The Wallis Formula can be used to approximate the value of the constant 𝛑. The formula is like so
Some more background is available at https://encyclopediaofmath.org/wiki/Wallis_formula
The estimates of PI increase as the number of terms in the equation
increase. When doing this using a computer, you will have a loop that
will run n times. The larger the value of n, the more terms there
will be and higher the accuracy of the estimate.
There is a file provided called wallis.c which has a prototype of
a function wallis_pi. It will take the number of iterations and
return the estimate of pi using the wallis formula.
Implement this function using the above formula.
Imagine a circle of radius 1. The area will be 𝛑11 which is 𝛑. Now imagine a square that circumscribes this circle. The edge of the square will be of length 2. Hence, the area of the square will be 4. This is illustrated in the diagram below.
The ratio of the areas of the two shapes will be
area of circle 𝛑
-------------- = -----
area of square 4
Hence the value of 𝛑 will be 4 times the ratio of the areas of the two figures.
4 x area of circle
𝛑 = ------------------
area of square
The ratio of the areas can be estimated using a monte carlo simulation.
Imagine throwing a dart randomly onto the square. It will either be inside or outside the circle. If we throw lots of darts, the ratio of the darts inside the circle to the total number of darts will start to approximate the ratio of the areas of the two figures.
We can use this to get the ratio and then multiply that by 4 to get the value of pi.
Here is a video of the process to understand how it works
There is a function prototype called mc_pi in the file
monte_carlo.c which should accept the number of iterations (the
number of darts thrown) as input and return the estimate of 𝛑.
You can use the provided frandom function to generate and return a
random number (to simulate dart throws). It will return numbers
between 0 and 1 (i.e. all in the first quadrant of the square).
Use the frandom twice per iteration to function to find a random
point (x,y) and then check whether it lies inside the circle by
calculating the distance between the point and (0,0).
You can run make test_mc and make test_wallis to check if your
implementations are correct before you push the code.