Dear Markus,
thank you for your availability. We are studying recurrence relations and, consequently, the polynomial equations that derive from them. For instance: x_n = -3*x_{n-1} + x_{n-4} -> x^4 + 3*x^3 - 1 = 0 x_n = -3 * x_{n-10} -> x^10 + 3 = 0
For the polynomial equations of degree 2, 3 or 4 there is a formula (which may involve the computation of square roots, or root of higher index) but for the polynomial equations of degree 5 or more the general formula does not exist and then we only want an approximation of the roots.
In conclusion, we always have to deal with square roots or roots with higher index, and the elementary functions sin, cos: we need to evaluate
them at points of complex plane which may be everywhere, including the negative real axis. At the moment we want to find a (possibly complex) interval that contains the solution of any polynomial equation (to consider polynomial equation will be probably only a first step of our work...)
ex: we consider again x^4 + 3*x^3 - 1 = 0.
We find the solution
x_1 = -3/4+1/4*sqrt(9+4*(-9/2+sqrt(2443/108))^(1/3)-4* *(9/2+sqrt(2443/108))^(1/3))+1/2*sqrt((3/2-1/2*sqrt(9+4*(-9/2+
+sqrt(2443/108))^(1/3)-4*(9/2+sqrt(2443/108))^(1/3)))^2-2*sqrt(4+
+((-9/2+sqrt(2443/108))^(1/3)-(9/2+sqrt(2443/108))^(1/3))^2)-2* *(-9/2+sqrt(2443/108))^(1/3)+2*(9/2+sqrt(2443/108))^(1/3))
If we estimate this solution we get
x_1 = -0.3068848065239627712+0.64263790881274810124*I
and, using your library, we can not find the interval because it throws an exception.
Thank you. Best regards
Tatiana Zolo