Newton's Method (also called the Newton-Raphson method) is a recursive algorithm for approximating the root of a differentiable function. We know simple formulas for finding the roots of linear and quadratic equations, and there are also more complicated formulae for cubic and quartic equations. At one time it was hoped that there would be formulas found for equations of quintic and higher-degree, though it was later shown byNeils Henrik Abel that no such equations exist. The Newton-Raphson method is a method for approximating the roots of polynomial equations of any order. In fact the method works for any equation, polynomial or not, as long as the function is differentiable in a desired interval.

In order to explain Newton's method, imagine that${\displaystyle x_0}$ is already very close to a 0 of${\displaystyle f(x)}$ . We know that if we only look at points very close to${\displaystyle x_0}$ then${\displaystyle f(x)}$ looks like its tangent line. If${\displaystyle x_0}$ was already close to the place where${\displaystyle f(x)}$ was 0, and near${\displaystyle x_0}$ we know that${\displaystyle f(x)}$ looks like its tangent line, then we hope the 0 of the tangent line at${\displaystyle x_0}$ is a better approximation then${\displaystyle x_0}$ itself.

The equation for the tangent line to${\displaystyle f(x)}$ at${\displaystyle x_0}$ is given by

${\displaystyle y=f^{\prime }(x_0)\cdot (x-x_0)+f(x_0)}$

Now we set${\displaystyle y=0}$ and solve for${\displaystyle x}$ .

${\displaystyle 0=f^{\prime }(x_0)\cdot (x-x_0)+f(x_0)}$

${\displaystyle -f(x_0)=f^{\prime }(x_0)\cdot (x-x_0)}$

${\displaystyle \frac{-f(x_0)}{f^{\prime }(x_0)}=(x-x_0)}$

${\displaystyle x=\frac{-f(x_0)}{f^{\prime }(x_0)}+x_0}$

This value of${\displaystyle x}$ we feel should be a better guess for the value of${\displaystyle x}$ where${\displaystyle f(x)=0}$ . We choose to call this value of${\displaystyle x_1}$ , and a little algebra we have

${\displaystyle x_1=x_0-\frac{f(x_0)}{f^{\prime }(x_0)}}$

If our intuition was correct and${\displaystyle x_1}$ is in fact a better approximation for the root of${\displaystyle f(x)}$ , then our logic should apply equally well at${\displaystyle x_1}$ . We could look to the place where the tangent line at${\displaystyle x_1}$ is zero. We call${\displaystyle x_2}$ , following the algebra above we arrive at the formula

${\displaystyle x_2=x_1-\frac{f(x_1)}{f^{\prime }(x_1)}}$

And we can continue in this way as long as we wish. At each step, if your current approximation is${\displaystyle x_n}$ our new approximation will be${\displaystyle x_{n+1}=x_n-\frac{f(x_n)}{f^{\prime }(x_n)}}$ .

Find the root of the function${\displaystyle f(x)=x^2}$.

As you can see${\displaystyle x_n}$ is gradually approaching 0 (which we know is the root of${\displaystyle f(x)}$) . One can approach the function's root with arbitrary accuracy.

Answer:${\displaystyle f(x)=x^2}$ has a root at${\displaystyle x=0}$.

This method fails when${\displaystyle f^{\prime }(x)=0}$ . In that case, one should choose a new starting place. Occasionally it may happen that${\displaystyle f(x)=0}$ and${\displaystyle f^{\prime }(x)=0}$ have a common root. To detect whether this is true, we should first find the solutions of${\displaystyle f^{\prime }(x)=0}$ , and then check the value of${\displaystyle f(x)}$ at these places.

Newton's method also may not converge for every function, take as an example:

For this function choosing any${\displaystyle x_1=r-h}$ then${\displaystyle x_2=r+h}$ would cause successive approximations to alternate back and forth, so no amount of iteration would get us any closer to the root than our first guess.

Newton's method may also fail to converge on a root if the function has a local maximum or minimum that does not cross the x-axis. As an example, consider${\displaystyle f(x)=x^5-x+1}$ with initial guess${\displaystyle x_0=0}$ . In this case, Newton's method will be fooled by the function, which dips toward the x-axis but never crosses it in the vicinity of the initial guess.

• Wikipedia:Newton's method
• Wikibooks
• Wikipedia:Abel–Ruffini theorem

Wikimedia Commons has media related to:Newton Method

• Book:Calculus
• Newton's method