha'*ot]HF* 7$$\"3H+pdvMZ@pF*$\"3a+\"=jump4)F*7$$\"3')*f$yy2X!f$F*$\"3u*z:9;4.v&F* 7$$\"3')*f9A-fsk%F*$\"30+M?A*HcS$F?7$$\"3=+&QmLe%R&)F*$\"3E+vb&3u(3vF* 7$$\"3))*\\Vvvd0`$F*$\"3++^Bj*eF2&F*7$$\"3M+dDL2c.dF*$\"3/+`9BMT$H&F*7 $$\"3?+fc'4om\"fF*$\"3&**H#\\%3F)4BF*7$$\"3&**f\"[!)zD.;F*$\"3L+zoi%[n !zF*7$$\"3a*zb!*[h=x&F*$\"3')*fUJKwbc#F*7$$\"3!**H3sY$)fD)F*$\"3m*41*y Wy)4'F?7$$\"3M+w#*zX[ZwF*$\"3#)*fpr<<^k*F*7$$\"3b+u!o1OAT(F*$\"30+Xh.@ pVXF*7$$\"3i*zch'4]enF*$\"35+BE6Em,BF*7$$\"37+s@>8mFsF*$\"37+<@S!z5o\" F*7$$\"3(**f-`[!*zj#F?$\"3?+Nyk93nMF*7$$\"3@+y:B9a:yF*$\"38+%\\JSVsV*F *7$$\"3Q+%e>/qkO)F*$\"32+8H\"Hu$>8F*7$$\"3++d4s.70dF*$\"3u*RtsN#*QN$F* 7$$\"3v*zAhivE!GF*$\"3-+MEX(RrS$F*7$$\"3%)*p24rb)>xF?$\"3M+y!HF-4x(F*7 $$\"3\"**>C*e/=CUF*$\"3/+C!oiabW#F?7$$\"3_+*Hag!ft)*F*$\"3-+=Jh8D'y\"F *-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fed^lFdd^l-%'SYMBOLG6$%'CIRCLEG\"\" %-%&STYLEG6#%&POINTG-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6$Q!6\"Ff e^l-%%VIEWG6$%(DEFAULTGF[f^l" 1 5 4 1 4 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "This does not look very uniform. Does the coverage improve when " }{TEXT 19 1 "N" }{TEXT -1 33 " is increased? Look for yourself." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 196 "Other se quences are obtained by changing the seed. The seed value of the pseud orandom sequence can be chosen on the basis of the system clock. This \+ yields new sequences which are not reproducible." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "#randomize(Seed)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Histo:=proc(S) local N,M,H,i,nu; N:=nops(S): M:=10: H :=Vector(M,datatype=float[8]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "f or i from 1 to N do: nu:=trunc(S[i]*M): H[nu+1]:=H[nu+1]+1: od: H; end :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "H1:=Histo(S1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H1G-%'RTABLEG6%\")sm!G#-%'MATRIXG6# 7,7#$\"$.\"\"\"!7#$\"#))F07#$\"##*F07#$\"#$*F07#$\"$2\"F0F47#$\"$F\"F0 F-7#$\"#**F07#$\"#'*F0&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "H2:=Histo(S2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%#H2G-%'RTABLEG6%\"'3Xd-%'MATRIXG6#7,7#$\"#$*\"\"!7#$\"$.\"F07#$\"$1 \"F07#$\"#\"*F07#$\"$3\"F07#$\"#%)F07#$\"$7\"F07#$\"$4\"F07#$\"#**F07# $\"#&*F0&%'VectorG6#%'columnG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "P1:=plot([seq(op([[nu-1,H1[nu]],[nu,H1[nu]]]),nu=1..10)]): P2:= plot([seq(op([[nu-1,H2[nu]],[nu,H2[nu]]]),nu=1..10)],color=blue): P3:= plot([seq(op([[nu-1,0.5*(H1[nu]+H2[nu])],[nu,0.5*(H1[nu]+H2[nu])]]),nu =1..10)],color=green): plots[display](P1,P2,P3,view=[0..10,0..130]);" }}{PARA 13 "" 1 "" {GLPLOT2D 899 340 340 {PLOTDATA 2 "6'-%'CURVESG6$76 7$$\"\"!F)$\"$.\"F)7$$\"\"\"F)F*7$F-$\"#))F)7$$\"\"#F)F07$F3$\"##*F)7$ $\"\"$F)F67$F9$\"#$*F)7$$\"\"%F)F<7$F?$\"$2\"F)7$$\"\"&F)FB7$FEF67$$\" \"'F)F67$FI$\"$F\"F)7$$\"\"(F)FL7$FOF*7$$\"\")F)F*7$FS$\"#**F)7$$\"\"* F)FV7$FY$\"#'*F)7$$\"#5F)Ffn-%'COLOURG6&%$RGBG$Fjn!\"\"F(F(-F$6$767$F( F<7$F-F\"Fjr7$FOF_s7$FOFho7$FSFhoFUFX 7$FYF`r7$FinF`r-F\\o6&F^oF(FeqF(-%+AXESLABELSG6%Q!6\"F[t-%%FONTG6#%(DE FAULTG-%%VIEWG6$;F(Fin;F($\"$I\"F)" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "The fluctuations about the expected valu e of 100 have decreased a little when combining the two sequences of 1 000 points each." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 0 "" }} {PARA 0 "" 0 "" {TEXT -1 106 "We now look into measuring the deviation s from the average and observe how these decrease with increasing " } {TEXT 19 1 "N" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "sqrt(add((H1[nu]-100)^2,nu=1 ..10))/10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/l4[;\\nL!#8" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "sqrt(add((H2[nu]-100)^2,nu=1 ..10))/10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/&\\n0+8t#!#8" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "sqrt(add((0.5*(H1[nu]+H2[nu] )-100)^2,nu=1..10))/10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/Ea_PH'p #!