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brownian_motion_simulation.f
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brownian_motion_simulation.f
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subroutine brownian_displacement_display ( k, n, d, t, dsq,
& header )
c*********************************************************************72
c
cc BROWNIAN_DISPLACEMENT_DISPLAY displays average Brownian motion displacement.
c
c Licensing:
c
c This code is distributed under the GNU LGPL license.
c
c Modified:
c
c 30 September 2012
c
c Author:
c
c John Burkardt
c
c Parameters:
c
c Input, integer K, the number of repetitions.
c
c Input, integer N, the number of time steps.
c
c Input, double precision D, the diffusion coefficient.
c
c Input, double precision T, the total time.
c
c Input, double precision DSQ(K,N), the displacements over time for
c each repetition.
c
c Input, character * ( * ) HEADER, an identifier for the output files.
c
implicit none
integer k
integer n
character * ( 80 ) command_filename
integer command_unit
double precision d
character * ( 80 ) data_filename
integer data_unit
double precision dsq(k,n)
double precision dsq_ave
double precision dsq_ideal
character * ( * ) header
integer i
integer i4_uniform
integer ii(5)
integer j
integer seed
double precision t
double precision ti
seed = 123456789
c
c Choose 5 paths at random.
c
do j = 1, 5
ii(j) = i4_uniform ( 1, k, seed )
end do
c
c Create the data file.
c
call get_unit ( data_unit )
data_filename = trim ( header ) // '_data.txt'
open ( unit = data_unit, file = data_filename,
& status = 'replace' )
do j = 1, n
ti = dble ( j - 1 ) * t / dble ( n - 1 )
dsq_ave = 0.0D+00
do i = 1, k
dsq_ave = dsq_ave + dsq(i,j)
end do
dsq_ave = dsq_ave / dble ( k )
dsq_ideal = d * ti
write ( data_unit, '(8(2x,g14.6))' )
& ti, dsq(ii(1:5),j), dsq_ave, dsq_ideal
end do
close ( unit = data_unit )
write ( *, '(a)' ) ' '
write ( *, '(a)' ) ' BROWNIAN_DISPLACEMENT data stored in "'
& // trim ( data_filename ) // '".'
c
c Create the command file.
c
call get_unit ( command_unit )
command_filename = trim ( header ) // '_commands.txt'
open ( unit = command_unit, file = command_filename,
& status = 'replace' )
write ( command_unit, '(a)' ) '# ' // trim ( command_filename )
write ( command_unit, '(a)' ) '#'
write ( command_unit, '(a)' ) '# Usage:'
write ( command_unit, '(a)' ) '# gnuplot < ' //
& trim ( command_filename )
write ( command_unit, '(a)' ) '#'
write ( command_unit, '(a)' ) 'set term png'
write ( command_unit, '(a)' ) 'set output "'
& // trim ( header ) // '.png"'
write ( command_unit, '(a)' ) 'set xlabel "T"'
write ( command_unit, '(a)' ) 'set ylabel "D^2"'
write ( command_unit, '(a)' ) 'set title "Squared ' //
& 'displacement (Red), Predicted (Black), Samples (Blue)"'
write ( command_unit, '(a)' ) 'set grid'
write ( command_unit, '(a)' ) 'set style data lines'
write ( command_unit, '(a)' ) 'plot "'
& // trim ( data_filename ) //
& '" using 1:2 title "sample 1" linecolor rgb "blue", \'
write ( command_unit, '(a)' ) ' "'
& // trim ( data_filename ) //
& '" using 1:3 title "sample 2" linecolor rgb "blue", \'
write ( command_unit, '(a)' ) ' "'
& // trim ( data_filename ) //
& '" using 1:4 title "sample 3" linecolor rgb "blue", \'
write ( command_unit, '(a)' ) ' "'
& // trim ( data_filename ) //
& '" using 1:5 title "sample 4" linecolor rgb "blue", \'
write ( command_unit, '(a)' ) ' "'
& // trim ( data_filename ) //
& '" using 1:6 title "sample 5" linecolor rgb "blue", \'
write ( command_unit, '(a)' ) ' "'
& // trim ( data_filename ) //
& '" using 1:7 title "Averaged" lw 3 linecolor rgb "red", \'
write ( command_unit, '(a)' ) ' "'
& // trim ( data_filename ) //
& '" using 1:8 title "Ideal" lw 3 linecolor rgb "black"'
write ( command_unit, '(a)' ) 'quit'
close ( unit = command_unit )
write ( *, '(a)' )
& ' BROWNIAN_DISPLACEMENT plot commands stored in "'
& // trim ( command_filename ) // '".'
