Interval Interpolating Polynomial

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.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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].
[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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+