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Math Help - Polynomials for the General Iterative Method

  1. #1
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    Exclamation Polynomials for the General Iterative Method

    "Using the General Iterative Method estimate the greatest positive solution to the following equation: x^4 - 2x^2 - x - 3 = 0"

    ( ^ means to the power of , e.g x^2 is the same as x squared)

    I need help simplifying the equation. I'm fine with the General Iterative Method, and I've attempted to simplify the equation so that I can then use it in the General Iterative Method, but I have doubts about whether my answer is right. Please help! I need to learn this before Monday 30th April (Australian EST).
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  2. #2
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    Quote Originally Posted by loquaci View Post
    "Using the General Iterative Method estimate the greatest positive solution to the following equation: x^4 - 2x^2 - x - 3 = 0"

    ( ^ means to the power of , e.g x^2 is the same as x squared)

    I need help simplifying the equation. I'm fine with the General Iterative Method, and I've attempted to simplify the equation so that I can then use it in the General Iterative Method, but I have doubts about whether my answer is right. Please help! I need to learn this before Monday 30th April (Australian EST).
    Could you tell us what the "General Iterative Method" is?

    Descartes rule of signs tells us that x^4-2x^2-x-3=0 has exactly one positive real solution.
    For large x, f(x) = x^4-2x^2-x-3 is positive, so if we put x=10, we find that f(x)~=9000, so
    the root is less that 10.

    Rearrange the equation as:

    x = FourthRoot(x^2+x+3)

    and use the iterative scheme:

    x_{n+1} = FourthRoot(x_n^2+x_n+3)

    and we find:

    Code:
                n       x_n
                0       10.0000
                1       3.82028 
                2       2.44965 
                3       2.04388 
                4       1.91323 
                5       1.87022 
                6       1.85598 
                7       1.85125 
                8       1.84969 
                9       1.84916 
               10       1.84899 
               11       1.84893 
               12       1.84891 
               13       1.84891 
               14       1.84891
    So the (largest and only) positive root of x^4-2x^2-x-3=0 is ~1.84891

    RonL
    Last edited by CaptainBlack; April 28th 2007 at 05:12 AM.
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