The rate of convergence of Newton Raphson method is :

Explanation:

Rate of Convergence of Newton-Raphson Method

Definition: The Newton-Raphson method is a widely used iterative numerical technique for finding approximations to the roots (or zeros) of a real-valued function. It is particularly renowned for its fast convergence properties when compared to other root-finding algorithms.

Working Principle: Given a function f(x) and its derivative f'(x), the Newton-Raphson method uses the formula:

x_n+1 = x_n - f(x_n)/f'(x_n)

to iteratively converge to the root of the function. Starting from an initial guess x₀, the method generates a sequence {x_n} that ideally converges to a solution x such that f(x) = 0.

Rate of Convergence: The rate of convergence refers to how quickly the sequence {x_n} approaches the root. For the Newton-Raphson method, the rate of convergence is quadratic. This means that the error in the approximation decreases proportionally to the square of the error in the previous step. Mathematically, if e_n represents the error at the n-th iteration, then:

e_n+1 ≈ C * (e_n)²

where C is a constant. This quadratic convergence implies that the number of correct digits approximately doubles with each iteration, leading to very rapid convergence to the root.

Correct Option Analysis:

The correct option is:

Option 2: 2

This option correctly identifies the rate of convergence of the Newton-Raphson method as quadratic (order 2). The method's efficiency in rapidly approaching the root makes it a popular choice for solving non-linear equations