Newton-Raphson Method

[src10003]

I am struggling to understand the concept of when this method would be most appropriate to use, or when it would be inappropriate? Any ideas?

[eas15111]

Newton-Raphson is a method that converges more quickly than the bisection method, as shown from the error plot in M3_NB4:

However, since Newton-Raphson involves taking the derivative of the function, the function must be continuously differentiable / the derivative must exist at the points at which the derivative are taken. This adds for some additional requirements to the function in order to be able to use the Newton-Raphson method.

This article is pretty helpful in understanding the uses and shortcomings of the Newton-Raphson method: https://en.wikipedia.org/wiki/Newton%27s_method

[src10003]

If I'm understanding correctly and getting the animation on the link you sent, Newton-Raphson method would probably mean that it would also be an inappropriate method for very high order polynomials with high order derivatives? am I tracking right or still off the trail?

[arr16112]

As long as the function is continuous and differentiable the Newton-Raphson method should work. However, if your initial guess is the max or min of the original function this will cause errors because in the function we divide by the derivative function with the input value, which, if at max or min, would be zero and dividing by zero is a no no.
If you want to think about it geometrically the tangent line would be horizontal at these points so the line would never intersect with the x axis to be the next guess.
It's also inappropriate when there is no root like an exponential function.