To the core developers (of numpy.polynomial e.g.): Skip the mess and
read the last paragraph.
The other things I will post back to the list, where they belong to.
I just didn't want to have off-topic discussion there.
> I wanted to stress that one can do arithmetic on Taylor polynomials in
> a very similar was as with complex numbers.
I do not understand completely. What is the analogy with complex
numbers which one cannot draw to real numbers? Or, more precise: What
/is/ actually the analogy besides that there are operations? With i,
i * i = -1, thus one has not to discard terms, contrary to the
polynomial product as you defined it, no?
> I guess there are also situations when you have polynomials z =
> \sum_{d=0}^{D-1} z_d t^d, where z_d are complex numbers.
>> I like it more to implement operators as overloads of the __mul__
>
> I thought about something like
> def __mul__(self, other):
> return mul(self,other)
Yes, I know! And in fact, it may symmetrise the thing a bit.
>> etc., but this is a matter of taste.
>> In fact, you /have/ to provide
>> external binary operators, because I guess you also want to have
>> numpy.ndarrays as left operand. In that case, the undarray will have
>> higher precedence, and will treat your data structure as a scalar,
>> applying it to all the undarray's elements.
>
> well, actually it should treat it as a scalar since the Taylor
> polynomial is something like a real or complex number.
Maybe I misunderstood you completely, but didn't you want to implement
arrays of polynomials using numpy? So I guess you want to apply a
vector from numpy pairwise to the polynomials in the P-object?
> [z]_{D+1} &=_{D+1}& [x]_{D+1} y_{D+1} \\
> \sum_{d=0}^D z_d t^d & =_{D+1} & \left( \sum_{d=0}^D x_d t^d \right)
> \left( \sum_{c=0}^D y_c t^c \right)
Did your forget the [] around the y in line 1, or is this intentional?
Actually, I had to compile the things before I could imagine what you
could mean.
Why don't you use multidimensional arrays? Has it reasons in the C
accessibility? Now, as I see it, you implement your strides manually.
With a multidimensional array, you could even create arrays of shape
(10, 12) of D-polynomials by storing an ndarray of shape (10, 12, D)
or (D, 10, 12).
Just because of curiosity: Why do you set X_{0, 1} = ... X_{0, P} ?
Furthermore, I believe there is some elegant way formulating the
product for a single polynomial. Let me think a bit ...
For one entry of [z]_E, you want to sum up all pairs:
x_{0} y{E} + ... + x{D} y{E - D} , (1)
right? In a matrix, containing all mutual products x_{i} y{j}, this
are diagonals. Rotating the matrix by 45° counterclockwise, they are
sums in columns. Hmmm... how to rotate a matrix by 45°?
Another fresh look: (1) looks pretty much like the discrete
convolution of x_{i} and y_{i} at argument E. Going into Fourier
space, this becomes a product. x_i and y_i have finite support,
because one sets x_{i} and y_{i} = 0 outside 0 <= i <= D. The support
of (1) in the running index is at most [0, D]. The support in E is at
most [0, 2 D]. Thus you don't make mistakes by using DFT, when you
pad x and y by D zeros on the right. My strategy is: Push the padded
versions of x and y into Fourier space, calculate their product,
transform back, cut away the D last entries, and you're done!
I guess you mainly want to parallelise the calculation instead of
improving the singleton calculation itself? So then, even non-FFT
would incorporate 2 D explicit python sums for Fourier transform, and
again 2 D sums for transforming back, is this an improvement over the
method you are using currently? And you could code it in pure Python
:-)
I will investigate this tonight. I'm curious. Irrespective of that I
have also other fish to fry ... ;-)
iirc, in fact DFT is used for efficient polynomial multiplication.
Maybe Chuck or another core developer of numpy can tell whether
numpy.polynomial does it actually that way?
Friedrich
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