Roots of Polynomials Examples - GNU Scientific Library -- Reference Manual
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### 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 = -0.809016994374947451 +0.587785252292473137
         z1 = -0.809016994374947451 -0.587785252292473137
         z2 = +0.309016994374947451 +0.951056516295153642
         z3 = +0.309016994374947451 -0.951056516295153642
         z4 = +1.000000000000000000 +0.000000000000000000

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

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