I have a system of $2n$ linear equations in $2n$ unknowns represented by the standard matrix equation:
$$Ax = b$$
Where the solution vector $x = (p_1, ..., p_n, q_1, ..., q_n)$ represents real coefficients of the function:
$$ g(x) = \frac {P(x)} {Q(x)} \qquad P(x) = \sum_{i = 0}^n p_i tan^{2i}(x) \qquad Q(x) = \sum_{i = 0}^n q_i tan^{2i}(x) $$
Note: $p_0$ and $q_0$ are known constants and so are not part of the solution vector $x$ and $g(x)$ is valid in the range $ x = [0, \frac {\pi} {2}] $.
Here's the really interesting part...
To generate my system of linear equations I take $2n$ reference points from a set of functions in order to build my $A$ and $b$ to solve $x$. Let a function in this set be $f(y)$ where y is positive and bounded and $f(y)$ is differentiable.
My experiments lead me to the following observations:
For a given $f(y)$, I can take $2n$ reference points (sensibly chosen to capture the characteristics of $f(y)$).
The majority of the time the reference points lead to well behaved $g(x)$ but every so often $P(x)$ and/or $Q(x)$ will have one or more zeros/roots in the region $ x = [0, \frac {\pi} {2}] $ leading to a bad $g(x)$ i.e. if a root of P(x) and/or Q(x) is positive real then $g(x)$ is bad.
It turns out that simply selecting different references points will correct the issue and the zeros go away.
So my problem is: it seems that there exist a set of reference points $y = (y_1, ..., y_{2n})$ so that $f(y)$ generates a system of linear equations $Ax = b$ that provides a solution vector $x = (p_1, ..., p_n, q_1, ..., q_n)$ where $P(x)$ and $Q(x)$ have no positive real roots in the region $ x = [0, \frac {\pi} {2}] $.
The question then: how to select these points?
I have so far concluded that if the solution vector $x$ satisfies the Hurwitz criterion as coefficients of a polynomial then the $g(x)$ would be stable but I do not need complex roots to have negative reals, I only need real roots to be negative (so Hurwitz seems overly restrictive in this case).
Is there any mathematics that will help me? I think this is an interesting problem because the references points $y$ of $f(y)$ can be chosen arbitrarily within a bounded positive region. The constraint is choosing points that don't produce roots/zeros for $P(x)$ and $Q(x)$.