[Numpy-discussion] Polynomial evaluation inconsistencies (original) (raw)

Maxwell Aifer maifer at haverford.edu
Sat Jun 30 17:33:22 EDT 2018

Interesting, I wasn't aware that both conventions were widely used.

Speaking of series with inverse powers (i.e. Laurent series), I wonder how useful it would be to create a class to represent expressions with integral powers from -m to n. These come up in my work sometimes, and I usually represent them with coefficient arrays ordered like this:

c[0]*x^0 + ... + c[n]*x^n + c[n+1]x^-m + ... + c[n+m+1]*x^-1

Because then with negative indexing you have:

c[-m]*x^-m + ... + c[n]*x^n

Still, these objects can't be manipulated as nicely as polynomials because they aren't closed under integration and differentiation (you get log terms).

Max

On Sat, Jun 30, 2018 at 4:56 PM, Charles R Harris <charlesr.harris at gmail.com

wrote:

On Sat, Jun 30, 2018 at 1:08 PM, Ilhan Polat <ilhanpolat at gmail.com> wrote: I think restricting polynomials to time series is not a generic way and quite specific.

I think more of complex analysis and it's use of series. Apart from the series and certain filter design actual usage of polynomials are always presented with decreasing order (control and signal processing included because they use powers of s and inverse powers of z if needed). So if that is the use case then probably it should go under a namespace of TimeSeries or at least require an option to present it in reverse. In my opinion polynomials are way more general than that domain and to everyone else it seems to me that "the intuitive way" is the decreasing powers.

In approximation, say by Chebyshev polynomials, the coefficients will typically drop off sharply above a certain degree. This has two effects, first, the coefficients that one really cares about are of low degree and should come first, and second, one can truncate the coefficients easily with c[:n]. So in this usage ordering by increasing degree is natural. This is the series idea, fundamental to analysis. Algebraically, interest centers on the degree of the polynomial, which determines the number of zeros and general shape, consequently from the point of view of the algebraist, working with polynomials of finite predetermined degree, arranging the coefficients in order of decreasing degree makes sense and is traditional. That said, I am not actually sure where the high to low ordering of polynomials came from. It could even be like the Arabic numeral system, which when read properly from right to left, has its terms arranged from small to greater. It may even be that the polynomial convention derives that of the Arabic numerals. Chuck

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