Newton Raphson Root Finding Method Hand Calculation

Introduction

The Newton-Raphson method is a root-finding algorithm that produces successively better approximations to the roots of a real-valued function. It's particularly useful when analytical solutions are difficult to find or when dealing with complex equations.

This method is based on linear approximation and requires the function to be differentiable. The key idea is to start with an initial guess and then repeatedly apply a refined version of the guess until the desired accuracy is achieved.

Method Overview

The Newton-Raphson method works by using the tangent line to the function at a given point to approximate the root. The formula for the next approximation is:

Where:

• x_n is the current approximation
• x_n+1 is the next approximation
• f(x) is the function for which we're finding roots
• f'(x) is the derivative of the function

The process continues until the difference between successive approximations is smaller than a predefined tolerance level.

Step-by-Step Calculation

1. Choose an initial guess x₀ for the root.
2. Calculate f(x₀) and f'(x₀).
3. Compute the next approximation using x₁ = x₀ - f(x₀)/f'(x₀).
4. Repeat steps 2-3 with x₁ as the new x₀ until the difference between successive approximations is less than the desired tolerance.

Worked Example

Let's find the root of the function f(x) = x² - 3 using the Newton-Raphson method.

Step 1: Initial Guess

Let's choose x₀ = 2 as our initial guess.

Step 2: First Iteration

Calculate f(2) = (2)² - 3 = 1

Calculate f'(2) = 2(2) = 4

x₁ = 2 - (1/4) = 1.75

Step 3: Second Iteration

Calculate f(1.75) = (1.75)² - 3 ≈ -0.0625

Calculate f'(1.75) = 2(1.75) = 3.5

x₂ = 1.75 - (-0.0625/3.5) ≈ 1.7688

Step 4: Third Iteration

Calculate f(1.7688) ≈ (1.7688)² - 3 ≈ -0.0001

Calculate f'(1.7688) ≈ 2(1.7688) ≈ 3.5376

x₃ ≈ 1.7688 - (-0.0001/3.5376) ≈ 1.7688

The root is approximately 1.7688, which is √3.