• The function cascade(S,L)  runs the cascade of homotopies in PHCpack to find a list witness sets C  for the component of the solution to the original system set in each dimension .
• After embedding S is constructed for the original system using embed , the list of solutions L  to the embedded system should be computed. Then the system S  and the solutions L  are passed to cascade .
• The returned witness sets may contain junk points, which can be filtered out by filter . In the example below only solutions 1 and 2 are actual 0-dimensional components.

>	with(phc): setPHCloc("C:\\PHCmaple"):

>	sys := makeSystem([x,y,z],[],[x^2+y^2+(z-1)^2-1, (z-0.5)*(z-1)*y,(z-0.5)*(z-1)*x]):

>	embsys := embed(sys,1):

>	embsols := solve(embsys):

>	comps := cascade(embsys,embsols):

>	dim0 := comps[1]: dim1 :=comps[2]:

>	printSolutions(dim0:-system,dim0:-points);

(1) [x = -.72143-.14303*I, y = -.72143+.14303*I, z = 1.0]

(2) [x = .72143+.14303*I, y = .72143-.14303*I, z = 1.0]

(3) [x = -.62478-.12387*I, y = -.62478+.12387*I, z = .50+.91088e-32*I]

(4) [x = 0., y = 0., z = 0.]

(5) [x = .62478+.12387*I, y = .62478-.12387*I, z = .50+.22187e-30*I]

(6) [x = 0., y = 0., z = 2.0]

phc , phc[embed] , phc[filter] .