#8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "S3:=[seq(randx(),i= 1..10000)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "H3:=Histo(S3 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#H3G-%'RTABLEG6%\")!=H=#-%'MAT RIXG6#7,7#$\"%+5\"\"!7#$\"$q*F07#$\"$%)*F07#$\"$u*F07#$\"$\"**F07#$\"% .5F07#$\"$()*F07#$\"%J5F07#$\"%;5F07#$\"%W5F0&%'VectorG6#%'columnG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "sqrt(add((H3[nu]-1000)^2,nu =1..10))/10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/F4'zY:C(!#8" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "The latter number is more impress ive than the previous ones, because it is the deviation from 1000, rat her than from 100." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 185 "We perform our first Monte-Carlo calculation with these \+ data points. We calculate the area of a circle enclosed by the unit sq uare. This is a circle of radius 0.5 centered on (0.5,0.5)." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "MC:=proc(S1,S2) local i,r0,x 0,y0,N,Nhit;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r0:=0.5; x0:=0.5; y 0:=0.5;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "N:=nops(S1): if N<>nops( S2) then RETURN(\"Problem with mismatched point sequences\",N,nops(S2) ) fi: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "Nhit:=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "for i from 1 to N do:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "if (S1[i]-x0)^2+(S2[i]-y0)^2 <= r0^2 then Nhit:=Nhit+ 1; fi: od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "evalf(Nhit/N); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "MC(S1,S2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"/+++++gz!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf(Pi*0.5^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$ \"/X(Rj\")R&y!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "S4:=[se q(randx(),i=1..10000)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " MC(S3,S4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/+++++.y!#9" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 258 6 "Tasks:" }}{PARA 0 "" 0 "" {TEXT -1 126 "Generate sequences with a larger number of points and observe \+ whether the estimated area converges towards the correct result." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 148 "Pick a c ircle with smaller radius, but keep using random numbers selected from the (0,1) range. The efficiency of the calculation deteriorates. Why? " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 261 22 "Error estimation in MC" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 0 "" }{TEXT -1 147 "In order to under stand whether the answer found by Monte-Carlo is good or not we need t o adopt the fundamental philosophy of experimental physics: " }}{PARA 0 "" 0 "" {TEXT 260 86 "The measurement of a quantity without an idea \+ of the errors involved is of limited use" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 383 "How can we measure the statistica l error in our MC evaluation? One approach is to break the measurement into subgroups, obtain averages for the subgroups and determine their deviations. We do this first for the sequences of data obtained for t he bar graph in the figure above. Each point was supposed to yield the same average number in the bin. By construction the average is exact: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "N_avg:=add(H1[n],n=1..1 0)/10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&N_avgG$\"/++++++5!#6" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Let us calculate the deviations by adding them in quadrature (to avoid fortuitous cancellations):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "N_dev:=sqrt(add((H1[n]-N_avg )^2,n=1..10))/10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&N_devG$\"/l4[; \\nL!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "There are individual \+ measurements which deviate from the average by more than this number, \+ but combined the data give us this amount of confidence." }}{PARA 0 " " 0 "" {TEXT -1 181 "By chance the other sequence had smaller deviatio ns. This reminds us of the fact that the determination of the deviatio n is a measurement itself, i.e., subject to statistical error." