What Method options does FindInstance accept?

17

4

While answering Trouble getting FindInstance to return multiple results for certain constraints it became clear that for this problem FindInstance would be better starting with small numbers, but it does not.

The function has the Method option:

Options[FindInstance, Method]   (*  Out: {Method -> Automatic}  *)

Yet, I cannot find documentation for the methods (and methods sub-options) that it accepts.

  • What Method values does FindInstance accept?
  • Will any of them scan an Integer search space small-values first?

Mr.Wizard

Posted 2012-06-14T12:36:49.103

Reputation: 259 163

A perusal of ??System`InstanceDump`* might provide a few clues, I think. – J. M.'s ennui – 2012-06-14T12:43:49.000

Related> Here there are a few options for FindInstance[] mentioned http://reference.wolfram.com/mathematica/tutorial/RealPolynomialSystems.html

– Dr. belisarius – 2012-06-14T12:59:56.267

@J.M. that command yields Information::nomatch: No symbol matching System`InstanceDump`* found. >> on my system. – Mr.Wizard – 2012-06-14T13:13:02.083

@belisarius I see the low-level (and wide effect) SystemOptions, but not Method options. Am I overlooking something? – Mr.Wizard – 2012-06-14T13:13:49.773

Ech, I keep forgetting that you're on version seven... – J. M.'s ennui – 2012-06-14T13:14:38.603

@Mr.Wizard No, you are right. That is the reason why I didn't add it as an answer and started my comment as "Related". I thought the info in the article could be of your interest, though. – Dr. belisarius – 2012-06-14T13:19:23.427

Answers

9

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[].

Michael E2

Posted 2012-06-14T12:36:49.103

Reputation: 190 928

At least some of those look like they're supposed to be options for CylindricalDecomposition[] – J. M.'s ennui – 2015-07-29T15:02:14.120

Thanks Michael! About time this old question got an answer. :-) +1 Necromancer for you. – Mr.Wizard – 2015-07-29T17:31:01.707

@Mr.Wizard Thanks for the accept. I added a little more information that might be used in other Method questions, if there are any. – Michael E2 – 2015-07-29T18:18:44.933