Here is a partial answer. I believe for Method -> {opts}
, opts
may be any of the following, and they will have whatever effect they have:
Internal`InequalitySolverOptions[]
Internal`ReduceOptions[]
Internal`NSolveOptions[]
(*
{"ARSDecision" -> False, "BrownProjection" -> True, "CAD" -> True,
"CADAlgebraicCoefficients" -> True, "CADBacksubstitution" -> Automatic,
"CADCombineCells" -> True, "CADConstruction" -> Automatic,
"CADDefaultPrecision" -> 30.103, "CADExtraPrecision" -> 30.103, "CADMethod" -> Automatic,
"CADNRootsMethod" -> Automatic, "CADSortVariables" -> Automatic,
"CADZeroTest" -> {0, ∞}, "EquationalConstraintsCAD" -> Automatic,
"FGLMBasisConversion" -> False, "FGLMElimination" -> Automatic,
"GenericCAD" -> True, "GroebnerCAD" -> True,
"LinearDecisionMethodCrossovers" -> {0, 30, 20, Automatic},
"LinearEquations" -> True, "LinearQE" -> True, "LWDecision" -> True,
"LWPreprocessor" -> Automatic, "ProjectAlgebraic" -> Automatic,
"ProveMultiplicities" -> True, "QuadraticQE" -> Automatic,
"QVSPreprocessor" -> False, "ReducePowers" -> Automatic,
"RootReduced" -> False, "Simplex" -> True,
"SimplifyInequalities" -> Automatic, "ThreadOr" -> True, "ZengDecision" -> False}
{"ADDSolveBound" -> 8, "AlgebraicNumberOutput" -> True,
"BDDEliminate" -> Automatic, "BooleanInstanceMethod" -> Automatic,
"BranchLinearDiophantine" -> False, "CacheReduceResults" -> Automatic,
"DiscreteSolutionBound" -> 10, "ExhaustiveSearchMaxPoints" -> {1000, 10000},
"FactorEquations" -> Automatic, "FactorInequalities" -> False,
"FinitePrecisionGB" -> False, "ImplicitIntegerSolutions" -> Automatic,
"IntervalRootsOptions" -> {"AllowIncomplete" -> True, "FailDepth" -> 20,
"MaxDepth" -> 50, "MaxFailures" -> 100, "MaxIncomplete" -> 1000,
"MaxSimplified" -> 1000, "MaxSteps" -> 100000, "MinPrecision" -> 12},
"LatticeReduceDiophantine" -> True, "LinearEliminationMaxDepth" -> ∞,
"MaxFrobeniusGraph" -> 1000000, "MaxModularPoints" -> 1000000,
"MaxModularRoots" -> 1000000, "MaxPrimeIndex" -> 1000000000,
"NIntegrateTimeConstraint" -> 60, "PresburgerQE" -> Automatic,
"QuickReduce" -> False, "RandomInstances" -> Automatic,
"RealRootsOptions" -> {"MaxExtensionDegree" -> 5, "MaxFactorSquareFreeDegree" -> 10000,
"MaxNestedRootsDegree" -> 100, "SparsityThreshold" -> 0.02},
"ReorderVariables" -> Automatic, "SieveMaxPoints" -> {10000, 1000000},
"SolveDiscreteSolutionBound" -> 1000000, "SyntacticSolveAssumptions" -> False,
"TranscendentalRecursionLimit" -> 12, "UseNestedRoots" -> Automatic,
"UseOldReduce" -> False, "UseTranscendentalRoots" -> Automatic,
"UseTranscendentalSolve" -> True}
{"ComplexEquationMethod" -> Automatic, "MonomialOrder" -> Automatic,
"ReorderVariables" -> True, "SelectCriterion" -> (True &),
"Tolerance" -> 0, "UseSlicingHyperplanes" -> True}
*)
I do not know if there are other settings that may be used. Building on comments by belisarius and Mr. Wizard, many (or maybe all) of the above are SystemOptions
that may be passed via the Method
option to FindInstance
. I do not know what all these options do. Here is an example to show they, or at least one of them, have an effect when passed through the Method
option (see the discussion of the "SieveMaxPoints"
option in Diophantine Polynomial Systems for an example that was the basis for this one):
FindInstance[x^2 + 21 y^3 - 17 z^4 == 632, {x, y, z}, Integers]
FindInstance::nsmet: The methods available to FindInstance are insufficient to find the requested instances or prove they do not exist. >>
FindInstance[x^2 + 21 y^3 - 17 z^4 == 632, {x, y, z}, Integers,
Method -> {"SieveMaxPoints" -> {10^4, 10^8}}]
(* {{x -> -944, y -> 22, z -> 16}} *)
Update - another example. This one is from Real Polynomial Systems mentioned by @belisarius. It is the last example in the tutorial and it is supposed to show a case in which "ZengDecision" -> True
improves the timing. In that respect things seem to have changed, but the option still has an effect.
FindInstance[
x^4 + y^4 + z^4 + w^4 - 5 x y z w + x^2 + y^2 + z^2 + w^2 + 1 < 0,
{x, y, z, w}, Reals] // AbsoluteTiming
(* {0.140519, {{x -> -6, y -> -5, z -> -6, w -> -4}}} *)
FindInstance[
x^4 + y^4 + z^4 + w^4 - 5 x y z w + x^2 + y^2 + z^2 + w^2 + 1 < 0,
{x, y, z, w}, Reals, Method -> "ZengDecision" -> True] // AbsoluteTiming
(* {0.4111, {{x -> -(7/2), y -> -4, z -> -3, w -> -3}}} *)
Note: There are other Internal`*Options
with corresponding Internal`Set*Options
, which might be used in certain cases of FindInstance
.
And if not in FindInstance
, then perhaps they can be used in other functions with an obscure Method
option. Each with one exception corresponds to a group of SystemOptions[]
.
??System`InstanceDump`*
might provide a few clues, I think. $\endgroup$FindInstance[]
mentioned reference.wolfram.com/mathematica/tutorial/… $\endgroup$Information::nomatch: No symbol matching System`InstanceDump`* found. >>
on my system. $\endgroup$SystemOptions
, but notMethod
options. Am I overlooking something? $\endgroup$