Is there an algorithm to find complex roots of equations of high degrees? Let's suppose I'm given an even function of degree greater than 6 that does not have real roots, how am I supposed to find its complex roots efficiently? (please no trial and error methods, and methods that needs computer calculations)

(This is basically for differential equations, solving the characteristic equation)

EDIT: Something like Newton's method, https://www.wikipedia.com/en/Newton's_method

  • $\begingroup$ A useful fact is that complex roots come in conjugate pairs. And as far as I'm aware, there is no formula for computing the roots of polynomials with degree 5 or more (you may want to check this though). $\endgroup$ – Mattos Nov 19 '15 at 12:20
  • $\begingroup$ As @Mattos suggests, there is no general formula for the roots of polynomials of degree $\geq 5$. Unless you can reduce the degree of the polynomials involved, e.g. by find a partial factorization of your polynomial or by writing it as $p(q(x))$ for some polynomials $p,q$ of degree $\geq 2$, you are out of luck. If you have a specific problem in mind you should ask about that instead. $\endgroup$ – A.P. Nov 19 '15 at 12:51
  • $\begingroup$ ahh so it really boils down to reducing the polynomial? I'm thinking about some algorithm like Newton's (the successive iterations for roots) wikipedia.com/en/Newton's_method $\endgroup$ – I_Love_Eng Nov 19 '15 at 13:34
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    $\begingroup$ Sure, there are many root-find algorithms out there, some of which can efficiently give you a good approximation of your roots... but you ruled those out in your question because (in general) the result is not exact and they need computer calculations — that holds for Newton's method, too. $\endgroup$ – A.P. Nov 19 '15 at 14:15

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