return
end
subroutine brownian_displacement_simulation ( k, n, m, d, t,
& seed, dsq )
c*********************************************************************72
c
cc BROWNIAN_DISPLACEMENT_SIMULATION simulates Brownian displacement.
c
c Discussion:
c
c This function computes the square of the distance of the Brownian
c particle from the starting point, repeating this calculation
c several times.
c
c Licensing:
c
c This code is distributed under the GNU LGPL license.
c
c Modified:
c
c 30 September 2012
c
c Author:
c
c John Burkardt
c
c Parameters:
c
c Input, integer K, the number of repetitions.
c
c Input, integer N, the number of time steps to take, plus 1.
c
c Input, integer M, the spatial dimension.
c
c Input, double precision D, the diffusion coefficient.
c Computationally, this is simply a scale factor between time and space.
c
c Input, double precision T, the total time.
c
c Input/output, integer SEED, a seed for the random
c number generator.
c
c Output, double precision DSQ(K,N), the displacements over time for each
c repetition. DSQ(:,1) is 0.0, because we include the displacement at the
c initial time.
c
implicit none
integer k
integer m
integer n
double precision d
double precision dt
double precision dsq(k,n)
integer i
integer i2
integer j
double precision s
integer seed
double precision t
double precision x(m,n)
do i = 1, k
call brownian_motion_simulation ( m, n, d, t, seed, x )
do j = 1, n
s = 0.0D+00
do i2 = 1, m
s = s + x(i2,j)**2
end do
dsq(i,j) = s
end do
end do
return
end
subroutine brownian_motion_display ( m, n, x, header )
c*********************************************************************72
c
cc BROWNIAN_MOTION_DISPLAY displays successive Brownian motion positions.
c
c Licensing:
c
c This code is distributed under the GNU LGPL license.
c
c Modified:
c
c 30 September 2012
c
c Author:
c
c John Burkardt
c
c Parameters:
c
c Input, integer M, the spatial dimension.
c M should be 1 or 2.
c
c Input, integer N, the number of time steps.
c
c Input, double precision X(M,N), the particle positions.
c
c Input, character ( len = * ) HEADER, an identifier for the output files.
c
implicit none
integer m
integer n
character * ( 80 ) command_filename
integer command_unit
character * ( 80 ) data_filename
integer data_unit
character * ( * ) header
integer i
double precision t
double precision x(m,n)
if ( m .ne. 1 .and. m .ne. 2 ) then
write ( *, '(a)' ) ' '
write ( *, '(a)' ) 'BROWNIAN_MOTION_DISPLAY - Fatal error!'
write ( *, '(a)' ) ' This routine can only handle M = 1 or 2.'
stop
end if
c
c Create the data file.
c
call get_unit ( data_unit )
data_filename = trim ( header ) // '_data.txt'
open ( unit = data_unit, file = data_filename,
& status = 'replace' )
if ( m .eq. 1 ) then
do i = 1, n
t = dble ( i - 1 ) / dble ( n - 1 )
write ( data_unit, '(2x,g14.6,2x,g14.6)' ) t, x(1,i)
end do
else if ( m .eq. 2 ) then
do i = 1, n
write ( data_unit, '(2x,g14.6,2x,g14.6)' ) x(1,i), x(2,i)
end do
end if
close ( unit = data_unit )
write ( *, '(a)' ) ' '
write ( *, '(a)' ) ' BROWNIAN_MOTION data stored in "'
& // trim ( data_filename ) // '".'
c
c Create the command file.
