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Interval Computing

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Granular, Fuzzy, and Soft Computing

Part of the book series: Encyclopedia of Complexity and Systems Science Series ((ECSSS))

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  • R. A. Meyers (ed.), Encyclopedia of Complexity and Systems Science, © Springer Science+Business Media LLC 2021

Abstract

The basic idea and underlying motivation for interval arithmetic are presented, along with operational definitions, properties, applications, and how pitfalls can be avoided. Interval arithmetic’s history, starting in the early nineteenth century, is outlined with cited references. The arguably most prominent current uses are given. A section on selected currently available reference resources, including introductory books, application descriptions, software, and web resources then appears. Finally, some thoughts on the state of and direction of interval computing are given.

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References

  • Adams E, Kulisch U (eds) (1993) Scientific computing with automatic result verification, mathematics in science and engineering, vol 189. Academic Press Inc., New York

    Google Scholar 

  • Alefeld G, Frommer A, Lang B (eds) (1996) Scientific computing and validated numerics: proceedings of the international symposium on scientific computing, computer arithmetic and validated numerics SCAN-95 held in Wuppertal, Germany, September 26–29, 1995, mathematical research = Mathematische Forschung: Wissenschaftliche Beitrage herausgegeben von der Akademie der Wissenschaften der DDR, Zentralinstitut fur Mathematik und Mechanik, vol 90. Akademie-Verlag, Berlin

    Google Scholar 

  • Alefeld G, Grigorieff RD (eds) (1980) Fundamentals of numerical computation (computer-oriented numerical analysis), computing supplementum, vol 2. Springer Verlag, Wien/New York

    Google Scholar 

  • Alefeld G, Herzberger J (1974) Einführung in die Intervallrechnung. Springer-Verlag, Berlin/Heidelberg/London/etc

    MATH  Google Scholar 

  • Alefeld G, Herzberger J (1983) Introduction to interval computations. Academic Press Inc., New York, transl. by J. Rokne from the original German ‘Einführung In Die Intervallrechnung’

    Google Scholar 

  • Alefeld G, Herzberger J (eds) (1996) Numerical methods and error bounds: proceedings of the IMACS GAMM international symposium on numerical methods and error bounds held in Oldenburg, Germany, July 9–12, 1995, mathematical research, vol 89. Akademie-Verlag, Berlin

    MATH  Google Scholar 

  • Alt R, Frommer A, Kearfott RB, Luther W (eds) (2004) Numerical software with result verification: International Dagstuhl Seminar, Dagstuhl Castle, Germany, January 19–24, 2003. Revised papers, lecture notes in computer science, vol 2991, Springer-Verlag, Berlin/Heidelberg/London/etc., URL http://www.springeronline.com/3-540-21260-4

  • Apostolatos N, Kulisch U (1967a) Approximation der Erweiterten Intervallarithmetik Durch die einfache Maschinenintervallarithmetik. Computing 2:181–194

    Article  MathSciNet  MATH  Google Scholar 

  • Apostolatos N, Kulisch U (1967b) Grundlagen einer Maschinenintervallarithmetik. Computing 2:89–104

    Article  MathSciNet  MATH  Google Scholar 

  • Atanassova L, Herzberger J (eds) (1992) Computer arithmetic and enclosure methods: proceedings of the third international IMACS-GAMM symposium on computer arithmetic and scientific computing (SCAN-91), Oldenburg, Germany, 1–4 October 1991, North-Holland. The Netherlands, Amsterdam

    MATH  Google Scholar 

  • Benet L, Sanders DP (2020) JuliaIntervals web page. Online, open, URL https://github.com/JuliaIntervals/ValidatedNumerics.jl

  • Berz M, Makino K, Shamseddine K, Hoffstätter GH, Wan W (1996) COSY INFINITY and its applications to nonlinear dynamics. In: Berz M, Bischof C, Corliss G, Griewank A (eds) Computational differentiation: techniques, applications, and tools. SIAM, Philadelphia, pp 363–365

