[PURRS-devel] Extended arithmetic

[Tatiana Zolo]

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