Newton-Raphson MethodDate: 02/28/2000 at 09:47:19 From: James Dyer Subject: Direct-Search / Newton-Raphson Method Using direct-search method and Newton-Raphson method, find the 1st, 2nd, 3rd iterations and the parameters of the following: x^3-13.1x^2+48.48x-46.62 I'm not sure how to do this. Help, please? Date: 02/28/2000 at 10:50:59 From: Doctor Mitteldorf Subject: Re: Direct-Search / Newton-Raphson Method Dear James, You can help us help you by telling us what you know and exactly where you get stuck. Don't be afraid to flood us with information - let's see what you've tried already, even things that you figured out along the way weren't going to work. I'll do the best I can, but I can't be sure I'm aiming at the kind of answer that will be most helpful to you. I don't know what your teacher means by the direct-search method. Perhaps you do. I do know Newton-Raphson, however, and here's an explanation of it that I recently wrote for another questioner: Inventing an Operation to Solve x^x = y http://mathforum.org/dr.math/problems/redsting.02.19.00.html In your case, f(x) = x^3 - 13.1x^2 + 48.48x - 46.62 and y isn't specified - presumably it's zero: x^3 - 13.1x^2 + 48.48x - 46.62 = 0 - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ Date: 02/28/2000 at 10:53:40 From: Doctor Jerry Subject: Re: Direct-Search / Newton-Raphson Method Hi James, Because it's a cubic, all you have to do is to determine one real root. You can locate that root by finding an interval in which the sign changes. At 0 and 1 this polynomial is negative and at x = 2 it is positive. So, there's a root between 1 and 2. We'll try x_1 = 1.5, where by x_1 I mean "x sub 1," the first guess for Newton's method. x_2 = 1.26679325896 x_3 = 1.26102002205 etc. where x_{n+1} = x_n-f(x_n)/f'(x_n), n = 1, 2, 3, ... Now you can use synthetic division to remove this root, leaving a quadratic, which you can solve by hand if you like. Or you can locate roots near 4 and 7 (as above) and then use Newton's method. - Doctor Jerry, The Math Forum http://mathforum.org/dr.math/ |
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