Re: [Numpy-discussion] Polynomial evaluation inconsistencies

[Maxwell Aifer]

Ok I see what you mean. If people really want math-like symbolic
representations for everything it’s probably better to use sympy or
something
On Mon, Jul 2, 2018 at 2:59 AM Eric Wieser 
wrote:

> I think the `x` is just noise there, especially if it's ignored (that is,
> `T[0](x*2)` doesn't do anything reasonable).
>
> Chebyshev.literal(lambda T: 1*T[0] + 2*T[1] + 3*T[2])
>
> Would work, but honestly I don't think that provides much clarity. I think
> the value here is mainly for "simple" polynomials.
>
> On Sun, 1 Jul 2018 at 23:42 Maxwell Aifer  wrote:
>
>> Say we add a constructor to the polynomial base class that looks
>> something like this:
>>
>>
>> ---
>>@classmethod
>> def literal(cls, f):
>> def basis_function_getter(self, deg):
>> coefs = [0]*deg + [1]
>> return lambda _: cls(coefs)
>> basis = type('',(object,),{'__getitem__':
>> basis_function_getter})()
>> return f(basis, None)
>>
>> ---
>>
>>
>> Then the repr for, say, a Chebyshev polynomial could look like this:
>>
>> >>> Chebyshev.literal(lambda T,x: 1*T[0](x) + 2*T[1](x) + 3*T[2](x))
>>
>> Does this sound like a good idea to anyone?
>>
>> Max
>>
>>
>> On Sat, Jun 30, 2018 at 6:47 PM, Charles R Harris <
>> charlesr.har...@gmail.com> wrote:
>>
>>>
>>>
>>> On Sat, Jun 30, 2018 at 4:42 PM, Charles R Harris <
>>> charlesr.har...@gmail.com> wrote:
>>>
>>>>
>>>>
>>>> On Sat, Jun 30, 2018 at 3:41 PM, Eric Wieser <
>>>> wieser.eric+nu...@gmail.com> wrote:
>>>>
>>>>> Since the one of the arguments for the decreasing order seems to just
>>>>> be textual representation - do we want to tweak the repr to something like
>>>>>
>>>>> Polynomial(lambda x: 2*x**3 + 3*x**2 + x + 0)
>>>>>
>>>>> (And add a constructor that calls the lambda with Polynomial(1))
>>>>>
>>>>> Eric
>>>>>
>>>>
>>>> IIRC there was a proposal for that. There is the possibility of adding
>>>> renderers for latex and html that could be used by Jupyter, and I think the
>>>> ordering was an option.
>>>>
>>>
>>> See https://github.com/numpy/numpy/issues/8893 for the proposal. BTW,
>>> if someone would like to work on this, go for it.
>>>
>>> Chuck
>>>
>>>> ​
>>>>>
>>>>
>>>
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Re: [Numpy-discussion] Polynomial evaluation inconsistencies

[Maxwell Aifer]

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  wrote:

>
>
> On Sat, Jun 30, 2018 at 1:08 PM, Ilhan Polat  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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Re: [Numpy-discussion] Polynomial evaluation inconsistencies

[Maxwell Aifer]

Thanks, that explains a lot! I didn't realize the reverse ordering actually
originated with matlab's polyval, but that makes sense given the one-based
indexing. I see why it is the way it is, but I still think it would make
more sense for np.polyval() to use conventional indexing (c[0] * x^0 + c[1]
* x^1 + c[2] * x^2). np.polyval() can be convenient when a polynomial
object is just not needed, but if a single program uses both np.polyval()
and np.polynomail.Polynomial, it seems bound to cause unnecessary confusion.

Max

On Fri, Jun 29, 2018 at 11:23 PM, Eric Wieser 
wrote:

