# Bisection Method

## Definition:

In mathematics, the bisection method is an easy way to obtain numerical solutions for an equation with a single unknown. For example, if you find yourself stuck for the solution of the transcendental equation, you can always resort to the bisection method for the solution. The bisection method or the Bolzano theorem is utilized to find the roots of any polynomial equation. Roots in mathematics mean the numeric value which, when substituted in the equation in place of the variable, satisfies it or makes the equation zero. Other names of this method in the mathematics world are ‘the binary search method’, ‘the interval halving method’, and ‘the dichotomy method’.

The basics of this method go back to the intermediate value theorem. This method is extremely accurate, robust, and straightforward to use. In this method, the interval or distance between the points is considered a line segment. For analyzing the root, the line segment is bisected or cut into two halves. These are known as subintervals.

Further, the roots are analyzed inside these subintervals, which makes the guessing easy. The actual solution is either on the left of the bisection or right to it. This method splits the interval repeatedly in proportion and replaces a centre point to break the root.

## Bolzano Theorem:

Bolzano theorem states that if there exists an interval (x, y) and the dot product of the functions of both the intervals is less than zero (f (x). f (y) < 0), and there exists a function f(a) which is continuous, then there prevails a value ‘z’ which belongs in between the interval (x, y) such that the function of ‘z’, f(z) = 0. Please note that the bisection method works only when the function between the intervals is continuous. The Bolzano theorem is also known as the Bisection Method formula.

By applying the bisection method formula, the z value of the f(z) function that passes through the x-axis is solved for the issues of the bisection method. In this example, the value z is the root of the function f. (z). The value of the tolerance we have chosen for the algorithm defines the value in this bisection method program where c reaches the true root.

Bisection Method Procedure

To solve any problem related to the bisection method or the problem where a continuous transcendental equation is given, follow the steps mentioned below:

• Find out two distinct points and name them x and y in a way that x is smaller than y and the value of their functions is less than 0. (f(a)* f(b) < 0, Bolzano Theorem)
• Next, determine the middle point of the line segment or interval (x, y), say ‘z.’
• z will be the root of the function if and only if f(z) = 0; if not the root, follow the steps further.
• Now we need to further divide the interval. If f(z) x f(x) <0, there exists a root or solution between z and x, else if f(z) x f (y) < 0, the root will exist between y and z.
• Carry on these steps until you get f(z) = 0.
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