some interesting examples and families

PHCpack has been tested on many examples of polynomial systems taken from the research literature. The module examples exports some of those examples. Running python examples.py at the command prompt performs a regression test, solving all examples.

Polynomial systems often occur in families and are defined for any number of equations and variables.

interactive regression testing

An interactive use of examples.py at the Python prompt can go as follows:

>>> from phcpy.examples import noon3
>>> f = noon3()
>>> for p in f: print(p)
...
x1*x2^2 + x1*x3^2 - 1.1*x1 + 1;
x2*x1^2 + x2*x3^2 - 1.1*x2 + 1;
x3*x1^2 + x3*x2^2 - 1.1*x3 + 1;

The functions in examples.py returns the polynomials as lists of strings. If we want to solve the system defined by f, we continue the above session as

>>> from phcpy.solver import solve
>>> s = solve(f,silent=True)
>>> len(s)
21
>>> print(s[0])
t :  1.00000000000000E+00   0.00000000000000E+00
m : 1
the solution for t :
 x1 : -1.65123467890611E-01  -7.61734168646636E-01
 x2 :  8.98653694263692E-01  -3.48820047576431E-01
 x3 :  8.98653694263692E-01  -3.48820047576431E-01
== err :  3.034E-16 = rco :  2.761E-01 = res :  5.974E-16 =

The example session continues in the description of the module solutions.

the cyclic n-roots problem

One such noteworthy family is the cyclic n-roots problem:

>>> from phcpy.families import cyclic
>>> c4 = cyclic(4)
>>> for p in c4: print(p)
...
x0 + x1 + x2 + x3;
x0*x1 + x1*x2 + x2*x3 + x3*x0;
x0*x1*x2 + x1*x2*x3 + x2*x3*x0 + x3*x0*x1;
x0*x1*x2*x3 - 1;