Specification and Implementation
Specification of the scalar Basic Interval Arithmetic Subroutines (BIAS):
is generalized to support interval arithmetic on directed intervals:
- Knueppel, O.:
BIAS - Basic Interval Arithmetic Subroutines.
Bericht 93.3, Technische Universitaet Hamburg-Harburg, Hamburg, 1993.
Generalized specifications are sufficiently precise and complete to include
everything a user needs such as subroutine's purpose, method of invocation,
details of its behaviour and communication with the environment.
- Popova, E. D.; Ullrich, C. P.:
Generalising BIAS Specification.
Journal of Universal Computer Science,
Vol. 3, no. 1,
1997, pp. 23-41.
Full Text - 240K PDF)
Hierarchical consistency of interval arithmetic specification with
floating-point arithmetic, specified by the IEEE Floating-Point Standard
754/854, is provided by specification of operations and functions on intervals
involving Not-A-Numbers (NaNs) and a model of interval arithmetic exceptions
and their handling:
- Popova, E.:
Interval Operations Involving NaNs.
Reliable Computing, 2 (2), 1996, pp. 161-165.
- PASCAL-XSC Module
exi_ari.p is a self-contained PASCAL-XSC module for directed interval arithmetic.
Using data type INTERVAL which is part of the language core of
PASCAL-XSC, exi_ari module supplies all interval arithmetic operations,
functions and procedures overloaded to produce correct results for directed
intervals, too, as well as many other functions necessary for computations in
extended interval arithmetic spaces.
Short overview of the directed interval arithmetic and description of
the exi_ari module is given in
- Popova, E.:
Extended Interval Arithmetic in IEEE Floating-Point Environment.
Interval Computations, No. 4, 1994, pp. 100-129.
(FullText - 200K PDF, 385K PS)
- Mathematica Package
An attractive goal is to make use of the algebraic completeness of the
directed interval arithmetic embedding it in a computer algebra system
and investigating how the algebraic and other properties of this
arithmetic can be exploited for algebraic and symbolic manipulation of
interval expressions, development of explicit algebraic interval
methods, their effective implementation, as well as of studying other
aspects of the symbolic-numeric computations.
Mathematica package directed.m
extends Mathematica interval capabilities by providing a new data
object (Directed) representing directed multi-intervals, as well as
operations and functions for basic arithmetic on them.
A full description of the package and many application functions are
- Popova, E; Ullrich, C.:
Directed Interval Arithmetic in Mathematica: Implementation and
Technical Report 96-3, Universitaet Basel, January 1996.
tr96-3.ps (305800 bytes PostScript file)
Y. Akyildiz, E. Popova, Ch. Ullrich: Computer Algebra
Support for the Completed Set of Intervals. MISC'99: Preprints of
the Workshop on Applications of Interval Analysis to Systems and
Control, Girona, Spain, 1999, pp. 3-12.
(Full Text - PDF)
Interval Model of Linear Equilibrium Equations in Mechanics
- Popova, E.: Improved solution to the generalized Galilei’s problem with interval loads, Archive of Applied Mechanics 87 (2017) (1):115-127.
- Popova, E.D.: Equilibrium equations in interval models of structures. Int. J. Reliability and Safety, 12, 1/2, 2018, 218-235.
- Popova, E.D.: Algebraic solution to interval equilibrium equations of truss structures. Applied Mathematical Modelling 65 (2019) 489-506.
- Popova, E.D., Elishakoff, I.:
Novel interval model applied to derived variables in static and structural problems. Archive of Applied Mechanics (2019).
- Algebraic Manipulation of Interval Formulas
- Popova, E. D.; Ullrich, C.: Simplification of Symbolic-Numerical Interval Expressions
In O. Gloor (Ed.):
Proceedings of the 1998 International Symposium on Symbolic and Algebraic Computation, ACM Press, 1998,
(Full Text - 180K PS, 220K PDF)
- Algebraic Solutions of Linear Equations
- Markov, S. M.; Popova, E. D.; Ullrich, C.: On the Solution
of Linear Algebraic Equations Involving Interval Coefficients.
In S. Margenov, P. Vassilevski (Eds.):
Iterative Methods in Linear Algebra, II, IMACS Series in
Computational and Applied Mathematics, 3 1996, pp. 216-225.
- Popova, E. D.: Algebraic Solutions to a Class of Interval
Equations Journal of Universal
Computer Science, Vol. 4, no. 1,
1998, pp. 48-67.
- Efficient Inner Bounds for the solution set hull of parametric and non-parametric
interval linear systems
- Popova, E. and W. Kraemer: Inner and Outer Bounds for Parametric Linear Systems.
J. Computational and Applied Mathematics
Full Text )
- Parametric Gauss-Seidel Iteration
- Popova, E. D.: On the Solution of Parametrised Linear Systems.
In: W. Kraemer, J. Wolff von Gudenberg (Eds.): Scientific Computing,
Validated Numerics, Interval Methods. Kluwer Acad. Publishers,
2001, pp. 127-138.
(Full Text - PS 151K, PDF 200K)
- Solving Linear Systems whose Input Data are
Rational Functions of Interval Parameters
- Popova, E.: Solving Linear Systems whose Input Data are Rational Functions of Interval Parameters.
In: T. Boyanov et al. (Eds.) Numerical Methods and Applications, LNCS 4310, 2007, Springer Berlin/Heidelberg,
Expanded version in: Preprint No. 3/2005, Institute of Mathematics and Informatics, BAS, Sofia, December 2005.
(Full Text - PDF 550K)
- Popova, E., R. Iankov, Z. Bonev: Bounding the Response of Mechanical Structures with Uncertainties in all the Parameters
. In R.L.Muhannah, R.L.Mullen (Eds): Proceedings of the NSF Workshop on Reliable Engineering Computing
Svannah, Georgia USA, Feb. 22-24, 2006, 245-265.
(Full Text - PDF 352K)
- Applications in Discrete Mechanics
- F. Tonon, Using Extended Interval Algebra in Discrete Mechanics.
NSF Workshop “Reliable Engineering Computing 2006: Modeling Errors and Uncertainty in Engineering
Computations”, Rafi L. Muhanna and Robert L. Mullen eds., February 22-24, 2006,
Georgia Institute of Technology, Savannah.
- Yan Wang's articles on Uncertainty Quantification
See also the relevant articles of S. Shary and A. Goldsztejn.
Link to the Interval Computations website.
Any questions and/or comments are welcome.
Please contact Evgenija D. Popova at
Created: September 10, 1996, Last modified: Jan. 2020.