    Google Scholar 

  • Bnhelyi B, Csendes T, Krisztin T, Neumaier A (2014) Global attractivity of the zero solution for Wright’s equation. SIAM J Appl Dyn Syst 13(1):537–563. https://doi.org/10.1137/120904226

    Article  MathSciNet  MATH  Google Scholar 

  • Burat E (1885) Traité D’Arithmétique a L’usage des élèves des lycées et colléges. Eugéne Belin et Fils., Paris, found November 16, 2020 at https://gallica.bnf.fr/ark:/12148/bpt6k202612w.texteImage

  • Ceberio M, Kreinovich V (eds) (2008) SCAN ‘08: proceedings of the 13th IMACS-GAMM international symposium on scientific computing, computer arithmetic and validated numerics, reliable computing 15 (journal special volume), USA, http://www.cs.utep.edu/interval-comp/rc.html or https://interval.louisiana.edu/reliable-computing-journal/tables-of-contents.html. Accessed on 28 Nov 2020

  • Corliss GF, Kearfott RB (eds) (1993) Abstracts for an international conference on numerical analysis with automatic result verification: mathematics, application and software, February 25–March 1, 1993, Lafayette, LA, 1993, available at https://interval.louisiana.edu/NAARV-93-abstracts.pdf. Last accessed 16 Dec 2020)

  • Csendes T (ed) (1999) Developments in reliable computing: papers presented at the international symposium on scientific computing, computer arithmetic, and validated Numerics, SCAN-98, in Szeged, Hungary, reliable computing, vol 5(3), Kluwer Academic Publishers Group, Norwell/Dordrecht

    MATH  Google Scholar 

  • Desrochers B, Jaulin L (2017) Computing a guaranteed approximation of the zone explored by a robot. IEEE Trans Autom Control 62(1):425–430. https://doi.org/10.1109/TAC.2016.2530719

    Article  MathSciNet  MATH  Google Scholar 

  • Dwyer PS (1951) Computation with approximate numbers. In: Dwyer PS (ed) Linear computations. Wiley & Sons Inc, New York, pp 11–35

    MATH  Google Scholar 

  • Gauss C (1809) Theoria motus corporum coelestium in sectionibus conicis solem ambientium. Carl Friedrich Gauss Werke, Hamburgi sumptibus Frid. Perthes et I.H.Besser, URL https://books.google.com/books?id=ORUOAAAAQAAJ, eBook and PDF currently available free of charge from Google at https://books.google.com/books/about/Theoria_motus_corporum_coelestium_in_sec.html?id\unhbox\voidb@x\hbox{$=$}ORUOAAAAQAAJ

  • Gustafson J (2015) The end of error: Unum computing. CRC Press/Chapman and Hall, Boca Raton. https://doi.org/10.1201/9781315161532

    Book  MATH  Google Scholar 

  • Hales TC (2005) A proof of the Kepler conjecture. Ann Math 162(3):1065–1185. URL https://annals.math.princeton.edu/wp-content/uploads/annals-v162-n3-p01.pdf, together with Samuel P. Ferguson, winner of the 2007 Robbins prize

  • Hansen E (ed) (1969) Symposium on interval analysis (1968: Culham, England): topics in interval analysis. Clarendon Press, Oxford, UK

    Google Scholar 

  • Hansen E, Walster GW (2003) Global optimization using interval analysis. Marcel Dekker, Inc., New York

    Book  MATH  Google Scholar 

  • Hansen ER (1983) Global optimization using interval analysis. Marcel Dekker, New York

    Google Scholar 

  • Henrici P (1971) Circular arithmetic and the determination of polynomial zeros. In: Conference on application of numerical analysis. Lecture notes in mathematics, vol 228. Springer Verlag, Berlin/Heidelberg/New York, pp 86–92

    Chapter  Google Scholar 

  • Herzberger J (ed) (1994) Topics in validated computations: proceedings of IMACS-GAMM international workshop on validated computation, Oldenburg, Germany, 30 august–3 September 1993, studies in computational mathematics, vol 5. Elsevier, Amsterdam