c
call get_unit ( command_unit )
command_filename = trim ( header ) // '_commands.txt'
open ( unit = command_unit, file = command_filename,
& status = 'replace' )
write ( command_unit, '(a)' ) '# ' // trim ( command_filename )
write ( command_unit, '(a)' ) '#'
write ( command_unit, '(a)' ) '# Usage:'
write ( command_unit, '(a)' ) '# gnuplot < '
& // trim ( command_filename )
write ( command_unit, '(a)' ) '#'
write ( command_unit, '(a)' ) 'set term png'
write ( command_unit, '(a)' ) 'set output "'
& // trim ( header ) // '.png"'
write ( command_unit, '(a)' ) 'set xlabel "X"'
write ( command_unit, '(a)' ) 'set ylabel "T"'
write ( command_unit, '(a)' ) 'set title "Brownian motion in 1D"'
write ( command_unit, '(a)' ) 'set grid'
write ( command_unit, '(a)' ) 'set style data lines'
write ( command_unit, '(a)' ) 'plot "' // trim ( data_filename )
& // '" using 1:2'
write ( command_unit, '(a)' ) 'quit'
close ( unit = command_unit )
write ( *, '(a)' ) ' BROWNIAN_MOTION commands stored in "'
& // trim ( command_filename ) // '".'
return
end
subroutine brownian_motion_simulation ( m, n, d, t, seed, x )
c*********************************************************************72
c
cc BROWNIAN_MOTION_SIMULATION simulates Brownian motion.
c
c Licensing:
c
c This code is distributed under the GNU LGPL license.
c
c Modified:
c
c 30 September 2012
c
c Author:
c
c John Burkardt
c
c Parameters:
c
c Input, integer M, the spatial dimension.
c
c Input, integer N, the number of time steps to take, plus 1.
c
c Input, double precision D, the diffusion coefficient.
c
c Input, double precision T, the total time.
c
c Input/output, integer SEED, a seed for the random
c number generator.
c
c Output, double precision X(M,N), the initial position at time 0.0, and
c the N-1 successive locations of the particle.
c
implicit none
integer m
integer n
double precision d
double precision dt
double precision dx(m)
integer i
integer j
double precision norm_dx
double precision r8_normal_01
double precision s
integer seed
double precision t
double precision x(m,n)
c
c Set the time step.
c
dt = t / dble ( n - 1 )
c
c Start at the origin.
c
do i = 1, m
x(i,1) = 0.0D+00
end do
c
c Take N - 1 steps.
c
do j = 2, n
c
c S is the stepsize.
c
s = sqrt ( d * dt ) * r8_normal_01 ( seed )
c
c Direction DX is random, unit norm.
c
if ( m .eq. 1 ) then
dx(1) = s
else
call r8vec_normal_01 ( m, seed, dx )
norm_dx = 0.0D+00
do i = 1, m
norm_dx = norm_dx + dx(i)**2
end do
norm_dx = sqrt ( norm_dx )
do i = 1, m
dx(i) = s * dx(i) / norm_dx
end do
end if
c
c Add the step to the current position.
c
do i = 1, m
x(i,j) = x(i,j-1) + dx(i)
end do
end do
return
end
subroutine get_unit ( iunit )
c*********************************************************************72
c
cc GET_UNIT returns a free FORTRAN unit number.
c
c Discussion:
c
c A "free" FORTRAN unit number is a value between 1 and 99 which
c is not currently associated with an I/O device. A free FORTRAN unit
c number is needed in order to open a file with the OPEN command.
c
c If IUNIT = 0, then no free FORTRAN unit could be found, although
c all 99 units were checked (except for units 5, 6 and 9, which
c are commonly reserved for console I/O).
c
c Otherwise, IUNIT is a value between 1 and 99, representing a
c free FORTRAN unit. Note that GET_UNIT assumes that units 5 and 6
c are special, and will never return those values.
c
c Licensing:
c
c This code is distributed under the GNU LGPL license.
c
c Modified:
c
c 02 September 2013
c
c Author:
c
c John Burkardt
c
c Parameters:
c
c Output, integer IUNIT, the free unit number.
c
implicit none
integer i
integer iunit
logical value
iunit = 0
do i = 1, 99
if ( i .ne. 5 .and. i .ne. 6 .and. i .ne. 9 ) then
inquire ( unit = i, opened = value, err = 10 )
if ( .not. value ) then
iunit = i
return
end if
end if
10 continue
end do
return
end
function i4_uniform ( a, b, seed )
c*********************************************************************72
c
cc I4_UNIFORM returns a scaled pseudorandom I4.
c
c Discussion:
c
c An I4 is an integer value.
