Deriving the Divide and Average Method from Newton-Raphson Formula

[thedc]

Homework Statement

[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

Homework Equations

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]

Homework Statement

[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

Homework Equations

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)
= ??

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

 

[thedc]

I still don't get it, do i take the derivative of 2/x?

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

 

[CEL]

I still don't get it, do i take the derivative of 2/x?

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

What is the Newton-Raphson method?

 

[Hidemons]

What is the Newton-Raphson method?

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]

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

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 x for x = \sqrt{2}. Consider the more general solution for x with x = \sqrt{A} for some positive A.

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

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

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

<br /> x_{\nu+1} = x_{\nu}-\frac{F(x_{\nu})}{F&#039;(x_{\nu})}<br />

Here, F&#039;(x) is shorthand to mean \frac{d\,}{dx}F(x). Also, in your case, the vale of A is A=2. You will need an initial estimate x_{0} 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.