## General Polynomial Equations

The roots of polynomial equations cannot be found analytically beyond the special cases of the quadratic, cubic and quartic equation. The algorithm described in this section uses an iterative method to find the approximate locations of roots of higher order polynomials.

Function: gsl_poly_complex_workspace * gsl_poly_complex_workspace_alloc (size_t n)
  This function allocates space for a `gsl_poly_complex_workspace` struct and a workspace suitable for solving a polynomial with n coefficients using the routine `gsl_poly_complex_solve`.

  The function returns a pointer to the newly allocated `gsl_poly_complex_workspace` if no errors were detected, and a null pointer in the case of error.

Function: void gsl_poly_complex_workspace_free (gsl_poly_complex_workspace * w)
  This function frees all the memory associated with the workspace w.

Function: int gsl_poly_complex_solve (const double * a, size_t n, gsl_poly_complex_workspace * w, gsl_complex_packed_ptr z)
  This function computes the roots of the general polynomial P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1} using balanced-QR reduction of the companion matrix. The parameter n specifies the length of the coefficient array. The coefficient of the highest order term must be non-zero. The function requires a workspace w of the appropriate size. The n-1 roots are returned in the packed complex array z of length 2(n-1), alternating real and imaginary parts.

  The function returns `GSL_SUCCESS` if all the roots are found and `GSL_EFAILED` if the QR reduction does not converge.