Bertini -- software for numerical algebraic geometry

Description

Interfaces the functionality of the software Bertini to solve polynomial systems and perform calculations in numerical algebraic geometry. The software is available at http://bertini.nd.edu/. Bertini is under ongoing development by D. Bates, J. Hauenstein, A. Sommese, and C. Wampler.

The user may place the executable program bertini in the execution path. Alternatively, the path to the executable needs to be specified, for instance,

i1 : needsPackage("Bertini", Configuration=>{"BERTINIexecutable"=>"/folder/subfolder/bertini"})

o1 = Bertini

o1 : Package

Below is a simple example using the most popular function, a basic zero-dimensional solve with no special options.

i2 : R = CC[x,y]

o2 = R

o2 : PolynomialRing

i3 : F = {x^2-1,y^2-2}

       2       2
o3 = {x  - 1, y  - 2}

o3 : List

i4 : solns = bertiniZeroDimSolve(F)

o4 = {{1, 1.41421}, {1, -1.41421}, {-1, 1.41421}, {-1, -1.41421}}

o4 : List

Authors

Version

This documentation describes version 2.1.2.3 of Bertini.

Citation

If you have used this package in your research, please cite it as follows:

@misc{BertiniSource,
  title = {{Bertini: interface to Bertini. Version~2.1.2.3}},
  author = {Elizabeth Gross and Jose Israel Rodriguez and Dan Bates and Anton Leykin},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Types
- B'Section -- a mutable hash table that gives information about a hyperplane used to slice a numerical variety.
- B'Slice -- a mutable hash table that gives information about a linear space used to slice a numerical variety.
Functions and commands
- bertiniComponentMemberTest -- a main method to test whether points lie on a given variety
- bertiniParameterHomotopy -- a main method to perform a parameter homotopy in Bertini
- bertiniPosDimSolve -- a main method that is used to produce witness sets
- bertiniRefineSols -- sharpen solutions to a prescribed number of digits
- bertiniSample -- a main method to sample points from an irreducible component of a variety
- bertiniTrackHomotopy -- a main method to track using a user-defined homotopy
- bertiniUserHomotopy -- a main method to track a user-defined homotopy
- bertiniZeroDimSolve -- a main method to solve a zero-dimensional system of equations
- importIncidenceMatrix -- Imports an incidence matrix file after running makeMembershipFile.
- importMainDataFile -- This function imports points from the main data file form a Bertini run.
- importParameterFile -- Imports parameters from a Bertini parameter file.
- importSolutionsFile -- Imports coordinates from a Bertini solution file.
- makeB'InputFile -- write a Bertini input file in a directory
- makeB'Section -- makeB'Section creates a hash table that represents a hyperplane.
- makeB'Slice -- makeB'Slice creates a hash table that represents a linear slice.
- moveB'File -- Move or copy files.
- NumberToB'String -- Translates a number to a string that Bertini can read.
- radicalList -- A support function that removes multiplicities of numbers in a list up to a tolerance.
- subPoint -- This function evaluates a polynomial or matrix at a point.
- valueBM2 -- This function makes a number in CC from a string.
- writeStartFile -- Writes the list of list of coordinates to a file that Bertini can read.
Methods
- bertiniComponentMemberTest(List,NumericalVariety) -- see bertiniComponentMemberTest -- a main method to test whether points lie on a given variety
- bertiniParameterHomotopy(List,List,List) -- see bertiniParameterHomotopy -- a main method to perform a parameter homotopy in Bertini
- bertiniPosDimSolve(Ideal) -- see bertiniPosDimSolve -- a main method that is used to produce witness sets
- bertiniPosDimSolve(List) -- see bertiniPosDimSolve -- a main method that is used to produce witness sets
- bertiniRefineSols(String,ZZ,List,String) -- see bertiniRefineSols -- sharpen solutions to a prescribed number of digits
- bertiniRefineSols(ZZ,List) -- see bertiniRefineSols -- sharpen solutions to a prescribed number of digits
- bertiniRefineSols(ZZ,List,String) -- see bertiniRefineSols -- sharpen solutions to a prescribed number of digits
- bertiniSample(ZZ,WitnessSet) -- see bertiniSample -- a main method to sample points from an irreducible component of a variety
- bertiniTrackHomotopy(RingElement,List,List) -- see bertiniTrackHomotopy -- a main method to track using a user-defined homotopy
- bertiniUserHomotopy(Thing,List,List,List) -- see bertiniUserHomotopy -- a main method to track a user-defined homotopy
- bertiniZeroDimSolve(Ideal) -- see bertiniZeroDimSolve -- a main method to solve a zero-dimensional system of equations
- bertiniZeroDimSolve(List) -- see bertiniZeroDimSolve -- a main method to solve a zero-dimensional system of equations
- importIncidenceMatrix(String) -- see importIncidenceMatrix -- Imports an incidence matrix file after running makeMembershipFile.
- importMainDataFile(String) -- see importMainDataFile -- This function imports points from the main data file form a Bertini run.
- importParameterFile(String) -- see importParameterFile -- Imports parameters from a Bertini parameter file.
- importSolutionsFile(String) -- see importSolutionsFile -- Imports coordinates from a Bertini solution file.
- makeB'InputFile(String) -- see makeB'InputFile -- write a Bertini input file in a directory