    MATH  Google Scholar 

  • IEEE (2015) 1788–2015 – IEEE standard for interval arithmetic. IEEE, New York, https://doi.org/10.1109/IEEESTD.2015. 7140721, URL http://ieeexplore.ieee.org/servlet/opac?punumber=7140719. Approved 11 June 2015 by IEEE-SA Standards Board

  • IEEE (2017) IEEE Std 1788.1-2017 – IEEE standard for interval arithmetic (simplified). IEEE, New York, URL https://standards.ieee.org/findstds/standard/1788.1-2017.html

    Google Scholar 

  • IEEE-754 (2019) IEEE 754-2019, standard for floating-point arithmetic. IEEE, New York. https://doi.org/10.1109/IEEESTD.2019.876622

  • Institute for New Technologies (1991-present) Reliable Computing journal. https://interval.louisiana.edu/reliable-computing-journal/RC.html and http://www.cs.utep.edu/interval-comp/rcjournal. html. Accessed 3 Dec 2020

  • Jaulin L (2009) Robust set-membership state estimation; application to underwater robotics. Autom 45(1):202–206. https://doi.org/10.1016/j.automatica.2008.06.013

    Article  MathSciNet  MATH  Google Scholar 

  • Jaulin L, Keiffer M, Didrit O, Walter E (2001) Applied interval analysis. SIAM, Philadelphia

    Book  Google Scholar 

  • Kaucher EW, Markov SM, Mayer G (eds) (1991) Computer arithmetic, scientific computation and mathematical modelling: proceedings of the second international conference on computer arithmetic, scientific computation and mathematical modelling, Albena, Bulgaria, September 24–28, 1990, IMACS annals on computing and applied mathematics, vol 12. J. C. Baltzer AG, Scientific Publishing Company, Basel

    Google Scholar 

  • Kearfott RB (1996) Rigorous global search: continuous problems. No. 13 in nonconvex optimization and its applications. Kluwer Academic Publishers, Dordrecht

    Book  Google Scholar 

  • Kearfott RB (2020) Ramon Moore’s early papers. Online, open, URL https://interval.louisiana.edu/Moores_early_papers/bibliography.html. Accessed 3 Dec 2020

  • Kearfott RB, Kreinovich V (eds) (1996) Applications of interval computations: papers presented at an international workshop in El Paso, Texas, February 23–25, 1995, applied optimization, vol 3. Kluwer Academic Publishers Group, Norwell/Dordrecht

    Google Scholar 

  • Knowles S, McAllister WH (eds) (1995) Proceedings of the 12th symposium on computer arithmetic, July 19–21, 1995, Bath, England, symposium on computer arithmetic, vol 12. IEEE Computer Society Press, Silver Spring

    Google Scholar 

  • Krämer W, von Gudenberg JW (eds) (2001) Scientific computing, validated numerics, interval methods, Kluwer Academic Publishers Group, Norwell/Dordrecht., URL http://www.wkap.nl/prod/b/0-306-46706-2, sCAN 2000, the GAMM–IMACS International symposium on scientific computing, computer arithmetic, and validated numerics and interval 2000, the international conference on interval methods in science and engineering were jointly held in Karlsruhe, September 19–22, 2000

  • Kreinovich V (2008) Relations between interval and soft computing. In: Hu C, Kearfott RB, de Korvin A (eds) Knowledge processing with interval and soft computing, Advanced information and knowledge processing. Springer-Verlag, Berlin/Heidelberg/London/etc, pp 75–97. https://doi.org/10.1007/BFb0085718

    Chapter  Google Scholar 

  • Kreinovich V (2020) UTEP interval computations web site. Online, open, URL http://www.cs.utep.edu/interval-comp/main.html. Accessed 3 Dec 2020

