next | previous | forward | backward | up | top | index | toc | Macaulay2 website
Bertini :: bertiniZeroDimSolve

bertiniZeroDimSolve -- a main method to solve a zero-dimensional system of equations

Synopsis

Description

This method finds isolated solutions to the system F via numerical polynomial homotopy continuation by (1) building a Bertini input file from the system F, (2) calling Bertini on this input file, (3) returning solutions from a machine readable file that is an output from Bertini.

i1 : R = CC[x,y];
i2 : F = {x^2-1,y^2-2};
i3 : S = bertiniZeroDimSolve F

o3 = {{1, 1.41421}, {1, -1.41421}, {-1, 1.41421}, {-1, -1.41421}}

o3 : List

Each solution is of type Point. Additional information about the solution can be accessed by using peek.

i4 : peek S_0

o4 = Point{AccuracyEst => 5.55221e-14        }
           AccuracyEstInternal => 1.09532e-14
           ConditionNumber => 3.19545
           Coordinates => {1, 1.41421}
           CycleNumber => 1
           FinalTValue => .000390625
           FunctionResidual => 1.11886e-16
           MaxPrecisionUtilized => 52
           Multiplicity => 1
           NewtonResidual => 8.23628e-17
           PathNumber => 0
           PathsWithSameEndpoint => {}
           PrecisionIncreased => 0
           SolutionNumber => 1

Bertini uses a multihomogeneous homotopy as a default, but regeneration can be deployed with the option UseRegeneration=>1 .

i5 : R = CC[x];
i6 : F = {x^2*(x-1)};
i7 : S = bertiniZeroDimSolve F

o7 = {{1}, {3.51449e-14+6.41131e-14*ii}}

o7 : List
i8 : B = bertiniZeroDimSolve(F,UseRegeneration=>1)

o8 = {{1}}

o8 : List

Variables must begin with a letter (lowercase or capital) and can only contain letters, numbers, underscores, and square brackets. Regeneration in bertiniZeroDimSolve only finds nonsingular isolated points.

Ways to use bertiniZeroDimSolve :

For the programmer

The object bertiniZeroDimSolve is a method function with options.