Root Finding Algorithm (original) (raw)

Last Updated : 23 Jul, 2025

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.

Different types of root finding algorithms are bisection method, Regula-Falsi method, Newton-Raphson method, and secant method. These algorithms are essential in various fields of science and engineering because they help solve equations that cannot be easily rearranged or solved analytically.

Root-Finding-Algorithm

Table of Content

Types of Root Finding Algorithms

Root-finding algorithms can be broadly categorized into Bracketing Methods and Open Methods.

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

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:

\bold{N \approx \log_2 \left( \frac{b - a}{\varepsilon} \right)}

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:

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

Read more about Regula Falsi method.

Open Methods

Open methods are root-finding algorithms that don't necessarily require an interval containing the root. They start with one or more initial guesses and iteratively refine them until a root is found. These methods are generally faster but may not always converge.

In this section we will further learn about the classification of open method, that are:

Newton-Raphson Method

Newton-Raphson method is an iterative algorithm that uses the derivative of the function to find the root. It’s faster than the bisection method but requires a good initial guess and the calculation of derivatives. Procedure is given as below:

**Step 1: Start with an initial guess x0.

**Step 2: Use the formula, x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} to find the next approximation, where f'(xn) is the derivative of f(x) at xn.

**Step 3: Repeat the iteration until the change between xn and xn+1​ is smaller than a predefined tolerance.

**Note: Newton-Raphson method converges quickly when the initial guess is close to the root, but it can fail if f′(x) is zero or if the function is not well-behaved near the root.

Secant Method

Secant method is similar to the Newton-Raphson method but does not require the calculation of derivatives. Instead, it uses a secant line to approximate the root. Procedure of secant method is given as:

**Step 1: Start with two initial guesses __x_0​ and x1​.

**Step 2: Use the formula x_{n+1} = x_n - f(x_n) \frac{x_n - x_{n-1}}{f(x_n) - f(x_{n-1})} to find the next approximation.

**Step 3: Repeat the iteration until the change between xn and xn+1​ is smaller than a predefined tolerance.

Secant method can be faster than the bisection method and does not require the derivative of the function, but it can be less reliable than the Newton-Raphson method, especially if the initial points are not well chosen.

Comparison of Root Finding Methods

The comparison between the root finding methods are being showed below, on the basis of advantages and disadvantages.

Method Description Advantage Disadvantage
Bisection Method It divides interval in half, and guarantees convergence Simple and faster method Slow convergence
False Position Method It uses linear interpolation, faster than bisection It maintains bracketing, faster than bisection It may fail due to roundoff errors
Newton's Method It uses function and derivative, fast convergence It is a quadratic convergence, works in higher dimensions It may not converge if initial guess is far
Secant Method It is a derivative-free variant of Newton's, simpler It doesn't require derivative, faster than bisection Slower convergence (order ~1.6)

Comparison of Root Finding Methods with Example

Solving the equation f(x)= x3−4x−9= 0 using Bisection Method, Regula-Falsi Method, Newton-Raphson Method, and Secant Method with 10 iterations. The computed root approximations are displayed in the table.

Method Root Approximation
Bisection Method 2.7060546875
Regula-Falsi Method 2.7065276119801087
Newton-Raphson Method 2.7065279765747587
Secant Method 2.7065278974619447

Comparison of Methods:

Observations:

This comparison highlights that if derivatives are available, Newton-Raphson is preferred. If derivatives are difficult to compute, Secant Method is a good alternative.

How to Choose a Root Finding Algorithm?

Choosing a root finding algorithm depends on several factors:

Applications of Root Finding Algorithms

The various applications of root-finding algorithms are:

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