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I realize that the initial nonlinear equation needs to be linearly convergent and that we then use Steffensen's Method (a modification of Aitken's $\Delta ^2$ Method) to speed up the convergence, but I'm unsure if having the possibility of a singularity would influence Steffensen's Method in any way.

I have been asked this question, but I am unsure how to answer it, because there doesn't seem to be any mention of singularities in my textbook.

I also realize that singularities in math are defined as "In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability." (from Singularity in Mathematics).

I just need some clarification on the actual nature of the question as well as possibly an answer and explanation.

Many thanks.

Richard Slabbert
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  • I think that Steffensen's Method cannot be applied when there is a possibility of a singularity since taking the Aitken's Method formula, if $\Delta ^2 p_n$ in the denominator of the equation is equal to 0 the algorithm would given an error and exit prematurely. – Richard Slabbert Feb 26 '16 at 16:22

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