Newton-Raphson help!

thedc

1. The problem statement, all variables and given/known data

[Divide and average Method] Square root of 2 was computed using the formula

Xi+1 = 1/2 ( Xi + 2/Xi).------------ (1)

Derive this method from the Newton-Raphson Formula

2. Relevant equations



3. The attempt at a solution

Im boggled at how to derive this solution.

The equation for Newton Raphson is

F'(Xi) = (F(Xi) - 0) / Xi -(Xi+1) ------------- (2)

which can be rearranged to

Xi+1 = Xi - F(Xi) / F'(Xi)--------- (3)

does this mean that i take the derivative of the equation (1)?

(Xi+1)' =1/2(Xi+2/Xi)
= ??

CEL

You have
[tex]x^2 = 2[/tex]
or
[tex]x = \frac{2}{x}[/tex]
So,
[tex]f(x) = \frac{2}{x}[/tex]
Derive your equations from there.

thedc

I still dont get it, do i take the derivative of 2/x?

that would be f(x)'=-2(1/x^2)

CEL

What is the Newton-Raphson method?

Hidemons

Good job guy.

/s

Hidemons

I am having the same problem.

Newton Raphson method: Xof(i+1) = xi - f(x)/f(x)'

it is used to find roots by iteration

CEL

Write your equation in the form y = f(x).
Calculate f'(x).
Choose a starting value for x0.
If y - f(x0) < tolerance then end
else
Calculate x1 using Newton-Raphson formula.
Iterate

TheoMcCloskey

I think there is sufficient confusion amoung these posts to warrent another (hopefully non-confusing) post

thedc: For Newton-Raphson, you are looking for the zero of a function (F), hence, you need to express the function (F) such that F(x) = 0.

In your original post, you desire to find the answer to [itex]x[/itex] for [itex]x = \sqrt{2}[/itex]. Consider the more genral solution for [itex]x[/itex] with [itex]x = \sqrt{A}[/itex] for some positive [itex]A[/itex].

Question: How can we express a function, [itex]F(x)[/itex], such that it results in [itex]F(x)=0[/itex] for this problem?

Answer: Look at the [itex]x = \sqrt{A}[/itex]. This is really the same as finding [itex]x^2[/itex] such that [itex]x^2 = A[/itex]. Hence, one selection of [itex]F(x)[/itex] might be [itex]F(x) = x^2 - A=0[/itex].

This is the "F" that is needed in the N-R method. The iterates for the solution of x are as follows:

[tex]
x_{\nu+1} = x_{\nu}-\frac{F(x_{\nu})}{F'(x_{\nu})}
[/tex]

Here, [itex]F'(x)[/itex] is shorthand to mean [itex]\frac{d\,}{dx}F(x)[/itex]. Also, in your case, the vale of A is A=2. You will need an initial estimate [itex]x_{0}[/itex] to start this procedure.



The key to achieve the end goal of your exercise is to do some algebra on the resulting iterate expresion once you take the derivative of F and substitute it into the expression.

Hope this helps.