Numerical algorithms based on the theory of complex variable

ABSTRACT

Since its introduction in the early part of the nineteenth century, the theory of complex variables has played a steadily increasing role in mathematics, and in scientific research. In some fields complex algebra is used to simplify the description of a physical system. The use of a complex impedance Z in network theory is an example of this. In other fields complex algebra seems to be a basic ingredient of the physical laws. In Wave Mechanics for example a probability density P(x,t) is related to the square modulus of a wave function &psgr;(x,t) which is itself complex, being obtained from a wave equation whose coefficients may be complex. In mathematical research itself, it is rare to find a topic which is naturally restricted to real variables, and in many topics the extension to complex variables results in a simpler theory. For example a polynomial of degree n has exactly n zeros in the field of complex numbers.

REFERENCES

• 1: L M DELVES and J N LYNESS A numerical method for locating the zeros of an analytic function Math. Comp. 21, October 1967
• 2: D. H. Lehmer, A Machine Method for Solving Polynomial Equations, Journal of the ACM (JACM), v.8 n.2, p.151-162, April 1961 [doi>10.1145/321062.321064]
• 3: J N LYNESS Differentiation formulas for analytic functions Submitted to Math. Comp.
• 4: J N LYNESS and L M DELVES On numerical contour integration round a closed contour Math. Comp. 21, October 1967
• 5: J N LYNESS and C B MOLER Vander monde systems and numerical differentiation Numer. Math 8 pp 458-464 1966
• 6: J N LYNESS and C B MOLER Numerical differentiation of analytic functions SIAM Journal for Num An 4 pp 202-210 1967

INDEX TERMS

Primary Classification:
  G. Mathematics of Computing
  G.1 NUMERICAL ANALYSIS
      G.1.0 General
          Subjects: Numerical algorithms

General Terms:
Algorithms, Theory