phc_track.html

phc[track] - tracks the continuation paths

Calling Sequence

track(T,S,L)

track(T,S,L,n)

Parameters

T - target system

S - start system

L - list of solutions to the start system

n - number of subdivisions (default = 1)

{max_steps::integer} - maximum number of steps taken along one section of the path

{condition_hom::integer} - condition of homotopy
{hom_parameter_k::integer} - homotopy parameter k

{a_const::complex} - homotopy parameter a
{block_size::integer} - number of solutions tracked simultaneously

{max_step_size::float} - limit on the step size of the continuation parameter

Return value

R - a list of lists of solutions with R[1]=L and R[n+1] containing the corresponding solutions of T

The function track(T,S,L,n) tracks the homotopy paths originating in the solutions of the start system S specified in L to find the corresponding solutions of the target system T . The paths are subdivided in n sections.
The optional parameters hom_parameter_k = k (default = 2) and a_const = a (default = 1) result in homotopy H(x,t) = a (1-t)^k S(x) + t^k T(x) being tracked.
The tuning parameter condition_hom should be used according to the general expectancy of how difficult the solution paths will be. Low values of this parameter lead to a loose path tracking, whereas higher values tighten up the path-following.

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

> T := makeSystem([x,y], [], [x^2+y^2-1, y^3+x^3-1]):

> S := makeSystem([x,y], [], expand((1+I)*[x^2-1, y^3-1])):

> startSolutions := [makeSolution([1,1]), makeSolution([1,evalf(-0.5+sqrt(3)/2*I)])]:

> R := track(T,S,startSolutions,2):

> map(s->printSolutions(T,s), R):

(1) [x = 1, y = 1]

(2) [x = 1, y = -.5+.8660254040*I]

(1) [x = .821434762572163+.950946565477868e-1*I, y = .908266702344702+.120183300746065e-1*I]

(2) [x = 1.09194892917596+.849383617594522e-1*I, y = -.447909551450832+.620807649117482*I]

(1) [x = -.403397134924552e-6-.208818637592402e-6*I, y = 1.00000000000000-.123782607980647e-18*I]

(2) [x = 1.00000000000000-.422780400349326e-19*I, y = .263261288857061e-6-.337617040437438e-6*I]

>

phc , phc[solve] , phc[track] , phc[embed] , phc[drawPaths]