Algorithm 763: INTERVAL_ARITHMETIC: a Fortran 90 module for an interval data type

ABSTRACT

Interval arithmetic is useful in automatically verified computations, that is, in computations in which the algorithm itself rigorously proves that the answer must lie within certain bounds. In addition to rigor, interval arithmetic also provides a simple and sometimes sharp method of bounding ranges of functions for global optimization and other tasks. Convenient use of interval arithmetic requires an interval data type in the programming language. Although various packages supply such a data type, previous ones are machine specific, obsolete, and unsupported, for languages other than Fortran, or commercial. The Fortran 90 module INTERVAL_ARITHMETIC provides a portable interval data type in Fortran 90. This data type is based on two double-precision real Fortran storage units. Module INTERVAL_ARTHMETIC uses the Fortran 77 library INTLIB (ACM TOMS Algorithm 737) as a supporting library. The module has been employed extensively in the author's own research.

REFERENCES

• 1: CRARY, F. 1976. The AUGMENT precompiler. Tech. Rep. 1470, MRC, Univ. of Wisconsin, Madison, Wisc.
• 2: HAMMER, R., NEAGA, M., AND RATZ, D. 1993. PASCAL-XSC, New concepts for scientific computation and numerical data processing. In Scientific Computing with Automatic Result Verification, E. Adams, and U. Kulisch, Eds. Academic Press, New York, 15-44.
• 3: HUSUNG, D. 1989. Precompiler for scientific computation (TPX). Tech. Rep. 91.1, Inst. of Computer Science III, Technical Univ. Hamburg, Hamburg, Germany.
• 4: R. Baker Kearfott, A Fortran 90 environment for research and prototyping of enclosure algorithms for nonlinear equations and global optimization, ACM Transactions on Mathematical Software (TOMS), v.21 n.1, p.63-78, March 1995 [doi>10.1145/200979.200991]
• 5: R. B. Kearfott , M. Dawande , K. Du , C. Hu, Algorithm 737: INTLIB---a portable Fortran 77 interval standard-function library, ACM Transactions on Mathematical Software (TOMS), v.20 n.4, p.447-459, Dec. 1994 [doi>10.1145/198429.198433]
• 6: G. F. Corliss , Rudi Klatte, C-XSC: A C++ Class Library for Extended Scientific Computing, Springer-Verlag New York, Inc., Secaucus, NJ, 1993
• 7: KNi~PPEL, O. 1994. PROFIL-BIAS--A fast interval library. Computing 53, 277-287.
• 8: LECLERC, A.1993. Parallel interval global optimization in C++. Interval Comput. 1993, 3, 148 -163.
• 9: NEUMAIER, A. 1990. Interval Methods for Systems of Equations. Cambridge University Press, Cambridge, U.K.
• 10: WALTER, W.V. 1993a. ACRITH-XSC: A Fortran-like language for verified scientific computing. In Scientific Computing with Automatic Result Verification, E. Adams and U. Kulisch, Eds. Academic Press, New York, 45-70.
• 11: WALTER, W.V. 1993b. FORTRAN-XSC: A portable Fortran 90 module library for accurate and reliable scientific computing. Computing 9, 265-286. Supplement.
• 12: J. M. Yohe, Software for Interval Arithmetic: A Reasonably Portable Package, ACM Transactions on Mathematical Software (TOMS), v.5 n.1, p.50-63, March 1979 [doi>10.1145/355815.355819]

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.0 General
          Subjects: Computer arithmetic

Additional Classification:
  D. Software
  D.3 PROGRAMMING LANGUAGES
      D.3.2 Language Classifications

          Nouns: Fortran 90

  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.0 General
          Subjects: Numerical algorithms; Error analysis

General Terms:
Algorithms, Languages

Keywords:
interval arithmetic, operator overloading, portability