Numerical Analysis Help

ThornMartin

So, I'm studying for a numerical analysis midterm, and I've come across one of the example problems that the professor gave that's confusing me.

Here's the question:

"x(n+1) = .5 * x(n) + 1/x(n). Identify as Newton's method for certain function, f(x)."

(Note that (n+1) and (n) refer to the subscripts of x)

Now, I know the steps to follow:

x(n+1) = x(n) - .5 * x(n) + 1/x(n)
Since Newton's method looks like x(n+1) = x(n) - f(x(n))/f'(x(n)), I'm looking for an f(x) s.t. f(x)/f'(x) = .5 * x - 1/x.

I know the answer is f(x) = x^2 - 2, but I have no idea how to solve that kind of problem in general.

For example, in another study problem, I need to find f(x) s.t. f(x)/f'(x) = 3x - y * 3(x^(2/3)), for some constant y, and I have no idea how to proceed.

Any advice would be appreciated, as I'm positive that I'll have this type of problem on my midterm.

JoeUser

It's been a while since I did any of this, but it looks like the key is to just get a common denominator.

Well here are some steps I did:
f(x)/f'(x) = .5 * x - 1/x

= .5*x*(x/x) - 1/x

= (.5*x^2)/x - 1/x

= (.5*x^2 - 1)/x

I don't know why your answer key has f(x) = x^2 - 2, but that's just 2/2 times my result.

For the second:
f(x)/f'(x) = 3x - y * 3(x^(2/3))

= 3x - y * 3 / (x^(3/2)) [just flipped to get a denominator]

= 3x * [(x^(3/2)) / (x^(3/2))] - y * 3 / (x^(3/2))

= (3x^(5/2)) / x^(3/2) - y * 3 / (x^(3/2))

= [3x^(5/2)) - y*3] / (x^(3/2))

When I got to this point, it didn't look like it would work out, since f' of the top is (15/2)*x^(3/2) = 7.5*x^(3/2)

However! We can go back to step one and multiply y * 3(x^(2/3)) by 7.5/7.5

So at the end we have:
= [3x^(5/2)) - 7.5*y*3] / 7.5*(x^(3/2))

Since the constant term is thrown away in the derivative, we can do that.

Sorry I can't give you an exact way to do it; I just played around to get common denominators and went from there.

Does that many any sense?