Order of Convergence of Newton Raphson Method

The Newton Raphson method is an iterative numerical method used to find the roots of a given equation. It is a popular method due to its fast convergence rate. The order of convergence of the Newton Raphson method determines how quickly the method converges to the root.

The order of convergence can be defined as the rate at which the error in the approximate solution decreases as the number of iterations increases. In other words, it measures how fast the method converges to the root.

Explanation:

The order of convergence of the Newton Raphson method is determined by the behavior of the error term in the method. The error term is given by the difference between the current approximation and the true root of the equation.

In general, the order of convergence of the Newton Raphson method is 2. This means that the error term decreases quadratically as the number of iterations increases. The error at each iteration is roughly squared compared to the previous iteration.

However, there are cases where the order of convergence can be different. This occurs when the derivative of the function becomes zero or when the derivative changes sign near the root. In such cases, the order of convergence can be reduced.

Example:

Let's consider an example to illustrate the order of convergence of the Newton Raphson method. Suppose we want to find the root of the equation f(x) = x^2 - 4.

1. Initialize the initial guess x0 = 3.
2. Calculate the derivative of the function f'(x) = 2x.
3. Update the approximation using the Newton Raphson formula: x1 = x0 - f(x0)/f'(x0).
4. Repeat the process until the desired accuracy is achieved.

After a few iterations, we can observe that the error term decreases quadratically. The error at each iteration is roughly the square of the previous error, indicating a second-order convergence.

Hence, the correct answer is option 'A' - 2. The order of convergence of the Newton Raphson method is 2.