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IFORT Coding Problem (Simpson's Rule)

  1. May 11, 2009 #1
    1. The problem statement, all variables and given/known data

    For a function f(x) over the interval [a,b], simpson's rule approximates the defenite integral:


    [tex]\int[/tex]f(x) dx = h/3*[f0 + 4(f1 + f3+ ...+ f2n-1) + 2(f2 + f4 + ... + f2n-2) + f2n]

    where h = (b-a)/n
    fi is f(a + i*h)

    Given an error epsilon, I need to write an executable program using IFORT that will find this integral using the simpson's method for f(x) = exp(-x)*sin(x) on [0,10] which converges if
    new integral - old integral < new integral*eps
    obvioulsy a loop will be in this program. What you have to do is cut the h in half every time to get a better approximation. And each time you have to change the odd fi to the even fi, and vice versa.


    2. Relevant equations

    see the above equation

    3. The attempt at a solution

    Here is my program.

    PROGRAM Simpson_Rule
    IMPLICIT NONE
    REAL(8) :: eps, a, b, integral, f
    EXTERNAL f
    eps = 1.0d-10
    a = 0.0d0
    b = 10.0d0
    CALL Simpson_Integrate(f, a, b, eps, integral)
    PRINT *, integral
    STOP
    END PROGRAM Simpson_Rule

    SUBROUTINE Simpson_Integrate(f, a, b, eps, integral)
    IMPLICIT NONE
    REAL(8) :: a, b, integral, f, eps, h, x, sum, sumeven, sumodd,
    & integral_old, end_points
    INTEGER :: i, n_sum_add, ne
    h = (b-a)/2.0d0
    sum = f(a) + f(b)
    end_points = sum/3.0d0
    x = (a+b)/2.0d0
    sumodd = f(x)
    integral_old = h*end_points + 4.0d0*sumodd*h/3.0d0
    n_sum_add = 2
    do ne = 1, 40
    sumeven = sumeven + sumodd
    sumodd = 0.0d0
    x = a + h/2.0d0
    do i = 1, n_sum_add
    sumodd = f(x)
    x = x+h
    end do
    h = h/2.0d0
    integral = h*end_points + (4.0d0*sumodd + 2.0d0*sumeven)*h/3.0d0
    print*, 2**ne, integral
    if (abs(integral-integral_old) < abs(integral)*eps) return
    integral_old = integral
    n_sum_add = n_sum_add*2
    end do
    print*, "Integration has not converged"
    return
    END SUBROUTINE Simpson_Integrate

    real(8) function f(x)
    IMPLICIT NONE
    REAL(8) :: x
    f = exp(-x)*sin(x)
    return
    end function f

    When I execute, i don't get sensible results. Any help is appreciated
     
  2. jcsd
  3. May 11, 2009 #2

    minger

    User Avatar
    Science Advisor

    The first thing I noticed is that fortran might not like having a function named the same as a variable.

    I was confused when you wrote f(a) + f(b); I originally thought that f was an array. Try changing the name of that function.
     
  4. May 11, 2009 #3
    I don't believe that is the problem. I've done this same type problem with Trapezoid rule and it works just fine with the same function defined as I have done in the program shown above.
     
  5. May 11, 2009 #4
    Ok I actually solved this problem on my own. In the loop I need:

    sumodd = f(x) + sumodd

    and then all is swell.

    :D
     
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