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(jω)P(j\omega), at the sequence of points where it is real and draw appropriate conclusions.

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

For case 1 : As we are interested in values of PR(ω2P_R(\omega^2) at those ω\omega, where PI(ω2)=0P_I(\omega^2)=0, it is obvious that we are interested ONLY in positive and real solutions to PI(x)=0P_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
5 years, 3 months ago

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