Evaluating polynomials


Evaluating polynomials is not a thing I do very often. When I do, it’s for interpolation and splines; and traditionally those are done with relatively low degree polynomials—cubic at most. There are a few rather simple tricks you can use to evaluate them efficiently, and we’ll have a look at them.

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Horner’s Method


It’s not like evaluating polynomial is something that comes up that frequently in the average programmer’s day. In fact, almost never, but it brings up a number of interesting questions, such as, how do we evaluate it very efficiently and how much of a simplification in the computation is actually a simplification?

The generic polynomial has the form


If we evaluate this naïvely, we end up doing O(n^2) multiply and O(n) additions. Even using the fast exponentiation algorithm, we still use O(n \lg n) multiplies. But we can, very easily, get down to O(n).

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