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "N_dev:=sqrt(add((H2[n]-N_avg )^2,n=1..10))/10;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&N_devG$\"/&\\n 0+8t#!#8" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Now we look at the re al measurement using these two sets of random numbers." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "MC:=proc(S1,S2) local i,r0,x0,y0,N, Nhit,i0,nu,Nhit_a,Nhit_d;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "r0:=0. 5; x0:=0.5; y0:=0.5;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "N:=nops(S1) : if N<>nops(S2) then RETURN(\"Problem with mismatched point sequences \",N,nops(S2)) fi: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Nhit:=Vector (10,datatype=float[8]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "for i0 f rom 1 to 10 do: for i from 1 to N/10 do: nu:=(i0-1)*(N/10)+i;" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "if (S1[nu]-x0)^2+(S2[nu]-y0)^2 <= r 0^2 then Nhit[i0]:=Nhit[i0]+1; fi: od: od: Nhit:=Nhit/(N/10);" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 140 "Nhit_a:=evalf(add(Nhit[i0],i0=1..1 0)/10); Nhit_d:=sqrt(add((Nhit[i0]-Nhit_a)^2,i0=1..10)/(9*10)); [Nhit_ a-Nhit_d,Nhit_a,Nhit_a+Nhit_d]; end:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "The procedure MC has been modified to break the two given sequ ences of random numbers (used for the two cartesian components) into 1 0 groups of " }{TEXT 19 4 "N/10" }{TEXT -1 255 " points. The number of hits is recorded within each subgroup and appropriately turned into a ratio of successful hits to the total. Then the result is averaged, a nd the deviation from the average is estimated. The result is returned as a list containing " }{TEXT 19 11 "r-d, r, r+d" }{TEXT -1 9 " , wh ere " }{TEXT 19 1 "r" }{TEXT -1 29 " is the measured result, and " } {TEXT 19 1 "d" }{TEXT -1 274 " is the standard deviation. We divided b y 10-1 = 9 when calculating the deviation, because the average value w as deduced from the data (loss of one degree of freedom). We find that the measured range includes the correct value. There was a 95% probab ility for this to occur." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 157 "Suppose it had not made the range (it is actually rather close to the left-hand bracket. Would this mean that our estim ation procedure for the area is wrong?" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 90 "The answer is NO. The deviation has b een estimated only. The probability to be inside the " }{TEXT 19 9 "r \+ +/- 2*d" }{TEXT -1 488 " window (2-sigma interval) is 99%, which is cl oser to but still not 100%. In physics one is aware of statistical flu ctuations, seqences of random data can be skewed in one direction for \+ a while, only to be corrected later (when the total number of samples \+ is much increased). One trusts 5-sigma windows to the point where devi ations outside such a window are believed to signal 'new' physics, i.e ., something that is not explained by the model (to which the experime nt is being compared)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "r es:=MC(S1,S2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$resG7%$\"/LLLLLLy !#9$\"/+++++gzF($\"/nmmmm'3)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "evalf(Pi*0.5^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"/X(Rj\" )R&y!#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 133 "Let is see how the er ror decreases with 10 times longer sequences of randoms. First, record the deviation for the current simulation:" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 19 "dev:=res[2]-res[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$devG$\".nmmmmE\"!#9" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "N ow define long sequences:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "S1_l:=[seq(randx(),i=1..10000)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "S2_l:=[seq(randx(),i=1..10000)]:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 19 "res:=MC(S1_l,S2_l);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$resG7%$\"/WuYZV_y!