The database of polynomial systems

The demonstration database is an essential part of PHCpack. Motivations are plentiful:

Examples of data formats: As part of the user's guide, we provide a bounty of examples, which show the input formats and how the results of calculations can be preserved.
Capabilities of the program: We are proud to list some systems which are really hard for other solvers, although execution times are very transient thanks to rapid advances in hardware.
Relevance to practical applications: One important criterion for inclusion in the collection is the connection with an application field. We strived to provide accurate references to the literature, documenting the source of the application.

The database is tailored towards PHCpack. Besides the peculiar format of solution lists, this implies that the systems often have floating-point input coefficients and the focus is on approximating the isolated roots. Moreover, one criterion for entering the database is that it can be solved with PHCpack.

We tried to find original references, sometimes a piece of Maple code accompanies the derivation of the polynomial equations. An exciting part of the field of polynomial system solving is that one encounters applications as diverse as game theory, mechanical design, chemical reactions, control of linear systems, etc... We hope that the relevance of these applications will help to elevate our field to approximate the status currently enjoyed by areas such as Linear Algebra, Optimization, or Differential Equations.

We warn that a general-purpose solver will in general not be competitive against methods tailored for a specific application. Homotopies unable to exploit structure usually fail and end up squandering computer resources.

boon comb3000s eco6 gaukwa4 mickey
rbpl24s s9_1 conform1 eco7 geneig
noon3 redcyc5 sendra butcher cpdm5
eco8 heart noon4 redcyc6 solotarev
butcher8 cyclic5 extcyc5 i1 noon5
redcyc7 sparse5 camera1s cyclic6 extcyc6
ipp proddeco redcyc8 speer caprasse
cyclic7 extcyc7 ipp2 puma redeco5
trinks cassou cyclic8 extcyc8 katsura5
quadfor2 redeco6 virasoro chemequ d1
fbrfive12 kin1 quadgrid redeco7 wood
chemequs des18_3 fbrfive4 kinema rabmo
redeco8 wright cohn2 des22_24 fourbar
ku10 rbpl rediff3 cohn3 discret3
gaukwa2 lorentz rbpl24 reimer5 comb3000
eco5 gaukwa3 lumped rbpl24es rose

Some new test examples...

pb601 pb601es pb601vs chemkin filter9
katsura6 katsura7 katsura8 katsura9 katsura10
utbikker kotsireas chandra4 chandra5 chandra6
pole27sys pole34sys pole43sys pole28sys pltp34sys
robspat stewgou40 tangents0 tangents1 tangents2
game4two game5two game6two game7two assur44
fbremb2 butemb3 cyc8emb1 rcyc8emb1 rcyc9emb2
cyclic8e1 cyclic9e2 cyclic9 cyclic10 cyclic11

Some examples to illustrate Newton's method with deflation for isolated singularities.

The files in the left column below contain the original formulation of the polynomial system, along with a list of initial approximations for the isolated solutions. These approximations have been found at the end of solution paths defined by a homotopy. The other columns below contain the deflated systems along with their solutions. The number following the "_d" in the name is the deflation step while the number following the "_R" in the name is the numerical rank of the Jacobian matrix at the isolated root before the deflation. Multiple deflations may be needed to characterize a multiple root.
simple simple_d1R0
eg1 eg1_d1R0
eg2 eg2_d1R1
eg3 eg3_d1R1
eg4 eg4_d1R1_d2R3
eg5 eg5_d1R1_d2R3_d3R7
baker1 baker1_d1R1
cbms1 cbms1_d0 cbms1_d1R0
cbms2 cbms2_d0 cbms2_d1R0
mth191 mth191_d0 mth191_d1R1
decker1 decker1_d0 decker1_d1R1_d2R3
decker2 decker2_d0 decker2_d1R1_d2R3_d3R7
decker3 decker3_d0 decker3_d1R1
kss3 kss3_d0 kss3_d1R1
ojika1 ojika1_d0 ojika1_d1R1_d2R3
ojika2 ojika2_d0 ojika2_d1R2
ojika3 ojika3_d1R1 ojika3_d1R2
ojika4 ojika4_d1R2_d2R5
caprasse caprasse_d0 caprasse_d1R2
cyclic9 cyclic9_d0 cyclic9_d1R7
tangents1 tangents1_d1R4

Some examples to illustrate the monodromy breakup algorithm to decompose algebraic sets into irreducible components.

This material is based upon work supported by the National Science Foundation under Grants No. 9804846, 0105739, and 0134611. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.