#7644: generic power series reversion
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Reporter: was | Owner: AlexGhitza
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-4.6.1
Component: algebra | Keywords: lagrange, reversion
Author: Niles Johnson | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Changes (by newvalueoldvalue):
* author: niles => Niles Johnson
Old description:
> From this [http://groups.google.com/group/sage-
> support/browse_thread/thread/34fdf02add8100b6 sage-support] thread: Make
> the following work over any base ring:
> {{{
> sage: R.<x> = QQ[[]]
> sage: f = 1/(1-x) - 1; f
> x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12
> + x^13 + x^14 + x^15 + x^16 + x^17 + x^18 + x^19 + O(x^20)
> sage: g = f.reversion(); g
> x - x^2 + x^3 - x^4 + x^5 - x^6 + x^7 - x^8 + x^9 - x^10 + x^11 - x^12
> + x^13 - x^14 + x^15 - x^16 + x^17 - x^18 + x^19 + O(x^20)
> sage: f(g)
> x + O(x^20)
> }}}
>
> Matt Bainbridge says about power series reversion, which uses pari in
> some cases, and maybe isn't there in others:
> {{{
> Its easy enough to code this in sage. This seems to work over any
> field:
>
> def ps_inverse(f):
> if f.prec() is infinity:
> raise ValueError, "series must have finite precision for
> reversion"
> if f.valuation() != 1:
> raise ValueError, "series must have valuation one for
> reversion"
> t = parent(f).gen()
> a = 1/f.coefficients()[0]
> g = a*t
> for i in range(2, f.prec()):
> g -= ps_coefficient((f + O(t^(i+1)))(g),i)*a*t^i
> g += O(t^f.prec())
> return g
>
> def ps_coefficient(f,i):
> if i >= f.prec():
> raise ValueError, "that coefficient is undefined"
> else:
> return f.padded_list(f.prec())[i]
> }}}
New description:
From this [http://groups.google.com/group/sage-
support/browse_thread/thread/34fdf02add8100b6 sage-support] thread: Make
the following work over any base ring:
{{{
sage: R.<x> = QQ[[]]
sage: f = 1/(1-x) - 1; f
x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + x^10 + x^11 + x^12
+ x^13 + x^14 + x^15 + x^16 + x^17 + x^18 + x^19 + O(x^20)
sage: g = f.reversion(); g
x - x^2 + x^3 - x^4 + x^5 - x^6 + x^7 - x^8 + x^9 - x^10 + x^11 - x^12
+ x^13 - x^14 + x^15 - x^16 + x^17 - x^18 + x^19 + O(x^20)
sage: f(g)
x + O(x^20)
}}}
Matt Bainbridge says about power series reversion, which uses pari in some
cases, and maybe isn't there in others:
{{{
Its easy enough to code this in sage. This seems to work over any
field:
def ps_inverse(f):
if f.prec() is infinity:
raise ValueError, "series must have finite precision for
reversion"
if f.valuation() != 1:
raise ValueError, "series must have valuation one for
reversion"
t = parent(f).gen()
a = 1/f.coefficients()[0]
g = a*t
for i in range(2, f.prec()):
g -= ps_coefficient((f + O(t^(i+1)))(g),i)*a*t^i
g += O(t^f.prec())
return g
def ps_coefficient(f,i):
if i >= f.prec():
raise ValueError, "that coefficient is undefined"
else:
return f.padded_list(f.prec())[i]
}}}
== Apply ==
1. [attachment:trac_7644_reversion_lagrange_2.2.patch]
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/7644#comment:8>
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