Interval Methods

A collection of links to interval methods and researcher. Much more information on intervals can be found in Vladik Kreinovich's comprehensive archive on Interval Computations.

My Books

A. Neumaier
Introduction to Numerical Analysis
Cambridge Univ. Press, Cambridge 2001
ISBN-0-521-333237 (hardback), ISBN 0-521-336104 (paperback)
Table of Contents

A. Neumaier
Interval Methods for Systems of Equations
Encyclopedia of Mathematics and its Applications 37,
Cambridge Univ. Press, Cambridge 1990
(out of print)

Some of my Papers

R.B. Kearfott, M.T. Nakao, A. Neumaier, S.M. Rump, S.P. Shary, and P. van Hentenryck, Standardized notation in interval analysis

Slides on robotics applications

A. Neumaier, Certified error bounds for uncertain elliptic equations

A. Neumaier, Computer-assisted proofs

A. Neumaier and A. Pownuk, Linear systems with large uncertainties, with applications to truss structures

D. Daney, Y. Papegay and A. Neumaier, Interval methods for certification of the kinematic calibration of parallel robots

H. Schichl and A. Neumaier, Interval Analysis on Directed Acyclic Graphs for Global Optimization

H. Schichl and A. Neumaier, Exclusion regions for systems of equations

A. Neumaier, A Gerschgorin-type theorem for zeros of polynomials

A. Neumaier, Taylor forms - use and limits, Reliable Computing 9 (2002), 43-79.

A. Neumaier, Grand challenges and scientific standards in interval analysis, Reliable Computing 8 (2002), 313-320.

A. Neumaier, Generalized Lyapunov-Schmidt reduction for parametrized equations at near singular points, Linear Algebra Appl. 324 (2001), 119-131.

R.B. Kearfott, J. Dian and A. Neumaier, Existence verification for singular zeros of nonlinear systems, SIAM J. Numer. Anal. 38 (2000), 360-379.

A. Neumaier, On Shary's algebraic approach for linear interval equations, SIAM J. Matrix Anal. Appl. 21 (2000), 1156-1162.

A. Neumaier, A simple derivation of the Hansen-Bliek-Rohn-Ning-Kearfott enclosure for linear interval equations, Reliable Computing 5 (1999), 131-136.

T. Rage, A. Neumaier and C. Schlier, Rigorous verification of chaos in a molecular model, Phys. Rev. E. 50 (1994), 2682-2688.

A. Neumaier, Global, rigorous and realistic bounds for the solution of dissipative differential equations. Part I: Theory, Computing 52 (1994), 315-336.

A. Neumaier, The wrapping effect, ellipsoid arithmetic, stability and confidence regions, Computing Supplementum 9 (1993), 175-190.

A. Neumaier, The enclosure of solutions of parameter-dependent systems of equations. In: Reliability in Computing (ed. by R.E. Moore). Acad. Press, San Diego 1988, pp. 269-286.
scanned text

Online Introductions

Interval Analysis - Basics (ps.gz-file, 1813K; slides by Hermann Schichl and Arnold Neumaier)

Auto-validating numerical methods (A course by Warwick Tucker)

Interval analysis in MATLAB (Hargreaves)

Interval Arithmetic (Van Emden)

Interval Computations: Introduction, Uses, and Resources (Kearfott)

Some Applications of Interval Computations (Vladik Kreinovich)

Vorlesung globale Optimierung (Dietmar Ratz, in German)

Self-Validated Numerical Methods and Applications (ps.gz, 538K; de Figueiredo and Stolfi)

Interval methods for bounding the range of polynomials and solving systems of nonlinear equations (ps, Ph.D. thesis by Volker Stahl)

Fields Institute Informal Working Group on Validated Methods for Optimization (May 27-31, 2002, with notes by George Corliss)

Lists and Bibliographies

Interval Computations
This is Vladik Kreinovich's comprehensive interval archive. It contains much more information than I have on this page. )

Intervals and Probability Distributions (by Dan Berleant)
The comprehensive archive on this subtopic

My page on uncertainty modeling

Bibliography of the old Freiburg Interval Library, a complete collection of over 2000 papers on interval analysis until 1987, until that time collected by Jürgen Garloff (

A bibtex version (1068KB; currently unavailable for downloading, but a copy of it can be searched) including more recent papers is being prepared by Nelson Beebe, who also has links to bibliographies on other Math/CS topics.

Karlsruhe Interval Bibliography

Reliable Computing: Table of Contents


INTLAB interval toolbox for Matlab5

VERSOFT: Verification software in MATLAB / INTLAB

(by Jiri Rohn)
includes an INTLAB primer, verified solvers for linear systems, eigenvalues and eigenvectors, linear programming, polynomial zeros, matrix functions, and the matrix equation A*X*B+C*X*D=F

b4m, Interval Arithmetic Toolbox for Matlab, based on PROFIL/BIAS (fast interval arithmetic)

Interval Arithmetic in Fortran 90

FI_LIB, fast interval library in ANSI-C, with C++ interface

FDLIBM (Freely Distributable LIBM) in C from netlib
produces elementary function values with <1ulp relative error, for machines that support IEEE 754 floating-point arithmetic (readme file)

FADBAD and TADIFF C++ packages for performing automatic differentiation of functions implemented as C/C++ programs (including interval enclosures, based on PROFIL/BIAS)

XSC Software Homepage (free download of Pascal-XSC or C-XSC)

Online Interval Calculator (Hu and Hung)

BLAS Technical Workshop 1995

BLAS Technical Workshop 1996

libaa, affine arithmetic (Stolfi)

Some Home Pages

Götz Alefeld, Karlsruhe, Germany

Fernando Alvarado, Madison, Wisconsin

Daniel Berleant (interval based arithmetic on random variables)

Gerd Bohlender, Karlsruhe, Germany

George F. Corliss, Milwaukee, Wisconsin

Tibor Csendes, Szeged, Hungary

Maarten van Emden, Victoria, Canada

R. Baker Kearfott, Lafayette, Louisiana

Vladik Kreinovich, El Paso, Texas

Ulrich Kulisch, Karlsruhe, Germany
and his interval crew

Wolfgang Kühn, ZIB, Berlin

Oscar E. Lanford III

Bruno Lang

Weldon Lodwick, Denver, Colorado

Rudolf Lohner, Karlsruhe, Germany

Early Interval Work of R. E. Moore (pdf files)

Marian Mrozek, Krakow , Poland
(computer-assisted proofs)

Rafi Muhanna
(intervals in finite element computations)

Knut Petras
(rigorous integration)

Andrzej Pownuk
(intervals in mechanical structures)

Dietmar Ratz, Karlsruhe, Germany

Siegfried Rump, Hamburg-Harburg, Germany

Peter Wittwer

Jürgen Wolff von Gudenberg, Würzburg, Germany


Two disasters caused by computer arithmetic errors (Patriot Missile Failure, and Explosion of the Ariane 5)

Interval Computations/Reliable Computing (table of contents) and a local copy (upto 1995)

Reliable Computing (table of contents)

Solving interval quadratic equations (Dimitrova and Markov)

The Kepler Conjecture
local copy of the 2002 paper
local copy of the 2004 paper (containing a complete proof, apart from the computer calculations)

Zonotopes for reduced wrapping (by Wolfgang Kühn)

Affine Arithmetic

Modal Interval Analysis

Set Estimation using Constraints and Interval Analysis

uncertainty modeling
FMathL - Formal mathematical language
recent papers and preprints
global optimization
mathematics links
my home page (

Arnold Neumaier (