The Newton-Raphson method (also known as Newton's method) is a way to quickly find a good approximation for the root of a real-valued function . It uses the idea that a continuous and differentiable function can be approximated by a straight line tangent to it.
Suppose you need to find the root of a continuous, differentiable function , and you know the root you are looking for is near the point . Then Newton's method tells us that a better approximation for the root is This process may be repeated as many times as necessary to get the desired accuracy. In general, for any -value , the next value is given by
Note: the term "near" is used loosely because it does not need a precise definition in this context. However, should be closer to the root you need than to any other root (if the function has multiple roots).
Here is a picture to demonstrate what Newton's method actually does:
We draw a tangent line to the graph of at the point . This line has slope and goes through the point . Therefore it has the equation . Now, we find the root of this tangent line by setting and for our new approximation. Solving this equation gives us our new approximation, which is .
Find the root of the equation near to the nearest thousandth.
We have our . In order to use Newton's method, we also need to know the derivative of . In this case, , and .
Using Newton's method, we get the following sequence of approximations:
We can stop now, because the thousandth and ten-thousandth digits of and are the same. If we were to continue, they would remain the same because we have gotten sufficiently close to the root:
Our final answer is therefore 5.317.
Newton's method may not work if there are points of inflection, local maxima or minima around or the root.
For example, suppose you need to find the root of which is near .
The correct answer is However, Newton's method will give you the following:
This is very clearly not helpful. That's because the graph of the function around looks like this:
As you can see, this graph has a local maximum, a local minimum and a point of inflection around . To see why Newton's method isn't helpful here, imagine choosing a point at random between and and drawing a tangent line to the function at that point. That tangent line will have a negative slope, and therefore will intersect the -axis at a point that is farther away from the root.
In a situation like this, it will help to get an even closer starting point, where these critical points will not interfere.