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Math Help - Newton Method - HELP!

  1. #1
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    Newton Method - HELP!

    How would I carry out the newton method on the non-linear system below:

    x12 - 10x1 + x22 + 8 = 0

    x1x22 + x1 - 10x2 + 8= 0

    Perform one iteration using (0,0)T

    What are the steps i have to do?
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  2. #2
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    Re: Newton Method - HELP!

    Quote Originally Posted by NFS1 View Post
    How would I carry out the newton method on the non-linear system below:

    x12 - 10x1 + x22 + 8 = 0

    x1x22 + x1 - 10x2 + 8= 0

    Perform one iteration using (0,0)T

    What are the steps i have to do?
    Newton's method - Wikipedia, the free encyclopedia
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  3. #3
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    Re: Newton Method - HELP!

    I find it surprising that you would have a problem like this if you do not know "the steps".

    In any case, the "Newton-Raphson" method for solving, say, f(x)= 0, is to start with some "test value", x_0, and approximate y= f(x) with its tangent line approximation, y= f'(x_0)(x- x_0)+ f(x_0). Setting that equal to 0 and solving for x, giving x= x_0- \frac{f(x_0)}{f'(x_0)}, will, under reasonable conditions, give a new value of x, closer to the root. Then we can repeat to get even closer.

    Here, f is "vector" function f(x_1, x_2)= \begin{pmatrix}x_1^2- 10x_1+ x_2^2+ 8 \\ x_1x_2^2+ x_1- 10x_2+ 8\end{pmatrix} and its derivative is the matrix \begin{pmatrix}2x_1- 10 & 2x_2 \\ x_2^2+ 1 & 2x_1x_2- 10\end{pmatrix} so that, for a given starting point (x_1, x_2) the next point would be \begin{pmatrix}x_1\\ x_2\end{pmatrix}- \begin{pmatrix}2x_1- 10 & 2x_2 \\ x_2^2+ 1 & 2x_1x_2- 10\end{pmatrix}^{-1}\begin{pmatrix}x_1^2- 10x_1+ x_2^2+ 8 \\ x_1x_2^2+ x_1- 10x_2+ 8\end{pmatrix}.
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