GenericCylindricalDecomposition

GenericCylindricalDecomposition[ineqs,{x1,x2,…}]

finds the full-dimensional part of the decomposition of the region represented by the inequalities ineqs into cylindrical parts whose directions correspond to the successive xi, together with any hypersurfaces containing the rest of the region.

Details and Options

Examples

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Basic Examples  (1)

Find a cylindrical decomposition of the full-dimensional part of the unit disk:

Scope  (3)

GenericCylindricalDecomposition gives a full‐dimensional set of solutions and a hypersurface:

The hypersurface contains the difference between all solutions and the full‐dimensional set:

There is no remaining hypersurface here:

Here the whole solution set is lower-dimensional:

Generalizations & Extensions  (1)

Find solutions that are generic with respect to the first two variables:

The difference between the generic and full solution is contained in the blue cylindrical surface:

Options  (1)

Method  (1)

By default, GenericCylindricalDecomposition returns cylindrical decomposition represented as a Boolean combination of equations and inequalities:

Use Method{"CylindricalDecompositionFunctionOutput"True} to get CylindricalDecompositionFunction results:

Applications  (1)

Plot a region described by inequalities:

The curves on the surface correspond to cylinder boundaries:

Computing full cylindrical decomposition takes longer and is not necessary to plot the picture:

RegionPlot3D uses a numerical method and gives a somewhat less-precise picture:

Properties & Relations  (3)

GenericCylindricalDecomposition finds the solution set up to lower‐dimensional parts:

CylindricalDecomposition finds the complete solution set:

Reduce uses cylindrical decomposition to solve inequalities:

Possible Issues  (1)

When the solution set is lower-dimensional, no solutions are found:

To find lower-dimensional solution sets, use CylindricalDecomposition:

Wolfram Research (2007), GenericCylindricalDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/GenericCylindricalDecomposition.html.

Text

Wolfram Research (2007), GenericCylindricalDecomposition, Wolfram Language function, https://reference.wolfram.com/language/ref/GenericCylindricalDecomposition.html.

CMS

Wolfram Language. 2007. "GenericCylindricalDecomposition." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GenericCylindricalDecomposition.html.

APA

Wolfram Language. (2007). GenericCylindricalDecomposition. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GenericCylindricalDecomposition.html

BibTeX

@misc{reference.wolfram_2024_genericcylindricaldecomposition, author="Wolfram Research", title="{GenericCylindricalDecomposition}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/GenericCylindricalDecomposition.html}", note=[Accessed: 22-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_genericcylindricaldecomposition, organization={Wolfram Research}, title={GenericCylindricalDecomposition}, year={2007}, url={https://reference.wolfram.com/language/ref/GenericCylindricalDecomposition.html}, note=[Accessed: 22-December-2024 ]}