krawczykTest -- certify the interval box for square polynomial system

Description

For given interval and polynomial system, this function computes the Krawczyk operator and check that the operator is contained in the input interval.

i1 : R = RR[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 : (I1, I2, I3, I4) = (interval(.94,.96), interval(.31,.33), interval(-.31,-.29), interval(.94,.96));

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.

i4 : M = matrix{{I1,I2,I3,I4}}

o4 = | [.94,.96] [.31,.33] [-.31,-.29] [.94,.96] |

                 1          4
o4 : Matrix RRi    <-- RRi
               53         53

If the Krawczyk operator is contained in the input interval, then the function returns the result that the input interval (or the Krawczyk operator) contains a unique root of the system.

i5 : krawczykTest(f,M)

o5 = true

If the function encounters a AbstractPoint as an input, then it computes a proper interval box for the given point using pointToInterval function.

i6 : p = point {{.95437+0.0001*ii, .318445, -.298627, .947941}}

o6 = p

o6 : Point

i7 : krawczykTest(f,p)

o7 = false

Ways to use krawczykTest :

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

For the programmer

The object krawczykTest is a method function.