Early Papers on Interval Computations
Origins of Interval Computations: from Archimedes to 1960s
Achimedes used two-sided bounds to compute Pi:
Archimedes, "On the measurement of the
circle", In: Thomas L. Heath
(ed.), The Works of Archimedes, Cambridge University Press,
Cambridge, 1897; Dover edition, 1953, pp. 91-98.
The concept of a function having values which are bounded within
limits was discussed by W. H. Young:
W. H. Young, "Sull due funzioni a piu valori constituite dai limiti
d'una funzione di variable reale a destra ed a sinistra di ciascun
punto", Rendiconti Academia di Lincei, Classes di Scienza Fiziche,
1908, Vol. 17, No. 5, pp. 582-587.
For positive quantities, rules for the arithmetic of intervals were
explicitly stated and applied to evaluation of rational expressions by
Vladimir M. Bradis*:
V. M. Bradis, "An experience of the verification of practically useful
operations with approximate numbers", Proceedings of Tver Pedagogical Institute,
1927, No. 3, in Russian.
The concept of operations with a set of multi-valued numbers was
introduced by R. C. Young, who developed a formal algebra of
The special case of closed intervals (not necessarily positive)
was further developed by P. S. Dwyer:
Interval mathematics was further developed by
by T. Sunaga:
T. Sunaga, "Theory of interval algebra and its
application to numerical analysis", In:
Research Association of Applied Geometry (RAAG) Memoirs,
Ggujutsu Bunken Fukuy-kai. Tokyo, Japan, 1958, Vol. 2, pp. 29-46 (547-564);
Japan Journal on Industrial and Applied Mathematics, 2009,
Vol. 26, No. 2-3, pp. 126-143.
and by R.
For an early history of interval computations, see also
S. Markov and K. Okumura, "The Contribution of T. Sunaga to
Interval Analysis and Reliable Computing", In: T. Csendes (ed.),
Developements in Reliable Computing, Kluwer, Dordrecht,
1999, pp. 167-188.
Selected Papers from the 1970s and 1980s
*This approach was further developed in several other publications of
V. M. Bradis:
Back to Bibliographies on Interval
- V. M. Bradis, "Theory and Practice of Computations", Moscow, Uchpedgiz Publ.,
- V. M. Bradis, "Tools and Methods of Elementary Computations", Moscow, Russian
Academy of Pedagogical Sciences, 1948 (in Russian).
- V. M. Bradis, Oral and Written
Computations: Tools and Techniques for Computations,
in: "Encyclopedia of Elementary Mathematics", Moscow, 1951
(in Russian), see Sections 6 and 8; German translation:
H. Grell, K. Maruhn, and W. Rinow (eds.),
Enzyklopaedie der Elementarmathematik, Band I Arithmetik, VEB Deutcsher Verlag der
Wissenschaften, Berlin, 1966, pp. 346 ff.
Back to the main menu of the Interval