Fast series method for ring series

[shivamvats]

This PR implements a series_fast method that takes an arbitrary Basic expression and expands it using sparse ring manipulations. We tried using EX domain for all expansion in #9614, but that proved to be too slow for our liking. This PR starts expanding over the QQ domain and keeps adding generators as and when required. Thus, we always work on the simplest (almost) possible domain, resulting in considerable speed-up.

The timings are extremely encouraging:

In [31]: %timeit series_fast(sin(a**2) + cos(a)*sin(a),a,100)
1 loops, best of 3: 303 ms per loop

In [32]: %timeit (sin(a**2) + cos(a)*sin(a)).series(a,0,100)
1 loops, best of 3: 5.42 s per loop

Because of the way we add generators, there is a constant overhead in expanding a given expression(irrespective of the asked order), depending on the complexity of the expression. So, for small orders, series_fast (yes, we need a new name!) might seem slow, but soon far outperforms series for even medium sized orders:

In [34]: %timeit series_fast(p, a, 10)
1 loops, best of 3: 764 ms per loop

In [35]: %timeit p.series(a,0,10)
1 loops, best of 3: 561 ms per loop

Here series_fast is actually slower than series for 10 terms, but once we expand till the 100th order:

In [36]: %timeit series_fast(p, a, 100)
1 loops, best of 3: 917 ms per loop

In [37]: %timeit p.series(a,0,100)
1 loops, best of 3: 6.67 s per loop

Clearly, the performance advantage will grow larger with larger orders.

There is a lot of scope for optimising the code (my priority was to make it work). It also needs to be played with to find bugs. However, I feel this is the way to go.

cc @certik @pernici @jksuom @thilinarmtb

ToDo
Fix wrong expansion of

• sin(a**2 - a + 2)**5*cos(a**3 - a/2)
• sin(x + a + b)
• Rational exponent in argument: sin(x**QQ(1,2)). _parallel_dict_from_expr_if_gens etc need to be modified

[certik]

Great job! How does this compare to @pernici's fastest code? Per his recent comment, it looks like this might be faster? I thought @pernici's code was performing well, but that was only for cases where general coefficients (EX) were not needed?

[shivamvats]

I benchmarked some expressions here. The benchmarked expressions are mostly random. Is there a standard set of series for proper benchmarking?

series_fast is faster than or as fast as taylor in most cases and slower in some. Though I don't fully understand how taylor works, my guess is that taylor is much slower than series_fast when it uses EX. As long as taylor uses QQ, there can't be more than a constant difference in performance as series_fast uses QQ at all times (i.e, the backend is the same).

In cases in which series_fast is slower than taylor by more than 100%, it seems to be because series_fast had to re-expand as it did not get the required precision the first time. We can use the methods used in taylor to predict the precision functions need to be expanded upto.

[certik]

Can you point me to which taylor you were using?

[certik]

Also, some tests fail.

[shivamvats]

This one

I had to change from sympy.polys.monomialtools to from sympy.polys.orderings at line 10 in rs_taylor.py to make it work.

[certik]

I see. The benchmarks are good enough. I am fine with this. Let's finish this up and get it in.

[shivamvats]

@certik @pernici The series function now works for taylor series of sin, cos, exp and tan. I have fixed the known bugs. I will gradually add other functions.

For puiseux series, we need to modify polys (as I explained in the email). Other than that this is ready for review.

[certik]

You do not need a parenthesis around rs_series:

rs_series(a, a, 5).as_expr()

[certik]

Since this is a higher level function, it needs a docstring with documentation and a few doctests showing how to use it.

[certik]

I left some comments. In general this looks good, let's fix the issues I pointed out and merge.

[shivamvats]

I've made the changes.

[shivamvats]

@certik Anything else to be added here?

[certik]

I think it looks good, thanks!

[shivamvats]

Thanks for the review!