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:

$$x_{\nu+1} = x_{\nu}-\frac{F(x_{\nu})}{F'(x_{\nu})}$$

Here, $F'(x)$ 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.