# Math Help - Root finding problem

1. ## 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. $[A(t)][X] = 0$

$
\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 $t$ 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

2. 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 $\lambda.$

3. Thank you.

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

$
\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
$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?