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Lab 9: Accumulator

As usual, create a directory to hold today's files. All programs that you write today should be stored in this directory. 

$ cd ~/cs120/labs
$ mkdir lab9
$ cd lab9

Factorial

Some mathematical expressions are bit harder to evaluate than others. You could probably add numbers all day, but comuting square roots is much harder. Factorials are another good example of this; Factorials require an iterated evaluation of mathematical expressions. While there's no single operator to compute factorials in Python, you can write a function that can compute the value of factorials.

Details

Create the function compute_factorial(operand) in a file called factorial.py. This function takes an integer as a parameter, which is the number the user wants to compute the factorial of. It should use this parameter and return the factorial of the input.

Recall that a factorial can be computed in the following manner:

$$ n! = n \times (n - 1) \times (n - 2) \times \ldots \times 1$$

Make sure to test your function by calling the function multiple times with different parameters. Make sure your code follows the course's code conventions.

Test Cases

Function Parameters Expected Output
0 1
1 1
2 2
5 120
10 3628800

Hint

  • You will need a for loop that uses the accumulator pattern to accomplish this goal. Create a variable (defaulted value to 1, the multiplicitive identity) before the for loop. Then you just need to update the value of that variable inside the for loop.
  • Your accumulator is going to store the product of the first i terms in the factorial function definition. You simply need to multiply the accumulator by the current term in this for loop.

Challenge

Another equation that got a lot of publicity recently is the Ramanujuan Summation. It states that the infinite sum of the positive integers is the value: $$-\frac{1}{12}$$ Depending on how you define your operations this summation is correct. This result is a big deal in String Theory. Does Python follow these rules?

Write a function that uses a for loop to sum the integers in the range [0, n). Try this function for arbitrarily large values. Do you ever get to the specified value?

Leibniz Approximation

The mathematical constant π is an irrational number that represents the ratio between a circle's diameter and its circumference. The first algorithm designed to approximate π was developed by Archimedes in 250 BC, and was able to approximate its value within 3 significant figures. Today we know that there are an infinite number of digits in π, so lets use a little more sophisticated approach to estimate its value.

Details

Create the function estimate_pi(duration) in a file called leibniz.py. This function takes an integer as a parameter, which is the number of terms to compute in the infinite series. It should use this parameter to compute the Leibniz approximation, discussed below.

The Leibniz approximation relies on the fact that:

$$ \frac{\pi}{4} = \frac{1}{1} - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} - \ldots $$

Which can be rewritten to approximate the true value of π:

$$ \pi = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} - \ldots $$

Make sure to test your function by calling the function multiple times with different parameters. Make sure your code follows the course's code conventions.

Test Cases

Function Parameters Expected Output
0 0
1 4
10 3.0418396189294032
100 3.1315929035585537
1000 3.140592653839794

Hint

  • You will need a for loop that uses the accumulator pattern to accomplish this goal. Create a variable (defaulted value to 0) before the for loop. Then you just need to update the value of the variable inside the for loop.
  • As you progress through your for loop, the denominator of your fraction increases by a factor of two. Some simple algebra will reveal that during loop iteration i, the denominator is: $$(2 \times i) + 1$$
  • You need to alternate adding and subtracting. However, subtracting is the same as adding a negative number. So we can accomplish this by multiplying every other fraction by -1. The easy way to do this is to compute: $$-1^{i}$$ Where \(i\) is your loop variable.
  • The accumulator be the sum of all of the terms in the sequence, so you only need to add one term to the accumulator during each loop iteration.

Challenge

The value you get from the Leibniz formula will be incredibly close. However, it will never meet the true definition of π. To see how close it does get, compute the approximations for all equations from length 1 to length 1000. For each approximation, subtract the value of math.pi from it to compute the error. Sum up all of these errors, and print the average error.