Using the Newton-Raphson Method to Calculate Square Roots

The Newton-Raphson method is a powerful approach for finding the roots of a function. In this article, we will explore how to use this method to calculate the square root of a number. We'll discuss the mathematical principles behind the method, provide a step-by-step guide, and demonstrate how to apply the method using a specific example.

Introduction to the Newton-Raphson Method

The Newton-Raphson method is an iterative numerical technique used to find the roots of a real-valued function. It can be applied to find the square root of a number by solving the equation ( f(x) x^2 - N 0 ).

Convergence of a Sequence

Convergence of a sequence can be analyzed in two ways. The first method involves using the standard #948;-#945; definition, while the second method involves the Cauchy criterion. However, for our purposes, the second method is more suitable. Here, the sequence is represented by a function, and the convergence of this function can provide insights into the convergence of the original sequence.

Graphical Representation and Analytical Treatment

The Newton-Raphson method can be visualized as a series of steps where a tangent line is drawn at each step to approximate the root. Let's dive into a step-by-step analysis.

Mathematical Analysis

Consider the function ( f(x) x^2 - N ). We want to find the value of ( x ) such that ( f(x) 0 ). This value will be the square root of ( N ).

Newton-Raphson Update Formula

The Newton-Raphson update formula is:

Step 1: Compute the Derivative

The derivative of the function ( f(x) ) is:

[ f'(x) 2x ]

Step 2: Apply the Newton-Raphson Method

Substituting ( f(x) ) and ( f'(x) ) into the Newton-Raphson formula:

[ x_{n 1} x_n - frac{f(x_n)}{f'(x_n)} x_n - frac{x_n^2 - N}{2x_n} ]

This simplifies to:

[ x_{n 1} frac{x_n}{2} frac{N}{2x_n} ]

Example: Calculating (sqrt{5})

To find the square root of 5 using the Newton-Raphson method, we start with an initial guess and iteratively refine it.

Step-by-Step Calculation

Let's use the initial guess ( x_0 2 ).

Iteration 1:

[ x_1 frac{2}{2} frac{5}{2 cdot 2} 2.25 ]

Iteration 2:

[ x_2 frac{2.25}{2} frac{5}{2 cdot 2.25} approx 2.2361 ]

Iteration 3:

[ x_3 frac{2.2361}{2} frac{5}{2 cdot 2.2361} approx 2.2361 ]

Since ( x_3 ) is already very close to the previous result, we can conclude that the value of (sqrt{5}) correct to four decimal places is 2.2361.

Conclusion

The Newton-Raphson method provides a powerful and efficient way to find the square root of a number. By iteratively refining our guess, we can approximate the root to a desired level of accuracy.

Key Points

The method is based on the convergence of a sequence. It uses the derivative of the function to approximate the root. Starting with an initial guess and iteratively applying the formula improves accuracy. Convergence can be observed visually through successive approximations.

By understanding the theory behind the Newton-Raphson method and following these steps, you can calculate the square root of any number with high precision.