Calculus
posted by Elizabeth .
Use newtons method and the function f(x)=x^2a to derive the mechanics rule.
Mechanics Rule for approaching sqrt(a), a>0, is
Xn+1=1/2(Xn+a/Xn), n=1,2,3,...

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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