RegularChains - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Mathematics : Factorization and Solving Equations : RegularChains : SemiAlgebraicSetTools Subpackage : RegularChains/ConstructibleSetTools/Intersection

RegularChains

  

ConstructibleSetTools[Intersection]

  

compute the intersection of two constructible sets

  

SemiAlgebraicSetTools[Intersection]

  

compute the intersection of two semi-algebraic sets

 

Calling Sequence

Parameters

Description

Examples

References

Compatibility

Calling Sequence

Intersection(cs1, cs2, R)

Intersection(lrsas1, lrsas2, R)

Parameters

cs1, cs2

-

constructible sets

lrsas1, lrsas2

-

lists of regular semi-algebraic systems

R

-

polynomial ring

Description

• 

This command computes the set-theoretic intersection of two constructible sets, or two semi-algebraic set, depending on the input type of its arguments.

• 

A constructible set must be encoded as an constructible_set object, see the type definition in ConstructibleSetTools.

• 

A semi-algebraic set must be encoded by a list of regular_semi_algebraic_system, see the type definition in RealTriangularize.

• 

The command Intersection(cs1, cs2, R) returns the intersection of two constructible sets.  The polynomial ring may have characteristic zero or a prime characteristic.

• 

The command Intersection(lrsas1, lrsas2, R) returns the intersection of two semi-algebraic sets, encoded by list of regular_semi_algebraic_system. The polynomial ring must have characteristic zero.

• 

This command is available once RegularChains[ConstructibleSetTools] submodule or RegularChains[SemiAlgebraicSetTools] submodule have been loaded. It can always be accessed through one of the following long forms: RegularChains:-ConstructibleSetTools:-Intersection or RegularChains:-SemiAlgebraicSetTools:-Intersection.

Examples

withRegularChains:

withConstructibleSetTools:

withSemiAlgebraicSetTools:

First, define the polynomial ring R and two polynomials of R.

RPolynomialRingx,y,t

R:=polynomial_ring

(1)

p5t+5xy10t+7

p:=5t+5xy10t7

(2)

q5t5xt+2y7t+11

q:=5t5xt+2y7t+11

(3)

Using the GeneralConstruct command and adding one inequality, you can build a constructible set. Using the polynomials xt and x+t for defining inequations, the two constructible sets cs1 and cs2 are different.

cs1GeneralConstructp,q,xt,R

cs1:=constructible_set

(4)

cs2GeneralConstructp,q,x+t,R

cs2:=constructible_set

(5)

The intersection of cs1 and cs2 is a new constructible set cs.

csIntersectioncs1,cs2,R

cs:=constructible_set

(6)

Check the result in another way.

cs3GeneralConstructp,q,x+t,xt,R

cs3:=constructible_set

(7)

IsContainedcs3,cs,RandIsContainedcs,cs3,R

true

(8)

The results are as desired.

Consider now the semi-algebraic case:

lrsas1RealTriangularizep,q,,,xt,R

lrsas1:=regular_semi_algebraic_system,regular_semi_algebraic_system

(9)

lrsas2RealTriangularizep,q,,,x+t,R

lrsas2:=regular_semi_algebraic_system,regular_semi_algebraic_system

(10)

lrsasIntersectionlrsas1,lrsas2,R

lrsas:=regular_semi_algebraic_system,regular_semi_algebraic_system

(11)

lrsas12RealTriangularizep,q,,,x+t,xt,R

lrsas12:=regular_semi_algebraic_system,regular_semi_algebraic_system

(12)

Verify the results

Differencelrsas,lrsas12,R

(13)

Differencelrsas12,lrsas,R

(14)

References

  

Chen, C.; Golubitsky, O.; Lemaire, F.; Moreno Maza, M.; and Pan, W. "Comprehensive Triangular Decomposition". Proc. CASC 2007, LNCS, Vol. 4770: 73-101. Springer, 2007.

  

Chen, C.; Davenport, J.-D.; Moreno Maza, M.; Xia, B.; and Xiao, R. "Computing with semi-algebraic sets represented by triangular decomposition". Proceedings of 2011 International Symposium on Symbolic and Algebraic Computation (ISSAC 2011), ACM Press, pp. 75--82, 2011.

Compatibility

• 

The RegularChains[SemiAlgebraicSetTools][Intersection] command was introduced in Maple 16.

• 

The lrsas1 parameter was introduced in Maple 16.

• 

For more information on Maple 16 changes, see Updates in Maple 16.

See Also

Complement

ConstructibleSet

ConstructibleSetTools

Difference

GeneralConstruct

RealTriangularize

RegularChains

RegularChains

SemiAlgebraicSetTools

 


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam