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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]](../../art/inf.gif) |
1.259921049894... |
Source: Jeff Yates, et al.
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