Bracketing Methods

A bracketing method finds the root of a function by progressively narrowing down an interval that contains the root. It uses the intermediate value theorem, which states that if a continuous function changes signs over an interval, a root exists within that interval. Starting with such an interval, the method repeatedly reduces the interval size until it is small enough to identify the root.

For polynomials, additional techniques like Descartes’ rule of signs, Budan’s theorem, and Sturm’s theorem can determine the number of roots in an interval, ensuring all real roots are found accurately.

The bracketing method is further classified into:

• Bisection Method
• False Position (Regula Falsi) Method

Bisection Method

Bisection method is one of the simplest and most reliable root finding algorithms. It works by repeatedly narrowing down an interval that contains the root. We can use the bisection method using following methods:

Step 1: Start with two points, a and b, such that f(a) and f(b) have opposite signs. This guarantees that there is at least one root between a and b.

Step 2: Calculate the midpoint, c, of the interval [a,b] using c = (a + b)/2.

Step 3: Determine the sign of f(c). If f(c) is close enough to zero (within a predefined tolerance), c is the root. Otherwise, replace a or b with c depending on the sign of f(c), ensuring that the new interval still brackets the root.

Step 4: Repeat the process until the interval is sufficiently small or f(c) is close enough to zero.

Here, number of iterations needed to achieve an ε-approximate root using the bisection method is given by:

[Tex]\bold{N \approx \log_2 \left( \frac{b – a}{\varepsilon} \right)}[/Tex]

False Position (Regula Falsi) Method

False Position method, also known as the Regula Falsi method, is a numerical technique used to find the roots of a function, where the function equals zero. It is similar to the bisection method but often converges faster. The False Position method combines the concepts of the bisection method and the secant method, making it both simple and efficient for solving equations.

Here’s a step-by-step explanation of how it works:

Step 1: Start with two points, a and b, such that f(a) and f(b) have opposite signs. This guarantees that there is at least one root between a and b.

Step 2: Calculate the midpoint, c, of the interval [a,b] using c = a – [f(a).(b – a)]/[f(b) – f(a)].

Step 3: Evaluate f(c). If f(c) is close enough to zero (within a predefined tolerance), then c is the root.

Step 4: Depending on the sign of f(c), update the interval:

• If f(a) and f(c) have opposite signs, set b = c.
• If f(b) and f(c) have opposite signs, set a = c.

Step 5: Repeat the process until the interval is sufficiently small or f(c) is close enough to zero.

Root-finding algorithms are tools used in mathematics and computer science to locate the solutions, or “roots,” of equations. These algorithms help us find solutions to equations where the function equals zero. For example, if we have an equation like f(x) = 0, a root-finding algorithm will help us determine the value of x that makes this equation true.

In this article, we will explore different types of root finding algorithms, such as the bisection method, Regula-Falsi method, Newton-Raphson method, and secant method. We’ll explain how each algorithm works, and how to choose the appropriate algorithm according to the use case.