Section 9.3: Difference Between Sum of Squares and Square of Sums

Adapted from: “Introduction to Programming Using Fortran 95/2003/2008” by Ed Jorgensen (March 2018 / Version 3.0.51).

Program to Calculate the Difference Between the Sum of Squares and the Square of Sums

program SOSdifference

    ! declare variables
    implicit none
    integer :: i, n, SumOfSqrs = 0, SqrOfSums = 0, difference

    ! display initial header

    !write (*,*) "Example Program"
    print *
    write (*,'(a)') "This program calculates the"
    write (*,'(a)') "difference between sum of squares "
    write (*,'(a)') "and square of sums"

    ! prompt for and read the n value
    !write (*,*) "Enter N value: "
    !read  (*,*) n
    ! Set n = 100 for Jupyter Notebook
    n = 100

    ! loop from 1 to n

    do i = 1, n
        ! compute sum of squares
        SumOfSqrs = SumOfSqrs + i ** 2

        ! compute square of sums
        SqrOfSums = SqrOfSums + i
    end do

    ! square the sums
    SqrOfSums = SqrOfSums ** 2

    ! compute the difference between sum of squares and square of sums  
    difference = SqrOfSums - SumOfSqrs

    ! display results
    print *
    write (*,'(a)') "Results..."
    print *
    write (*, '(a, i5, a, i10)') "The square of the sums from 1 to ", n, " is -> ", SqrOfSums
    print *
    write (*, '(a, i5, a, i10)') "The sum of the squares from 1 to ", n, " is -> ", SumOfSqrs
    print *
    write (*,'(a, i10)') "The difference between the square of the sums and sum of the squares is -> ", difference
    print *
end program SOSdifference

The following explanation is given by “Code Copilot” within ChatGPT 4.

The Fortran program SOSdifference is designed to calculate and display the difference between two mathematical quantities: the square of the sum of the first n natural numbers and the sum of the squares of the first n natural numbers. Here is a step-by-step explanation of how the program works:

1. Variable Declaration: The program begins by declaring the necessary variables:

   • i and n are integers used for looping and setting the upper limit, respectively.

   • SumOfSqrs and SqrOfSums are initialized to zero. They are used to store the sum of squares and the square of sums, respectively.

   • difference is declared to store the computed difference between the two calculated values.

2. User Interface Messages: The program uses write statements to print messages to the user explaining what the program does. These messages are:

   • “This program calculates the difference between sum of squares and square of sums.”

3. Input Handling: Originally, the program likely had code to prompt the user to enter the value of n (number of terms). In this specific version, the user input is commented out, and n is set to 100 directly in the code. This adjustment might be for simplification, especially in environments like Jupyter Notebooks where interactive inputs can be cumbersome.

4. Computation Loop:

   • The program uses a do loop from 1 to n. Inside this loop:

     • It calculates the sum of squares (SumOfSqrs) by adding the square of each number i to the existing sum.

     • It also calculates the sum of the numbers (SqrOfSums) by adding each number i to the cumulative sum.

5. Final Calculation:

   • After exiting the loop, it squares the total sum (SqrOfSums) to get the square of the sum of the numbers from 1 to n.

   • It calculates the difference by subtracting the sum of squares from the square of sums.

6. Result Display: Finally, the program prints the results:

   • It displays the square of the sums.

   • It displays the sum of the squares.

   • It displays the computed difference between these two values.

This program highlights a classic mathematical problem often used to demonstrate algebraic identities or computational capabilities in programming.

Program Compilation and Execution

The above program is compiled and run using Fortran Package Manager (fpm):

import os
root_dir = os.getcwd()

code_dir = root_dir + "/" + "Fortran_Code/Section_9_3_SOS_Difference"

os.chdir(code_dir)

build_status = os.system("fpm build 2>/dev/null")

exec_status = os.system("fpm run 2>/dev/null")

This program calculates the
difference between sum of squares 
and square of sums

Results...

The square of the sums from 1 to   100 is ->   25502500

The sum of the squares from 1 to   100 is ->     338350

The difference between the square of the sums and sum of the squares is ->   25164150