krawczykRealnessTest -- certify the realness of the associated solution for the square polynomial system from the given interval box

Description

For given interval and polynomial system, this function determines if the associated solution is real or not.

i1 : R = CC[x1,x2,y1,y2];

i2 : f = polySystem {3*y1 + 2*y2 -1, 3*x1 + 2*x2 -7/2,x1^2 + y1^2 -1, x2^2 + y2^2 - 1};

i3 : p = point {{.95437+0.0001*ii, .318445, -.298627, .947941}} -- a numerical solution over the complex number

o3 = p

o3 : Point

i4 : I = pointToInterval(p, 1e-2) -- an interval box centered at p with radius 1e-2

o4 = |  [.94437,.96437] + [-.0099,.0101]*ii [.308445,.328445] + [-.01,.01]*ii
     ------------------------------------------------------------------------
     [-.308627,-.288627] + [-.01,.01]*ii [.937941,.957941] + [-.01,.01]*ii |

o4 : CCiMatrix

Intervals for certification should be given as a Matrix, and we set the relationships between variables and intervals by aligning them in the order of variables of the polynomial ring. For constructing a proper interval box from a given point, see the function pointToInterval. If the given interval box passes the Krawczyk test and its associated solution is real, then the function returns true

i5 : krawczykRealnessTest(f,I)

o5 = true

Ways to use krawczykRealnessTest :

krawczykRealnessTest(Matrix,AbstractPoint)
krawczykRealnessTest(Matrix,CCiMatrix)
krawczykRealnessTest(Matrix,List)
krawczykRealnessTest(PolySystem,AbstractPoint)
krawczykRealnessTest(PolySystem,CCiMatrix)
krawczykRealnessTest(PolySystem,List)

For the programmer

The object krawczykRealnessTest is a method function.