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I don’t have enough reputation to comment, this is not a complete answer. I have a necessary and sufficient condition for consecutive ni, which is that the terms of the nth level difference are divisible by n! (We define the kth level difference to be the differences between consecutive terms of the (k-1)th level difference, with the 0th level difference ...


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Inverse Quadratic Interpolation (IQI) is described on the Wiki and in Section 4.5. Note some of the pitfalls mentioned there. Both of the references give the iteration formula. This method can be used to find all roots of a polynomial, whether they are real or complex. As an example, take $f(x) = 4 x^3-16 x^2+17 x-4$. I usually start by plotting the ...


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