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Shifting nth root algorithm
The '''shifting ''n''th root algorithm''' is an [[algorithm]] for extracting the [[nth root|''n''th root]] of a positive [[real number]] which proceeds iteratively by shifting in ''n'' [[numerical digit|digits]] of the radicand, starting with the most significant, and produces one digit of the root on each iteration, in a manner similar to [[long division]].

==Algorithm==

===Notation===

(contracted; show full)               2 79 75

===[[Cube root]] of 5===

      1.  7   0   9   9   7
     ----------------------
   3/ 5.000 000 000 000 000
 /\/  1 = 300
*×(0^2)*×1+30*0*×0×(1^2)+1^3
      -
      4 000
      3 913 = 300*×(1^2)*×7+30*1*×1×(7^2)+7^3
      -----
         87 000
              0 = 300*×(17^2)*0+30*×17*×(0^2)+0^3
        -------
         87 000 000
         78 443 829 = 300*×(170^2)*×9+30*×170*×(9^2)+9^3
         ----------
          8 556 171 000
          7 889 992 299 = 300*×(1709^2)*×9+30*×1709*×(9^2)+9^3
          -------------
            666 178 701 000
            614 014 317 973 = 300*×(17099^2)*×7+30*×17099*×(7^2)+7^3
            ---------------
             52 164 383 027

===Fourth root of 7===

      1.   6    2    6    5    7
     ---------------------------
 _ 4/ 7.0000 0000 0000 0000 0000
  \/  1 = 4000×(0^3)×1+400×(0^2)×(1^2)+40×0×(1^3)+1^4
      -
      6 0000
      5 5536 = 4000×(1^3)×6+600×(1^2)×(6^2)+40×1×(6^3)+6^4
      ------
        4464 0000
        3338 7536 = 4000×(16^3)×2+600*(16^2)×(2^2)+40×16×(2^3)+2^4
        ---------
        1125 2464 0000
        1026 0494 3376 = 4000×(162^3)×6+600×(162^2)×(6^2)+40×162×(6^3)+6^4
        --------------
          99 1969 6624 0000
          86 0185 1379 0625 = 4000×(1626^3)×5+600×(1626^2)×(5^2)+
          -----------------   40×1626×(5^3)+5^4
          13 1784 5244 9375 0000
          12 0489 2414 6927 3201 = 4000×(16265^3)×7+600×(16265^2)×(7^2)+
          ----------------------   40×16265×(7^3)+7^4
           1 1295 2830 2447 6799

[[Category:Root-finding algorithms]]

[[de:Schriftliches Wurzelziehen]]
[[fr:Algorithme de décalage n-racines]]
[[nl:Worteltrekken]]