c
c The pseudorandom number should be uniformly distributed
c between A and B.
c
c Licensing:
c
c This code is distributed under the GNU LGPL license.
c
c Modified:
c
c 12 November 2006
c
c Author:
c
c John Burkardt
c
c Reference:
c
c Paul Bratley, Bennett Fox, Linus Schrage,
c A Guide to Simulation,
c Springer Verlag, pages 201-202, 1983.
c
c Pierre L'Ecuyer,
c Random Number Generation,
c in Handbook of Simulation,
c edited by Jerry Banks,
c Wiley Interscience, page 95, 1998.
c
c Bennett Fox,
c Algorithm 647:
c Implementation and Relative Efficiency of Quasirandom
c Sequence Generators,
c ACM Transactions on Mathematical Software,
c Volume 12, Number 4, pages 362-376, 1986.
c
c Peter Lewis, Allen Goodman, James Miller
c A Pseudo-Random Number Generator for the System/360,
c IBM Systems Journal,
c Volume 8, pages 136-143, 1969.
c
c Parameters:
c
c Input, integer A, B, the limits of the interval.
c
c Input/output, integer SEED, the "seed" value, which should NOT be 0.
c On output, SEED has been updated.
c
c Output, integer I4_UNIFORM, a number between A and B.
c
implicit none
integer a
integer b
integer i4_uniform
integer k
real r
integer seed
integer value
if ( seed .eq. 0 ) then
write ( *, '(a)' ) ' '
write ( *, '(a)' ) 'I4_UNIFORM - Fatal error!'
write ( *, '(a)' ) ' Input value of SEED = 0.'
stop
end if
k = seed / 127773
seed = 16807 * ( seed - k * 127773 ) - k * 2836
if ( seed .lt. 0 ) then
seed = seed + 2147483647
end if
r = real ( seed ) * 4.656612875E-10
c
c Scale R to lie between A-0.5 and B+0.5.
c
r = ( 1.0E+00 - r ) * ( real ( min ( a, b ) ) - 0.5E+00 )
& + r * ( real ( max ( a, b ) ) + 0.5E+00 )
c
c Use rounding to convert R to an integer between A and B.
c
value = nint ( r )
value = max ( value, min ( a, b ) )
value = min ( value, max ( a, b ) )
i4_uniform = value
return
end
function r8_normal_01 ( seed )
c*********************************************************************72
c
cc R8_NORMAL_01 returns a unit pseudonormal R8.
c
c Discussion:
c
c Because this routine uses the Box Muller method, it requires pairs
c of uniform random values to generate a pair of normal random values.
c This means that on every other call, the code can use the second
c value that it calculated.
c
c However, if the user has changed the SEED value between calls,
c the routine automatically resets itself and discards the saved data.
c
c Licensing:
c
c This code is distributed under the GNU LGPL license.
c
c Modified:
c
c 08 January 2007
c
c Author:
c
c John Burkardt
c
c Parameters:
c
c Input/output, integer SEED, a seed for the random number generator.
c
c Output, double precision R8_NORMAL_01, a sample of the standard normal PDF.
c
implicit none
double precision pi
parameter ( pi = 3.141592653589793D+00 )
double precision r1
double precision r2
double precision r8_normal_01
double precision r8_uniform_01
integer seed
integer seed1
integer seed2
integer seed3
integer used
double precision v1
double precision v2
save seed1
save seed2
save seed3
save used
save v2
data seed2 / 0 /
data used / 0 /
data v2 / 0.0D+00 /
c
c If USED is odd, but the input SEED does not match
c the output SEED on the previous call, then the user has changed
c the seed. Wipe out internal memory.
c
if ( mod ( used, 2 ) .eq. 1 ) then
if ( seed .ne. seed2 ) then
used = 0
seed1 = 0
seed2 = 0
seed3 = 0
v2 = 0.0D+00
end if
end if
c
c If USED is even, generate two uniforms, create two normals,
c return the first normal and its corresponding seed.
c
if ( mod ( used, 2 ) .eq. 0 ) then
seed1 = seed
r1 = r8_uniform_01 ( seed )
if ( r1 .eq. 0.0D+00 ) then
write ( *, '(a)' ) ' '
write ( *, '(a)' ) 'R8_NORMAL_01 - Fatal error!'
