Question

The order of convergence of the Newton's Raphson's method is

• A. linear
• B. quadratic
• C. biquadratic
• D. cubic

Hint:

• Newton-Raphson method generally has quadratic convergence (order 2) for simple roots, provided \(f'(root) \neq 0\).
• This means \( |x_{n+1} - \alpha| \le K |x_n - \alpha|^2 \).
• Quadratic convergence is considered fast.
• If the root has multiplicity \(m>1\), the convergence of Newton's method reduces to linear.

Answer 1

The Correct Option is B

The Newton-Raphson method is an iterative method for finding the roots of a real-valued function \(f(x)=0\). The iterative formula is: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] The order of convergence of an iterative method describes how quickly the sequence of approximations \(x_n\) converges to the true root \(\alpha\).
If \(e_n = x_n - \alpha\) is the error at the nth iteration, then for a method with order of convergence \(p\), we have \(|e_{n+1}| \approx C|e_n|^p\) for some constant \(C \neq 0\), as \(n \to \infty\). For the Newton-Raphson method, under certain conditions (such as the root being simple and \(f'(x)\) being non-zero near the root, and \(f''(x)\) being continuous), the order of convergence is \(p=2\). 
This is known as quadratic convergence. 
This means that the number of correct decimal places roughly doubles with each iteration, once the approximation is close enough to the root. 
Linear convergence: \(p=1\). Cubic convergence: \(p=3\). 
Biquadratic convergence: \(p=4\). \[ \boxed{\text{quadratic}} \]