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
• Secant Method

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, [Tex]x_{n+1} = x_n – \frac{f(x_n)}{f'(x_n)}[/Tex] to find the next approximation, where f'(x_n) is the derivative of f(x) at x_n.

Step 3: Repeat the iteration until the change between x_n and x_n+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 x₀​ and x₁​.

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

Step 3: Repeat the iteration until the change between x_n and x_n+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.

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.