Newton Raphson Method: Definition, Formula, Convergence & Examples

The Newton-Raphson method, named after Isaac Newton and Joseph Raphson, is an inherently root-finding algorithm, which means that its goal is to find the value of x such that f(x )=0 . Geometrically, we think of this as the value of x at which the function intersects the x-axis. This algorithm is versatile with a wide range of use cases that span many domains. For example, if we reframe the function of interest we can search for the value of x which will give us the selected value instead of being constrained to 0.

Newton Raphson method is also called the tangent method or Newton's method. In this math article, we will learn how to use the Newton-Raphson method to find the roots, Formula, geometric interpretation, convergence, and limitations of this method.

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What is Newton Raphson Method? 

The Newton-Raphson method is a way to quickly find a good approximation to the root of a real function 

f(x )=0. It is based on the idea that a continuous and differentiable function can be approximated by a straight line tangent to it. Let a single root, , of the function f(x). The Newton-Raphson method starts with an initial root estimate, denoted as, and uses the tangent of f(x) at to improve the root estimate. The improvement denoted by which is obtained by determining where the tangent to f(x) at intersects the x-axis.

Newton Raphson method formula is used to find out the roots of a polynomial equation. If we have any root of the equation we can calculate successive roots of the equation using this method. 

Let we need to find the root of a continuous, differentiable function f(x).

is the initial approximate root of

is the value of the function at .

  is the first derivative of the function at .

  is a better approximation than 

This process may be repeated as many times as necessary to get the desired accuracy.

The successive approximations are given by

This is known as Newton Raphson's formula.

Newton Raphson Method Example

Let us find the cube root of 11 using Newton Raphson method.

The cube root of 11 =

We can write in the equation form .

Assuming

Applying Newton Raphson's formula.

Putting

For

Putting

For

Putting

For

Putting

 

Therefore, 2.224 is a good approximation to cube root 11.

Thus, the cube root of 11 is 2.224 approx.

Geometrical Interpretation of Newton Raphson Formula

The below figure represents the Newton Raphson Formula.

A tangent line is drawn to the graph of the function f(x) at point and has the slope which is represented by . Hence, the equation of the tangent line is . The root of the tangent line which is found by setting and is a new approximation. Solving this equation provides the new approximation and the value of the new approximation is 

 .

Example: To calculate the root of equation x^2 - 4x - 7 = 0 near to x = 5 and the nearest thousandth.

Given:

Applying Newton Raphson's formula.

\(x_{1} =5-(\frac{-2}{6})=\frac{16}{3}=5.333 \(

For

Putting

For

Putting

.

For

Putting

We can stop because the values of x_{3} and x_{4} are the same. Hence, the root of the equation is 5.31662 .

Convergence of Newton Raphson Method 

The order of convergence of Newton Raphson method is 2 or the convergence is quadratic.

Let is a root of and is an estimate of such that .

 We know that by Taylor series expansion 

—------------ equation no.01.

Where is between and .

 We know that by Newton's -Raphson method 

.

—---------- equation no.02.

Putting the value of from equation no.02 to equation no.01,we get.

Let and

Where and denote the error in the solution at nth and (n+1)th iterations.

Therefore,

Hence,Newton Raphson Method is called to have quadratic convergence.

Applications of Newton Raphson Method

The Newton-Raphson method can be used as an optimal solution to obtain estimates in regression models, the maxima or minima of many functions in both the one-dimensional and multidimensional case, or to solve systems of equations with many unknowns in both the one-dimensional as well as in the multidimensional case.In real life, the Newton-Raphson method is often used for flow analysis in water distribution networks.

Advantages of Newton Raphson Method

• The method has one of the fastest convergences to the root.
• It is easy to program as it has a simple formula
• It is used to further improve a root found by other methods.
• It requires only one guess to find the root.
• Derivation of the method is more intuitive, so it is easier to understand its behaviour, like when it is to converge and when it is to diverge.

Properties of the Newton-Raphson Method

1. Fast Convergence
   The method converges very quickly (quadratic convergence) if the initial guess is close to the actual root.

