2016 Update
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Tina Müller
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7965 changed files with 139854 additions and 31002 deletions
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@ -3,20 +3,24 @@ where calculating the actual value is difficult or impossible. <br>
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It uses random sampling to define constraints on the value
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and then makes a sort of "best guess."
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A simple Monte Carlo Simulation can be used to calculate the value for π.
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A simple Monte Carlo Simulation can be used to calculate the value for <big><math>\pi</math></big>.
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If you had a circle and a square where the length of a side of the square
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was the same as the diameter of the circle, the ratio of the area of the circle
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to the area of the square would be π/4.
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to the area of the square would be <big><math>\pi/4</math></big>.
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So, if you put this circle inside the square and select many random points
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inside the square, the number of points inside the circle
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divided by the number of points inside the square and the circle
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would be approximately π/4.
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would be approximately <big><math>\pi/4</math></big>.
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;Task:
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Write a function to run a simulation like this, with a variable number of random points to select.
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Write a function to run a simulation like this, with a variable number
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of random points to select. <br>
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Also, show the results of a few different sample sizes.
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For software where the number π is not built-in,
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we give π to a couple of digits:
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3.141592653589793238462643383280
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For software where the number <big><math>\pi</math></big> is not built-in,
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we give <big><math>\pi</math></big> as a number of digits:
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3.141592653589793238462643383280
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<br><br>
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@ -1,9 +1,8 @@
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defmodule MonteCarlo do
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def pi(n) do
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:random.seed(:os.timestamp)
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count = Enum.count(1..n, fn _ ->
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x = :random.uniform
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y = :random.uniform
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x = :rand.uniform
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y = :rand.uniform
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:math.sqrt(x*x + y*y) <= 1
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end)
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4 * count / n
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16
Task/Monte-Carlo-methods/Fortran/monte-carlo-methods-2.f
Normal file
16
Task/Monte-Carlo-methods/Fortran/monte-carlo-methods-2.f
Normal file
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@ -0,0 +1,16 @@
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program mc
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integer :: n,i
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real(8) :: pi
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n=10000
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do i=1,5
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print*,n,pi(n)
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n = n * 10
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end do
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end program
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function pi(n)
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integer :: n
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real(8) :: x(2,n),pi
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call random_number(x)
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pi = 4.d0 * dble( count( hypot(x(1,:),x(2,:)) <= 1.d0 ) ) / n
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end function
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@ -1,32 +1,30 @@
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/*REXX program computes pi ÷ 4 using the Monte Carlo algorithm. */
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parse arg times chunks . /*does user want a specific number? */
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if times=='' then times=1000000000 /*one billion should do it, me thinks. */
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if chunks=='' then chunks=10000 /*do Monte Carlo in 10,000 chunks. */
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limit=10000-1 /*REXX random generates only integers. */
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limitSq=limit**2 /*··· so, instead of one, use limit**2.*/
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!=0 /*the number of "pi hits" (so far). */
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accur=0 /*accuracy of Monte Carlo pi (so far). */
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if 1=='f1'x then piChar='pi' /*if EBCDIC, then use literal. */
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else piChar='e3'x /* " ASCII, " " pi glyph, */
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pi=3.14159265358979323846264338327950288419716939937511 /*this, da real McCoy*/
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numeric digits length(pi) /*this program uses these decimal digs.*/
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say 'real pi='pi"+" /*we might as well brag about it. */
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say /*a blank line, just for the eyeballs. */
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do j=1 for times%chunks
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do chunks /*do Monte Carlo, one chunk-at-a-time.*/
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if random(0,limit)**2 + random(0,limit)**2 <=limitSq then !=!+1
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end /*chunks*/
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reps=chunks*j /*compute the number of repetitions. */
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piX=4*!/reps /*let's see how this puppy does so far.*/
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_=compare(piX,pi) /*compare apples and ··· crabapples. */
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if _<=accur then iterate /*if not better accuracy, keep going. */
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say right(commas(reps),20) 'repetitions: Monte Carlo' piChar,
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"is accurate to" _-1 'places.' /*subtract one for decimal point.*/
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accur=_ /*use this accuracy for the baseline. */
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end /*j*/
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exit /*stick a fork in it, we're all done. */
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/*────────────────────────────────────────────────────────────────────────────*/
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commas: procedure; parse arg _; n=_'.9'; #=123456789; b=verify(n,#,"M")
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e=verify(n,#'0',,verify(n,#"0.",'M'))-4
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do j=e to b by -3; _=insert(',',_,j); end /*j*/; return _
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/*REXX program computes and displays the value of pi÷4 using the Monte Carlo algorithm*/
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pi=3.141592653589793238462643383279502884197169399375105820974944592307816406 /*true pi.*/
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say ' 1 2 3 4 5 6 7 '
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say 'scale: 1·234567890123456789012345678901234567890123456789012345678901234567890123'
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say /* [↑] a two-line scale for showing pi*/
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say 'true pi='pi"+" /*we might as well brag about true pi.*/
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numeric digits length(pi) - 1 /*this program uses these decimal digs.*/
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parse arg times chunk . /*does user want a specific number? */
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if times=='' | times=="," then times=1000000000 /*one billion should do it, hopefully. */
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if chunk=='' | chunk=="." then chunk= 10000 /*perform Monte Carlo in 10k chunks.*/
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limit=10000-1 /*REXX random generates only integers. */
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limitSq=limit**2 /*··· so, instead of one, use limit**2.*/
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accur=0 /*accuracy of Monte Carlo pi (so far). */
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!=0; @reps='repetitions: Monte Carlo pi is' /*pi decimal digit accuracy (so far).*/
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say /*a blank line, just for the eyeballs.*/
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do j=1 for times%chunk
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do chunk /*do Monte Carlo, one chunk at-a-time.*/
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if random(0,limit)**2 + random(0,limit)**2 <=limitSq then !=!+1
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end /*chunk*/
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reps=chunk*j /*calculate the number of repetitions. */
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_=compare(4*! / reps, pi) /*compare apples and ··· crabapples. */
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if _<=accur then iterate /*if not better accuracy, keep trukin'.*/
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say right(commas(reps),20) @reps 'accurate to' _-1 "places." /*-1 for dec. pt.*/
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accur=_ /*use this accuracy for next baseline. */
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end /*j*/
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exit /*stick a fork in it, we're all done. */
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/*──────────────────────────────────────────────────────────────────────────────────────*/
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commas: procedure; parse arg _; n=_'.9'; #=123456789; b=verify(n,#,"M")
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e=verify(n, #'0', , verify(n, #"0.", 'M') ) - 4
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do j=e to b by -3; _=insert(',',_,j); end /*j*/; return _
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