Applying the Newton-Raphson Method to Solve ( f(x) e^{-x^2} - x ) with Initial Guess ( x_0 0.5 )

The Newton-Raphson method is a powerful iterative algorithm designed to find the roots of a real-valued function. In this guide, we will walk through the process of using this method to solve the function ( f(x) e^{-x^2} - x ) starting from an initial guess of ( x_0 0.5 ). The goal is to approximately locate where this function intersects the x-axis, resulting in a solution correct to four decimal places.

Understanding the Function

To begin, let's examine the function in question:

[ f(x) e^{-x^2} - x ]

This function involves an exponential term and a linear term, making it a non-trivial task to solve algebraically. We will use the Newton-Raphson method to approximate the solution.

Newton-Raphson Method Overview

The Newton-Raphson method can be summarized with the following formula:

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

where ( f'(x_n) ) is the derivative of the function evaluated at ( x_n ).

Derivative of the Function

To apply the Newton-Raphson method, we need to compute the derivative of ( f(x) ).

[ f(x) e^{-x^2} - x ]

[ f'(x) frac{d}{dx}(e^{-x^2} - x) -2xe^{-x^2} - 1 ]

Iterative Process with Initial Guess ( x_0 0.5 )

We start with the initial guess ( x_0 0.5 ). The iterative process continues until the difference between consecutive approximations is less than or equal to ( 0.00001 ).

Iterating with the Newton-Raphson Formula

Given ( x_0 0.5 ), let's find ( x_1 ). [ f(x_0) e^{-(0.5)^2} - 0.5 e^{-0.25} - 0.5 ] [ f(0.5) approx 0.778800783 Next, use ( x_1 approx 0.652951807 ) to find ( x_2 ). [ f(x_1) e^{-(0.652951807)^2} - 0.652951807 approx 0.000000027 ] [ f'(x_1) -2 cdot 0.652951807 cdot e^{-(0.652951807)^2} - 1 approx -1.333333333 ] [ x_2 0.652951807 - frac{0.000000027}{-1.333333333} approx 0.652951808 ]

Conclusion

After several iterations, the approximate solution converges to ( x approx 0.6529 ) with a difference between successive approximations less than ( 0.00001 ).

This process highlights the effectiveness of the Newton-Raphson method in solving complex equations iteratively. Further applications of this method can be explored for similar and even more challenging functions.

For those interested in learning more about the Newton-Raphson method and its applications, explore resources on root finding, function approximation, and numerical methods for solving non-linear equations.