2. Iterative Approach
   It uses an iterative process to refine guesses and reach the root of a function.

3. Requires Derivatives
   The method needs both the function f(x) and its derivative f′(x) for calculations.

4. Highly Accurate
   When it converges, it provides highly accurate root approximations in fewer steps.

5. Sensitive to Initial Guess
   The success of the method depends heavily on how close the initial guess is to the actual root.

6. Fails if Derivative is Zero
   If f′(x)=0, the method breaks down due to division by zero.

7. Not Always Convergent
   For functions with sharp curves or discontinuities, the method may not converge.

8. Can be Used for Complex Roots
   It can be extended to find complex roots of functions, using complex numbers.

Limitations of Newton's Method 

If there are points of inflection, local maxima or minima around or the root then Newton's method may not work.

For example, we need to find the root of the function which is near x = 0.

The correct answer is -0.44157265… However, by Newton's method, we get the following:

x_{1}=\frac{1}{3},x_{2}=\frac{1}{6},x_{3}=1,x_{4}= 0.679,x_{5}=0.463,x_{6}=0.3035

x_{7}=0.114, x_{8}=0.473.

From the above results, it is clear that these results are not helpful. These results are not helpful because the graph of the function around x=0 looks like as per given figure:

As we can observe, this graph has a local maximum, a local minimum, and a point of inflection around point x = 0. Newton's method is not helpful here because if we imagine choosing a point at random between x = -0.19 and x = 0.19 and drawing a tangent line to the function at that point. The tangent line has a negative slope, and so, the tangent line intersects the y-axis at a point that is farther away from the root.With the Bisection method, the rate of convergence is linear and therefore it is slow.

In this type of situation, Newton's method helps to get an even closer starting point, where these critical points do not interfere.

Bisection and Newton Raphson method

The following is the difference between Bisection and Newton Raphson method,

S.No.	Bisection Method	Newton Raphson Method
1.	With the Bisection method, the rate of convergence is linear and therefore it is slow.	With the Newton-Raphson method, the rate of convergence is second order or quadratic.
2.	In this method, we take two initial approximations of the root in which the root is expected to lie.	In this method we take an initial root approximation.
3.	In Bisection's method,the following formula is used.	In Newton Raphson's method,the following formula is used.
4.	The calculation of the function per iteration is 1.	The calculation of the function per iteration is 2.
5	The initial approximation is less sensitive.	The initial approximation is very sensitive.
6.	It is not necessary to find the derivatives in the bisection method.	In Newton Raphson's method we have to find the derivatives.
7.	This method is not applicable to find two complex, multiple and nearly equal roots.	This procedure is applied to the search for two complex, multiple and nearly equal roots.

Summary of Newton Raphson Method

• The Newton-Raphson method is a way to quickly find a good approximation to the root of a real function .
• The successive approximations are given by.This is known as Newton Raphson's formula.
• The geometric interpretation of Newton's Raphson method is that a tangent is drawn at the point [x0, f(x0)] to the curve y = f(x). It cuts the x-axis at x1, which will be a better approximation of the root.
• if there are points of inflection, local maxima or minima around x_0 or the root then Newton's method may not work.
• The Newton-Raphson method is not always convergent and this method fails when f'(x) is equal to 0.

Solved Examples

Ex-1: To find the root of equation and the nearest thousandth.

A1. Given:

 

Applying Newton Raphson's formula.

For

Putting

For

Putting

For

Putting

We can stop because the value of x_{3} and x_{4} are the same. 

Answer: Hence, the root of the equation is 1.32472 .

 Ex-2. Find the real root of the equation , by Newton Raphson method up to four decimal places, assuming .

A2.Given equation:

Let  

 

Now,

 

Therefore, a root lies between 0 and 1.

Let us find the first approximation.

Ex-3 The cube root of 5 =

A3. We can write in the equation form .

Assuming

Applying Newton Raphson's formula.

 

 

Putting

 

 

For

 

Putting

 

 

For

 

Putting

 

 

For

 

Putting

 

 

Therefore, 1.7099 is a good approximation to cube root 5. 

Thus, the cube root of 5 is 1.7099 approx.

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