pointToInterval -- finds an interval box from a given point

Description

This function finds an interval box from a given point. There are two ways to find an interval box. If a user has a desired radius for the interval box, then running the function with the point and the radius returns an interval centered at the point with the given radius.

i1 : p = point{{.151879*ii, -.142332-.358782*ii, .142332-.358782*ii}};

i2 : I = pointToInterval(p, 1e-3) -- returns an interval box centered at the point with the radius 1e-3

o2 = |  [-.001,.001] + [.150879,.152879]*ii [-.143332,-.141332] +
     ------------------------------------------------------------------------
     [-.359782,-.357782]*ii [.141332,.143332] + [-.359782,-.357782]*ii |

o2 : CCiMatrix

If a user doesn't know what radius to choose, but there is a given system, the function computes a proper interval box with the radius estimating the distance between the input point and its Newton convergence limit via the epsilon-inflation method (see "Interval Analysis").

i3 : R = CC[x,y,z];

i4 : f = polySystem {(x-y)^3 - z^2, (z-x)^3 - y^2, (y-z)^3 - x^2};

i5 : p = point{{.151879*ii, -.142332-.358782*ii, .142332-.358782*ii}};

i6 : I = pointToInterval(f,p)

o6 = |  [-1.43243e-12,1.43243e-12] + [.15092,.152838]*ii [-.143688,-.140976]
     ------------------------------------------------------------------------
     + [-.361002,-.356562]*ii [.140976,.143688] + [-.361002,-.356562]*ii |

o6 : CCiMatrix

Ways to use pointToInterval :

pointToInterval(AbstractPoint,Number)
pointToInterval(PolySystem,AbstractPoint)

For the programmer

The object pointToInterval is a method function.