Question

Use newtons method and the function f(x)=x^2-a 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,...

Answer 1

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.