- makeB'Section(List) -- see makeB'Section -- makeB'Section creates a hash table that represents a hyperplane.
- makeB'Slice(Thing,List) -- see makeB'Slice -- makeB'Slice creates a hash table that represents a linear slice.
- makeMembershipFile(String) (missing documentation)
- makeSampleSolutionsFile(String,Number) (missing documentation)
- moveB'File(String,String,String) -- see moveB'File -- Move or copy files.
- NumberToB'String(Thing) -- see NumberToB'String -- Translates a number to a string that Bertini can read.
- radicalList(List) -- see radicalList -- A support function that removes multiplicities of numbers in a list up to a tolerance.
- radicalList(List,Number) -- see radicalList -- A support function that removes multiplicities of numbers in a list up to a tolerance.
- readFile(String) (missing documentation)
- readFile(String,Number) (missing documentation)
- readFile(String,String,Number) (missing documentation)
- runBertini(String) (missing documentation)
- sortMainDataComponents(List) (missing documentation)
- subPoint(Thing,List,Thing) -- see subPoint -- This function evaluates a polynomial or matrix at a point.
- valueBM2(String) -- see valueBM2 -- This function makes a number in CC from a string.
- writeParameterFile(String,List) (missing documentation)
- writeStartFile(String,List) -- see writeStartFile -- Writes the list of list of coordinates to a file that Bertini can read.
Symbols
- AdditionalFiles (missing documentation)
- B'Constants -- an option to designate the constants for a Bertini Input file
- CoeffBound -- see Bertini input configuration -- a configuration option for a Bertini input file
- CondNumThreshold -- see Bertini input configuration -- a configuration option for a Bertini input file
- DegreeBound -- see Bertini input configuration -- a configuration option for a Bertini input file
- EndGameNum -- see Bertini input configuration -- a configuration option for a Bertini input file
- FinalTol -- see Bertini input configuration -- a configuration option for a Bertini input file
- ImagThreshold -- see Bertini input configuration -- a configuration option for a Bertini input file
- MaxCycleNum -- see Bertini input configuration -- a configuration option for a Bertini input file
- MaxNewtonIts -- see Bertini input configuration -- a configuration option for a Bertini input file
- MaxNorm -- see Bertini input configuration -- a configuration option for a Bertini input file
- MaxNumberSteps -- see Bertini input configuration -- a configuration option for a Bertini input file
- MaxStepSize -- see Bertini input configuration -- a configuration option for a Bertini input file
- MinStepSizeBeforeEG -- see Bertini input configuration -- a configuration option for a Bertini input file
- MinStepSizeDuringEG -- see Bertini input configuration -- a configuration option for a Bertini input file
- MPType -- see Bertini input configuration -- a configuration option for a Bertini input file
- ODEPredictor -- see Bertini input configuration -- a configuration option for a Bertini input file
- OutputLevel -- see Bertini input configuration -- a configuration option for a Bertini input file
- PRECISION -- see Bertini input configuration -- a configuration option for a Bertini input file
- RandomSeed -- see Bertini input configuration -- a configuration option for a Bertini input file
- RegenStartLevel -- see Bertini input configuration -- a configuration option for a Bertini input file
- ScreenOut -- see Bertini input configuration -- a configuration option for a Bertini input file
- SecurityLevel -- see Bertini input configuration -- a configuration option for a Bertini input file
- SingValZeroTol -- see Bertini input configuration -- a configuration option for a Bertini input file
- StepsForIncrease -- see Bertini input configuration -- a configuration option for a Bertini input file
- TrackTolBeforeEG -- see Bertini input configuration -- a configuration option for a Bertini input file
- TrackTolDuringEG -- see Bertini input configuration -- a configuration option for a Bertini input file
- RandomComplex -- see Bertini input file declarations: random numbers -- an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
- RandomReal -- see Bertini input file declarations: random numbers -- an option which designates symbols/strings/variables that will be set to be a random real number or random complex number
- CopyB'File -- an optional argument to specify whether make a copy of the file.
- IsProjective -- optional argument to specify whether to use homogeneous coordinates
- MainDataDirectory (missing documentation)
- NameB'Slice -- see makeB'Slice -- makeB'Slice creates a hash table that represents a linear slice.
- PathList (missing documentation)
- TopDirectory -- Option to change directory for file storage.
- UseRegeneration -- an option specifying when to use regeneration
- AffVariableGroup -- see Variable groups -- an option to group variables and use multihomogeneous homotopies
- HomVariableGroup -- see Variable groups -- an option to group variables and use multihomogeneous homotopies
Mutable symbols
- storeBM2Files (missing documentation)
Other things
- storeBM2Files (missing documentation)

For the programmer

The object Bertini is a package, defined in Bertini.m2, with auxiliary files in Bertini/.

The source of this document is in Bertini.m2:2624:0.