  • Kreinovich V, Tucker W (eds) (2012) SCAN ‘08: Proceedings of the 17th IMACS-GAMM international symposium on scientific computing, computer arithmetic and validated numerics, Reliable Computing 25 (journal special volume), USA, http://www.cs.utep.edu/interval-comp/rc.html or https://interval.louisiana.edu/reliable-computing-journal/tables-of-contents.html. Accessed 28 Nov 2020

  • Kulisch U, Miranker WL (eds) (1983) A new approach to scientific computation: proceedings of a symposium held at IBM Thomas J. Watson Research Center, Yorktown Heights, New York. Academic Press, New York

    MATH  Google Scholar 

  • Kulisch U, Stetter HJ (eds) (1988) Scientific computation with automatic result verification, computing. Supplementum, vol 6, Springer, Wien/New York, based on papers presented at the conference on computer arithmetic and scientific computation held Sep. 30–Oct. 2, 1987 in Karlsruhe, FRG

    Google Scholar 

  • Kulisch UW, Miranker WL (1981) Computer arithmetic in theory and practice. Computer science and applied mathematics. Academic Press Inc., New York

    MATH  Google Scholar 

  • Luther W, Otten W (eds) (2006) SCAN 2006: proceedings of the 12th GAMM - IMACS international symposium on scientific computing, computer arithmetic and validated Numerics, IEEE Computer Society, USA., https://www.computer.org/csdl/proceedings/scan/2006/12OmNwwd2WZ. Accessed 28 Nov 2020

  • Mayer G (2017) Interval analysis: and automatic result verification. De Gruyter, Berlin, Boston. https://doi.org/10.1515/9783110499469, URL https://www.degruyter.com/view/title/523499

  • Miranker WL, Toupin RA (eds) (1985) Accurate scientific computations: symposium, Bad Neuenahr, FRG, March 12–14, 1985: proceedings, lecture notes in computer science, vol 235. Springer-Verlag, Berlin/Heidelberg/London/etc

    Google Scholar 

  • Moore RE (1959) Automatic error analysis in digital computation. Technical Report Space Div. Report LMSD84821, Lockheed Missiles and Space Co., Sunnyvale, CA, USA, currently available at http://interval.louisiana.edu/Moores_early_papers/Moore_Lockheed.pdf

  • Moore RE (1962) Interval arithmetic and automatic error analysis in digital computing. Ph.D. dissertation, Department of Mathematics, Stanford University, Stanford, CA, USA, URL http://interval.louisiana.edu/Moores_early_papers/disert.pdf, also published as applied mathematics and statistics laboratories technical report no. 25

  • Moore RE (1966) Interval analysis. Prentice-Hall, Upper Saddle River

    MATH  Google Scholar 

  • Moore RE (1979) Methods and applications of interval analysis. SIAM, Philadelphia

    Book  MATH  Google Scholar 

  • Moore RE, Kearfott RB, Cloud MJ (2009) Introduction to interval analysis. SIAM, Philadelphia, URL. http://www.loc.gov/catdir/enhancements/fy0906/2008042348-b.html. http://www.loc.gov/catdir/enhancements/fy0906/2008042348-d.html;, http://www.loc.gov/catdir/enhancements/fy0906/2008042348-t.html

  • Nehmeier M, von Gudenberg JW, Tucker W (eds) (2016) Scientific computing, computer arithmetic, and validated numerics – 16th international symposium, SCAN 2014, Würzburg, Germany, September 21-26, 2014. Revised selected papers, lecture notes in computer science, vol 9553, Springer, https://doi.org/10.1007/978-3-319-31769-4

  • Neumaier A (1990) Interval methods for Systems of equations, encyclopedia of mathematics and its applications, vol 37. Cambridge University Press, Cambridge, UK

    Google Scholar 

  • Nickel K (1966) Über die Notwendigkeit einer Fehlerschranken- Arithmetik für Rechenautomaten. Numer Math 9:69–79

    Article  MathSciNet  MATH  Google Scholar 

  • Nickel K (1973) The contraction mapping fixed point theorem in interval analysis. MRC technical summary 1334. University of Wisconsin, Madison

    Google Scholar 

  • Nickel K (ed) (1975) Interval mathematics: Proceedings of the International Symposium, Karlsruhe, West Germany, May 20–24, 1975, no. 29 in Lecture Notes In Computer Science, Springer Verlag, Berlin/Heidelberg

    Google Scholar 

  • Nickel K (ed) (1980) Interval mathematics 1980: proceedings of an international symposium on interval mathematics, held at the Institut für Angewandte Mathematik, Universität Freiburg i. Br., Germany, May 27–31, 1980. Academic Press Inc., New York

    MATH  Google Scholar 

  • Nickel K (ed) (1986) Interval mathematics 1985: proceedings of the international symposium, Freiburg i. Br., Federal Republic of Germany, September 23–26, 1985. Lecture notes in computer science, vol 212, Springer-Verlag, Berlin/Heidelberg/London/etc

    Google Scholar 

  • Rao TRN, Matula DW (eds) (1975) 3rd symposium on computer arithmetic, November 19–20, 1975, Southern Methodist University, Dallas, Texas, IEEE Computer Society Press, Silver Spring, iEEE Cat. No 75 CH1017-3C

    Google Scholar 

  • Rohn J (1981) Strong solvability of interval linear programming problems. Computing 26:79–82

    Article  MathSciNet  MATH  Google Scholar 

  • Rump SM (1981) Kleine, exakte Fehlerschranken für die Lösung linearer Gleichungssysteme. Z Angew Math Mech 61:T313–T315

    MATH  Google Scholar 

  • Rump SM (1999) INTLAB–INTerval LABoratory. In: Csendes (1999), pp 77–104, uRL: http://www.ti3.tu-harburg.de/rump/intlab/

  • Sahinidis NV (1996) BARON: a general purpose global optimization software package. J Glob Optim 8(2):201–205

    Article  MathSciNet  MATH  Google Scholar 

  • Schnellinger J (1879) Das gekürzte Rechnen (basically, truncated arithmetic or inexact arithmetic). Zeitschrift für das Realschulwesen 1879:257–284, (currently available in Google books)

    Google Scholar 

  • Shary S, Corliss G (eds) (2012) SCAN ‘08: Proceedings of the 15th IMACS-GAMM International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics, Reliable Computing 19 (journal special volume), USA, http://www.cs.utep.edu/interval-comp/rc.html or https://interval.louisiana.edu/reliable-computing-journal/tables-of-contents.html. Accessed on 28 Nov 2020)

  • Sunaga T (1958) Theory of interval algebra and its application to numerical analysis. RAAG memoirs 2:29–46, URL http://www.cs.utep.edu/interval-comp/sunaga.pdf

  • Szpiro GG (2003) Kepler’s conjecture: how some of the greatest minds in history helped solve one of the oldest math problems in the world. Wiley

    MATH  Google Scholar 

  • Tucker W (1999) The Lorenz attractor exists. C R Acad Sci Paris 328:1197–1202

    Article  MathSciNet  MATH  Google Scholar 

  • Tucker W (2002) A rigorous ODE solver and Smale’s 14th problem. Found Comput Math 24:53–117

    Article  MathSciNet  MATH  Google Scholar 

  • Warmus MM, Steinhaus H (1956) Calculus of approximations. Bulletin de l’Academie Polonaise des Sciences, Cl III IV(5):253–259

    Google Scholar 

  • Wikipedia (2020) Interval arithmetic. Online, open, URL https://en.wikipedia.org/wiki/Interval_arithmetic. Accessed 20 Dec 2020

  • Young RC (1931) The algebra of many-valued quantities. Math Ann 104:260–290

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Ralph Baker Kearfott .

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Kearfott, R.B. (2023). Interval Computing. In: Lin, TY., Liau, CJ., Kacprzyk, J. (eds) Granular, Fuzzy, and Soft Computing. Encyclopedia of Complexity and Systems Science Series. Springer, New York, NY. https://doi.org/10.1007/978-1-0716-2628-3_722

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