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).