Newton-Raphson's theorem, commonly known as Newton-Raphson method, is an iterative numerical method used to find the approximate roots (solutions) of a real-valued function $f(x)$. It's particularly useful when finding exact roots is challenging. The method is based on linear approximation (tangent line) of the function and refines guesses to converge towards an accurate solution.

Formula:

Given a differentiable function $f(x)$, the Newton-Raphson iteration formula is:

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

Where:

• $x_n$ is the current approximation of the root.
• $f'(x_n)$ is the derivative of $f(x)$ at $x_n$.
• $x_{n+1}$ is the next approximation.

Steps of the Newton-Raphson Method:

1. Start with an initial guess $x_0$.
2. Compute the next approximation using the formula.
3. Repeat the process until the difference between successive approximations is sufficiently small, indicating convergence to a root.

Example:

Suppose we want to find the root of the equation $f(x) = x^2 - 2$ (the square root of 2).

1. Let the initial guess $x_0 = 1$.
2. Calculate the next approximation using the formula:

$x_1 = x_0 - \frac{f(x_0)}{f'(x_0)} = 1 - \frac{1^2 - 2}{2 \cdot 1} = 1 - \frac{-1}{2} = 1.5$

3. Repeat the process until $x_n$ converges to a solution.

This method converges quickly if the initial guess is close to the actual root.

Key Points:

• The method works best when the function is well-behaved and differentiable near the root.
• If the initial guess is far from the root, the method may fail to converge or converge to a wrong root.
• It is widely used in calculus, numerical analysis, and solving engineering problems.

Would you like more details or have any questions?

Related Questions:

1. How does the Newton-Raphson method compare to other root-finding methods like bisection or secant method?
2. What are the conditions required for the Newton-Raphson method to converge?
3. How can the method fail, and what strategies can be used to avoid failure?
4. What are some real-world applications of the Newton-Raphson method?
5. How can Newton-Raphson be extended to systems of nonlinear equations?

Tip: Always check the derivative $f'(x)$ doesn't vanish, as division by zero would cause failure of the method.