# To really find a solution, Positively

During my research in Engineering, I came across this problem.

There is a complex valued function, with the domain as the imaginary axis.

$P(j\omega) = P_R(\omega^2)+j\omega P_I(\omega^2)$

I need to find the value of $P(j\omega)$, at the sequence of points where it is real and draw appropriate conclusions.

There are two scenarios here. (1) Both $P_R(x)$ and $P_I(x)$ are polynomials in $x$. (2) Not case 1.

For case 1 : As we are interested in values of $P_R(\omega^2$) at those $\omega$, where $P_I(\omega^2)=0$, it is obvious that we are interested ONLY in positive and real solutions to $P_I(x)=0$.

My question to the community is this.

• Are there any methods to find the real roots (or better positive real roots) of a polynomial, without actually finding all the roots?*

If there is no simpler solution, than finding all the roots; then the procedure which I envision would become more computationally complex than a conventional technique that is more than 80 years old.

If case 1 is worth pursuing, then case 2 could be an interesting extension.

Note by Janardhanan Sivaramakrishnan
3 years, 8 months ago

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