!   From: "COMPUTATIONAL PHYSICS, 2nd Ed" 
!   by RH Landau, MJ Paez, and CC Bordeianu 
!   Copyright Wiley - VCH, 2007.
!   Electronic Materials copyright: R Landau, Oregon State Univ, 2007;
!   MJ Paez, Univ Antioquia, 2007; and CC Bordeianu, Univ Bucharest, 2007.
!   Supported by the US National Science Foundation
!   
! integrate.f90: Integrate exp(-x) using trap, Simp and Gauss rules 
!                Need to add in Gauss.f95

Program integrate

  Implicit none
  Real*8 :: trapez, simpson, quad, r1, r2, r3         ! declarations
  Real*8 :: theo, vmin, vmax
  Integer :: i
	
  theo  =  0.632120558829    ! theoretical result, integration range
  vmin  =  0.
  vmax  =  1.
  open(6, File  =  'integ.dat', Status  =  'Unknown')
         ! calculate integral using both methods for steps  =  3..501
  Do  i  =  3, 501 , 2
    r1  =  trapez(i, vmin, vmax)
    r1  =  abs(r1 - theo)
    r2  =  simpson(i, vmin, vmax)
    r2  =  abs(r2 - theo)
    r3  =  quad(i, vmin, vmax)
    r3  =  abs(r3 - theo)
    write(6, *) i, r1, r2, r3      
  End Do
  close(6)
  Stop 'data saved in integ.dat'
End Program integrate
                                      ! Function we want to integrate
Function f(x)                               
  Implicit none
  Real*8 :: f, x
  f  =  exp( - x)
  Return
End

Function trapez(i, min, max)! trapezoid rule
  Implicit none
  Integer :: i, n        
  Real*8 :: f, interval, min, max, trapez, x
  trapez  =  0     
  interval  =  ((max - min) / (i- 1))
  Do  n  =  2, (i- 1)                                ! sum midpoints
    x  =  interval * (n - 1)
    trapez  =  trapez + f(x)*interval
  End Do 
  trapez  =  trapez + 0.5*(f(min) + f(max))*interval ! add Endpoints
  Return
End
                                                      ! Simpson rule
Function simpson(i, min, max)                                
  Implicit none
  Integer :: i, n        
  Real*8 :: f, interval, min, max, simpson, x
  simpson  =  0    
  interval  =  ((max - min) / (i- 1))
  Do  n  =  2, (i- 1), 2                       ! loop for odd points
    x  =  interval * (n - 1)
    simpson  =  simpson + 4*f(x)
  End Do 
  Do  n  =  3, (i- 1), 2                      ! loop for even points
     x  =  interval * (n - 1)
     simpson  =  simpson + 2*f(x)
  End Do  
  simpson  =  simpson + f(min) + f(max)          ! add the Endpoints
  simpson  =  simpson*interval/3
  Return
End

Function quad(i, min, max)                       ! uses Gauss points
  Implicit none
  Real*8 :: w(1000), x(1000)
  Real*8 :: f, min, max, quad
  Integer :: i, job, n
  quad  =  0
  job  =  0
  call gauss(i, job, min, max, x, w)
  Do  n  =  1, i
    quad  =  quad + f(x(n))*w(n)
  End Do
  Return 
End