# Advantages and Disadvantages of Newton Raphson (NR) Method

Updated on Thursday, 24 October 2020

Move towards advantages of nr method.

Read advantages of n-r method

The main drawback of nr method is that its slow convergence rate and thousands of iterations may happen around critical point.

Here are the disadvantages of Newton-Raphson Method or we can say demerits of newton's method of iteration.

∴ f(X

Therefore expanding f(X

Now we have

By continuing in this way, (n+1)

The curve intersects X-axis at A, so at A it has the true root. We first take X

This process has a second order rate of convergence. If we take the initial approximation sufficiently closer to the true root, then the formula converges. This means that with each iteration, the number of correct decimals is approximately doubled.

Let X

The point of intersection of this tangent with X-axis gives the second approximation X

∴ (X

∴ X

∴ X

f(1)=-2, f(2)=64-16-4-1=43

Since f(1).f(2)<0, the root lies between 1 and 2.

∴ x

Since f'(1)≠0 is not satisfied we take an initial value other than 1. Let it be x=1.5. Then

x

Continuing in this way we get the final answer as 1.4036 .

## Introduction

This article is about Newton's Method which is used for finding roots. In numerical analysis, this method is also know as Newton-Raphson Method named after Isaac Newton and Joseph Raphson. This method is used for finding successively better approximations to the roots (or zeroes) of a real-valued function.Move towards advantages of nr method.

Read advantages of n-r method

## Newton-Raphson Method Drawbacks

**What is the main drawback of nr method?**

The main drawback of nr method is that its slow convergence rate and thousands of iterations may happen around critical point.

Here are the disadvantages of Newton-Raphson Method or we can say demerits of newton's method of iteration.

- We must find the derivative to use this method.
- Poor global convergence properties.
- Dependent on initial guess
- May be too far from local root
- May encounter a zero derivative
- May loop indefinitely

#### Index

- Introduction
- Newton-Raphson Method
- Algebraic Interpretation of Newton-Raphson Method
- Geometrical Interpretation of Newton-Raphson Method
- Derivation of Newton-Raphson Formula Without Using Taylor's Expansion
- Procedure for Newton-Raphson Method to find the Root of the Equation f(X)=0
- Solved Example on Newton-Raphson Method
- Advantages of Newton-Raphson Method
- Disadvantages of Newton-Raphson Method
- Examples for practice
- Related Articles

## Newton-Raphson Method

In false position method, geometrically we use two points between which the root lies. We then used a chord joining two points. In Newton-Raphson method, however we use only one point close which is close to the root and a tangent instead of a chord. This method is also know as iterative method. It is generally used to improve the solution obtained by the previous methods.## Algebraic Interpretation of Newton-Raphson Method

Let f(x)=0 be a given (algebraic or transcendental) equation. Let X_{0}be the approximate root of f(x)=0. Suppose the correct root is X_{1}. Therefore we can write X_{1}=X_{0}+h.∴ f(X

_{1})=f(X_{0})+hTherefore expanding f(X

_{0}+h) by Taylor's series,
f(X

By neglecting the higher and second higher order derivatives we have f(X_{0})+hf'(X_{0})+
h

f''(X^{2}2!_{0})+......=0_{0})+hf'(X_{0})=0
∴ h=

-f(X

_{0}) f'(X_{0})
∴ X

where X_{1}=X_{0}+h=X_{0}-
f(X

_{0}) f'(X_{0})_{1}is the better approximation than X_{0}. Let X_{2}be still a better approximation than X_{1}.Now we have

X

_{2}=X_{1}-
f(X

_{1}) f'(X_{1})By continuing in this way, (n+1)

^{th}approximation is Newton-Raphson formula is
X

_{n+1}=X_{n}-
f(X

_{n}) f'(X_{n})## Geometrical Interpretation of Newton-Raphson Method

Geometrically in this method, we replace that part of the curve between the point (X_{0},f(X_{0})) and the X-axis by tangent to the curve at the point which is shown below._{0}as the approximate value of this root. Let A_{0}(X_{0},f(X_{0})) be a point on the curve. Draw a tangent at A_{0}. Its intersection with X-axis gives the next approximation X_{1}. Continue the process. Observe that we are approaching the true root at A.This process has a second order rate of convergence. If we take the initial approximation sufficiently closer to the true root, then the formula converges. This means that with each iteration, the number of correct decimals is approximately doubled.

If f(X)=0 has a root with simplicity k then we use the following formula, X

_{n+1}=X_{n}-k
f(X

_{n}) f'(X_{n})## Derivation of Newton-Raphson Formula Without Using Taylor's Expansion

Consider a graph of y=f(x) as shown below._{0}be an approximate root of f(x)=0. Draw a tangent to the curve at X=X_{0}as shown in the figure above.The point of intersection of this tangent with X-axis gives the second approximation X

_{1}to the root.
Then tan ɑ=

f(X

=f'(X_{0}) X_{0}-X_{1}_{0})=slope of tangent at X=X_{0}∴ (X

_{0}-X_{1})f'(X_{0})=f(X_{0})∴ X

_{0}f'(X_{0})-X_{1}f'(X_{0})=f(X_{0})∴ X

_{1}f'(X_{0})=X_{0}f'(X_{0})-f(X_{0})
X

which is Newton-Raphson Formula.
_{1}=
X

=X_{n}f'(X_{0})-f(X_{0}) f'(X_{0})_{0}
f(X

_{0}) f'(X_{0})## Procedure for Newton-Raphson Method to find the Root of the Equation f(X)=0

This is the procedure for solving examples using Newton-Raphson formula.**Step I:**Let X_{0}be initial approximate root of f(X)=0. Find f(X_{0}) and f'(X_{0}).**Step II:**Let X_{1}be the next approximate root. Find it using the formula:X_{1}=X_{0}-f(X_{0}) f'(X_{0})**Step III:**Find successive approximations using the formula:X_{n+1}=X_{n}-f(X, n=1,2,3,..._{n}) f'(X_{n})**Step IV:**End the process when |X_{n+1}-X_{n}|<ε, where ε is the required accuracy.

## Solved Example on Newton-Raphson Method

Here is the solved example based on Newton-Raphson method.### Problem:

Find the real root of x^{6}-x^{4}-x^{2}-1=0 which lies between 1 and 2.### Solution:

f(x)=x^{6}-x^{4}-x^{2}-1f(1)=-2, f(2)=64-16-4-1=43

Since f(1).f(2)<0, the root lies between 1 and 2.

∴ x

_{2}=1 be the initial approximation. Then
X

_{1}=X_{0}-
f(X

=1-
_{0}) f'(X_{0})
f(X

=1-
_{1}) f'(X_{1})
-2
0

Since f'(1)≠0 is not satisfied we take an initial value other than 1. Let it be x=1.5. Then

X

_{1}=X_{0}-
f(X

=1.5-
_{0}) f'(X_{0})
f(X

=1.5-
_{1.5}) f'(X_{1.5})
3.0781
29.5625

=1.395878
X

_{2}=X_{1}-
f(X

=1.395878-
_{1}) f'(X_{1})
f(1.395878)
f'(1.395878)

=1.395878-
0.652464
(-8.371562)

x

_{2}=1.4738Continuing in this way we get the final answer as 1.4036 .

## Advantages of nr Method

Here are the advantages of Newton Raphson Method or we can say merits of newton's method of iteration.- One of the fastest methods which converges to root quickly.
- Converges on the root quadratically i.e rate of convergence is 2.
- As we go near to root, number of significant digits approximately doubles with each step.
- It makes this method useful to get precise results for a root which was previously obtained from some other convergence method.
- Easy to convert to multiple dimension.

## Applications of Newton Raphson Method

- Newton Raphson Method can be used to optimally design water distribution network.
- NR method is used in solving transcendental equations.
- It is used to solve minimization and maximization problems.
- It is used for numerical verification for solutions of nonlinear equations.
- It is used to obtain zeroes of special functions.
- It is used to find the reciprocal of a number, using only multiplication and subtraction.

## Examples For Practice

- Find the root of the equation x
^{5}+5x^{4}+1=0. - Find the approximate root of x
^{3}-20=0 by using Newton-Raphson method. - Solve the equation logx=cosx where the root lies between 1 and 2.
- Find the real root of the equation x=e
^{-x}with x_{0}=0.