Lazy laurent series

[klee]

Introduce lazy Laurent series to Sage.

A lazy Laurent series computes coefficients only when demanded or needed. In a sense, lazy Laurent series are Laurent series of infinite precision.

A generating function example from the code:

Generating functions are laurent series over the integer ring::

    sage: from sage.rings.lazy_laurent_series_ring import LazyLaurentSeriesRing
    sage: L = LazyLaurentSeriesRing(ZZ, 'z')

This defines the generating function of Fibonacci sequence::

    sage: def coeff(s, i):
    ....:     if i in [0, 1]:
    ....:         return 1
    ....:     else:
    ....:         return s.coefficient(i - 1) + s.coefficient(i - 2)
    ....:
    sage: f = L.series(coeff, valuation=0); f
    1 + z + 2*z^2 + 3*z^3 + 5*z^4 + 8*z^5 + 13*z^6 + ...

The 100th element of Fibonacci sequence can be obtained from the generating
function::

    sage: f.coefficient(100)
    573147844013817084101 

Lazy Laurent series is of course useful for other things. This will be used to implement infinite precision Laurent series expansion of algebraic functions in function fields, as a sequel to #27418.

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​5d24a34

Introduce lazy laurent series

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​ace4282

pyflakes fixes

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​389798e

Add series method

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Branch pushed to git repo; I updated commit sha1. New commits:

​43ba785

Small fixes on docstrings

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Ticket retargeted after milestone closed (if you don't believe this ticket is appropriate for the Sage 8.8 release please retarget manually)

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Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​e7ff999

Add lazy laurent series

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Branch pushed to git repo; I updated commit sha1. This was a forced push. New commits:

​15e1b1c

Add lazy laurent series

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Branch pushed to git repo; I updated commit sha1. New commits:

​f70fca8

Add __bool__

[tscrim]

Some comments:

Is there a reason why you do not use the LazyPowerSeriesRing for the main element (with a non-zero constant term until the whole series is zero) and then just store the valuation? This is how things are done for the Laurent polynomials. That way there is less duplication of code and improvements to one improve both.

Why do you override __eq__ and not use the one provided by Element and the coercion framework? It is more flexible and allows you do implement comparisons (if you want) all in one function of _richcmp_.

Do not have bare except: statements (see coefficient).

Does the elements pickle? I am a little worried that the addition of two power series will not pickling because of the little helper add function. You should add a loads(dumps(foo)) test if it does work.

Typo laurent -> Laurent (it is proper noun).

if len(s) == 0: -> if not s:

[klee]

Replying to tscrim:

Is there a reason why you do not use the LazyPowerSeriesRing for the main element (with a non-zero constant term until the whole series is zero) and then just store the valuation? This is how things are done for the Laurent polynomials. That way there is less duplication of code and improvements to one improve both.

I looked into the LazyPowerSeriesRing many years ago:

​https://groups.google.com/forum/#!searchin/sage-support/lazy$20power$20series%7Csort:date/sage-support/JZwDWKq3DtM/tyfxxdxPyXYJ

I think I didn't like it in several ways :-) Still I don't understand this behavior:

sage: L.<t>=LazyPowerSeriesRing(QQ)
sage: s=L([0,0,1,2])
sage: s.coefficient(0)
0
sage: s.coefficient(1)
0
sage: s.coefficient(2)
1
sage: s.coefficient(3)
2
sage: s
t^2 + 2*t^3 + O(x^4)
sage: s.get_order()
1
sage: s.get_aorder()
1

The reason was that it was easy to implement lazy laurent series with an interface and behaviors that I prefer, while I could not understand well enough LazyPowerSeriesRing...

I would reconsider basing lazy laurent series on LazyPowerSeriesRing, if you help me understand the strange behaviour above.

[tscrim]

So here is what refine_aorder does. It first checks if self.order has been set, if it has, do not do anything. Then it goes through all of the n computed entries if self.aorder is 0 and we have already computed at least one entry. Then find the first nonzero entry. If we have found one, set the order.

The problem is the bold part. After the first iteration when n = 1, the self.aorder gets set to 1, then on the second run through, that block does not get run, which then means self.aorder < n and we set the order to be 1.

sage: s.coefficient(1)
0
sage: s.aorder
1
sage: s.order
Unknown series order
sage: s.coefficient(2)
1
sage: s.aorder
1
sage: s.order
1

IMO, this is a bug.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​2f2c59c	Capitalize laurent as it is a proper noun
​146205a	No bare except clause
​fb4d65d	Use _richcmp_ instead of primitive __eq__

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​41f3a42

Lazy Laurent series is not picklable

[klee]

Why do you override __eq__ and not use the one provided by Element and the coercion framework? It is more flexible and allows you do implement comparisons (if you want) all in one function of _richcmp_.

Ok. Fixed.

Do not have bare except: statements (see coefficient).

Fixed.

Does the elements pickle? I am a little worried that the addition of two power series will not pickling because of the little helper add function. You should add a loads(dumps(foo)) test if it does work.

No. Your worry is real; lazy Laurent series is not picklable in general. This seems an inherent nature of my implementation...

I added a test to show this.

