# Newton Raphson method and Fixed Point Iteration method ?

• atomic_light11
In summary, the fixed point iteration method and Newton raphson method work by starting with an initial value and using the slope of the function at that point to find a new point where the function would equal zero if it were a linear function. This process is repeated until a point is found that is closer to the zero of the function.
atomic_light11
Hi everyone, I has been learning numerical method recently, i am very wonder how fixed point iteration method and Newton raphson method works (a more insight explanation rather than mathematical proof ) thanks!

Intuitively it works as follows:

You start with x0, and you know what f(x0) and f'(x0) is. You also know that the graph of f(x) is approximately a line passing through (x0,f(x0) with slope f'(x0). The first step is to find x1 which is the place where f(x) would equal zero IF f(x) was actually a linear function.

Most likely f(x1) is not equal to zero, but since f(x) kind of looked like the line whose zero you calculated, you expect that x1 is closer to the zero of f(x) than x0 is. Then you just repeat the whole process with x1 in place of x0 to find x2 which is even closer, etc.

## What is the difference between Newton-Raphson method and Fixed Point Iteration method?

The main difference between the Newton-Raphson method and the Fixed Point Iteration method is the way in which they approach finding the root of a function. The Newton-Raphson method uses the derivative of the function to approximate the root, while the Fixed Point Iteration method uses an iterative process to find the root. Additionally, the Newton-Raphson method typically converges faster than the Fixed Point Iteration method.

## What are the advantages of using the Newton-Raphson method?

The Newton-Raphson method has several advantages over other root-finding methods. It is a fast and efficient algorithm, converging quickly to the root of a function. It also has a high accuracy, meaning it can find roots with a high degree of precision. Additionally, the method can be applied to a wide range of functions, making it a versatile tool for solving mathematical problems.

## What types of problems can be solved using the Fixed Point Iteration method?

The Fixed Point Iteration method is commonly used to find the roots of nonlinear equations that cannot be solved algebraically. It can also be used to solve optimization problems, such as finding the minimum or maximum of a function. This method is particularly useful when the function is continuous and has a convergent sequence of values.

## What are some potential drawbacks of using the Newton-Raphson method?

One potential drawback of the Newton-Raphson method is that it requires knowledge of the derivative of the function. This can be a limitation in cases where the derivative is difficult or impossible to compute. Additionally, the method may fail to converge if the initial guess is far from the actual root or if the function has multiple roots in the same vicinity.

## How can one determine which method is more appropriate for a given problem?

The choice between the Newton-Raphson method and the Fixed Point Iteration method depends on the problem at hand. If the function is differentiable and the derivative can be easily computed, the Newton-Raphson method is generally a good choice. However, if the function is not differentiable or the derivative is difficult to compute, the Fixed Point Iteration method may be a better option. It is also important to consider the initial guess and potential for convergence when choosing a method.

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