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Moved Gaussian RNG in generalized metropolis section
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QMC.org
@ -1297,16 +1297,9 @@ print(f"E = {E} +/- {deltaE}")
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#+END_SRC
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#+RESULTS:
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: E = -0.4773221805255284 +/- 0.0022489426510479975
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: E = -0.48363807880008725 +/- 0.002330876047368999
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*Fortran*
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#+begin_note
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When running Monte Carlo calculations, the number of steps is
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usually very large. We expect =nmax= to be possibly larger than 2
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billion, so we use 8-byte integers (=integer*8=) to represent it, as
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well as the index of the current step.
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#+end_note
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#+BEGIN_SRC f90 :exports code
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subroutine uniform_montecarlo(a,nmax,energy)
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implicit none
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@ -1703,58 +1696,6 @@ gfortran hydrogen.f90 qmc_stats.f90 qmc_metropolis.f90 -o qmc_metropolis
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: E = -0.47948142754168033 +/- 4.8410177863919307E-004
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: A = 0.50762633333333318 +/- 3.4601756760043725E-004
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** Gaussian random number generator
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To obtain Gaussian-distributed random numbers, you can apply the
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[[https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform][Box Muller transform]] to uniform random numbers:
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\begin{eqnarray*}
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z_1 &=& \sqrt{-2 \ln u_1} \cos(2 \pi u_2) \\
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z_2 &=& \sqrt{-2 \ln u_1} \sin(2 \pi u_2)
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\end{eqnarray*}
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Below is a Fortran implementation returning a Gaussian-distributed
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n-dimensional vector $\mathbf{z}$. This will be useful for the
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following sections.
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*Fortran*
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#+BEGIN_SRC f90 :tangle qmc_stats.f90
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subroutine random_gauss(z,n)
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implicit none
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integer, intent(in) :: n
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double precision, intent(out) :: z(n)
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double precision :: u(n+1)
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double precision, parameter :: two_pi = 2.d0*dacos(-1.d0)
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integer :: i
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call random_number(u)
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if (iand(n,1) == 0) then
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! n is even
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do i=1,n,2
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z(i) = dsqrt(-2.d0*dlog(u(i)))
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z(i+1) = z(i) * dsin( two_pi*u(i+1) )
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z(i) = z(i) * dcos( two_pi*u(i+1) )
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end do
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else
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! n is odd
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do i=1,n-1,2
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z(i) = dsqrt(-2.d0*dlog(u(i)))
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z(i+1) = z(i) * dsin( two_pi*u(i+1) )
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z(i) = z(i) * dcos( two_pi*u(i+1) )
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end do
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z(n) = dsqrt(-2.d0*dlog(u(n)))
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z(n) = z(n) * dcos( two_pi*u(n+1) )
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end if
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end subroutine random_gauss
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#+END_SRC
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In Python, you can use the [[https://numpy.org/doc/stable/reference/random/generated/numpy.random.normal.html][~random.normal~]] function of Numpy.
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** Generalized Metropolis algorithm
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:PROPERTIES:
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:header-args:python: :tangle vmc_metropolis.py
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@ -1782,7 +1723,7 @@ end subroutine random_gauss
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\[
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T(\mathbf{r}_{n} \rightarrow \mathbf{r}_{n+1}) = T(\mathbf{r}_{n+1} \rightarrow \mathbf{r}_{n})
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\text{constant}\,,
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= \text{constant}\,,
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\]
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so the expression of $A$ was simplified to the ratios of the squared
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@ -1844,6 +1785,58 @@ end subroutine random_gauss
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5) else, reject the move : set $\mathbf{r}_{n+1} = \mathbf{r}_n$
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6) evaluate the local energy at $\mathbf{r}_{n+1}$
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*** Gaussian random number generator
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To obtain Gaussian-distributed random numbers, you can apply the
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[[https://en.wikipedia.org/wiki/Box%E2%80%93Muller_transform][Box Muller transform]] to uniform random numbers:
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\begin{eqnarray*}
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z_1 &=& \sqrt{-2 \ln u_1} \cos(2 \pi u_2) \\
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z_2 &=& \sqrt{-2 \ln u_1} \sin(2 \pi u_2)
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\end{eqnarray*}
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Below is a Fortran implementation returning a Gaussian-distributed
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n-dimensional vector $\mathbf{z}$. This will be useful for the
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following sections.
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*Fortran*
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#+BEGIN_SRC f90 :tangle qmc_stats.f90
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subroutine random_gauss(z,n)
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implicit none
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integer, intent(in) :: n
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double precision, intent(out) :: z(n)
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double precision :: u(n+1)
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double precision, parameter :: two_pi = 2.d0*dacos(-1.d0)
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integer :: i
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call random_number(u)
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if (iand(n,1) == 0) then
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! n is even
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do i=1,n,2
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z(i) = dsqrt(-2.d0*dlog(u(i)))
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z(i+1) = z(i) * dsin( two_pi*u(i+1) )
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z(i) = z(i) * dcos( two_pi*u(i+1) )
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end do
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else
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! n is odd
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do i=1,n-1,2
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z(i) = dsqrt(-2.d0*dlog(u(i)))
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z(i+1) = z(i) * dsin( two_pi*u(i+1) )
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z(i) = z(i) * dcos( two_pi*u(i+1) )
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end do
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z(n) = dsqrt(-2.d0*dlog(u(n)))
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z(n) = z(n) * dcos( two_pi*u(n+1) )
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end if
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end subroutine random_gauss
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#+END_SRC
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In Python, you can use the [[https://numpy.org/doc/stable/reference/random/generated/numpy.random.normal.html][~random.normal~]] function of Numpy.
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*** Exercise 1
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