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Bertini :: bertiniParameterHomotopy

bertiniParameterHomotopy -- run parameter homotopy in Bertini

Synopsis

Description

This method numerically solves several polynomial systems from a parameterized family at once. The list F is a system of polynomials in ring variables and the parameters listed in P. The list T is the set of parameter values for which solutions to F are desired. Both stages of Bertini’s parameter homotopy method are called with bertiniParameterHomotopy. First, Bertini assigns a random complex number to each parameter and solves the resulting system, then, after this initial phase, Bertini computes solutions for every given choice of parameters using a number of paths equal to the exact root count in the first stage.
i1 : R=CC[u1,u2,u3,x,y];
i2 : f1=u1*(y-1)+u2*(y-2)+u3*(y-3); --parameters are u1, u2, and u3
i3 : f2=(x-11)*(x-12)*(x-13)-u1;
i4 : paramValues0={{1,0,0}};
i5 : paramValues1={{0,1+2*ii,0}};
i6 : bPH=bertiniParameterHomotopy( {f1,f2}, {u1,u2,u3},{paramValues0 ,paramValues1 })
Temporary directory for input and output files:/var/folders/46/9b86vqxj4hjcngvy7kd7sb140000gn/T/M2-66055-0/0

The version of Bertini 
            you have installed on your computer 
            was used for this run. 
Bertini is under ongoing development by 
            D. Bates, J. Hauenstein, A. Sommese, and C. Wampler.


o6 = {{{11.3376+.56228*ii, 1}, {13.3247, 1}, {11.3376-.56228*ii, 1}}, {{12,
     ------------------------------------------------------------------------
     2}, {13, 2}, {11, 2}}}

o6 : List
i7 : bPH_0--the solutions to the system with parameters set equal to paramValues0

o7 = {{11.3376+.56228*ii, 1}, {13.3247, 1}, {11.3376-.56228*ii, 1}}

o7 : List
i8 : R=CC[x,y,z,u1,u2]

o8 = R

o8 : PolynomialRing
i9 : f1=x^2+y^2-z^2

      2    2    2
o9 = x  + y  - z

o9 : R
i10 : f2=u1*x+u2*y

o10 = x*u1 + y*u2

o10 : R
i11 : finalParameters0={{0,1}}

o11 = {{0, 1}}

o11 : List
i12 : finalParameters1={{1,0}}

o12 = {{1, 0}}

o12 : List
i13 : bPH=bertiniParameterHomotopy( {f1,f2}, {u1,u2},{finalParameters0 ,finalParameters1 },ISPROJECTIVE=>1)
Temporary directory for input and output files:/var/folders/46/9b86vqxj4hjcngvy7kd7sb140000gn/T/M2-66055-0/2

The version of Bertini 
            you have installed on your computer 
            was used for this run. 
Bertini is under ongoing development by 
            D. Bates, J. Hauenstein, A. Sommese, and C. Wampler.


o13 = {{{.231211+.490643*ii, -1.04639e-16-7.6165e-18*ii, .231211+.490643*ii},
      -----------------------------------------------------------------------
      {-.972753+.285112*ii, -2.82435e-17+1.87567e-17*ii,
      -----------------------------------------------------------------------
      .972753-.285112*ii}}, {{-2.1684e-18+6.85487e-16*ii, 1.04288-.251215*ii,
      -----------------------------------------------------------------------
      1.04288-.251215*ii}, {4.62683e-17+5.48064e-17*ii, -.252659-.491864*ii,
      -----------------------------------------------------------------------
      .252659+.491864*ii}}}

o13 : List
i14 : bPH_0--The two solutions for finalParameters0

o14 = {{.231211+.490643*ii, -1.04639e-16-7.6165e-18*ii, .231211+.490643*ii},
      -----------------------------------------------------------------------
      {-.972753+.285112*ii, -2.82435e-17+1.87567e-17*ii, .972753-.285112*ii}}

o14 : List

Caveat

Variables must begin with a letter (lowercase or capital) and can only contain letters, numbers, underscores, and square brackets.

Ways to use bertiniParameterHomotopy :