Take the 2-minute tour ×
Mathematica Stack Exchange is a question and answer site for users of Mathematica. It's 100% free, no registration required.

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?
share|improve this question
    
A perusal of ??System`InstanceDump`* might provide a few clues, I think. –  Guess who it is. Jun 14 '12 at 12:43
    
Related> Here there are a few options for FindInstance[] mentioned reference.wolfram.com/mathematica/tutorial/… –  belisarius Jun 14 '12 at 12:59
    
@J.M. that command yields Information::nomatch: No symbol matching System`InstanceDump`* found. >> on my system. –  Mr.Wizard Jun 14 '12 at 13:13
    
@belisarius I see the low-level (and wide effect) SystemOptions, but not Method options. Am I overlooking something? –  Mr.Wizard Jun 14 '12 at 13:13
    
Ech, I keep forgetting that you're on version seven... –  Guess who it is. Jun 14 '12 at 13:14

1 Answer 1

up vote 6 down vote accepted

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[].

share|improve this answer
    
At least some of those look like they're supposed to be options for CylindricalDecomposition[] –  Guess who it is. 2 days ago
    
Thanks Michael! About time this old question got an answer. :-) +1 Necromancer for you. –  Mr.Wizard 2 days ago
    
@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 2 days ago

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.