write ( *, '(a)' ) ' R8_UNIFORM_01 returned a value of 0.'
stop
end if
seed2 = seed
r2 = r8_uniform_01 ( seed )
seed3 = seed
v1 = sqrt ( -2.0D+00 * log ( r1 ) ) * cos ( 2.0D+00 * pi * r2 )
v2 = sqrt ( -2.0D+00 * log ( r1 ) ) * sin ( 2.0D+00 * pi * r2 )
r8_normal_01 = v1
seed = seed2
c
c If USED is odd (and the input SEED matched the output value from
c the previous call), return the second normal and its corresponding seed.
c
else
r8_normal_01 = v2
seed = seed3
end if
used = used + 1
return
end
function r8_uniform_01 ( seed )
c*********************************************************************72
c
cc R8_UNIFORM_01 returns a unit pseudorandom R8.
c
c Discussion:
c
c This routine implements the recursion
c
c seed = 16807 * seed mod ( 2^31 - 1 )
c r8_uniform_01 = seed / ( 2^31 - 1 )
c
c The integer arithmetic never requires more than 32 bits,
c including a sign bit.
c
c If the initial seed is 12345, then the first three computations are
c
c Input Output R8_UNIFORM_01
c SEED SEED
c
c 12345 207482415 0.096616
c 207482415 1790989824 0.833995
c 1790989824 2035175616 0.947702
c
c Licensing:
c
c This code is distributed under the GNU LGPL license.
c
c Modified:
c
c 11 August 2004
c
c Author:
c
c John Burkardt
c
c Reference:
c
c Paul Bratley, Bennett Fox, Linus Schrage,
c A Guide to Simulation,
c Springer Verlag, pages 201-202, 1983.
c
c Pierre L'Ecuyer,
c Random Number Generation,
c in Handbook of Simulation,
c edited by Jerry Banks,
c Wiley Interscience, page 95, 1998.
c
c Bennett Fox,
c Algorithm 647:
c Implementation and Relative Efficiency of Quasirandom
c Sequence Generators,
c ACM Transactions on Mathematical Software,
c Volume 12, Number 4, pages 362-376, 1986.
c
c Peter Lewis, Allen Goodman, James Miller,
c A Pseudo-Random Number Generator for the System/360,
c IBM Systems Journal,
c Volume 8, pages 136-143, 1969.
c
c Parameters:
c
c Input/output, integer SEED, the "seed" value, which should NOT be 0.
c On output, SEED has been updated.
c
c Output, double precision R8_UNIFORM_01, a new pseudorandom variate,
c strictly between 0 and 1.
c
implicit none
double precision r8_uniform_01
integer k
integer seed
if ( seed .eq. 0 ) then
write ( *, '(a)' ) ' '
write ( *, '(a)' ) 'R8_UNIFORM_01 - Fatal error!'
write ( *, '(a)' ) ' Input value of SEED = 0.'
stop
end if
k = seed / 127773
seed = 16807 * ( seed - k * 127773 ) - k * 2836
if ( seed .lt. 0 ) then
seed = seed + 2147483647
end if
r8_uniform_01 = dble ( seed ) * 4.656612875D-10
return
end
subroutine r8vec_normal_01 ( n, seed, x )
c*********************************************************************72
c
cc R8VEC_NORMAL_01 returns a unit pseudonormal R8VEC.
c
c Discussion:
c
c An R8VEC is a vector of R8's.
c
c The standard normal probability distribution function (PDF) has
c mean 0 and standard deviation 1.
c
c This routine can generate a vector of values on one call. It
c has the feature that it should provide the same results
c in the same order no matter how we break up the task.
c
c The Box-Muller method is used, which is efficient, but
c generates an even number of values each time. On any call
c to this routine, an even number of new values are generated.
c Depending on the situation, one value may be left over.
c In that case, it is saved for the next call.
c
c Licensing:
c
c This code is distributed under the GNU LGPL license.
c
c Modified:
c
c 17 July 2006
c
c Author:
c
c John Burkardt
c
c Parameters:
c
c Input, integer N, the number of values desired. If N is negative,
c then the code will flush its internal memory; in particular,
c if there is a saved value to be used on the next call, it is
c instead discarded. This is useful if the user has reset the
c random number seed, for instance.
