Implementing complex elementary functions using exception handling

ABSTRACT

Algorithms are developed for reliable and accurate evaluations of the complex elementary functions required in FORTRAN 77 and FORTRAN 9, namely, cabs, csqrt, cexp, clog, csin, and ccos. The algorithms are presented in a pseudocode that has a convenient exception-handling facility. A tight error bound is derived for each algorithm. Corresponding FORTRAN programs for an IEEE environment have also been developed to illustrate the practicality of the algorithms, and these programs have been tested very carefully to help confirm the correctness of the algorithms and their error bounds. The results of these tests are included in the paper, but the FORTRAN programs are not.

REFERENCES

• 1: ANSI. 1978 American National Standard Programming Language FORTRAN. ANSI X3 9- 1978, American National Standards Institute, Inc., New York.
• 2: HULL, T. E., F^mCmEVE, T. F, AND TAN(~, P. T. P 1993. Implementing complex elementary functions using exception handhng. Preprint MCS-P338-1192, Argonne National Laboratory, Argonne, Ill., Jan.
• 3: IEEE. 1985 IEEE Standard for Binary Floating~Point Arithmetic. ANSI/IEEE Standard 754 1985, IEEE, New York
• 4: ISO. 1991. Information Technolo~r Programming Languages--FORTRAN ISO/IEC 1539: 1991, International Standards Organization, Geneva, Switzerland.
• 5: KAHAN, W. 1987. Branch cuts for complex elementary functmns, or much ado about nothing's sign bit. In The State of the Art m Numerzcal Analysts. Proceedings of the Joint IMA/SIAM Conference, A. Iserles and M J. D Powell, Eds. Clarendon Press, Oxford, England, pp. 165 211.
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INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Algorithm design and analysis

Additional Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.0 General
          Subjects: Error analysis; Numerical algorithms
      G.1.2 Approximation
          Subjects: Elementary function approximation
  G.4 MATHEMATICAL SOFTWARE
      Subjects: Reliability and robustness; Verification**

General Terms:
Algorithms, Design

Keywords:
complex elementary functions, implementation