#1956: implement multivariate truncated power series arithmetic
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Reporter: was | Owner: pernici
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-4.7
Component: commutative algebra | Keywords: multivariate power
series
Author: Niles Johnson | Upstream: N/A
Reviewer: Martin Albrecht, Simon King | Merged:
Work_issues: |
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Comment(by niles):
Replying to [comment:97 niles]:
> Replying to [comment:96 hlaw]:
> > There seems to be a problem with how coercion/evaluation works between
two multivariate power series rings when the number of variables in each
differs.
I tracked down this bug -- it was unique to the case where one of the
power series rings is univariate, and comes from the fact that
(univariate) power series rings automatically convert any univariate power
series from a given variable to another:
{{{
sage: R.<a> = PowerSeriesRing(ZZ)sage: S.<b> = PowerSeriesRing(ZZ)
sage: b in R
False
sage: R.has_coerce_map_from(S)
False
sage: R(b)
a
}}}
I was thinking this should be filed as a separate bug, but polynomials
have the same behavior, so perhaps it is intentional. In any case, I've
fixed multivariate power series to take this into account, so an error is
raised when the variables don't match:
{{{
sage: B.<s, t> = PowerSeriesRing(QQ)
sage: C.<z> = PowerSeriesRing(QQ)
sage: B(z)
...
TypeError: Cannot coerce input to polynomial ring.
}}}
> > Another problem, possibly related, is the following...
I also implemented a `._subs_formal` method, which is called automatically
by `.__call__`. This method substitutes *anything* into the power series,
requiring only that multiplication and exponentiation be defined for the
inputs. This gives all of the "expected" results from above. Some
further examples (from the new doctests):
{{{
sage: B.<s, t> = PowerSeriesRing(QQ)
sage: C.<z> = PowerSeriesRing(QQ)
sage: s(z,z)
z
sage: f = -2/33*s*t^2 - 1/5*t^5 - s^5*t + s^2*t^4
sage: f(z,z)
-2/33*z^3 - 1/5*z^5
sage: f(z,1)
-1/5 - 2/33*z + z^2 - z^5
sage: RF = RealField(10)
sage: f(z,RF(1))
-0.20 - 0.061*z + 1.0*z^2 - 0.00*z^3 - 0.00*z^4 - 1.0*z^5
sage: m = matrix(QQ,[[1,0,1],[0,2,1],[-1,0,0]])
sage: m
[ 1 0 1]
[ 0 2 1]
[-1 0 0]
sage: f(m,m)
[ 2/33 0 1/5]
[ 131/55 -1136/165 -24/11]
[ -1/5 0 -23/165]
sage: f(m,m) == -2/33*m^3 - 1/5*m^5
True
sage: f = f.add_bigoh(10)
sage: f(z,z)
-2/33*z^3 - 1/5*z^5 + O(z^10)
sage: f(m,m) # cannot substitute arbitrary elements when f has finite
precision
Traceback (most recent call last):
...
AttributeError: 'sage.matrix.matrix_rational_dense.Matrix_rational_dense'
object has no attribute 'add_bigoh'
}}}
Please test it out and let me know how it goes!
-Niles
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/1956#comment:98>
Sage <http://www.sagemath.org>
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