Implementing the complex arcsine and arccosine functions using exception handling

ABSTRACT

We develop efficient algorithms for reliable and accurate evaluatins of the complex arcsine and arccosine functions. A tight error bound is derived for each algorithm; the results are valid for all machine-representable points in the complex plane. The algorithms are presented in a pseudocode that has a convenient exception-handling facility. Corresponding Fortran 77 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 article, but the Fortran 77 programs are not (these programs are available from Fairgrieve). Tests of other widely available programs fail at many points in the complex plane, and otherwise are slower and produce much less accurate results.

REFERENCES

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INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.0 General
          Subjects: Error analysis

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

General Terms:
Algorithms, Design

Keywords:
complex elementary functions, implementation