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Math Help - Error Formula

  1. #1
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    Error Formula

    For the recursive algorithm p0=1, p1=1/3, pn=(10/3)(pn-1 - pn-2) for n >1 let the error be denoted by p'=pn+en for initial error e0, and assume the error has the form

    p'0=p0+e0
    p'1=p1+3e0
    p'n=(10/3)(p'n-1 - p'n-2) for n>1

    so that e1=3e0
    Find an equation for en in terms of e0, n, and a constant.
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  2. #2
    Grand Panjandrum
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    Re: Error Formula

    Quote Originally Posted by veronicak5678 View Post
    For the recursive algorithm p0=1, p1=1/3, pn=(10/3)(pn-1 - pn-2) for n >1 let the error be denoted by p'=pn+en for initial error e0, and assume the error has the form

    p'0=p0+e0
    p'1=p1+3e0
    p'n=(10/3)(p'n-1 - p'n-2) for n>1

    so that e1=3e0
    Find an equation for en in terms of e0, n, and a constant.
    p'_n=(10/3)[p'_{n-1}-p'_{n-2}]

    is a linear difference equation, so suppose p'_n=A^n for some real number A, then find the characteristic equation and solve for A. Form the general solution as a linear combination of the two solutions you will have found and fit to the initial conditions.

    CB
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  3. #3
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    Re: Error Formula

    Could you please explain how to find the characteristic equation?
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  4. #4
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    Re: Error Formula

    Quote Originally Posted by veronicak5678 View Post
    Could you please explain how to find the characteristic equation?
    You make the substitution p'_n=A^n in the difference equarion to get:

    A^n=\frac{10}{3}[A^{n-1}-A^{n-2}}]

    Now divide through by A^{n-2} to get:

    A^2=\frac{10}{3}[A-1]

    which is a quadratic in A which rearranges to:

    A^2-(10/3)A+(10/3)=0

    which is the charateristic equation which you solve for A.

    CB
    Last edited by CaptainBlack; November 12th 2011 at 08:11 PM.
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  5. #5
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    Re: Error Formula

    I see. Thanks!
    Last edited by veronicak5678; November 12th 2011 at 03:08 PM.
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