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November 13th, 2008, 11:47 AM   #1
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Newton's Method

I need help on a Newton's Method problem. I have to "Use Newton's method to approximate the given number correct to eight decimal places".

The number is the hundredth root of 100 (100^(1/100)). When done on calculator, the result is 1.04712855. I've tried the approximation but am not able to get close.

The method is Xn+1=Xn-(f(n)/f'(n))

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November 13th, 2008, 01:15 PM   #2
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Re: Newton's Method

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Originally Posted by djo201
I've tried the approximation but am not able to get close.

The method is Xn+1=Xn-(f(n)/f'(n))

Thanks
Can you not show more precisely the work you have done; i.e. the steps to see where you might have made an error? This is just a description of the method, of which we are already aware.

[You might want to multiply your calculator answer by 100?] First, what equation did you set up to use for the calculations?

Do note also, from the Wicki:
"Newton's method makes no guarantee on convergence, and depending on the shape of the function and the starting point it may or may not converge."
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November 13th, 2008, 01:26 PM   #3
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Re: Newton's Method

From the examples I've seen on the book, f(x) = x^(100)-100 and f'(x) = 100x^99

I've tried using n=1, n=2, n=3, also tried big numbers and cannot seem to come up with the true approximation.
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November 13th, 2008, 01:50 PM   #4
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Re: Newton's Method

Quote:
Originally Posted by djo201
From the examples I've seen on the book, f(x) = x^(100)-100 and f'(x) = 100x^99

I've tried using n=1, n=2, n=3, also tried big numbers and cannot seem to come up with the true approximation.
What do you use for the initial guess (X0)? Since you know the answer from the calculator, you could start with something close by, like 1.047.
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November 13th, 2008, 02:08 PM   #5
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Re: Newton's Method

For the initial guess I use the Intermediate Value Theorem. But in this case it isn't helping much. Someone else told me to to try 1.1 and I think it is the best choice in this case, even though I have to repeat more than 9 times to get the 8 decimal places.
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November 13th, 2008, 02:24 PM   #6
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Re: Newton's Method

Anyone know how to delete a message already sent? I am logged in and have cookies on.
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November 15th, 2008, 01:32 PM   #7
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Re: Newton's Method

To find the first approximation, you can use the value from the calculator.
If you do not want to "cheat", you can repeatedly take the square root of 100 9 times until you get
100^(1/512)=100^.001953125=1.009035,
raising it to the fifth power gives you 100^(.009765625)= 1.045999, which is close enough for an initial "guess".
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