I need an example of a failure of Newton-Raphson method that occurs because it's close to an equation's curve?

Question

What I'm asking is, when a point is selected near the curve of the graph of an equation, it diverges from the root you are trying to find, I just need an example of an equation that diverges from the required root, I honestly don't know how to explain this so it might not make sense but if you can help, please do.

Answer 1

I don't understand what you mean by "close to an equation's curve".

There's 3 things that I can think of that would cause failure.
1. The slope is zero at one of the iteration points.
2. The function is undefined at one of the iteration points.
3. The points oscillate in a repeating patten.

Answer 2

Take the flow parameter F, which is a function of Mach number.
F = M / [ 1 + (k-1)/2 * M^2]^3, where k = 1.4 (for air).
At the sonic condition, M = 1, and F = .5787.
Suppose you want the Mach number, M, for which F = .56, and assume an initial guess of, say, 0.97. This might cause the problem you mention.