
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_{n1}(t) & 0 & ... & 1 & 0 \\
A_n(t) & 0 & ... & 0 & 1 \\
\end{array}
\right)
\left(
\begin{array}{c}
x_0 \\
x_1 \\
... \\
x_{n1} \\
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

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 $\displaystyle \lambda.$

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_{n1}(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?