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 })
o6 = {{{11.3376+.56228*ii, 1}, {13.3247, 1}, {11.3376-.56228*ii, 1}}, {{12,
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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 },HomVariableGroup=>{x,y,z})
o13 = {{{1, -2.30252e-16+1.70802e-16*ii, -1}, {1,
-----------------------------------------------------------------------
-5.73444e-16+3.53557e-16*ii, 1}}, {{-2.9577e-16-1.70683e-16*ii, 1, 1},
-----------------------------------------------------------------------
{-2.1372e-17-6.80607e-17*ii, 1, -1}}}
o13 : List
|
i14 : bPH_0--The two solutions for finalParameters0
o14 = {{1, -2.30252e-16+1.70802e-16*ii, -1}, {1, -5.73444e-16+3.53557e-16*ii,
-----------------------------------------------------------------------
1}}
o14 : List
|
i15 : finParamValues={{1},{2}}
o15 = {{1}, {2}}
o15 : List
|
i16 : bPH1=bertiniParameterHomotopy( {"x^2-u1"}, {u1},finParamValues,AffVariableGroup=>{x})
o16 = {{{1}, {-1}}, {{1.41421}, {-1.41421}}}
o16 : List
|
i17 : bPH2=bertiniParameterHomotopy( {"x^2-u1"}, {u1},finParamValues,AffVariableGroup=>{x},OutputStyle=>"OutSolutions")
o17 = {{{1}, {-1}}, {{1.41421}, {-1.41421}}}
o17 : List
|
i18 : class bPH1_0_0 o18 = Point o18 : Type |
i19 : class bPH2_0_0 o19 = List o19 : Type |
i20 : dir1 := temporaryFileName(); -- build a directory to store temporary data |
i21 : makeDirectory dir1; |
i22 : bPH5=bertiniParameterHomotopy( {"x^2-u1"}, {u1},{{1},{2}},AffVariableGroup=>{x},OutputStyle=>"OutNone",TopDirectory=>dir1)
|
i23 : B0=importSolutionsFile(dir1,NameSolutionsFile=>"ph_jade_0")
o23 = {{1}, {-1}}
o23 : List
|
i24 : B1=importSolutionsFile(dir1,NameSolutionsFile=>"ph_jade_1")
o24 = {{1.41421}, {-1.41421}}
o24 : List
|
The object bertiniParameterHomotopy is a method function with options.