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Recursive Formulas
(Math | Advanced | Recursive Formulas | y1/n)

Recursive Formulas for y1/n

Explicit form:

y1/n = x
Recursive form:
xk+1 = (xk + y / (xk)n-1) / 2

where y ³ 0 and n > 0
or y Î Â and n is odd, positive, and integer.
(for negative n, evaluate the above formula with n positive, then invert your answer).

Example: 21/3 = 1.259921049894...
iteration value
x0 1.0000000000
x1 1.5000000000
x2 1.1944444444
x3 1.2981416381
x4 1.2424821566
x5 1.2690093603
x6 1.2554742937
x7 1.2621680807
x8 1.2588035314
x9 1.2604812976
x10 1.2596412994
x20 1.2599207765
x50 1.259921049894
x[infinity] 1.259921049894...
Source: Jeff Yates, et al.

  
 
  

 
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