Bonus Coding Exercises: Array Algorithms
Here are some fun and challenging bonus exercises to illustrate some mathematical applications. In these exercises you will use loops to calculate approximations to infinite series.
Approximating \(\pi\)
Write a function to approximate \(\pi\) by using the following infinite series:
\(\pi = 4 - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \frac{4}{9} - \cdots\)
Your function should take a single integer n as
input, representing the number of terms to include in the sum:
public double approximate_pi(int n);
Note: This series converges very slowly, so you’ll need a lot of iterations to get a good estimate of \(\pi\) (e.g. ~ 1 million iterations for 5 decimal places).
Approximating \(e\)
Write a function to approximate the natural exponential base \(e = 2.718...\) by using the following infinite series:
\(e = 1 + 1 + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots\)
Your function should take a single integer n as
input, representing the number of terms to include in the sum. You
will want to (re)write a factorial function to use in the
calculation of the sum.
Math Lesson
For those of you who have studied Taylor expansions in calculus, this is where these infinite series representations for \(\pi\) and \(e\) come from.
In particular, the formula for \(e\) comes from the Maclaurin series for \(e^x\) (evaluate at \(x=1\)):
\[ e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots \]
The formula for \(\pi\) comes from the Maclaurin series for \(\tan^{-1}\):
\[ \tan^{-1}(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \\ \]
This series is known as the Madhava-Leibniz series, named after Madhava of Sangamagrama, an Indian mathematician from the 14th century, and Gottfried Leibniz, a German mathematician who is credited with developing calculus independently from Isaac Newton, and who studied the same series in the 17th century.
The Madhava-Leibniz series gives us an infinite series representation for \(\frac{\pi}{4}\) by evaluating at \(x=1\):
\[ \begin{aligned} \frac{\pi}{4} &= \tan^{-1}(1) \\ &= 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots \end{aligned} \]
Multiplying both sides by 4 gives a formula for \(\pi\).
Approximating \(\sin(x)\), \(\cos(x)\), and \(e^x\)
Write functions to approximate \(\sin(x)\), \(\cos(x)\), and \(\exp(x)\) using their Taylor expansions:
\[ \begin{aligned} \sin x &= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots \\ \cos x &= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots \\ e^x &= 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + \cdots \\ \end{aligned} \]
Your functions should take as input two variables: a double value
x, and an integer n representing the
number of terms to include in the sum:
public double approximate_sin(double x, int n);
public double approximate_cos(double x, int n);
public double approximate_exp(double x, int n); You should test your functions with known values. For example,
you should test your approximations for \(\sin\) and \(\cos\) at \(x
= 0, \frac{\pi}{6},
\frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}\), where you can
calculate approximations using the Math.sqrt()
function.
Approximating \(\sin(x)\), \(\cos(x)\), and \(e^x\) round two.
Of course, it’s an extra burden for clients of your functions to
have to include the integer n to tell the function how
many terms to include in the approximation.
It would be better to have your function automatically include as
many terms as is necessary to get a good approximation. To implement
this, you can accumulate terms as usual until the size of the term
falls below a certain threshold. For example, once the absolute
value of the term falls below 1e-6, you know that your
approximation is within 1e-6 of the actual value.
public double sin(double x);
public double cos(double x);
public double exp(double x);