Homework 2 (Due 6 pm, Friday, 05/11/2018)

Please submit your homework to your git repo by 6 pm, Friday, 05/11/2018.

Overview

In this homework, you are going to solve very popular mathematical problem called Basel Problem by writing a Fortran program.

The Basel problem asks for the summation of inverse square of all natural numbers, i.e.

(1)\[\sum^{\infty}_{n=1} = \frac{1}{n^2}\]

The solution of this problem remained unsolved for about a century, and finally Euler found the answer as \(\pi^2/6\).

(2)\[\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots = \frac{\pi^2}{6}\]

Note

If you want to see a proof of this answer in geometric view, please check a very nice video.

Direction

Make hw2 directory under homework directory in your git repository, and do the following;

  1. Make a module in a separate file, called param which contains a parameter variable named pi and store the real value of \(\pi\) by using acos(-1.0)

    This value will be used to check our calculation.

  2. Create a separate file which contains a subroutine named calc_basel which takes threshold variable as an input argument, and calculates

    1. a real solution \(\pi^2/6\), by using our param module. i.e.

      use param, only: pi
real_soln = pi**2.0/6.0

    2. a numerical solution by calculating

      (3)\[\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \dots\]

      1. Since it is an infinite summation, we need an exit condition for do loop. The proper exit condition would be

        (4)\[\text{Error} > \text{Desired threshold}\]

        So the loop will stop if the error is smaller than threshold.

        Specifically, in a code the infinite summation could be expressed as,

        error = 10000.0   ! error should be very large for initial step!
do while(error > threshold)
   numerical_soln = numerical_soln + ! something
   error = ABS(real_soln - numerical_soln) ! You need to calculate "error" every step
end do

      2. In the loop, prints error every 1000 steps. For example,

        n = 1000    error = 1.0005001666648639E-003
n = 2000    error = 5.0012502083029986E-004
n = 3000    error = 3.3338889505518665E-004
...

        The intrinsic function MOD would be very helpful. (see article-MOD)

    3. After finish the loop, prints;

      1. total number of cycles
      2. numerical solution
      3. error.

  3. In a main driver, take value of threshold variable as an user input, like

    $ ./basel
  enter the desire threshold as a real number:

    Thus, you will need to use read statement. see Fortran Input / Output

    After get a threshold value, calculate Basel problem by calling calc_basel subroutine, with threshold variable.

    call calc_basel(threshold)

  4. Use this makefile to compile your program. (You may need to edit this makefile)

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    FC = gfortran
EXE = basel
FFLAGS = -fdefault-real-8 -fdefault-double-8 -Wall -Wextra -Wconversion
OBJECTS = main.o param.o calc_basel.o

.PHONY: clean

$(EXE): $(OBJECTS)
	$(FC) $(FFLAGS) $(OBJECTS) -o $(EXE)

%.o : %.F90
	$(FC) $(FFLAGS) -c $<

main.o : param.o
calc_basel.o : param.o

clean:
	rm -f $(OBJECTS) $(EXE) *.mod

  5. Compile the code and run your program with 1.0E-7 threshold value. And save the results in results.txt file.

  6. Finally, you will have five files in your homework/hw2 directory,

    1. main.F90
    2. calc_basel.F90
    3. param.F90
    4. makefile
    5. results.txt

  7. Use the git frequently to save your work, and don’t forget to write your comments in your source codes.