Implementing Formal Power Series, rational algorithm

[leosartaj]

TODO

• Formal Power Series Class (without operations)
• rational algorithm

I have implemented first version of FormalPowerSeries and rational_algorithm. Still needs a lot of work.

Ping @jcrist @flacjacket @certik

[leosartaj]

This is built over code in #9523 which is under review.

[leosartaj]

Some results

>>> %timeit series(atan(x), n=200)
1 loops, best of 3: 3.51 s per loop
>>> %timeit fps(atan(x), full=True).truncate(n=200)
1 loops, best of 3: 721 ms per loop

>>> %timeit series(ln(1 + x), n=200)
1 loops, best of 3: 7.95 s per loop
>>> %timeit fps(ln(1 + x)).truncate(n=200)
1 loops, best of 3: 557 ms per loop

As, FormalPowerSeries is formula based, speedup comes when number of terms is high.

>>> %timeit series(ln(1 + x), n=500)
1 loops, best of 3: 39.4 s per loop
>>> %timeit fps(ln(1 + x)).truncate(n=500)
1 loops, best of 3: 1.34 s per loop

>>> %timeit series(atan(x), n=500)
1 loops, best of 3: 10.1 s per loop
>>> %timeit fps(atan(x), full=True).truncate(n=500)
1 loops, best of 3: 2.09 s per loop

[leosartaj]

I have written a lot of code on top of this, that is currently in my hyper2 branch. Should I open it is as well or it's better to first go through this?

There are some merge conflicts due to #9523. I will fix them.

[jcrist]

Can you break this line somewhere else? I didn't even realize it was legal to break a line before a period...

[leosartaj]

I have updated the docstrings and other comments except these 1, 2. I will fix them in #9639. As I have made some changes in the code in these particular areas. It's best if I change there and save myself from solving conflicts.

[leosartaj]

All the changes were merged in #9639.