### 6.5 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. If the QR reduction does not converge, the error handler is invoked with an error code of `GSL_EFAILED`. Note that due to finite precision, roots of higher multiplicity are returned as a cluster of simple roots with reduced accuracy. The solution of polynomials with higher-order roots requires specialized algorithms that take the multiplicity structure into account (see e.g. Z. Zeng, Algorithm 835, ACM Transactions on Mathematical Software, Volume 30, Issue 2 (2004), pp 218–236).