Posted by Elizabeth on Friday, July 20, 2012 at 10:56pm.
Newton's method for the root of f(x) is:
x_{n+1} = x_{n} - f(x_{n})/f'(x_{n})
= x_{n} - (x_{n}^2 - a)/(2 x_{n}) =
1/2 (x_{n} + a/(x_{n}))
This rule is not so convenient, because you need to divide by x_{n} at each step (division is slower than multiplication). A rule without division can be obtained by writing the square root as the root of the function:
f(x) = x^(-2) - 1/a
Newton's method then yields:
x_{n+1} = x_{n} -
( x_{n}^(-2) - 1/a)/(-2x_{n}^(-3)) =
1/2 x_{n} (3 - x_{n}^2/a)
So, in each step you now only have a division by a, so you only have to compute 1/a once.
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