Newton-Raphson Method Formula and Explanation Guide

The Newton-Raphson method is also referred to as the Newton’s method. It is a way to quickly find a good approximation for the root of the real-valued function. It is based on the idea that a continuous and differential function can be approximated by a straight line that is tangent to it. It has been observed in many cases there is an acute problem in finding the root of a complicated function algebraically. In these cases, the Newton Raphson method comes to the fore. With the usage of some basic concepts of calculus, one can find ways to evaluate the roots in case of complicated functions. It is an iterative process which usually follows a set of guidelines to calculate approximate one root. However, You have to consider the function as well as its derivative to an initial value of the unknown quantity x.

What is the Formula?

Unlike the earlier methods that are being used to calculate the root of a particular problem, this method requires only one appropriate starting point as an initial assumption of the root of the function. On the other hand, the point of intersection which can be termed as x₁ of the tangent with the y-axis (y=0) is taken to be the next approximation to the root, f(x)=0. So, while substituting y=0b in the tangent, we get an equation

As discussed earlier, the Newton Raphson method is a powerful technique for solving equations in a numerical manner. It is based on the simple idea of linear approximation. The Newton method, when properly used usually homes in on a root with a devastating efficiency.

Let’s take f(x) to be a well-behaved function and let r be a root of the equation. We start with an estimate x0 of r. From x_1, we produce a new estimate x_2. From x₂, we produce a new estimate in the form of x_3.

We carry on until we are close to r until it becomes clear that we are getting nowhere. This style indicates that the general style of proceeding is also known as iterative.

It is one of the most widely used methods to calculate roots of the function. The style of proceeding is also known as the iterative process. It is one of the quintessential attributes of a Newton Raphson method. It is one of the simplest and wider used means to calculate the roots of a particular equation.

Explanation Guide of the Newton-Raphson Method:

Let x be the good estimate of r and let’s assume that r equals x+h. Since, the true root is r, and h= r-x, the number h measures how far the equation estimate is from the truth. As h is small, then we can write the equation as 0 = f(r) = f(x0 + h) ≈ f(x0) + hf0 (x0)

Hence, unless f(x) =0, this linear equation cannot be solved. This is the reason that we have to take h ≈ − f(x0) divided by f0 (x0). The next estimate is also derived from x1 in the same manner as x1 was obtained from x0. Hence, the final equation is in the form of:

xn+1 = xn − f(xn) f0 (xn)

Some Advantages of the Newton Raphson Method:

There are various advantages that are associated with the Newton Raphson method. Here is the list of some advantages of the Newton Raphson method.

• The first advantage of this equation lies in the fact that apart from the first convergence, it also merges the quadratic feature of the root. This depicts that it can also work in case of the complexity of the equation is quite high.
• Another benefit of this method lies in the fact that it is one of the fastest convergences to the root. This feature makes the Newton Raphson method stand out in the crowd as one of the better methods to determine the root of the problem
• Another advantage of this method lies in the fact that it can basically polish a root from the other convergence techniques
• The Newton Raphson method is also popular due to the fact that the number of significant digits doubles with each step
• Another factor that makes this method popular is that it is quite flexible. It usually implies that it is easier to alter the method to various dimensions
• This method can also be used as a polishing medium for the roots of the quadratic equations. The polishing of root is quite important and is highly useful as it has advantageous propositions that can be applied during critical problems at the solution of the quadratic equations

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The main theme of the Newton Raphson method can be stated as it starts with an initial guess which is reasonably close to the actual root. It is also approximated as the tangent line that gets computed using the roots of the calculus. Another is able to calculate the x-intercept of the tangent line. This x-intercept would typically be a better adjustment to the root of the function when compared to the original guess. This is also the reason that this equation can also be iterated. Quite importantly, this Newton Raphson method can also be applied to the maximization and minimization problems. The derivative is usually kept at zero so that minimum and the maximum can be calculated in an easy manner. In many cases, this description differs substantially from the modern description that is mentioned in the quadratic equation form.

From the above-discussed points, it is quite clear that the Newton-Raphson method is very simple and is easy to implement where you have to calculate the roots of the quadratic equation. On the other hand, it has a large number of advantages which makes it possible for the equation to be implemented in a correct manner. However, one needs to understand this equation carefully as it can be useful for them in the future in case they want to use it. Hence, using a Newton Raphson equation with prior knowledge would fetch you good results.