Newton-Raphson Method Calculator

The Newton-Raphson method is a powerful technique used for finding successively better approximations to the roots (or zeroes) of a real-valued function.

Historical Background

Originally proposed by Isaac Newton in 1669 and later refined by Joseph Raphson in 1690, this method has become a cornerstone in numerical analysis for solving equations. It is appreciated for its simplicity and efficiency, especially in computational mathematics.

Calculation Formula

The Newton-Raphson formula is:

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

Where:

• \( x_n \) is the current approximation.
• \( f(x_n) \) is the value of the function at \( x_n \).
• \( f'(x_n) \) is the value of the derivative of the function at \( x_n \).

Example Calculation

Consider the function \( f(x) = x^2 - 4 \) with an initial guess of \( x_0 = 2 \).

1. Calculate \( f(x_0) = 2^2 - 4 = 0 \).
2. Calculate the derivative \( f'(x) = 2x \) and \( f'(x_0) = 4 \).
3. Apply the formula: \( x_1 = 2 - \frac{0}{4} = 2 \).

Since \( f(x_1) = 0 \), we have found the root.

Importance and Usage Scenarios

This method is essential for:

1. Solving Non-linear Equations: Where analytical solutions are not feasible.
2. Engineering and Science: For approximating solutions in various fields.
3. Optimization Problems: In machine learning and statistics.

Common FAQs

1. What happens if the derivative is zero?

   • The method fails as it leads to division by zero. A different starting point or method is needed.

2. Is convergence guaranteed?

   • Not always. Convergence depends on the function and the initial guess.

3. Can it find all roots of a function?

   • It finds one root based on the starting point. Other roots require different starting points or methods.