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Newton's Method

Introduction

Newton's method is a popular numerical method used for finding the roots of a differentiable function. It is named after Sir Isaac Newton, who first proposed the method in the 17th century. The method is widely used in various fields of science and engineering, including physics, computer science, and finance.

The Method

Given a differentiable function f(x)f(x), the goal of Newton's method is to find the root of the equation f(x)=0f(x)=0. The method starts with an initial guess x0x_0 and iteratively improves the guess using the formula:

xn+1=xn−f(xn)f′(xn)x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}

where f′(xn)f'(x_n) is the derivative of f(x)f(x) evaluated at xnx_n. This formula gives us the next approximation xn+1x_{n+1}, which is closer to the actual root than the previous approximation xnx_n. We can repeat this process until we obtain an approximation that is sufficiently close to the actual root.

Example

Let us consider the function f(x)=x3−5x+3f(x) = x^3 - 5x + 3. We want to find the root of the equation f(x)=0f(x)=0.

First, we need to find the derivative of f(x)f(x), which is f′(x)=3x2−5f'(x) = 3x^2 - 5. We choose an initial guess of x0=1x_0 = 1.

Using the formula for Newton's method, we get:

x1=x0−f(x0)f′(x0)=1−13−5(1)+33(1)2−5=1.8\begin{aligned} x_1 &= x_0 - \frac{f(x_0)}{f'(x_0)} \\ &= 1 - \frac{1^3 - 5(1) + 3}{3(1)^2 - 5} \\ &= 1.8 \end{aligned}

We can repeat this process, using x1x_1 as the new initial guess, to obtain a better approximation:

x2=x1−f(x1)f′(x1)=1.8−(1.8)3−5(1.8)+33(1.8)2−5=1.70795\begin{aligned} x_2 &= x_1 - \frac{f(x_1)}{f'(x_1)} \\ &= 1.8 - \frac{(1.8)^3 - 5(1.8) + 3}{3(1.8)^2 - 5} \\ &= 1.70795 \end{aligned}

We can continue this process until we obtain an approximation that is sufficiently accurate.

Convergence

Newton's method has a quadratic rate of convergence, which means that the error between the actual root and the approximation decreases quadratically with each iteration. However, the method may not converge for some functions or initial guesses. In addition, the method may converge to a wrong root or oscillate between two or more roots.

Conclusion

Newton's method is a powerful numerical method for finding the roots of a differentiable function. It is widely used in various fields of science and engineering. However, the method has some limitations, and care must be taken when applying it to different functions and initial guesses.

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