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MonodromySolver :: sparseMonodromySolve

sparseMonodromySolve -- an "out of the box" polynomial system solver

Synopsis

Description

Blackbox monodromy solver for a square polynomial system without parameters. The example below finds all six intersection of a generic cubic with its quadratic polar curve. Note: this method requires PHCpack.

i1 : setRandomSeed 0;
i2 : R=CC[x,y,z];
i3 : F=random(3,R);
-- warning: experimental computation over inexact field begun
--          results not reliable (one warning given per session)
i4 : P=sum apply(gens R,g->diff(g,F)*random CC);
i5 : sparseMonodromySolve polySystem {F,P,random(1,R)-1}

o5 = ({.781262-.624203*ii, -.462954-.886382*ii, -.670948-.741504*ii,
     ------------------------------------------------------------------------
     .778029-.628229*ii, -.272832+.962062*ii, .9868+.161945*ii,
     ------------------------------------------------------------------------
     .118767+.992922*ii, -.219207-.975678*ii, -.971549+.236839*ii,
     ------------------------------------------------------------------------
     .157598-.987503*ii, 1.27972-2.17943*ii, .964676-1.04591*ii,
     ------------------------------------------------------------------------
     -.500957+3.56969*ii, .527086+.814973*ii, -2.94832-1.41706*ii,
     ------------------------------------------------------------------------
     1.2095-4.1219*ii, .360485-.932765*ii, -.651048+.759036*ii,
     ------------------------------------------------------------------------
     -.272372-.962192*ii, -1}, {PointArray( 3 points: 2 1 0 ),
     ------------------------------------------------------------------------
     {.212986+.505809*ii, -.0640254-.418811*ii, .174869+.144954*ii},
     ------------------------------------------------------------------------
     {-.147508-.127555*ii, -.071535-.796132*ii, .39264+.652917*ii},
     ------------------------------------------------------------------------
     {.453473+.301343*ii, -1.01938-.509842*ii, -.592697-.68244*ii}})

o5 : Sequence

For systems with dense support such as the above, the total number of paths tracked is generally not optimal, though timings may be comparable.

Ways to use sparseMonodromySolve :