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Math Help - 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. [A(t)][X] = 0

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

    I need to find solutions for t satisfying |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
    658
    Thanks
    131
    For a none trivial solution (i.e. a solution for which not all of x_0,x_1, x_2, ... , x_n are zero),
    it is necessary that the determinant of A

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

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


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

    and
    A_n(t) = \frac{p_{n2} t^2 + p_{n1} t + p_{n0}}{q_{n2} t^2 + q_{n1} t + q_{n0}}<br />

    Is it possible to give an analytical solution?
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