Typo laurent -> Laurent (it is proper noun).

Right. Fixed.

if len(s) == 0: -> if not s:

Done.

---

New commits:

​41f3a42

Lazy Laurent series is not picklable

[klee]

Replying to tscrim:

Some comments:

Is there a reason why you do not use the LazyPowerSeriesRing for the main element (with a non-zero constant term until the whole series is zero) and then just store the valuation? This is how things are done for the Laurent polynomials. That way there is less duplication of code and improvements to one improve both.

I thought about this. Now I think:

Lazy Laurent series is not just about Laurent series but also about how coefficients are computed lazily. A LazyPowerSeriesRing element gets its coefficients from a stream object, which essentially yields coefficients as required. On the other hand, lazy Laurent series s gets its coefficients from a python function that outputs n-th coefficient for input s and n. This allows coefficients to be computed recursively. For example, it is very easy to define the Fibonacci series.

So it is impossible to base my lazy Laurent series code on LazyPowerSeriesRing without abandoning the essential feature of the implementation.

Frankly, I would rather provide new lazy power series as a subclass of my lazy Laurent series. But this is of course highly biased opinion :-)

You may be reluctant to accept my implementation of lazy Laurent series into sage.rings as it then gets kind of standard status among different possible implementations of lazy Laurent series in Sage. Then I am willing to relocate it into sage.rings.function_field.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​b09cd84

Fix pickling doctest

[tscrim]

Replying to klee:

Replying to tscrim:

Some comments:

Is there a reason why you do not use the LazyPowerSeriesRing for the main element (with a non-zero constant term until the whole series is zero) and then just store the valuation? This is how things are done for the Laurent polynomials. That way there is less duplication of code and improvements to one improve both.

I thought about this. Now I think:

Lazy Laurent series is not just about Laurent series but also about how coefficients are computed lazily. A LazyPowerSeriesRing element gets its coefficients from a stream object, which essentially yields coefficients as required. On the other hand, lazy Laurent series s gets its coefficients from a python function that outputs n-th coefficient for input s and n. This allows coefficients to be computed recursively. For example, it is very easy to define the Fibonacci series.

Well, I am fairly certain you could do this with an RecursivelyEnumeratedSet and a bit of know-how, but I agree that it is more complicated and less intuitive. I also agree that having general functions as input would probably be a good thing for LazyPowerSeriesRing to have, but I believe that was designed for more specific use for the combinat/species code.

One benefit I do see is that you do not need to explicitly know the valuation at creation-time. With my suggested implementation, it does need to be computed at element creation.

Frankly, I would rather provide new lazy power series as a subclass of my lazy Laurent series. But this is of course highly biased opinion :-)

I don't have an opinion on this matter, but there are likely some considerations required to replace or extend the current implementation.

You may be reluctant to accept my implementation of lazy Laurent series into sage.rings as it then gets kind of standard status among different possible implementations of lazy Laurent series in Sage. Then I am willing to relocate it into sage.rings.function_field.

This I don't really care about. I just am a little unhappy with the large difference between the semantics (and syntax) between the two implementations. I guess that is a bit unavoidable here because this definitely is serving a purpose.

However, I do think we should at least try to address the pickling issues. I believe this means you cannot use this in parallel implementations (IIRC this uses pickling to communicate between the processes). It also means you cannot (easily) store the data you compute. I think it is sufficient to separate out the add and similar operations into functions, but maybe they need to be small little helper class, such as

class LaurentSeriesOperator(object):
    def __init__(self, lps, op):
        self.lps = lps
        self.op = op
    def __call__(self, s, n):
        return self.op(self.lps[n], s[n])
    def __reduce__(self):
        return (type(self), (self.lps, self.op), {})
    def __eq__(self, other):
        return (isinstance(other, LaurentSeriesOperator)
                and self.lps == other.lps and self.op == other.op)

where op is, e.g., operator.add. This way you might be able to do something with comparisons in some semi-reasonable capacity too.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​89992b8

Reimplement arithmetic operations for series to be picklable

[klee]

Replying to tscrim:

However, I do think we should at least try to address the pickling issues. I believe this means you cannot use this in parallel implementations (IIRC this uses pickling to communicate between the processes). It also means you cannot (easily) store the data you compute. I think it is sufficient to separate out the add and similar operations into functions, but maybe they need to be small little helper class, such as

class LaurentSeriesOperator(object):
    def __init__(self, lps, op):
        self.lps = lps
        self.op = op
    def __call__(self, s, n):
        return self.op(self.lps[n], s[n])
    def __reduce__(self):
        return (type(self), (self.lps, self.op), {})
    def __eq__(self, other):
        return (isinstance(other, LaurentSeriesOperator)
                and self.lps == other.lps and self.op == other.op)

where op is, e.g., operator.add. This way you might be able to do something with comparisons in some semi-reasonable capacity too.

Using module-level classes to define operators is a good idea. Hinted by your template class, I could reimplement lazy Laurent series to be picklable. Great!

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​989a3ac

Comparison now works in semi-reasonable capacity

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​ce4db37	Now pass _test_pickling
​e12fd5b	Minor fixes in string formatting