#10720: nth_root in power series
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Reporter: pernici | Owner: pernici
Type: PLEASE CHANGE | Status: new
Priority: minor | Milestone: sage-4.6.2
Component: commutative algebra | Keywords: power series
Author: mario pernici | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Changes (by newvalueoldvalue):
* author: => mario pernici
* component: PLEASE CHANGE => commutative algebra
* priority: major => minor
* owner: tbd => pernici
* milestone: => sage-4.6.2
* keywords: => power series
Old description:
> In this patch the nth-root of power series `y = x^n` is introduced using
> the Newton method for `y = x^-n`
> {{{
> x' = (1+1/n)*x - y*x^(n+1)/n (1)
> }}}
>
> {{{
> sage: R.<t> = QQ[]
> sage: p = (1 + 2*t + 5*t^2 + 7*t^3 + O(t^4))^3
> sage: p.nth_root(3)
> 1 + 2*t + 5*t^2 + 7*t^3 + O(t^4)
> sage: p = (1 + 2*t + 5*t^2 + 7*t^3 + O(t^4))^-3
> sage: p.nth_root(-3)
> 1 + 2*t + 5*t^2 + 7*t^3 + O(t^4)
> }}}
>
> The iterations are division-free;
> in the case `n=2` one can compare this division-free iteration
> with the iteration used in the Newton method for `sqrt`
> {{{
> x' = (x +y/x)/2 (2)
> }}}
>
> `nth_root` can be used to compute the square root of series which
> currently `sqrt` does not support
> {{{
> sage: R.<x,y> = QQ[]
> sage: S.<t> = R[[]]
> sage: p = 1 + x*t + (x^2+y^2)*t^2 + O(t^3)
> sage: p1 = p.nth_root(2); p1
> 1 + 1/2*x*t + (3/8*x^2 + 1/2*y^2)*t^2 + O(t^3)
> sage: p1^2
> 1 + x*t + (x^2 + y^2)*t^2 + O(t^3)
> }}}
>
> In particular it can be used in the multivariate series considered
> in ticket #1956 .
New description:
In this patch the nth-root of power series `y = x^n` is introduced using
the Newton method for `y = x^-n`
{{{
x' = (1+1/n)*x - y*x^(n+1)/n (1)
}}}
{{{
sage: R.<t> = QQ[]
sage: p = (1 + 2*t + 5*t^2 + 7*t^3 + O(t^4))^3
sage: p.nth_root(3)
1 + 2*t + 5*t^2 + 7*t^3 + O(t^4)
sage: p = (1 + 2*t + 5*t^2 + 7*t^3 + O(t^4))^-3
sage: p.nth_root(-3)
1 + 2*t + 5*t^2 + 7*t^3 + O(t^4)
}}}
The iterations are division-free;
in the case `n=2` one can compare this division-free iteration
with the iteration used in the Newton method for `sqrt`
{{{
x' = (x +y/x)/2 (2)
}}}
`nth_root` can be used to compute the square root of series which
currently `sqrt` does not support
{{{
sage: R.<x,y> = QQ[]
sage: S.<t> = R[[]]
sage: p = 1 + x*t + (x^2+y^2)*t^2 + O(t^3)
sage: p1 = p.nth_root(2); p1
1 + 1/2*x*t + (3/8*x^2 + 1/2*y^2)*t^2 + O(t^3)
sage: p1^2
1 + x*t + (x^2 + y^2)*t^2 + O(t^3)
}}}
In particular it can be used in the multivariate series considered
in ticket #1956 .
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10720#comment:1>
Sage <http://www.sagemath.org>
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