FindInstance
FindInstance[expr,vars]
finds an instance of vars that makes the statement expr be True.
FindInstance[expr,vars,dom]
finds an instance over the domain dom. Common choices of dom are Complexes, Reals, Integers, and Booleans.
FindInstance[expr,vars,dom,n]
finds n instances.
Details and Options
- FindInstance[expr,{x1,x2,…}] gives results in the same form as Solve: {{x1->val1,x2->val2,…}} if an instance exists, and {} if it does not.
- expr can contain equations, inequalities, domain specifications and quantifiers, in the same form as in Reduce.
- The statement expr can be any logical combination of:
-
lhs==rhs equations lhs!=rhs inequations lhs>rhs or lhs>=rhs inequalities expr∈dom domain specifications {x,y,…}∈reg region specification ForAll[x,cond,expr] universal quantifiers Exists[x,cond,expr] existential quantifiers - With exact symbolic input, FindInstance gives exact results.
- Even if two inputs define the same mathematical set, FindInstance may still pick different instances to return.
- The instances returned by FindInstance typically correspond to special or interesting points in the set.
- FindInstance[expr,vars] assumes by default that quantities appearing algebraically in inequalities are real, while all other quantities are complex.
- FindInstance[expr,vars,Integers] finds solutions to Diophantine equations.
- FindInstance[expr,vars,Booleans] solves Boolean satisfiability for expr.
- FindInstance[expr,vars,Reals] assumes that not only vars but also all function values in expr are real. FindInstance[expr&&vars∈Reals,vars] assumes only that the vars are real.
- FindInstance[…,x∈reg,Reals] constrains x to be in the region reg. The different coordinates for x can be referred to using Indexed[x,i].
- FindInstance may be able to find instances even if Reduce cannot give a complete reduction.
- By default, every time you run FindInstance with a given input, it will return the same output.
- FindInstance[expr,vars,dom,n] will return a shorter list if the total number of instances is less than n.
- The following options can be given:
-
Method Automatic method to use Modulus 0 modulus to assume for integers RandomSeeding 1234 how to seed randomness WorkingPrecision Infinity precision to use in internal computations
Examples
open all close allBasic Examples (6)
Scope (57)
Complex Domain (11)
A univariate polynomial equation:
Five roots of a polynomial of a high degree:
A multivariate polynomial equation:
Systems of polynomial equations and inequations:
This gives three solution instances:
If there are no solutions FindInstance returns an empty list:
If there are fewer solutions than the requested number, FindInstance returns all solutions:
Five out of a trillion roots of a polynomial system:
In this case there is no solution:
A solution in terms of transcendental Root objects:
Five roots of an unrestricted equation:
Real Domain (13)
A univariate polynomial equation:
A univariate polynomial inequality:
A multivariate polynomial equation:
A multivariate polynomial inequality:
Systems of polynomial equations and inequalities:
If there are no solutions FindInstance returns an empty list:
If there are fewer solutions than the requested number, FindInstance returns all solutions:
A quantified polynomial system:
A solution in terms of transcendental Root objects:
Integer Domain (12)
A linear system of equations and inequalities:
A univariate polynomial equation:
A univariate polynomial inequality:
If there are fewer solutions than the requested number, FindInstance returns all solutions:
A bounded system of equations and inequalities:
A high-degree system with no solution:
Modular Domains (5)
Finite Field Domains (4)
Mixed Domains (3)
Mixed real and complex variables:
Find a real value of 
and a complex value of 
for which 
is real and less than 
:
An inequality involving Abs[z]:
Geometric Regions (9)
Find instances in basic geometric regions in 2D:
Find instances in basic geometric regions in 3D:
Find a point in the projection of a region:
A parametrically defined region:
Regions dependent on parameters:
Find values of parameters 
, 
, and 
for which the circles contain the given points:
Options (3)
RandomSeeding (1)
Finding instances often involves random choice from large solution sets:
By default, FindInstance chooses the same solutions each time:
Use RandomSeedingAutomatic to generate potentially new instances each time:
WorkingPrecision (1)
Finding an exact solution to this problem is hard:
With a finite WorkingPrecision, FindInstance is able to find an approximate solution:
Applications (11)
Geometric Problems (6)
The region ℛ is a subset of if 
is empty. Show that Disk[{0,0},{2,1}] is a subset of Rectangle[{-2,-1},{2,1}]:
Show that Rectangle[] is a not a subset of Disk[{0,0},7/5]:
Show that Cylinder[]⊆Ball[{0,0,0},2]:
Show that Cylinder[]⊈Ball[{0,0,0},7/5]:
Find a point in the intersection of two regions:
Boolean Problems (2)
Prove that a statement is a tautology:
This can be proven with TautologyQ as well:
Show that a statement is not a tautology; get a counterexample:
This can be done with SatisfiabilityInstances as well:
Properties & Relations (10)
Solution instances satisfy the input system:
Use RootReduce to prove that algebraic numbers satisfy equations:
When there are no solutions, FindInstance returns an empty list:
If there are fewer solutions than the requested number, FindInstance returns all solutions:
To get a complete description of the solution set use Reduce:
To get a generic solution of a system of complex equations use Solve:
Solving a sum of squares representation problem:
Use SquaresR to find the number of solutions to sum of squares problems:
Solving a sum of powers representation problem:
Use PowersRepresentations to enumerate all solutions:
Find instances satisfying a Boolean statement:
Use SatisfiabilityInstances to obtain solutions represented as Boolean vectors:
FindInstance shows that the polynomial 
is non-negative:
Use PolynomialSumOfSquaresList to represent 
as a sum of squares:
The Motzkin polynomial is non-negative, but is not a sum of squares:
Text
Wolfram Research (2003), FindInstance, Wolfram Language function, https://reference.wolfram.com/language/ref/FindInstance.html (updated 2024).
CMS
Wolfram Language. 2003. "FindInstance." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2024. https://reference.wolfram.com/language/ref/FindInstance.html.
APA
Wolfram Language. (2003). FindInstance. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FindInstance.html