Changes between Version 1 and Version 2 of Ticket #10720


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Timestamp:
Feb 1, 2011, 5:23:32 PM (12 years ago)
Author:
pernici
Comment:

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  • Ticket #10720 – Description

    v1 v2  
    1 In this patch the nth-root of power series `y = x^n` is introduced using
    2 the Newton method for `y = x^-n`
    3 {{{
    4 x' = (1+1/n)*x - y*x^(n+1)/n               (1)
    5 }}}
     1computation of an nth root of a power series using the Newton method
    62
    7 {{{
    8 sage: R.<t> = QQ[]
    9 sage: p = (1 + 2*t + 5*t^2 + 7*t^3 + O(t^4))^3
    10 sage: p.nth_root(3)
    11 1 + 2*t + 5*t^2 + 7*t^3 + O(t^4)
    12 sage: p = (1 + 2*t + 5*t^2 + 7*t^3 + O(t^4))^-3
    13 sage: p.nth_root(-3)
    14 1 + 2*t + 5*t^2 + 7*t^3 + O(t^4)
    15 }}}
    16 
    17 The iterations are division-free;
    18 in the case `n=2` one can compare this division-free iteration
    19 with the iteration used in the Newton method for `sqrt`
    20 {{{
    21 x' = (x +y/x)/2                            (2)
    22 }}}
    23 
    24 
    25 `nth_root` can be used to compute the square root of series which
    26 currently `sqrt` does not support
    27 {{{
    28 sage: R.<x,y> = QQ[]
    29 sage: S.<t> = R[[]]
    30 sage: p = 1 + x*t + (x^2+y^2)*t^2 + O(t^3)
    31 sage: p1 = p.nth_root(2); p1
    32 1 + 1/2*x*t + (3/8*x^2 + 1/2*y^2)*t^2 + O(t^3)
    33 sage: p1^2
    34 1 + x*t + (x^2 + y^2)*t^2 + O(t^3)
    35 }}}
    36 
    37 In particular it can be used in the multivariate series considered
    38 in ticket #1956 .
     3Apply trac_10720_power_series_nth_root_2.patch