Proving Limit of y[n] for Newton-Raphson Method

In summary, the given function is the Newton-Raphson Method and can be proven analytically by setting y[n] = y[n-1] and taking the limit as n approaches infinity. This leads to a quadratic equation in y, which can be solved to show that y[n] approaches sqrt(a) in the infinite limit.
  • #1
dduardo
Staff Emeritus
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Can someone get me going in the right direction.

For the given function:

y[n] = (1/2)(y[n-1] + x[n]/y[n-1])

where x[n] = a * u[n] (u[n] is the unit step function)

and y[-1] = 1

prove that y[n] as n -> infinity is equal to sqrt(a)

-----------

I know that this is the Newton-Raphson Method, but how do I go about analytically proving the above.

I've tried writing out a few terms and seeing if there is a pattern, but couldn't find anything.

The inside looks like a accumulator and tried to do a subsitution, but that didn't work.

Any help would be appreciated.
 
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  • #2
When you see limiting behavior, it means that the differencebetween successive terms gets smaller and smaller. In the infinite limit, the difference between successive terms should be 0.

So, set y[n] = y[n-1] and see what happens.
 
  • #3
I think I need a little more information.

So you are saying lim n-> inf (y[n] - y[n-1]) = 0 ? But since I'm dealing with a circular function how would I go about taking the limit?
 
  • #4
dduardo said:
I think I need a little more information.

So you are saying lim n-> inf (y[n] - y[n-1]) = 0 ?
Yes, does it seem reasonable to you that this should be true ?

But since I'm dealing with a circular function how would I go about taking the limit?

There really isn't much to do. Let's call [itex]\lim_{n \rightarrow \infty} y[n] \equiv y [/itex]

Then you have [tex] y = \frac {y}{2} + \frac {x}{2y} [/tex]

This gives you a quadratic in y.
 
  • #5
Ok, now I understand. Thanks
 

1. What is the Newton-Raphson Method?

The Newton-Raphson Method is an iterative root-finding algorithm used to approximate solutions to equations of the form f(x) = 0. It is based on the concept of using a tangent line to approximate the root of a function.

2. How does the Newton-Raphson Method work?

The method starts with an initial guess, x0, and then iteratively improves the guess using the formula xn+1 = xn - f(xn)/f'(xn). This process is repeated until the desired level of accuracy is achieved.

3. What is the limit of y[n] for the Newton-Raphson Method?

The limit of y[n] for the Newton-Raphson Method is the root of the function f(x). In other words, as n approaches infinity, y[n] will converge to the actual root of the function.

4. How is the limit of y[n] for the Newton-Raphson Method proven?

The limit of y[n] for the Newton-Raphson Method can be proven using mathematical induction or by showing that the sequence {y[n]} is monotonically decreasing and bounded below. This ensures that the limit exists and is equal to the root of the function.

5. What are the limitations of the Newton-Raphson Method?

The Newton-Raphson Method may fail to converge if the initial guess is not close enough to the root or if there are multiple roots. It also requires the calculation of the derivative of the function, which can be difficult or impossible in some cases.

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