c
c Input/output, integer SEED, a seed for the random number generator.
c
c Output, double precision X(N), a sample of the standard normal PDF.
c
c Local parameters:
c
c Local, integer MADE, records the number of values that have
c been computed. On input with negative N, this value overwrites
c the return value of N, so the user can get an accounting of
c how much work has been done.
c
c Local, integer SAVED, is 0 or 1 depending on whether there is a
c single saved value left over from the previous call.
c
c Local, integer X_LO_INDEX, X_HI_INDEX, records the range of entries of
c X that we need to compute. This starts off as 1:N, but is adjusted
c if we have a saved value that can be immediately stored in X(1),
c and so on.
c
c Local, double precision Y, the value saved from the previous call, if
c SAVED is 1.
c
implicit none
integer n
integer i
integer m
integer made
double precision pi
parameter ( pi = 3.141592653589793D+00 )
double precision r(2)
double precision r8_uniform_01
integer saved
integer seed
double precision x(n)
integer x_hi_index
integer x_lo_index
double precision y
save made
save saved
save y
data made / 0 /
data saved / 0 /
data y / 0.0D+00 /
c
c I'd like to allow the user to reset the internal data.
c But this won't work properly if we have a saved value Y.
c I'm making a crock option that allows the user to signal
c explicitly that any internal memory should be flushed,
c by passing in a negative value for N.
c
if ( n .lt. 0 ) then
n = made
made = 0
saved = 0
y = 0.0D+00
return
else if ( n .eq. 0 ) then
return
end if
c
c Record the range of X we need to fill in.
c
x_lo_index = 1
x_hi_index = n
c
c Use up the old value, if we have it.
c
if ( saved .eq. 1 ) then
x(1) = y
saved = 0
x_lo_index = 2
end if
c
c Maybe we don't need any more values.
c
if ( x_hi_index - x_lo_index + 1 .eq. 0 ) then
c
c If we need just one new value, do that here to avoid null arrays.
c
else if ( x_hi_index - x_lo_index + 1 .eq. 1 ) then
r(1) = r8_uniform_01 ( seed )
if ( r(1) .eq. 0.0D+00 ) then
write ( *, '(a)' ) ' '
write ( *, '(a)' ) 'R8VEC_NORMAL_01 - Fatal errorc'
write ( *, '(a)' ) ' R8_UNIFORM_01 returned a value of 0.'
stop
end if
r(2) = r8_uniform_01 ( seed )
x(x_hi_index) =
& sqrt ( -2.0D+00 * log ( r(1) ) )
& * cos ( 2.0D+00 * pi * r(2) )
y = sqrt ( -2.0D+00 * log ( r(1) ) )
& * sin ( 2.0D+00 * pi * r(2) )
saved = 1
made = made + 2
c
c If we require an even number of values, that's easy.
c
else if ( mod ( x_hi_index - x_lo_index + 1, 2 ) .eq. 0 ) then
do i = x_lo_index, x_hi_index, 2
call r8vec_uniform_01 ( 2, seed, r )
x(i) =
& sqrt ( -2.0D+00 * log ( r(1) ) )
& * cos ( 2.0D+00 * pi * r(2) )
x(i+1) =
& sqrt ( -2.0D+00 * log ( r(1) ) )
& * sin ( 2.0D+00 * pi * r(2) )
end do
made = made + x_hi_index - x_lo_index + 1
c
c If we require an odd number of values, we generate an even number,
c and handle the last pair specially, storing one in X(N), and
c saving the other for later.
c
else
do i = x_lo_index, x_hi_index - 1, 2
call r8vec_uniform_01 ( 2, seed, r )
x(i) =
& sqrt ( -2.0D+00 * log ( r(1) ) )
& * cos ( 2.0D+00 * pi * r(2) )
x(i+1) =
& sqrt ( -2.0D+00 * log ( r(1) ) )
& * sin ( 2.0D+00 * pi * r(2) )
end do
call r8vec_uniform_01 ( 2, seed, r )
x(n) = sqrt ( -2.0D+00 * log ( r(1) ) )
& * cos ( 2.0D+00 * pi * r(1) )
y = sqrt ( -2.0D+00 * log ( r(2) ) )