Homework 1

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:

1. Finding the center point: the method finds the center of the given interval $$ c = \frac{a + b}{2}. $$
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

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

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.

abstract type BracketingMethod end
 
struct Bisection <: BracketingMethod end
struct RegulaFalsi <: BracketingMethod end

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 DomainError with with a meaningful error message.