Newton's Method

Throughout your course work in Mathematics, you've likely had to find the root, or zero, of a function. These functions have likely only been linear or quadratic. But how would we go about finding the roots of the equation of a higher-order polynomial? Or what about a cubic equation with a natural logarithm, such as

$f\left(x\right)=\ln \left(x\right)+x^3$

It is much more difficult to find the roots of higher-order functions like these algebraically. However, Calculus proposes a few methods for estimating the root of complex equations. This article will cover one method to help us solve the roots of functions like these nasty ones!

Newton's Method of Approximation

One method we can use to help us approximate the root(s) of a function is called Newton's Method (Yes, it was discovered by the same Newton you've studied in Physics)!

Newton's Method Formula

The Newton's Method formula states that for a differentiable function F(x) and an initial point x₀ near the root

$x_{n+1}=x_n-\frac{F\left(x_n\right)}{F\text{'}\left(x_n\right)}$ where n = 0, 1, 2, ...

With multiple iterations of Newton's Method, the sequence of x_n will converge to a solution for F(x) = 0.

As the derivative of F(x) is in the fraction's denominator, if F(x) is a constant function with the first derivative of 0, Newton's Method will not work. Additionally, as we must compute the derivative analytically, functions with complex first derivatives may not work for Newton's Method.

The Calculus behind Newton's Method

With the Newton's Method formula in mind, see the graphical representation below.

Newton's Method finds a line tangent to the initial point to find an approximation for the root of f(x) - StudySmarter Original

Newton's Method aims to find an approximation for the root of a function. In terms of the graph, the zero of the function is the green point, f(x) = 0. Newton's Method uses an initial point (the pink x₀ on the graph) and finds the tangent line at the point. The graph shows that the line tangent to $x_0$ touches the x-axis near the root.

On the second iteration, Newton's Method constructs a new tangent line based on the last approximation found by the tangent line - StudySmarter Original

The new point, x₁, found via the tangent line at x₀, is translated onto the graph of the function, and a new tangent line is found. This process is repeated until a plausible estimation is found for f(x) = 0.

When Newton's Method fails

In cases where we cannot solve a function's root directly, Newton's Method is an appropriate method to use. However, there are certain cases where Newton's Method may fail:

• The tangent line does not cross the x-axis

  • Occurs when f'(x) is 0

• Different approximations may approach different roots if there are multiple

  • This occurs when the initial x₀ isn't close enough to the root

• Approximations don't approach the root at all

  • Approximation oscillates back and forth

Let's consider one such example where Newton's Method fails.

Suppose we have the function

$f\left(x\right)=-\frac{1}{2}+\frac{1}{1+x^2}$

This function has roots at $x=-1$ and $x=1$. However, let's say you wanted to use Newton's Method to find the roots of $f\left(x\right)$. With an initial guess of $x=2$, Newton's Method will approach the $x=-1$ root rather than the $x=1$ root even though $x=2$ is closer to $x=1$. Try it for yourself and see!

Newton's Method Examples

Example 1

Newton's Method of Approximating Square Roots

It is also possible to use Newton's Method to approximate the square root of a number! The Newton's Method square root approximation formula is nearly identical to the Newton's Method formula.

To compute a square root $x=\sqrt{a}$ for $a>0$ and with an initial guess for $x$ of $x_0$

$x_{n+1}=\frac{1}{2}\left(x_n+\frac{a}{x_n}\right)$

Square root approximation using Newton's Method Example

Let's apply the Newton's Method square root approximation equation to an example!