Last week I told you why
rs_series doesn’t work with negative or fractional
powers because of the constraints of a polynomial back-end and why we need to
polys. The situation isn’t that hopeless actually. Let’s talk about
negative and fractional powers one by one.
The reason negative exponents work in
ring_series is because I modified
PolyElement to allow so. In hind sight, it wasn’t the right decision and needs
to be replaced with something that doesn’t alter
polys. It is rather
surprising that I came across a possible solution so late (Now we know why good
documentation is so important).
I already knew that
polys allows us to create a
FractionField. A fraction
field over a domain
R consists of elements of the form
R. In our case we are interested in the fraction field of polynomial
ring, i.e, fractions with polynomials as numerator and denominator. So
a * b**(-1) but is
a / b, where a and b are polynomials.
What was new to me was that just like
polys also has sparse
In effect, it allows us to create sparse rational functions without altering
I modified some functions in
ring_series to work with a rational function
field here, and it works quite well indeed.
In : from sympy.polys.fields import * In : F, a, b = field('a, b', QQ) In : p = rs_sin(a + b, a, 3)/a**5 In : p*a Out: (-3*a**2*b - 3*a*b**2 + 6*a - b**3 + 6*b)/(6*a**4)
Note that all these are field operations and I haven’t modified
any way. Elegant!
But then again, having a field increases the complexity as we need to evaluate the numerator and denominator separately.
Fractional powers are a much trickier case as there is no simple solution to it as
above. What we can do is optimise the option I had presented in my last post,
i.e, have each fractional power as a generator. But doing that opens up a
Pandora’s box. Simple things such as
sqrt(a)**2 == a do not hold true any
more. The current
sqrt(a) as a constant if we are expanding
with respect to
In : rs_series(sin(a**QQ(1,2)), a**QQ(1,2),5) Out: -1/6*(sqrt(a))**3 + (sqrt(a)) In : rs_series(sin(a**QQ(1,2)), a,5) Out: sin(sqrt(a))
So, if we indeed decide to tread this path, we would need to replace
sqrt(a)**2. This really complicates the situation as we need to figure
out what to replace with. In any calculation the powers change multiple times
and each time we’ll need to figure out how to rewrite the series.
It is now mostly a design decision whether we want the
ring_series to be
confined within the
polys module. The
polys environment allows efficient
manipulations of laurent series (with
FracField), but I doubt we can achieve
the speed we want with puiseux series without modifying
polys. One possible
solution is to separate the modified parts of
polys along with
polys. We are using
polys only because it has the data structure that
we want. Separating them would allow us to simultaneously make use of its
back-end and not introduce unnecessary complexity in our representation.
Other than that, documentation is another priority now. I had planned to do it earlier too, but couldn’t. This week’s discovery has reminded me of its importance.
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