#8972: Inversion and fraction fields for power series rings
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Reporter: SimonKing | Owner: AlexGhitza
Type: defect | Status: needs_review
Priority: major | Milestone: sage-4.4.2
Component: algebra | Keywords: power series ring, fraction field
Author: Simon King | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Comment(by SimonKing):
It was worth it!
First, I found one division bug for Laurent series left, which is now
fixed (and doc tested):
{{{
sage: L.<x> = LaurentSeriesRing(ZZ)
sage: 1/(2+x)
1/2 - 1/4*x + 1/8*x^2 - 1/16*x^3 + 1/32*x^4 - 1/64*x^5 + 1/128*x^6 -
1/256*x^7 + 1/512*x^8 - 1/1024*x^9 + 1/2048*x^10 - 1/4096*x^11 +
1/8192*x^12 - 1/16384*x^13 + 1/32768*x^14 - 1/65536*x^15 + 1/131072*x^16 -
1/262144*x^17 + 1/524288*x^18 - 1/1048576*x^19 + O(x^20)
}}}
This used to result in an error.
Second, I added more doc tests (and inserted my name to the author list).
Third, the new timings for sqrt are fully competitive (even very slightly
faster than before), and we still have the bug fix:
{{{
Loading Sage library. Current Mercurial branch is: powerseries
sage: P.<t> = QQ[[]]
sage: p = 9 - 33*t - 13*t^2 - 155*t^3 - 429*t^4 - 137*t^5 + 170*t^6 +
81*t^7 - 132*t^8 + 179*t^9 + O(t^10)
sage: p.sqrt()
3 - 11/2*t - 173/24*t^2 - 5623/144*t^3 - 174815/1152*t^4 -
8187925/20736*t^5 - 12112939/9216*t^6 - 7942852751/1492992*t^7 -
1570233970141/71663616*t^8 - 12900142229635/143327232*t^9 + O(t^10)
sage: timeit('q = p.sqrt()')
25 loops, best of 3: 13.2 ms per loop
sage: timeit('q = (p^2).sqrt()')
25 loops, best of 3: 13.5 ms per loop
sage: timeit('q = (p^4).sqrt()')
25 loops, best of 3: 14.5 ms per loop
sage: PZ = ZZ[['t']]
sage: timeit('q = (PZ(p)^2).sqrt()')
25 loops, best of 3: 16.4 ms per loop
sage: p = 9 - 33*t - 13*t^2 - 155*t^3 - 429*t^4 - 137*t^5 + 170*t^6 +
81*t^7 - 132*t^8 + 179*t^9
sage: timeit('q = p.sqrt()')
25 loops, best of 3: 20.6 ms per loop
sage: timeit('q = (p^2).sqrt()')
25 loops, best of 3: 21.7 ms per loop
}}}
The trick is that I do ``s*a.__invert__()`` instead of ``s/a``, where
``s`` and ``a`` are power series. This helps, because (after a little
change) the invert method returns a power series whenever possible -- and
since I know that ``a.valuation()>=0``, I can use fast multiplication of
power series rather than slow multiplication of Laurent series.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8972#comment:16>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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