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# krawczykOperator -- compute the Krawczyk operator

## Description

For given interval and polynomial system, this function computes the Krawczyk operator.

    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 we conclude that the input interval (or the Krawczyk operator) contains a unique root of the system.

    i5 : krawczykOperator(f,M)

    o5 = | [.954149,.954609] [.318086,.318777] [-.298824,-.298442]
         ------------------------------------------------------------------------
         [.947663,.948236] |

                     1          4
    o5 : Matrix RRi    <-- RRi
                   53         53

The function krawczykTest checks this criterion automatically.

## Ways to use krawczykOperator:

- `krawczykOperator(Matrix,AbstractPoint)`
- `krawczykOperator(Matrix,CCiMatrix)`
- `krawczykOperator(Matrix,Matrix)`
- `krawczykOperator(PolySystem,AbstractPoint)`
- `krawczykOperator(PolySystem,CCiMatrix)`
- `krawczykOperator(PolySystem,Matrix)`

## For the programmer

The object krawczykOperator is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NumericalCertification/Documents/DocNumericalCertification.m2:387:0.