Question

Please write a VBA program for 1b and for 2. I am lost. Thank you.

Numerically integrate the below integral doubling the number of intervals until the relative true error falls below 0.01%. Where 1. Using multiple application of the a. Trapezoidal Rule b. Simpson's 1/3 Rule 2. Using Romberg Integration with the Trapezoidal Rule 4 2N 3 3 and a table listing as coluns: number of intervas, approximate integral value, relative true error. That is, for part 1(a) and 1(b) Number of Intervals, N Integral Value Relative True Error % 16 32 0.01 And for part 2: Number of Intervals,Trapezoidal Integral Romberg Integration 9(h4) Relative True Error Value 4 <0.01

Answer 1

SOLUTION

program integration
! declaring the variable values
      real results,simp13_res,simp38_res,a,b,error
      real f
      integer n
      external f
         results=0.0
         n=0
         a=0.0
         b=1.0
         error=0.0
       write(*,*)achar(10),achar(10)
       write(*,*)"Intergration of (x^n*e^(x-1)) by dx range of 0 to 1"
10     write(*,*)achar(10)
       write(*,*)"Enter 0 to exit from the program"
       write(*,*)"Enter n : "
       read(*,'(i10)')n
!to exit      
       if(n==0)then
         goto 20
       end if
!factorial function calling
        results=factorial(n)   !calling the function factorial
        write(*,*)achar(10),"Recursion Solution = ",results   ! print the value of the function
!get simpson 1/3 solution
        call simpson13(f,a,b,simp13_res,real(n))
        write(*,*)achar(10),"Simpson 1/3 Solution= ",simp13_res
        error=(results-simp13_res)*100/results
        write(*,*)"Relative True Error in Simpson 1/3 (%) = ",abs(error)
       
        error=0.0
!get simpson 3/8 solution       
        call simpson38(f,a,b,simp38_res,real(n))
        write(*,*)achar(10),"Simpson 3/8 Solution = ",simp38_res
        error=(results-simp38_res)*100/results
        write(*,*)"Relative True Error in Simpson 3/8 (%) = ",abs(error)
!continue       
      goto 10
20    continue
      end program integration
! end of the main program, functions are below which is used in this program
!recursive algorithm
      recursive function factorial(n) result(results)
         real results,first,x
         integer n
            x=1
            first=1/exp(x)
            if(n<=0) then
              results = 0
              return
            else if(n==1) then
              results=1/exp(x)
              return
            else
              results=1-n*factorial(n-1)
              return
            end if
      end function factorial
!simpson 1/3 algorithm
      Subroutine simpson13(f,a,b,simp38_res,n)
         real a,b,x,n,h,f,simp38_res
         h=(b-a)/2.0
         simp38_res=h*(f(a,n)+f(a+h,n)+f(b,n))/3.0
         return
      end Subroutine simpson13
!simpson 3/8 algorithm
      Subroutine simpson38(f,a,b,simp13_res,n)
         real a,b,x,n,h,f,simp13_res
         h=(b-a)/3.0
         simp13_res=h*3.0*(f(a,n)+3.0*f(a+h,n)+3.0*f(a+2*h,n)+f(b,n))/8.0
         return
      end Subroutine simpson38
!function to integrate
      function f(x,n) result(res)
         real x,n,res
         res=(x**n)*exp(x-1)
         return
      end function f