Newton's Method: Finding Zeros of a Function - Part 1

Introduction

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.

Concept and Iterative Linearization

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 $f(x)$.

Mathematical Derivation

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 $x_{n+1}$:

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

Example

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 $f(x)$:

1. Start with an initial guess, say $x_0 = 1$.
2. Compute $x_1$:

$x_1 = 1 - \frac{1^2 - 2}{2 \cdot 1} = 1.5$

3. Compute $x_2$:

$x_2 = 1.5 - \frac{1.5^2 - 2}{2 \cdot 1.5} \approx 1.4167$.

(Note that after only 2 iteration we’re very close to $\sqrt(2)$.

4. Continue iterating until convergence.

Graphical Representation

Below is a graph illustrating the iterative process of Newton's Method:

1. Starting Point: Initial guess $x_0$.
2. Tangent Line: The tangent at $x_0$ intersects the x-axis, providing $x_1$.
3. Next Iterations: Repeat the process, moving closer to the root.

Graph 1: Function, Tangent Line and Iterative Convergence

Convergence

Newton's Method converges quadratically, meaning the number of correct digits roughly doubles with each iteration, assuming the initial guess is close to the actual root and $f'(x)$ is not zero. This makes Newton's Method highly efficient for well-behaved functions.

Conclusion

Newton's Method is an essential tool for finding zeros of a function due to its rapid convergence and simplicity. Understanding its iterative linearization process and practical implementation is crucial for solving nonlinear equations in various fields.