====== Homework 1 ======

==== Bisection method and regula falsi ====

Suppose we have a real function of one variable $f: \mathbb{R} \rightarrow \mathbb{R}$ continuous on the interval $[a, b] \in \mathbb{R}$, for which the function values ​​in the extreme points of this interval have the opposite sign. The bisection method is an iterative algorithm for finding the root of such a function on a specified interval. In each iteration, the bisection method performs the following two steps:

  - **Finding the center point:** the method finds the center of the given interval $$ c = \frac{a + b}{2}. $$
  - **Update of the search interval:** if the function values ​​$f(a)$ and $f(c)$ have the same sign, then the new interval for the root search is $[c, b]$. Otherwise, the new interval is $[a, c]$.

The above two steps are repeated until the desired accuracy is reached or the maximum number of iterations is reached. As a stopping criterion, we can use the functional value at point $c$, i.e. the algorithm will terminate if $|f(c)| < \varepsilon$ for the specified tolerance $\varepsilon \in \mathbb{R}$.

The regula falsi method differs from the bisection method only in how it selects the $c$ point. The following formula is used for this method

$$
c = \frac{a\cdot f(b) - b\cdot f(a)}{f(b) - f(a)}.
$$

==== Input ====

Implement a ''findroot'' function that finds the root of a given function on a given interval. The ''findroot'' function must have the following input arguments (in the order listed):

  * ''method'': the method that will be used to find the root,
  * ''f'': function of one variable whose root we want to find,
  * ''a'': lower limit of the interval,
  * ''b'': upper limit of the interval.

Choose the appropriate types for all input parameters of the ''findroot'' function. Use the following type hierarchy to differentiate the root search method.

<code julia>
abstract type BracketingMethod end

struct Bisection <: BracketingMethod end
struct RegulaFalsi <: BracketingMethod end
</code>

Additionally, the ''findroot'' function must accept the following keyword arguments (values ​​after ''='' are values):

  * ''atol = 1e-8'': algorithm tolerance,
  * ''maxiter = 1000'': maximum number of iterations.

When implementing, note that the bisection method and the regula falsi method differ only in the selection of a new point. Write a general ''findroot'' function for both methods. Use multiple-dispatch and write a ''midpoint'' function that will return a new point based on the method used. This function must have the following input arguments (in the order listed):

  * ''method'': the method that will be used to find the root,
  * ''f'': function of one variable whose root we want to find,
  * ''a'': lower limit of the interval,
  * ''b'': upper limit of the interval.

The ''findroot'' function must also meet the following properties:

  * The function must check that ''a < b'' holds, and if not, it must swap variables to satisfy this inequality.
  * If the function is given a root to look for, the function must return this root without any further calculation.
  * If the function values ​​at the endpoints of the specified interval have the same sign, the function must return [[https://docs.julialang.org/en/v1/base/base/#Core.DomainError|DomainError]] with with a meaningful error message.