[Numpy-discussion] Polynomial evaluation inconsistencies

Charles R Harris charlesr.harris at gmail.com
Sat Jun 30 16:56:14 EDT 2018

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.


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