PolynomialTestSystems

Introduction

PolynomialTestSystems.jl is a collection of systems of polynomials for benchmarking, testing etc. Most of the polynomial systems are obtained from the database of polynomial systems created by Jan Verschelde and available under http://homepages.math.uic.edu/~jan/.

Format

PolynomialTestSystems.TestSystem — Type.
TestSystem(equations;
    multi_bezout_number=nothing,
    mixed_volume=nothing,
    nsolutions=nothing,
    nreal_solutions=nothing)

Create a TestSystem.

source
PolynomialTestSystems.equations — Function.
equations(::TestSystem)

Get the the polynomial system.

source
PolynomialTestSystems.nvariables — Function.
nvariables(::TestSystem)

Obtain the number of variables of the test system.

source
PolynomialTestSystems.mixed_volume — Function.
mixed_volume(system)::Union{Nothing, Int}

Returns the number mixed volume of the system if known.

source
PolynomialTestSystems.nsolutions — Function.
nsolutions(system)::Union{Nothing, Int}

Returns the number of (complex) solutions of the system if known.

source
PolynomialTestSystems.nreal_solutions — Function.
nreal_solutions(system)::Union{Nothing, Int}

Returns the number of real solutions of the system if known.

source
PolynomialTestSystems.bezout_number — Function.
bezout_number(system)

Returns the bezout number of the system.

source
PolynomialTestSystems.multi_bezout_number — Function.
multi_bezout_number(system)::Union{Nothing, Tuple{Int, Vector{Vector{PolyVar{true}}}}}

Returns a tuple containing the multi-homogenous bezout number as well as the corresponding grouping of the variables if known.

source

Systems

PolynomialTestSystems.chandra — Function.
chandra(n)

The Chandrasekhar H-Equation for n.

References

Laureano Gonzalez-Vega: "Some examples on problem solving by using the symbolic viewpoint when dealing with polynomial systems of equations". in: "Computer Algebra in Science and Engineering", Editors: J. Fleischer, J. Grabmeier, F.W. Hehl and W. Kuechlin. Pages 102-116 World Scientific Publishing, 1995.

S. Chandrasekhar: "Radiative Transfer", Dover, NY, 1960.

C.T. Kelley: "Solution of the Chandrasekhar H-equation by Newton's method". J. Math. Phys., 21 (1980), pp. 1625-1628.

Jorge J. More': "A collection of nonlinear model problems" in: "Computational Solution of Nonlinear Systems of Equations", Editors: Eugene L. Allgower and Kurt Georg. Pages 723-762. Lectures in Applied Mathematics, Volume 26, AMS, 1990.

NOTE : the parameter c equals 0.51234. In general c can be any number in the interval (0,1]

source
PolynomialTestSystems.cyclic — Function.
cyclic(n)

The cyclic n-roots problems.

References

Göran Björck and Ralf Fröberg: `A faster way to count the solutions of inhomogeneous systems of algebraic equations, with applications to cyclic n-roots', in J. Symbolic Computation (1991) 12, pp 329–336.

source
PolynomialTestSystems.katsura — Function.
katsura(n)

A problem of magnetism in physics.

References

From the PoSSo test suite.

Shigetoshi Katsura: "Users posing problems to PoSSO", in the PoSSo Newsletter, no. 2, July 1994, edited by L. Gonzalez-Vega and T. Recio. Available at http://janet.dm.unipi.it/

S. Katsura, W. Fukuda, S. Inawashiro, N.M. Fujiki and R. Gebauer, Cell Biophysics, Vol 11, pages 309–319, 1987.

W. Boege, R. Gebauer, and H. Kredel: "Some examples for solving systems of algebraic equations by calculating Groebner bases", J. Symbolic Computation, 2:83-98, 1986.

Shigetoshi Katsura: "Spin Glass Problem by the Method of Integral Equation of the Effective Field". In "New Trends in Magnetism", edited by Mauricio D. Coutinho-Filho and Sergio M. Resende, pages 110-121, World Scientific, 1990.

source
PolynomialTestSystems.fourbar — Function.
fourbar()

A four-bar design problem, so-called 5-point problem.

References

See Morgan, A.P. and Wampler, C.W. : `Solving a planar four-bar design problem using continuation' in Transaction of the ASME, J. of Mechanical Design, Vol. 112 pages 544-550, 1990.

For the coefficients, see Table 2, a = (1,0). This is the start system, with five random precision points, that has been used to solve twenty other test systems.

source
PolynomialTestSystems.rps10 — Function.
rps10()

RPS Serial Chains for 10 positions on a circular hyperboloid

References

Hai-Jun Su and J. Micheal McCarthy: "Kinematics Synthesis of RPS Serial Chains", In Proceedings of the ASME Design Engineering Technical Conferences (CDROM). Paper DETC03/DAC-48813. Chicago, IL, Sept.02-06, 2003.

source
PolynomialTestSystems.ipp — Function.
ipp()

The six-revolute-joint problem of mechanics.

References

A. Morgan and A. Sommese `Computing all solutions to polynomial systems using homotopy continuation', Appl. Math. Comput., Vol. 24, pp 115-138, 1987.

source
PolynomialTestSystems.boon — Function.
boon()

Neurofysiology, posted by Sjirk Boon. Note that the system

References

The system has been posted to the newsgroup sci.math.num-analysis by Sjirk Boon.

P. Van Hentenryck, D. McAllester and D. Kapur: `Solving Polynomial Systems Using a Branch and Prune Approach' SIAM J. Numerical Analysis, Vol. 34, No. 2, pp 797-827, 1997.

source
PolynomialTestSystems.heart — Function.
heart()

The heart-dipole problem.

References

Nelsen, C.V. and Hodgkin, B.C.: `Determination of magnitudes, directions, and locations of two independent dipoles in a circular conducting region from boundary potential measurements' IEEE Trans. Biomed. Engrg. Vol. BME-28, No. 12, pages 817-823, 1981.

Morgan, A.P. and Sommese, A.J.: `Coefficient-Parameter Polynomial Continuation' Appl. Math. Comput. Vol. 29, No. 2, pages 123-160, 1989. Errata: Appl. Math. Comput. 51:207 (1992)

Morgan, A.P. and Sommese, A. and Watson, L.T.: `Mathematical reduction of a heart dipole model' J. Comput. Appl. Math. Vol. 27, pages 407-410, 1989.

source
PolynomialTestSystems.d1 — Function.
d1()

A a sparse system, known as benchmark D1.

References

H. Hong and V. Stahl: `Safe Starting Regions by Fixed Points and Tightening', Computing 53(3-4): 322-335, 1995.

source