### 6.6 Examples

To demonstrate the use of the general polynomial solver we will take the polynomial P(x) = x^5 - 1 which has the following roots,

    1, e^{2\pi i /5}, e^{4\pi i /5}, e^{6\pi i /5}, e^{8\pi i /5}

The following program will find these roots.

    #include <stdio.h>
    #include <gsl/gsl_poly.h>

    int
    main (void)
    {
      int i;
      /* coefficients of P(x) =  -1 + x^5  */
      double a[6] = { -1, 0, 0, 0, 0, 1 };
      double z[10];

      gsl_poly_complex_workspace * w
          = gsl_poly_complex_workspace_alloc (6);

      gsl_poly_complex_solve (a, 6, w, z);

      gsl_poly_complex_workspace_free (w);

      for (i = 0; i < 5; i++)
        {
          printf ("z%d = %+.18f %+.18f\n",
                  i, z[2*i], z[2*i+1]);
        }

      return 0;
    }

The output of the program is,

    $ ./a.out

    z0 = -0.809016994374947673 +0.587785252292473359
    z1 = -0.809016994374947673 -0.587785252292473359
    z2 = +0.309016994374947507 +0.951056516295152976
    z3 = +0.309016994374947507 -0.951056516295152976
    z4 = +0.999999999999999889 +0.000000000000000000

which agrees with the analytic result, z_n = \exp(2 \pi n i/5).