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Thread: Root finding problem

  1. #1
    Newbie
    Joined
    Apr 2009
    Posts
    16

    Root finding problem

    Hi,
    While trying to simplify a solution I came up with the following sparse matrix but don't know how to solve it. $\displaystyle [A(t)][X] = 0 $

    $\displaystyle
    \left(
    \begin{array}{ccccc}
    A_0(t) & 1 & 1 & ... & 1 \\
    A_1(t) & 1 & 0 & ... & 0 \\
    ... & ... & ... & ... & ... \\
    A_{n-1}(t) & 0 & ... & 1 & 0 \\
    A_n(t) & 0 & ... & 0 & 1 \\
    \end{array}
    \right)
    \left(
    \begin{array}{c}
    x_0 \\
    x_1 \\
    ... \\
    x_{n-1} \\
    x_n
    \end{array}
    \right)
    = 0


    $

    I need to find solutions for $\displaystyle t$ satisfying $\displaystyle |A(t)| = 0 $

    I would really appreciate if you would point me in finding out
    1. Whether there is actually an analytic solution for this
    2. If not a suitable numerical technique to find solutions

    Thanks

    Krindik
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  2. #2
    Super Member
    Joined
    Jun 2009
    Posts
    671
    Thanks
    136
    For a none trivial solution (i.e. a solution for which not all of $\displaystyle x_0,x_1, x_2, ... , x_n$ are zero),
    it is necessary that the determinant of $\displaystyle A$

    $\displaystyle \det{A(t)} = A_0(t) - A_1(t) - A_2(t) - ... - A_n(t) = 0,$

    and in that case there will be an infinite number of solutions.

    Let $\displaystyle x_0 = \lambda$, say, ($\displaystyle \lambda$ arbitrary) and solve for $\displaystyle x_i $ for $\displaystyle i = 1 ... n$ in terms of $\displaystyle \lambda.$
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  3. #3
    Newbie
    Joined
    Apr 2009
    Posts
    16
    Thank you.

    Infact, my problem is to find $\displaystyle t$ such that $\displaystyle |A(t)| = 0$ where,


    $\displaystyle
    \left(
    \begin{array}{ccccc}
    A_0(t) & 1 & 1 & ... & 1 \\
    A_1(t) & 1 & 0 & ... & 0 \\
    ... & ... & ... & ... & ... \\
    A_{n-1}(t) & 0 & ... & 1 & 0 \\
    A_n(t) & 0 & ... & 0 & 1 \\
    \end{array}
    \right) = [A(t)]

    $

    and
    $\displaystyle A_n(t) = \frac{p_{n2} t^2 + p_{n1} t + p_{n0}}{q_{n2} t^2 + q_{n1} t + q_{n0}}
    $

    Is it possible to give an analytical solution?
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