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Finding limits of sequences. #9836

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merged 13 commits into from Sep 1, 2015
Merged

Finding limits of sequences. #9836

merged 13 commits into from Sep 1, 2015

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leosartaj
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This PR implements the algorithm as defined in the poster "Computing Limits of Sequences" by Manuel Kauers.

This algorithm works fine for some simple functions and computes their limit at infinity which currently SymPy is unable to handle. However, it still needs work as it is not very robust and often fails. Admissible functions need to be carefully defined. This is a still a work in progress.

Ping @jcrist @flacjacket

@leosartaj
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The poster does not explain the case when there is no unique dominant term. I have tried to get it working but have failed to come to a good solution. Currently in this case, the function returns None.

To clarify as to what to do in this case. I wrote an email to Manuel Kauers.

Hi,
I am a contributor to the SymPy(https://github.com/sympy/sympy) project. I am currently working on an implementation of the algorithm as described in the poster on Computing Limits of Sequences. However, the poster does not explain the case when there is not a unique a dominant term in an expression. Any pointers as to what to do in this case will be very helpful.

I am still waiting for a reply. If anybody else has any ideas, feel free to discuss.

@leosartaj
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I got a reply to my email to Manuel Kauers.

Dear Sartaj,

sorry for the late reply. I am in China at the moment, and since China
blocks Google (and gmail), doing email is rather cumbersome.

It is nice to hear that you are implementing l'Hospital for sequences.
Unfortunately, there is no follow-up work for the case when there is
no unique dominant term. Of course when you have something like
(a+b)/c where a and b have the same order of growth, you can try to
compute the limits of a/c and b/c, and if you are lucky they are both
finite. But if they are not, you would need to compute some sort of
asymptotic expansion of a and b up to the first order at which they
differ, and then apply the algorithm to their difference.

Greetings from Beijing,
Manuel

@leosartaj
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I am stuck in the last two xfailed test-cases. According to the paper these should be solvable, but the function gets stuck in infinite recursion. I have been unable to pin-point the exact cause. Maybe it has to do with limitations in automatic simplification.

@leosartaj leosartaj changed the title [WIP]Finding limits of infinite sequences. Finding limits of infinite sequences. Aug 20, 2015
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This is mostly ready for review now. @jcrist @flacjacket

@leosartaj leosartaj changed the title Finding limits of infinite sequences. Finding limits of sequences. Aug 24, 2015
@leosartaj leosartaj self-assigned this Aug 24, 2015
jcrist
jcrist reviewed Aug 25, 2015
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sympy/series/__init__.py

O = Order
dd = difference_delta
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I'd rather this wasn't shortened (short functions/variables may clash with user things).

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jcrist commented Aug 25, 2015

Besides my comments above, this looks good to me.

@leosartaj
minor changes to limitseq.
979a8b8 
* Add paramter section in docstring of limit_seq.
* Remove dd from the global namespace.
* Simple code cleanup.
@leosartaj
allow n to be None
cc8f935
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I have addressed the comments. Please have a look @jcrist.

@leosartaj
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Can you please have a look @jcrist. I would like to submit this in my code sample.

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jcrist commented Sep 1, 2015

Sorry for the delay. Looks good to me, thanks for your hard work! Merging.

jcrist added a commit that referenced this pull request Sep 1, 2015
@jcrist
Merge pull request #9836 from leosartaj/limitseq
733da51 
Finding limits of sequences.
@jcrist jcrist merged commit 733da51 into sympy:master Sep 1, 2015
@leosartaj leosartaj deleted the limitseq branch September 2, 2015 07:40
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Thanks @jcrist for the review.

@leosartaj leosartaj mentioned this pull request Oct 7, 2015
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