Write a Fortran program to use Simpson\'s rule to evaluate the integral given by

Question

Write a Fortran program to use Simpson's rule to evaluate the integral given by

where , , and Indicate in the comment block (at the beginning of your code) what value you obtained for the integral as a function of the number of subintervals used. Make sure to clearly indicate what you think is the most accurate value for the integral.

Submit the source code file, i.e. the .f08 file containing the Fortran code, by uploading it into blackboard using the “attachments” button under the assignment. If you have any problems see the TA or the instructor. Submissions via email will not be accepted.

Note:

All programs must contain the implicit none statement. Programs that do not contain this statement will receive an automatic zero.

All programs should have a block of comment statements at the beginning of the code containing your name and a description of what the code does.

All programs must compile using the gfortran compiler on the Mathlab machines. Programs that do not compile will receive an automatic zero.

Programs must be uploaded into the blackboard assignment page. Programs may not be submitted via email or hardcopy. Programs that are not uploaded into the blackboard assignment page will not be graded.

Explanation / Answer

Simpsons solition

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

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