Newton Raphson Method

The Newton-Raphson Method is a powerful and widely used numerical technique for finding the roots of equations. This iterative method is particularly useful for solving non-linear equations, providing rapid convergence under suitable conditions. It is one of the most efficient root-finding algorithms employed in various scientific and engineering fields.

What is the Newton-Raphson Method?

The Newton-Raphson Method is an iterative technique for approximating the roots of a function. It refines an initial guess by following the function's tangent line, progressively leading to the actual root.

Newton-Raphson Method Formula

The Newton-Raphson iteration is defined by the formula:

Let x₀ be the approximate root of f(x) = 0 and let x₁ = x₀ + h be the correct root. Then f(x₁) = 0

⇒ f(x₀ + h) = 0….(1)

By expanding the above equation using Taylor’s theorem, we get:

f(x₀) + hf¹(x₀) + … = 0

⇒ h = -f(x₀) /f’(x₀)

Therefore, x₁ = x₀ – f(x₀)/ f’(x₀)

Now, x₁ is the better approximation than x₀.

Similarly, the successive approximations x₂, x₃, …., x_n+1 are given by

xn+1=xn−f(xn)f′(xn)

This is called Newton Raphson formula. 

Geometric Interpretation & Convergence of the Newton-Raphson Method

The geometric interpretation of the Newton-Raphson Method involves drawing a tangent to the function at an initial guess. The point where the tangent intersects the x-axis provides the next approximation of the root.

The rate of convergence of the Newton-Raphson Method is quadratic, meaning the error decreases significantly with each iteration. However, convergence is not guaranteed if:

• The derivative is zero or very small.
• The function has discontinuities or sharp bends.

Newton Raphson Method Solved Example

Example 1: Find the cube root of 12 using the Newton Raphson method assuming x₀ = 2.5.

Solution: xn+1=1b[(b−1)xn+axnb−1]

From the given, a = 12, b = 3

Let x₀ be the approximate cube root of 12, i.e., x₀ = 2.5.

So, x₁ = (⅓) [2x₀ + 12/x₀²]

= (⅓) [2(2.5) + 12/(2.5)²]

= (⅓) [5 + 12/6.25]

= (⅓)(5 + 1.92)

= 6.92/3

= 2.306

Now,

x₂ = (⅓)[2x₁ + 12/x₁²]

= (1/3) [2(2.306) + 12/(2.306)²]

= (⅓) [4.612 + 12/5.3176]

= (⅓) [4.612 + 2.256]

= 6.868/3

= 2.289

Therefore, the approximate cube root of 12 is 2.289.

Example 2: Find a real root of the equation -4x + cos x + 2 = 0, by Newton Raphson method up to four decimal places, assuming x₀ = 0.5.

Solution: Given equation: -4x + cos x + 2 = 0

x₀ = 0/5

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

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.

x₁ = x₀ – f(x₀)/f’(x₀)

= 0.5 – [-4(0.5) + cos 0.5 + 2]/ [-4 – sin 0.5]

= 0.5 – [(-2 + 2 + cos 0.5)/ (-4 – sin 0.4)]

= 0.5 – [cos 0.5/ (-4 – sin 0.5)]

= 0.5 – [0.8775/ (-4 – 0.4794)]

= 0.5 – (0.8775/-4.4794)

= 0.5 + 0.1958

= 0. 6958

Implementation of the Newton-Raphson Method

The Newton-Raphson Method is implemented in various programming languages to automate root-finding calculations.

Newton-Raphson Method in C++

#include <iostream>

#include <cmath>

using namespace std;

double func(double x) { return x * x - 4; }

double derivFunc(double x) { return 2 * x; }

void newtonRaphson(double x) {

double h = func(x) / derivFunc(x);

while (abs(h) >= 0.0001) {

h = func(x) / derivFunc(x);

x = x - h;

}

cout << "The root is: " << x << endl;

}

int main() {

double x0 = 3;

newtonRaphson(x0);

return 0;
}

Newton-Raphson Method in Python

def f(x):

return x**2 - 4

def df(x):

return 2*x

def newton_raphson(x):

while abs(f(x) / df(x)) >= 0.0001:

x = x - f(x) / df(x)

return x

x0 = 3

root = newton_raphson(x0)

print("The root is:", root)

Newton-Raphson Method in C

#include <stdio.h>

#include <math.h>

double func(double x) { return x*x - 4; }

double derivFunc(double x) { return 2*x; }

void newtonRaphson(double x) {

double h = func(x) / derivFunc(x);

while (fabs(h) >= 0.0001) {

h = func(x) / derivFunc(x);

x = x - h;

}

printf("The root is: %lf\n", x);

}

int main() {

double x0 = 3;

newtonRaphson(x0);

return 0;
}

Newton Raphson Method Flowchart

Advantages and Limitations of the Newton-Raphson Method

Advantages

• Fast convergence: The method has quadratic convergence, making it one of the fastest iterative methods.
• Widely used: Common in engineering, physics, and computer science for solving nonlinear equations.
• Simple implementation: Easy to program using basic calculus operations.

Limitations

• Requires a good initial guess: Poor choices may lead to divergence.
• Derivative dependency: If f'(x) is zero or undefined, the method fails.
• Doesn’t work for all functions: Discontinuous or highly oscillatory functions pose challenges.

Newton-Raphson Method Applications

• Engineering: Used in control systems, circuit analysis, and structural mechanics.
• Physics: Applied in thermodynamics and quantum mechanics for solving equations.
• Machine Learning: Optimizes cost functions in gradient-based algorithms.
• Finance: Helps in calculating internal rate of return (IRR) and risk assessments.