Given a set of
distinct real numbers
and a set of
whose end points
are real numbers. The pair
is a vertical segment in the plane
the set of
vertical segments. We assume that the input data belong to the interval
, such that
of all real algebraic polynomials of degree
interpolating all possible sets of points
can be represented explicitly by an interval function, called interval interpolating polynomial.
The interval interpolating polynomial has the explicit representation
are the usual coefficients in the Lagrangian interpolating polynomial,
At any point
of the interpolating polynomial is an interval.
Each vertical segment in the present Demonstration is generated as
where the point with coordinates
is represented by a locator. Drag the locators or create/delete locators to change the vertical segments and see how the interval interpolating polynomial (in its graphical or analytical representation) changes.
The boundary of the interval interpolating polynomial consists of piece-wise real interpolating polynomials defined by particular end points of the intervals
for each subinterval
subinterval slider to see the corresponding slice-boundary real interpolating polynomials—one colored in orange and the other in green.
The family of real polynomials interpolating all points in a given set of vertical segments can be investigated without using interval arithmetic . Other material related to the interval interpolating polynomial, including a more general setting, can be found, for example, in , . Another approach that is different to the interval function representation is called parameter set representation. Some material related to this approach can be found in .
 M. A. Crane, "A Bounding Technique for Polynomial Functions," SIAM J. Appl. Math.
 J. Garloff, "Optimale Schranken bei Intervallinterpolation mit Polynomen und mit Functionen
," Z. Angew. Math. Mech.
, 1979 pp. T59–T60.
 M. Milanese, J. P. Norton, H. Piet-Lahanier, and E. Walter (eds.), Bounding Approaches to System Identification
, London, N.Y.: Plenum Press, 1996.
 J. Rokne, "Explicit Calculation of the Lagrangian Interval Interpolating Polynomial," Computing
, 1972 pp. 149–157.