Newton's Method, also known as the Newton-Raphson method, is a powerful technique for finding the zeros of a real-valued function. It uses successive, iterative linearization to converge to a root with remarkable speed and efficiency.
Newton's Method starts with an initial guess $x_0$ for a root of the function $f(x)$. It then uses the tangent line at $f(x_0)$ to generate a better approximation. The iterative formula is derived from the linear approximation of $f(x)$ near $x_0$.
Given a function $f(x)$ and its derivative $f'(x)$, the next approximation $x_{n+1}$ is given by:
$$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $$
This process is repeated until $x_{n+1}$ converges to a root of $f(x)$.
Starting with the Taylor series expansion of $f(x)$ around $x_n$:
$$ f(x) \approx f(x_n) + f'(x_n)(x - x_n) $$
To find the root, set $f(x) = 0$:
$$ 0 \approx f(x_n) + f'(x_n)(x_{n+1} - x_n) $$
Solving for $x_{n+1}$:
$$ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} $$
Consider the function $f(x) = x^2 - 2$. The derivative is $f'(x) = 2x$. Using Newton's Method to find $\sqrt{2}$ , the root/zero of $f(x)$:
$$ x_1 = 1 - \frac{1^2 - 2}{2 \cdot 1} = 1.5 $$