Wolfram Demonstrations Project

An interval interpolating polynomial is the interval function that explicitly represents the set of all real polynomials of degree

interpolating through all points from a set of

vertical segments in the plane.

Contributed by: Evgenija D. Popova and Radostin Surilov

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Given a set of

distinct real numbers

and a set of

compact intervals

whose end points

are real numbers. The pair

is a vertical segment in the plane

Denote by

the set of

vertical segments. We assume that the input data belong to the interval

, such that

The set

of all real algebraic polynomials of degree

interpolating all possible sets of points

where

can be represented explicitly by an interval function, called interval interpolating polynomial.

The interval interpolating polynomial has the explicit representation

(

where

are the usual coefficients in the Lagrangian interpolating polynomial,

At any point

the value

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

Move the

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 [1]. Other material related to the interval interpolating polynomial, including a more general setting, can be found, for example, in [2], [4]. 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 [3].

References

[1] M. A. Crane, "A Bounding Technique for Polynomial Functions," SIAM J. Appl. Math., 29(4), 1975.

[2] J. Garloff, "Optimale Schranken bei Intervallinterpolation mit Polynomen und mit Functionen

," Z. Angew. Math. Mech., 59, 1979 pp. T59–T60.

[3] M. Milanese, J. P. Norton, H. Piet-Lahanier, and E. Walter (eds.), Bounding Approaches to System Identification, London, N.Y.: Plenum Press, 1996.

[4] J. Rokne, "Explicit Calculation of the Lagrangian Interval Interpolating Polynomial," Computing, 9, 1972 pp. 149–157.