Question

The order of convergence in Newton-Paphsen methed is (a) 2 (b) 3 (c) 0 (d) none.

Short answer

The order of convergence for the Newton-Raphson method is 2.

Step-by-step solution

Understanding the Newton-Raphson Method

The Newton-Raphson method is an iterative numerical method for finding approximations to the roots (or zeroes) of a real-valued function.

Formula of Newton-Raphson Method

The formula for the method is: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] In this formula, each approximation \(x_{n+1}\) is calculated using the current approximation \(x_n\), the function value \(f(x_n)\), and the derivative \(f'(x_n)\).

Analyzing Order of Convergence

The order of convergence is a concept that describes how quickly the iterations of a method converge to a solution. The Newton-Raphson method is known to have a quadratic convergence, which means it converges rapidly once sufficiently close to the solution.

Define Order of Quadratic Convergence

Quadratic convergence means that the error term \(e_n\) at step \(n\) is proportional to the square of the error term \(e_{n-1}\) from the previous step: \[ e_{n+1} \, \approx \, C \cdot e_n^2 \] This implies that the number of correct digits approximately doubles with each iteration.

Conclusion on Order of Convergence

Since the method exhibits quadratic convergence, the order of convergence is 2.

Key concepts

Iterative Numerical Methods

Iterative numerical methods are techniques used to approximate solutions for mathematical problems, often when an exact solution is difficult to obtain. These methods rely on repetition, producing a sequence of approximations that ideally become closer to the actual solution with each step.

The Newton-Raphson method is a prime example of an iterative method. It seeks to find the roots of a real-valued function, meaning the inputs which make this function equal to zero. By generating successive approximations, it incrementally narrows down to the solution. This method is especially useful in situations involving complicated functions where traditional algebraic solutions are not feasible.

With each iteration, a new approximation is calculated, refining the previous estimate using the formula:

• \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \]

Here, \( x_{n+1} \) represents the next approximation based on the current point \( x_n \), the function value at that point \( f(x_n) \), and its derivative \( f'(x_n) \).

Iterative numerical methods like Newton-Raphson are crucial in numerical analysis because they deal efficiently with complex problems offering solutions through a process of successive refinement.

Order of Convergence

The order of convergence of a numerical method tells us how fast the sequence of approximations approaches the true solution. It's a critical concept in understanding the efficiency and performance of an iterative method.

For any given method, if the error decreases significantly with each iteration, the method is said to have a higher order of convergence. The order is often represented with symbols, typically using the variable \( p \), in expressions like:

• \[ e_{n+1} = C \, e_n^p \]

where \( e_{n+1} \) is the approximation error at the next step, \( C \) is a constant, and \( e_n \) is the error at the current step.

The Newton-Raphson method, as detailed in this exercise, exhibits what is known as quadratic convergence, meaning its order is \( p = 2 \). This implies that once you're close to the true root, each iteration dramatically reduces the error, making the method highly efficient for suitably chosen initial guesses.

Quadratic Convergence

Quadratic convergence is a term used to describe the rate at which an iterative method converges when the error term decreases proportionally to the square of the previous term's error. When a method exhibits quadratic convergence, it demonstrates that the number of accurate digits in the approximation roughly doubles with each step, given a good initial guess.

In mathematical terms, a method with quadratic convergence satisfies:

• \[ e_{n+1} = C \, e_n^2 \]

This is significant in the Newton-Raphson method, as the method quickly homes in on the solution, provided the initial guess is near the actual root. The power of 2 in the formula indicates why it's called 'quadratic' convergence.

In practical terms, this rapid improvement in approximation means fewer iterations are needed, significantly reducing computation time in solving complex problems. It’s particularly advantageous in applications where high precision is required, making the Newton-Raphson method favored for its impressive efficiency among iterative methods. Understanding this concept helps in appreciating the Newton-Raphson method's capability to solve challenging numerical problems swiftly.