What is an inflection point?

Drawbacks of Newton-Raphson Method

1. Divergence at inflection points: If the selection of a guess or an iterated value turns out to be close to the inflection point of f(x), that is, near where f”(x)=0, the roots start to diverge away from the root. For example, to find the root of

Table 1. Divergence at inflection point

Iteration Number	xi	%	f(xi)
	-1 -0.33333 0.11111 0.40741 0.60494 0.73663 0.82442 0.88294 0.92196	72.73 32.65 17.88 10.65 6.629 4.232	-8 -2.3704 -0.70233 -0.2081 -0.06166 -0.01827 -5.4132x10^-3 -1.60389x10^-3 -4.75226x10^-3

Figure 3: Divergence at inflection point for .

For a function f(x) where the concavity changes from up-to-down or down-to-up are called inflection points of the graph. For example in the function, , the concavity changes at x=1. In fact, it also changes sign at x=1 and thus brings up the Inflection Point Theorem for a function f(x) that states: “If f’(c) exists and f’’(c) changes sign at x = c, then the point (c, f(c)) is an inflection point of the graph of f.

2. Division of zero or near zero: The formula of Newton-Raphson method is

Consequently if an iteration value, is such that , then one can face division by zero or a near-zero number. This will give a large magnitude for the next value, x_i+1. An example is finding the root of the equation

in which case

For or , division by zero occurs (Figure 4). For an initial guess close to 0.02, of , even after 9 iterations, the Newton-Raphson method is not converging.

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