Say you’re in a competition and you need to calculate the square root of two to several decimal places. Or perhaps you need to approximate the roots to an unwieldy polynomial like $x^3+7x+2$. Newton’s method can be used in these situations.

Newton’s method. The Taylor expansion of $f(x)$ around $a$ is $\displaystyle f(x)=f(a)+f^{(1)}(a)(x-a)+\frac{1}{2}f^{(2)}(x-a)^2 +\cdots .$ Taking the linear approximation, we have $\displaystyle f(x)\approx f(a)+f'(a)(x-a).$ Set $f(x)=0$ (because we are finding the roots of f), and the approximation above becomes $\displaystyle 0= f(a)+f'(a)(x-a)\implies x=a-\frac{f(a)}{f'(a)}.$

Next we plug in an initial estimate for the root (this number we just make up and hope it is close to the root) for a and call it $x_0$. So now we have a fair approximation $\displaystyle x_1=x_0-\frac{f(x_0)}{f'(x_0)}.$ To get a better approximation, we plug $x_1$ back into the equation in the place of $x_0$. In general, we can get an approximation $x_n$ that is better than $x_{n-1}$ by $\displaystyle x_n=x_{n-1}-\frac{f(x_{n-1})}{f'(x_{n-1})}.$

Example 1. Approximate $\sqrt{3}$.

This is the same as solving the equation $\displaystyle f(x)=x^2-3=0.$ Since $\displaystyle f'(x)=2x,$ and we make our initial approximation $x_0=2$, we have $\displaystyle x_1=2-\frac{1}{4}=\frac{7}{4}=1.75,$ whereas $\sqrt{3}\approx 1.732$. That’s already an excellent estimate, but let’s keep on going until we are accurate to the thousandths place.

Incredible!

Example 2. Estimate the roots of $x^3+7x+2$.

$f(0)=2$ is pretty close to $0$, so we set $x_0=2$. Also, $\displaystyle f'(x)=3x^2+7.$ $\displaystyle x_1=0-\frac{2}{7}\approx -0.286.$ $\displaystyle x_2=-\frac{2}{7}-\frac{-8/343}{355/49}=-\frac{702}{2485}\approx -0.28249.$ Indeed, one of the roots of $x^3+7x+2$ is $-0.28249.$ But after guessing more values for $x_0$ to find the other roots, you will be hard-pressed to find any that result in an $f(x)$ value that is close to $0$. This is because $x^3+7x+2$ only has one real root. In a later post, I will explain how to determine the number of real solutions to a polynomial.

By the way, Newton was born on Christmas (sort of).