# Reproducible Runs

The correctness of a polynomial homotopy relies on the choice of random complex constants. Except for an algebraic set of bad choices of complex constants, the solution paths are free of singularities, except at the end of the paths, when the system that is solved has singular solutions.

For correctness, it is important that the random constants are generated after the user has provided the input.

## solving the cyclic 5-roots system twice

The random choice of constants makes that the solutions are computed in a random order, as the experiment in this section shows.

    from phcpy.families import cyclic

The cyclic 5-roots problem belongs to an academic family of benchmark problems. The code

    c5 = cyclic(5)
    for pol in c5:
        print(pol)

prints the polynomials:

    x0 + x1 + x2 + x3 + x4;
    x0*x1 + x1*x2 + x2*x3 + x3*x4 + x4*x0;
    x0*x1*x2 + x1*x2*x3 + x2*x3*x4 + x3*x4*x0 + x4*x0*x1;
    x0*x1*x2*x3 + x1*x2*x3*x4 + x2*x3*x4*x0 + x3*x4*x0*x1 + x4*x0*x1*x2;
    x0*x1*x2*x3*x4 - 1;

Now let us solve the system twice. In each run we print the first solution.

    from phcpy.solver import solve

The outcome of the first run

    s1 = solve(c5)
    print(s1[0])

is

    t :  1.00000000000000E+00   0.00000000000000E+00
    m : 1
    the solution for t :
     x0 : -8.09016994374948E-01   5.87785252292473E-01
     x1 :  2.11803398874989E+00  -1.53884176858763E+00
     x2 :  3.09016994374947E-01  -2.24513988289793E-01
     x3 : -8.09016994374947E-01   5.87785252292473E-01
     x4 : -8.09016994374947E-01   5.87785252292473E-01
    == err :  1.152E-15 = rco :  1.034E-01 = res :  1.665E-15 =

and the second run

    s2 = solve(c5)
    print(s2[0])

shows

    t :  1.00000000000000E+00   0.00000000000000E+00
    m : 1
    the solution for t :
     x0 : -1.18033988749895E-01   3.63271264002680E-01
     x1 :  3.09016994374947E-01  -9.51056516295154E-01
     x2 :  3.09016994374947E-01  -9.51056516295154E-01
     x3 :  3.09016994374947E-01  -9.51056516295154E-01
     x4 : -8.09016994374947E-01   2.48989828488278E+00
    == err :  6.682E-16 = rco :  4.527E-02 = res :  1.874E-15 =

The cyclic 5-roots problem has 70 different solutions and with high likelihood, the first solution will be different in each run.

## fixing the seed

Fixing the seed of the random number generators makes the solver deterministic. Consider the following experiment.

    from phcpy.dimension import set_seed

We set the seed to the value `2024` and print the first solution again after solving the system:

    set_seed(2024)
    s3 = solve(c5)
    print(s3[0])

which shows

    t :  1.00000000000000E+00   0.00000000000000E+00
    m : 1
    the solution for t :
     x0 :  1.00000000000000E+00  -8.03364927637306E-17
     x1 :  3.09016994374947E-01   9.51056516295154E-01
     x2 : -8.09016994374947E-01   5.87785252292473E-01
     x3 : -8.09016994374947E-01  -5.87785252292473E-01
     x4 :  3.09016994374947E-01  -9.51056516295154E-01
    == err :  6.315E-16 = rco :  2.393E-01 = res :  4.631E-16 =

Let us do it again:

    set_seed(2024)
    s4 = solve(c5)
    print(s4[0])

which then shows

    t :  1.00000000000000E+00   0.00000000000000E+00
    m : 1
    the solution for t :
     x0 :  1.00000000000000E+00  -8.03364927637306E-17
     x1 :  3.09016994374947E-01   9.51056516295154E-01
     x2 : -8.09016994374947E-01   5.87785252292473E-01
     x3 : -8.09016994374947E-01  -5.87785252292473E-01
     x4 :  3.09016994374947E-01  -9.51056516295154E-01
    == err :  6.315E-16 = rco :  2.393E-01 = res :  4.631E-16 =

And of course, the point is that we see twice the same first solution.