bisection Algorithm
The bisection algorithm, also known as binary search or interval halving method, is a root-finding approach that narrows down the search interval by repeatedly dividing it in half. This algorithm is based on the intermediate value theorem and works on the premise that if a continuous function has values of opposite signs at two points within an interval, then it must have a root within that interval. The bisection method is especially effective for solving nonlinear equations with a single variable and is guaranteed to converge to a root if the function is continuous and the initial interval contains a root.
To implement the bisection algorithm, start by defining an initial interval [a, b] such that the function f(a) and f(b) have different signs. Then, compute the midpoint c = (a + b) / 2 and evaluate the function f(c). If f(c) is sufficiently close to zero or the specified tolerance, c is considered as the root. Otherwise, the algorithm checks the sign of f(c) and updates the interval [a, b] accordingly: if f(a) and f(c) have opposite signs, the new interval becomes [a, c], and if f(b) and f(c) have opposite signs, the interval becomes [c, b]. Repeat this process iteratively until the desired root is found or a maximum number of iterations is reached. The bisection algorithm is not the fastest root-finding method, but it is simple to implement and provides a robust and reliable means to find a root within a given interval.
"""
Given a function on floating number f(x) and two floating numbers ‘a’ and ‘b’ such that
f(a) * f(b) < 0 and f(x) is continuous in [a, b].
Here f(x) represents algebraic or transcendental equation.
Find root of function in interval [a, b] (Or find a value of x such that f(x) is 0)
https://en.wikipedia.org/wiki/Bisection_method
"""
def equation(x: float) -> float:
"""
>>> equation(5)
-15
>>> equation(0)
10
>>> equation(-5)
-15
>>> equation(0.1)
9.99
>>> equation(-0.1)
9.99
"""
return 10 - x * x
def bisection(a: float, b: float) -> float:
"""
>>> bisection(-2, 5)
3.1611328125
>>> bisection(0, 6)
3.158203125
>>> bisection(2, 3)
Traceback (most recent call last):
...
ValueError: Wrong space!
"""
# Bolzano theory in order to find if there is a root between a and b
if equation(a) * equation(b) >= 0:
raise ValueError("Wrong space!")
c = a
while (b - a) >= 0.01:
# Find middle point
c = (a + b) / 2
# Check if middle point is root
if equation(c) == 0.0:
break
# Decide the side to repeat the steps
if equation(c) * equation(a) < 0:
b = c
else:
a = c
return c
if __name__ == "__main__":
import doctest
doctest.testmod()
print(bisection(-2, 5))
print(bisection(0, 6))
