Gnuplot: save plot in jpeg

This are some useful commands:

set terminal jpeg

set output "fig.jpeg"

replot

set terminal x11

Log scale in gnuplot plots

The corresponding command is:

set log xy

Very important commands in Gnuplot

Plot two or more plots in the same window:

set multiplot

Plots with lines:

plot "file.dat" w l

Plot with yerrorbars. The third column of the input data is the lenght of the errorbar:

plot "file.dat" with errorbars

Hide the legend:

unset key

Fortran program to calculate an integral using "crude" Monte Carlo

The following program calculates the integral I

using "crude" Monte Carlo. In this case, n=3, x_ia=0, x_ib=1 and f(x)=exp(a·x).

        Program intmc
        
        double precision fsum,fsum2,a,xa,xb,V,x,f,I,IA,varf
        integer M,n,j,k
        dimension a(3),xa(3),xb(3),x(3)
c n: dimension of the array x
c M: number of random numbers we use
c V: volum of the hipervolume
        n=3
        M=1000000
        V=1 
        do j=1,n
            a(j)=rand()
            xa(j)=0
            xb(j)=1
            V=V*(xb(j)-xa(j))
        enddo
c let's find the value of the integral that we get using monte-carlo
        fsum=0
c fsum2 will help us to find the error
        fsum2=0
        do k=1,M
            do j=1,n
                x(j)=rand()
            enddo
            call func(a,x,f,n)
            fsum=fsum+f
            fsum2=fsum2+f**2
        enddo
c I is the value of the integral we wanted to find
c IA is the value of the integral calculated analiticaly
        I=V*fsum/M
        varf=(fsum2/M-((fsum/M)**2))
        sigma=sqrt(varf/M)
        IA=1
        do j=1,n
            IA=IA*((exp(a(j))-1)/a(j))
        enddo
        write(*,*) I,sigma*3,IA
        stop
        end


        SUBROUTINE func(a,x,f,n)
            double precision a,x,f,ax
            integer n
            dimension a(n),x(n)
            ax=0
            do j=1,n
                ax=ax+a(j)*x(j)
            enddo
            f=exp(ax)
        end

Fortran program to calculate an integral using the method of trapezes

The following program calculates the integral I

using the trapezes method. In this case, n=3, x_ia=0, x_ib=1 and f(x)=exp(a·x).

  Program inttr
  double precision isum,x,xa,xb,h,f,I,a,IA,error,e
  integer j,p,k1,k2,k3,n,l
  dimension x(3),xa(3),xb(3),h(3),a(3)
  n=3
  p=300
  do j=1,n
      a(j)=rand()
      xa(j)=0
      xb(j)=1
      h(j)=(xb(j)-xa(j))/p
  enddo
  isum=0
  error=0
c we have to use n do's one inside the other. In this case, n=3.
c to calculate the error we need to compute the second partial
c derivate of F
  do k1=1,p
      x(1)=xa(1)+(k1-1/2)*h(1)
      do k2=1,p                      
          x(2)=xa(2)+(k2-1/2)*h(2)
          do k3=1,p
              x(3)=xa(3)+(k3-1/2)*h(3)
              call func(a,x,f,n)
              isum=isum+f
              do l=1,n
                  error=error+f*(a(l)**2)
              enddo
          enddo
      enddo
  enddo
  I=isum*(h(1)*h(2)*h(3))
  e=(p**(-2/n))*error/(24*isum)
  IA=1
  do j=1,n
      IA=IA*((exp(a(j))-1)/a(j))
  enddo
  write(*,*) I,e,IA        
  stop
  end



  SUBROUTINE func(a,x,f,n)
      double precision a,x,f,ax
      integer n,j
      dimension a(n),x(n)
      ax=0
      do j=1,n
          ax=ax+a(j)*x(j)
      enddo
      f=exp(ax)
  end

Metropolis Algorithm

A very interesting link where we can understand easily the Metropolis Algorithm:

http://www.childrens-mercy.org/stats/weblog2007/MetropolisAlgorithm.asp

Calculate PI with a simple fortran program

In this link (Fer problemes), you can find a montecarlo problem that allows us to find the number PI.

Here, you have one code that could be implemented to simulate this program. It is done in fortran.

PROGRAM calculpi

integer b,d,R
real acirc, aquad
double precision div
b=12345
d=76545
R=1
acirc=0
aquad=0
do i=1,1000000
    a=rand(b)
    c=rand(d)
    b=a
    d=c
    if ((a**2+c**2)
        acirc=acirc+1
    endif
    if ((a
        aquad=aquad+1
    endif
    div=(acirc/aquad)*4

    if (mod(i,10)==0)then
        write (*,*) acirc, aquad, div
    endif
enddo
STOP
END