#9$\"/+++++6zF($\"/cD`_cpzF(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dev_l:=res[2]-res[1];" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&dev_lG$\"-cD`_ce!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dev/dev_l,evalf(sqrt(10));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$$\"/B&zlHG;#!#8$\"/%o,mxA;$F%" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The reduction in the deviation is \+ of the right magnitude." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 80 "Now repeat this with the built-in floating-point ran dom number generator of the " }{TEXT 19 5 "stats" }{TEXT -1 9 " packag e." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(stats);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7*%&anovaG%)describeG%$fitG%+importdat aG%'randomG%*statevalfG%*statplotsG%*transformG" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 14 "#?stats,random" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "S1:=[stats[random, uniform](1000)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "S2:=[stats[random, uniform](1000)]:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "res:=MC(S1,S2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$resG7%$\"/*)olj'=](!#9$\"/+++++?wF($\"/6JMO8 QxF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "dev:=res[2]-res[1]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$devG$\".6JMO8=\"!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "S1_l:=[stats[random, uniform](10000 )]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "S2_l:=[stats[random, uniform](10000)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "res:= MC(S1_l,S2_l);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$resG7%$\"/82rg<-y !#9$\"/+++++XyF($\"/(G*GR#y)yF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dev_l:=res[2]-res[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&de v_lG$\"-(G*GR#G%!#9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "dev/ dev_l,evalf(sqrt(10));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"/&45N*eeF !#8$\"/%o,mxA;$F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 5 "Tasks" }} {PARA 0 "" 0 "" {TEXT -1 225 "1) Try the Monte-Carlo integration metho d on some other area integrals where the boundaries of the area can be constructed by conditional statements. Make sure you understand the o verall rectangular area that bounds the area." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "2) Extend task 1 to the c ase of volume integrals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 197 "3) Calculate moments of inertia of simple objects , such as a thin disk spinning about the center, or about a point on t he boundary. Moments of inertia are discussed in your first-year physi cs text." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 6 "Caveat" }}{PARA 0 "" 0 "" {TEXT -1 153 "From the preceding section it seems that we can reach the desi red precision by just extending the length of the MC sequence. This is true to some extent." }}{PARA 0 "" 0 "" {TEXT -1 51 "The good news: t he effort grows only linearly with " }{TEXT 19 1 "N" }{TEXT -1 40 " (w hile the accuracy improves only with " }{TEXT 19 7 "sqrt(N)" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 589 "The bad news: pseudorandom g enerators run out of steam, their number sequences begin to repeat the mselves. The construction of excellent generators is a science in its \+ own right. The very best generators are truly random event based, not \+ programmable in higher-level programming languages but in assembly lan guage (to which C can interface, e.g.). They would rely on the compute r hardware to produce the random event (such as occurs when generating overflows in computations, etc.). There are electronics and physics h ardware devices which are built (and sold) to produce random sequences ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "61 0 0" 3 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }