Updated on Thursday, 24 October 2020

## 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.

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

## 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 X0 be the approximate root of f(x)=0. Suppose the correct root is X1. Therefore we can write X1=X0+h.

f(X1)=f(X0)+h

Therefore expanding f(X0+h) by Taylor's series,

f(X0)+hf'(X0)+
h2 2!
f''(X0)+......=0
By neglecting the higher and second higher order derivatives we have f(X0)+hf'(X0)=0
∴ h=
-f(X0) f'(X0)
∴ X1=X0+h=X0-
f(X0) f'(X0)
where X1 is the better approximation than X0. Let X2 be still a better approximation than X1.

Now we have
X2=X1-
f(X1) f'(X1)

By continuing in this way, (n+1)th approximation is Newton-Raphson formula is
Xn+1=Xn-
f(Xn) f'(Xn)

## Geometrical Interpretation of Newton-Raphson Method

Geometrically in this method, we replace that part of the curve between the point (X0,f(X0)) and the X-axis by tangent to the curve at the point which is shown below.

The curve intersects X-axis at A, so at A it has the true root. We first take X0 as the approximate value of this root. Let A0(X0,f(X0)) be a point on the curve. Draw a tangent at A0. Its intersection with X-axis gives the next approximation X1. 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, Xn+1=Xn-k
f(Xn) f'(Xn)

## Derivation of Newton-Raphson Formula Without Using Taylor's Expansion

Consider a graph of y=f(x) as shown below.

Let X0 be an approximate root of f(x)=0. Draw a tangent to the curve at X=X0 as shown in the figure above.

The point of intersection of this tangent with X-axis gives the second approximation X1 to the root.

Then tan ɑ=
f(X0) X0-X1
=f'(X0)=slope of tangent at X=X0

∴ (X0-X1)f'(X0)=f(X0)

∴ X0f'(X0)-X1f'(X0)=f(X0)

∴ X1f'(X0)=X0f'(X0)-f(X0)

X1=
Xnf'(X0)-f(X0) f'(X0)
=X0
f(X0) f'(X0)
which is Newton-Raphson Formula.

## 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.
1. Step I: Let X0 be initial approximate root of f(X)=0. Find f(X0) and f'(X0).
2. Step II: Let X1 be the next approximate root. Find it using the formula:
X1=X0-
f(X0) f'(X0)
3. Step III:Find successive approximations using the formula:
Xn+1=Xn-
f(Xn) f'(Xn)
, n=1,2,3,...
4. Step IV: End the process when |Xn+1-Xn|<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 x6-x4-x2-1=0 which lies between 1 and 2.

### Solution:

f(x)=x6-x4-x2-1

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

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

x2=1 be the initial approximation. Then

X1=X0-
f(X0) f'(X0)
=1-
f(X1) f'(X1)
=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

X1=X0-
f(X0) f'(X0)
=1.5-
f(X1.5) f'(X1.5)
=1.5-
3.0781 29.5625
=1.395878

X2=X1-
f(X1) f'(X1)
=1.395878-
f(1.395878) f'(1.395878)
=1.395878-
0.652464 (-8.371562)

x2=1.4738
Continuing in this way we get the final answer as 1.4036 .

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

You might want to check out regula falsi method and lagrange's interpolation method.

## Applications of Newton Raphson Method

1. Newton Raphson Method can be used to optimally design water distribution network.
2. NR method is used in solving transcendental equations.
3. It is used to solve minimization and maximization problems.
4. It is used for numerical verification for solutions of nonlinear equations.
5. It is used to obtain zeroes of special functions.
6. It is used to find the reciprocal of a number, using only multiplication and subtraction.
To know more about applications of Newton raphson Method please visit Newton's Method on Wikipedia.

## Examples For Practice

1. Find the root of the equation x5+5x4+1=0.
2. Find the approximate root of x3-20=0 by using Newton-Raphson method.
3. Solve the equation logx=cosx where the root lies between 1 and 2.
4. Find the real root of the equation x=e-x with x0=0.
Scilab Program For Newton-Raphson Method