Take a look at the new version of this page: [ beta version ]. Tell us what you think.

ABSTRACT

If we allow the magnitude and angle of a complex number (expressed in polar form) to range over an interval, it describes a semicircular region, similar to a fan; these regions are what we call complex fans. Complex numbers are a special case of complex fans, where the magnitude and angle are point intervals. Operations (especially addition) with complex numbers in polar form are complicated. What most applications do is to convert them to rectangular form, perform operations, and return the result to polar form. However, if the complex number is a Complex Fan, that transformation increases ambiguity in the result. That is, the resulting Fan is not the smallest Fan that contains all possible results. The need for minimal results took us to develop algorithms to perform the basic arithmetic operations with complex fans, ensuring the result will always be the smallest possible complex fan. We have developed the arithmetic operations of addition, negation, subtraction, product, and division of complex fans. The algorithms presented in this article are written in pseudocode, and the programs in Common Lisp, making use of CLOS (Common Lisp Object System). Translation to any other high-level programming language should be straightforward.

REFERENCES

• 1: ALEFELD, G. AND HERZBERGER, J. 1983. Introduction to Interval Computation. Academic Press, Inc., New York, NY.
• 2: Roger Conant, Engineering Circuit Analysis with PSpice and Probe, McGraw-Hill, Inc., New York, NY, 1993
• 3: Juan Jose Flores Romero, Reasoning about linear circuits in sinusoidal steady state, University of Oregon, Eugene, OR, 1998
• 4: G NEN, T. 1988. Modern Power System Analysis. John Wiley & Sons, Inc., New York, NY.
• 5: GRAINGER, J. J. AND STEVENSON, W. D. 1994. Power System Analysis. McGraw-Hill, Inc., New York, NY.
• 6: R. Baker Kearfott, Algorithm 763: INTERVAL_ARITHMETIC: a Fortran 90 module for an interval data type, ACM Transactions on Mathematical Software (TOMS), v.22 n.4, p.385-392, Dec. 1996 [doi>10.1145/235815.235816]
• 7: KERR, R. B. 1977. Electrical Network Science. Prentice-Hall, Englewood Cliffs, NJ.
• 8: KLATTE, R. AND ULLRICH, C. 1980. Complex sector arithmetic. Computing 24, 2-3, 139-148.
• 9: LANCASTER, G. 1974. DC and AC Circuits. Calendar Press, Oxford.
• 10: MOORE, R. E. 1996. Interval Analysis. Prentice-Hall, Inc., Englewood Cliffs, NJ.
• 11: Peter Struss, Problems of interval-based qualitative reasoning, Readings in qualitative reasoning about physical systems, Morgan Kaufmann Publishers Inc., San Francisco, CA, 1989
• 12: SWOKOWSKI, E.W. 1975. Fundamentals of Algebra and Trigonometry. Prindle, Weber & Schmidt, Inc., Boston, MA.
• 13: WALTON, A. K. 1987. Network Analysis and Practice. Cambridge University Press, New York, NY.

INDEX TERMS

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

Additional Classification:
  I. Computing Methodologies
  I.2 ARTIFICIAL INTELLIGENCE

  J. Computer Applications
  J.2 PHYSICAL SCIENCES AND ENGINEERING
      Subjects: Engineering

General Terms:
Algorithms

Keywords:
abstract data type, complex fans, complex numbers, interval computation, qualitative reasoning