describe a cell in terms of the real roots of its boundary polynomials - Maple Help

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RootFinding[Parametric][CellDescription] - describe a cell in terms of the real roots of its boundary polynomials

Calling Sequence

CellDescription(m, k)

Parameters

m

-

solution record, as returned by CellDecomposition

k

-

positive integer; the index of a cell

Description

• 

The CellDescription command returns a list of lists of the form p,i,a,q,j where

– 

a is a parameter

– 

p and q are polynomials in the parameters

– 

i and j are non-negative integers

• 

The CellDescription(m,k) calling sequence returns a description of the kth cell in m in terms of real roots of some projection polynomials.

• 

The solution record m must have been computed with the option output=cad or without using the output keyword.

• 

Each inner list, p,i,a,q,j, in the result is to be interpreted as follows: the a-coordinate of a point u lying in the interior of the kth cell is greater than the ith real root of the polynomial p and less than the jth real root of the polynomial q.

• 

If an inner list is of the form p,i,a,,0, then this means that the a-coordinate is unbounded from above, and similarly, if an inner list is of the form  ,0,a,q,j, then the a-coordinate is unbounded from below.

• 

The polynomials p and q in each inner list contain only the parameters from the current and all earlier lists. So the polynomials in the first inner list are univariate, the ones in the second inner list are bivariate, etc.

• 

The result, [[p1,i1,a1,q1,j1], [p2,i2,a2,q2,j2], ...], can be used to sample a cell as follows: Compute the i1th real root of the univariate polynomial p1 and the j1th real root of the univariate polynomial p2 (for example, using RootFinding[Isolate]), and pick a value for the a1-coordinate between those two roots. Then substitute that value of a1 into p2 and q2, turning these into univariate polynomials in a2. In the same way as above, compute their i2th and j2th roots, respectively, and pick a value for the a2-coordinate in between those two roots. Continue in a similar fashion.

• 

This command is part of the RootFinding[Parametric] package, so it can be used in the form CellDescription(..) only after executing the command with(RootFinding[Parametric]). However, it can always be accessed through the long form of the command by using RootFinding[Parametric][CellDescription](..).

Examples

withRootFinding[Parametric]:

m:=CellDecompositionx2+ay2b=0,ax2+by=0,x,y

m:=Equations=ay2+x2b,ax2+byInequalities=Filter=01Variables=x,yParameters=a,bDiscriminantVariety=a,b,4a3+bProjectionPolynomials=b,a,4a3+bSamplePoints=a=−1,b=−1,a=86581115277274877906944,b=−1,a=1,b=−1,a=−1,b=1,a=86581115277274877906944,b=1,a=1,b=1

(1)

CellDescriptionm,5

b,1,b,∞,0,4a3+b,1,a,a,1

(2)

This result is to be interpreted as follows: a point u1,u2 in the parameter space belongs to the 5th cell if and only if

– 

u2 is greater than the 1st (and only) real root of b&equals;0, that is, 0<u2; and

– 

u1 is greater than the first (and only) real root of 4a3&plus;u2&equals;0 and less than the first (and only) real root of a&equals;0, that is,

-u243<u1<0

Similarly, a point u1&comma;u2 belongs to the 4th cell if and only if

u2&gt;0andu1<-u243

CellDescriptionm&comma;4

b&comma;1&comma;b&comma;&infin;&comma;0&comma;&infin;&comma;0&comma;a&comma;4a3&plus;b&comma;1

(3)

CellPlotm&comma;4&comma;5&comma;&apos;samplepoints&apos;

See Also

CellDecomposition, CellPlot, Parametric, RootFinding, RootFinding[Isolate]


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