DifferentialElimination

Experimental implementations of algorithms based on the paper https://arxiv.org/abs/1610.04022

Usage

The main functions implemented in "DifferentialElimination.mpl" are

ComputeBound(eqs, vars_to_eliminate, radical)

Computes the bound for elimination of vars_to_eliminate in system eqs of differential-algebraic equations

• eqs is a list of equations

• vars_to_eliminate is a list of unknowns to be eliminated

• radical is an optional parameter (the default value is true). If the value is true, the algorithm uses the bound for radical ideals (from Theorem 2), otherwise the algorithm uses the bound from Theorem 1. The former bound is always smaller. If radical is set to be true but the ideal in not actually radical, the algorithm will return an error.

CheckPossibilityElimination(eqs, vars, vars_to_eliminate, p, radical, method)

Checks the possibility of elimination of vars_to_eliminate in system eqs of differential-algebraic equations. If the elimination is possible, returns the minimal number of sufficient prolongations, otherwise returns "no elimination".

• eqs is a list of equations

• vars is a list of all unknowns

• vars_to_eliminate is a list of unknowns to be eliminated

• p is the user-specified probability of correcteness of the result

• radical is an optional parameter (the default value is true). If the value is true, the algorithm uses the bound for radical ideals (from Theorem 2), otherwise the algorithm uses the bound from Theorem 1. The former bound is always smaller. If radical is set to be true but the ideal in not actually radical, the algorithm will return an error.

• method is an optional parameter (the default value is "groebner"). The parameter specifies the method used to determine the consistency of the resulting polynomial system (for the details, see Section 5 of the paper). In each individual case, one of the methods might be faster than the other one.

  • groebner - Groebner bases
  • triangular - Triangular decomposition (RegularChains library)

DifferentialElimination(eqs, vars, vars_to_eliminate, number_prolongations, radical)

Computes the result of elimination of vars_to_eliminate in system eqs of differential-algebraic equations.

• eqs is a list of equations

• vars is a list of all unknowns

• vars_to_eliminate is a list of unknowns to be eliminated

• number_prolongations is an optional parameter (the default value is -1).

  • -1, the algorithm performs the number of prolongtions given by the ComputeBound function
  • nonegative integer, the algorithm performs the specified number of prolongations. For efficiency, this number can be precomputed using the CheckPossibilityElimination function

• radical is an optional parameter (the default value is true), is used only if number_prolongations = -1. If the value is true, the algorithm uses the bound for radical ideals (from Theorem 2), otherwise the algorithm uses the bound from Theorem 1. The former bound is always smaller. If radical is set to be true but the ideal in not actually radical, the algorithm will return an error.

Structure

• DifferentialElimination.mpl contains the algorithms for elimination

• examples/ folder contains code for Examples 1-4 from the paper

Support

The software is partially supported by the National Science Foundation.