Friday

January 30, 2015

January 30, 2015

Posted by **Elizabeth** on Friday, July 20, 2012 at 10:56pm.

Mechanics Rule for approaching sqrt(a), a>0, is

Xn+1=1/2(Xn+a/Xn), n=1,2,3,...

- Calculus -
**Count Iblis**, Friday, July 20, 2012 at 11:16pmNewton'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.

**Answer this Question**

**Related Questions**

Calculus - Am I right so far? G(x)=e^x sqrt(1+x^2) =e^x(1+x^2)^1/2 =e^x(1/2)(1+x...

Calculus - Can anyone explain how to use the chain rule and power rule together ...

Calculus - Match the rule with the title: ____ 3. d/dx [f(x)/g(x) ]=(g(x) f^' (x...

pre-cal - Find the roots of the function f(x)= x^2+2x+2 Determine f(x) a) (x+1-i...

limit calc question - 2 questions!!! 1. Limit X approaching A (X^1/3-a^1/3)/ x-a...

Calculus - Use the General Power Rule or the Shifting and Scaling Rule to find ...

calculus - Find the derivative of the function below in two ways. F(x) = (x - 7 ...

CALCULUS!! OH DEATH!!! - I need to solve: Lim as x is approaching 1/4, for (4x-1...

calculus - how do not understand how to find a deriviative of this function?Do i...

Math - Calculus Question. - hey can someone explain to me the relationship ...