# Root finding problem

• Sep 6th 2009, 11:01 PM
krindik
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
• Sep 7th 2009, 02:08 PM
BobP
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 $\displaystyle \lambda.$
• Sep 7th 2009, 03:12 PM
krindik
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?