#12241: exp, log, derivative of multivariate power series
-----------------------------------+----------------------------------------
Reporter: vbraun | Owner: malb
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-5.0
Component: commutative algebra | Keywords: multivariate power series
Work_issues: efficiency | Upstream: N/A
Reviewer: Niles Johnson | Author: Volker Braun
Merged: | Dependencies:
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Changes (by niles):
* status: needs_review => needs_work
* reviewer: => Niles Johnson
* work_issues: => efficiency
Comment:
This is good -- thanks for working on it! Here are some comments:
== derivative ==
Looks great!
== pow ==
* I think the default behavior should be same as repeated multiplication
by self -- is there a good reason not to do this?
* Also, when precision is much larger than valuation, `pow` is slower
than repeated multiplication anyway:
{{{
sage: R.<a,b,c> = PowerSeriesRing(ZZ)
sage: f = a+R.random_element(100)
sage: f
a - a^23*c^11 + a^29*b^8*c^55 + 2*a^36*b^54*c^6 + O(a, b, c)^100
sage: %timeit f^10
5 loops, best of 3: 95.8 ms per loop
sage: %timeit f.pow(10)
5 loops, best of 3: 215 ms per loop
}}}
I think this should be addressed by rewriting `pow` to use `_bg_value`, as
has been done with `_mul_`.
* Unexpected behavior when requesting larger precision from `pow`:
{{{
sage: f = a*R.random_element(); f
2*a^3*b - 7*a^3*b^4*c^3 - 3*a^6*b^6 - 109*a^3*b^3*c^6 - 2*a*b^9*c^2 + O(a,
b, c)^13
sage: f^2
4*a^6*b^2 - 28*a^6*b^5*c^3 - 12*a^9*b^7 - 436*a^6*b^4*c^6 - 8*a^4*b^10*c^2
+ O(a, b, c)^17
sage: f.pow(2,prec=17)
4*a^6*b^2 + O(a, b, c)^13
}}}
== exp / log ==
* The warning about constant coefficients is confusing and vague. I
would prefer something more straightforward like: "If `f` has constant
coefficient `c`, and `exp(c)` is transcendental, then `exp(f)` would have
to be a power series over the Symbolic Ring. These are not yet
implemented and therefore such cases raise an error:"
* Another workaround for this limitation is to change base ring to one
which is closed under exponentiation, such as RR or CC:
{{{
sage: R.<a,b,c> = PowerSeriesRing(ZZ)
sage: f = 2+R.random_element()
sage: exp(f)
. . .
TypeError: unsupported operand parent(s) for '*': 'Symbolic Ring' and
'Multivariate Power Series Ring in a, b, c over Rational Field'
sage: S = R.change_ring(CC)
sage: f = S(f)
sage: exp(f)
7.38905609893065 + (-7.38905609893065)*a*b^3*c +
(-22.1671682967919)*a^6*b^2*c + (-7.38905609893065)*a^5*b^2*c^2 +
(-7.38905609893065)*b^2*c^7 + 3.69452804946533*a^2*b^6*c^2 +
(-7.38905609893065)*a*b^10 + O(a, b, c)^12
}}}
* For elements with infinite precision, consider using `default_prec` if
no precision is specified:
{{{
sage: R.<a,b,c> = PowerSeriesRing(ZZ)
sage: R.default_prec()
12
}}}
* The implementation seems unnecessarily slow, because of repeated calls
to `pow`: For precision `n`, calling `pow` for each term will require a
total of something like `n^2` multiplications of the input with itself.
But these functions should require only `n`. Here is a suggested
alternate implementation for `log`:
{{{
def fast_log(f):
prec = f.prec()
x = 1-f
r = 1
logx = 0
for i in range(1,prec):
r = r*x + R.O(prec)
log += r/i
return -logx
}}}
{{{
sage: f = 1+R.random_element(30); f
1 - 6*a^5*b^2*c^7 + a^14*b^13 + O(a, b, c)^30
sage: %timeit log(f)
5 loops, best of 3: 224 ms per loop
sage: %timeit fast_log(f)
25 loops, best of 3: 26.4 ms per loop
sage: fast_log(f) == log(f)
True
}}}
And this might be faster if you use `_bg_value` to do the multiplication
here too.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/12241#comment:3>
Sage <http://www.sagemath.org>
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