> Here's my take on this, but it may not be an accurate summary of the
> history.
>
> `np.poly` is part of the original matlab-style API, built around
> `poly1d` objects. This isn't a great design, because they represent:
>
> p(x) = c[0] * x^2 + c[1] * x^1 + c[2] * x^0
>
> For this reason, among others, the `np.polynomial` module was created,
> starting with a clean slate. The core of this is
> `np.polynomial.Polynomial`. There, everything uses the convention
>
> p(x) = c[0] * x^0 + c[1] * x^1 + c[2] * x^2
>
> It sounds like we might need clearer docs explaining the difference, and
> pointing users to the more sensible `np.polynomial.Polynomial`
>
> Eric
>
>
>
> On Fri, 29 Jun 2018 at 20:10 Charles R Harris 
> wrote:
>
>> On Fri, Jun 29, 2018 at 8:21 PM, Maxwell Aifer 
>> wrote:
>>
>>> Hi,
>>> I noticed some frustrating inconsistencies in the various ways to
>>> evaluate polynomials using numpy. Numpy has three ways of evaluating
>>> polynomials (that I know of) and each of them has a different syntax:
>>>
>>>-
>>>
>>>numpy.polynomial.polynomial.Polynomial
>>>
>>> <https://docs.scipy.org/doc/numpy-1.13.0/reference/generated/numpy.polynomial.polynomial.Polynomial.html#numpy.polynomial.polynomial.Polynomial>:
>>>You define a polynomial by a list of coefficients *in order of
>>>increasing degree*, and then use the class’s call() function.
>>>-
>>>
>>>np.polyval
>>>
>>> <https://docs.scipy.org/doc/numpy-1.13.0/reference/generated/numpy.polyval.html>:
>>>Evaluates a polynomial at a point. *First* argument is the
>>>polynomial, or list of coefficients *in order of decreasing degree*,
>>>and the *second* argument is the point to evaluate at.
>>>-
>>>
>>>np.polynomial.polynomial.polyval
>>>
>>> <https://docs.scipy.org/doc/numpy-1.12.0/reference/generated/numpy.polynomial.polynomial.polyval.html>:
>>>Also evaluates a polynomial at a point, but has more support for
>>>vectorization. *First* argument is the point to evaluate at, and
>>>*second* argument the list of coefficients *in order of increasing
>>>degree*.
>>>
>>> Not only the order of arguments is changed between different methods,
>>> but the order of the coefficients is reversed as well, leading to puzzling
>>> bugs (in my experience). What could be the reason for this madness? As
>>> polyval is a shameless ripoff of Matlab’s function of the same name
>>> <https://www.mathworks.com/help/matlab/ref/polyval.html> anyway, why
>>> not just use matlab’s syntax (polyval([c0, c1, c2...], x)) across the
>>> board?
>>> ​
>>>
>>>
>> The polynomial package, with its various basis, deals with series, and
>> especially with the truncated series approximations that are used in
>> numerical work. Series are universally written in increasing order of the
>> degree. The Polynomial class is efficient in a single variable, while the
>> numpy.polynomial.polynomial.polyval function is intended as a building
>> block and can also deal with multivariate polynomials or multidimensional
>> arrays of polynomials, or a mix. See the simple implementation of polyval3d
>> for an example. If you are just dealing with a single variable, use
>> Polynomial, which will also track scaling and offsets for numerical
>> stability and is generally much superior to the simple polyval function
>> from a numerical point of view.
>>
>> As to the ordering of the degrees, learning that the degree matches the
>> index is pretty easy and is a more natural fit for the implementation code,
>> especially as the number of variables increases. I note that Matlab has
>> ones based indexing, so that was really not an option for them.
>>
>> Chuck
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[Numpy-discussion] Polynomial evaluation inconsistencies

[Maxwell Aifer]

Hi,
I noticed some frustrating inconsistencies in the various ways to evaluate
polynomials using numpy. Numpy has three ways of evaluating polynomials
(that I know of) and each of them has a different syntax:

   -

   numpy.polynomial.polynomial.Polynomial
   
:
   You define a polynomial by a list of coefficients *in order of
   increasing degree*, and then use the class’s call() function.
   -

   np.polyval
   
:
   Evaluates a polynomial at a point. *First* argument is the polynomial,
   or list of coefficients *in order of decreasing degree*, and the *second*
   argument is the point to evaluate at.
   -

   np.polynomial.polynomial.polyval
   
:
   Also evaluates a polynomial at a point, but has more support for
   vectorization. *First* argument is the point to evaluate at, and *second*
   argument the list of coefficients *in order of increasing degree*.

Not only the order of arguments is changed between different methods, but
the order of the coefficients is reversed as well, leading to puzzling bugs
(in my experience). What could be the reason for this madness? As polyval
is a shameless ripoff of Matlab’s function of the same name
 anyway, why not
just use matlab’s syntax (polyval([c0, c1, c2...], x)) across the board?
​
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