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Complex Zeros of Complex Polynomials 35
opportunity that yesterday's student suffered upon encountering the instruction,
"In general, this will have to be done numerically."
The problem is to find the n values of z that make the following polynomial
equal to zero:
n
f(z)= 2: (ak+jb.)z·=O,
(3.1 )
k=O
The coefficients of this summation, a power series in z, may be complex.
Certainly the independent variable z and the roots in z may be complex, with
rectangular components
z=x+jy. (3.2)
Clearly, given a value of z, the polynomial may have a complex value with
components
f(z)=u+jv. (3.3)
To be explicit, the problem is to find the roots Zi so that the product form of
the summation in (3.1) is
f(z) = (an + jbn)(z- ZI)(Z - z,) ...(z- zn)' (3.4)
Polynomials in modern network synthesis commonly have only real coefficients,
a condition that results in roots being either real or in conjugate
complex pairs. Moore's root finder was formulated for the more general case
having complex coefficients, as in (3.1), which occurs, for example, in solving
the characteristic equations associated with complex matrices. The realcoefficient
polynomial will be solved more than twice as fast if the suggestions
that follow are incorporated. However, the more general case is retained for
instructional and practical reasons. Moore's method employs derivatives of
the polynomial. This causes some multiple-root inaccuracy not found in
nonderivative methods, such as the popular method of Muller (1956). There
are also root finders that utilize synthetic division in special ways, so that
convergence depends upon initial conditions (e.g., the Newton-Raphson, Lin,
an'd Bairstow methods). Some other methods that guarantee convergence are
not straightforward and are often slow, for example, the Lehmer-Schur and
Graeffe methods. See Ralston (1965) for descriptions of these six other
root-finding techniques.
There are two intriguing ideas central to Moore's method. The first is the
Cauchy-Riemann principle that defines the derivative of an analytic (regular)
complex function in terms of the partial derivatives of u and v (3.3) with
respect to x and y (3.2). Any student of complex-variable theory or its
application will find this worth knowing. The second idea is the Milrovic
method for evaluation of a polynomial and its derivatives. This is a much
more efficient means than the better-known "nesting" programming technique,
especially on computers where polar complex arithmetic is either slow
or nonexistent.