Introduction to Newton Raphson Method

Newton Raphson method is a numerical technique of finding the root of an equation by using derivatives. It required a function to be continuous and differentiable. If a function’s derivative is zero, the Newton Raphson method fails. It is also known as Newton method or iterative method and it can be denoted as NR method. Let us understand how to implement the NR method and what are the practical considerations for it.

Understanding of the Newton-Raphson Method

Derivative has many applications in calculus, numerical analysis, algebra, geometry and trigonometry. These are all main branches of mathematics that use derivative to solve problems. Newton Raphson method is a technique in numerical analysis which is used to approximate a function to find its root.

Newton Raphson method is named after two English mathematicians, Isaac Newton and Joseph Raphson. They developed this method to find successive approximations of a single-valued function defined on a real-valued variable. It uses the concept of continuity and differentiability and approximates a function by the slope of the tangent line.

Newton Raphson Method Formula

The Newton-Raphson method is an application of derivative that plays a major role in finding approximated root of an equation. Assume that f(x) is a continuous and differentiable function, then there will be a point x0 near to x such that,

$x_1=x_0-\frac{f(x_0)}{f'(x_0)}$

This method continuously repeats itself until we get the exact root of the function. For xn values, the Newton Raphson formula will be, 

$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$

Geometrical Interpretation of Newton Raphson Method

Since this method uses the concept of tangent line to a curve of a function, we can represent it geometrically. For this, suppose that x0 is the initial guess for the curve y=f(x) then the tangent line at the point [x0,f(x0)] will be drawn as,

 The tangent line intersect x-axis at x1, which means that this value is the approximation of f(x). This method will continue to repeat itself that can be seen by joining the function’s value with the root at x-axis.

How do you implement Newton’s method?

The Newton-Raphson method is based on the initial guess. So it is necessary to find out the initial value of x that satisfies the function. You can also use the following steps to implement this method easily. 

1. Find out the initial guess let say x0 of the function which is closer to the exact root.
2. Find the derivative of the given function and calculates $f'(x_0)$.
3. Now divide $f(x_0)$ with its derivative $f'(x_0)$.
4. Use the Newton Raphson method formula and substitute the values of $f'(x_0), f(x_0)$ and x₀.
5. Simplify to find the root. 

Let’s understand how to use Newton-Raphson method in the following examples.

Newton Raphson Method Example 1

Find the root of the equation $-4x + cos x + 2 = 0$ by using Newton Raphson method up to four decimal places and take the initial guess as 0.5

Given equation is,

$-4x + cos x + 2 = 0$

And the initial guess, 

$x_0=0.5$

Let $f(x) = -4x + cos x + 2$

Differentiating with respect to x,

$f’(x) = -4 – sin x$

Now,

$f(0) = -4(0) + cos 0 + 2 = 1 + 2 = 3 > 0$

$f(1) = -4(1) + cos 1 + 2 = -4 + 0.5403 + 2 = -1.4597 < 0$

Thus, a root lies between 0 and 1.

Let us find the first approximation.

$x_1= x_0 –\frac{f(x_0)}{f'(x_0)}$

$x_1= 0.5 – \frac{-4(0.5) + \cos 0.5 + 2}{-4 –\sin 0.5}$

$x_1= 0.5 –\frac{-2 + 2 +\cos 0.5}{-4 –\sin 0.5}$

$x_1= 0.5 – \frac{\cos 0.5}{-4-\sin 0.5}$

$x_1=0.5-\frac{0.8775}{-4-0.4794}$

$x_1=0.5-\frac{0.8775}{-4.4794}$

$x_1=0.5+0.1958=0.6958$

Applications of Newton Raphson Method

The Newton Method or NR method is one of the most important methods of determining the optimal solutions of many problems in different areas, including statistics, applied mathematics, numerical analysis, economics, management, finance and marketing. Let us discuss some of the most advanced applications of this method. 

• It is also known as an iterative method because it helps to solve nonlinear equations.
• It is used to analyse the flow in watch distribution networks. 
• It is used to calculate reactive/active power, voltage or current to get a complete understanding of a power flow system.

Practical Considerations of Newton Raphson Method

Although Newton method is one of the most efficient iterative methods that converges faster. Generally its convergence is quadratic as the method converges on the root. There are a few practical considerations that affect the convergence of the NR method. These are:

• The Newton method requires the derivative of a function to be calculated directly. If the derivative of a function cannot be easily calculated, the convergence of the NR method slows down.
• If the derivative of a function becomes zero, the NR method is unable to calculate the real root. Usually the derivative becomes zero on a stationary point. It is a point where the change in a function stops to increase or decrease.

Conclusion

The Newton Raphson Method is a fundamental concept of numerical analysis. It is also known as an application of derivative because, NR formula uses the tangent line slope. It is a powerful technique to find the fastest convergence of a function to its real root. But there are some practical considerations of this method such as it fails when the derivative of a function is zero at its initial guess. According to the above discussion, we can conclude that the Newton Raphson method is an application of derivative which is a power technique to converge a function faster unless the derivative of the function becomes zero.