— Function: int gsl_poly_solve_cubic (double a, double b, double c, double * x0, double * x1, double * x2)

This function finds the real roots of the cubic equation,

          x^3 + a x^2 + b x + c = 0

with a leading coefficient of unity. The number of real roots (either one or three) is returned, and their locations are stored in x0, x1 and x2. If one real root is found then only x0 is modified. When three real roots are found they are stored in x0, x1 and x2 in ascending order. The case of coincident roots is not considered special. For example, the equation (x-1)^3=0 will have three roots with exactly equal values. As in the quadratic case, finite precision may cause equal or closely-spaced real roots to move off the real axis into the complex plane, leading to a discrete change in the number of real roots.

— Function: int gsl_poly_complex_solve_cubic (double a, double b, double c, gsl_complex * z0, gsl_complex * z1, gsl_complex * z2)

This function finds the complex roots of the cubic equation,

          z^3 + a z^2 + b z + c = 0

The number of complex roots is returned (always three) and the locations of the roots are stored in z0, z1 and z2. The roots are returned in ascending order, sorted first by their real components and then by their imaginary components.