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36 Some Tools and Examples ofFilter Synthesis
Topics in this section include Moore's search algorithm, synthetic division
for linear and quadratic factors, the Mitrovic evaluation method, BASIC
program ROOTS, and polynomial scaling.
3././. Moore's Root Finder. Moore's root-finder method adjusts the components
of z=x+jy until the squared magnitude of f=u+jv is zero at z=z,. The
root factor (z-z,) is then removed from the polynomial by synthetic division,
and the process is repeated on the remainder polynomial. The adjustments on
x and yare made by the Newton-Raphson method. The method now will be
developed in detail.
The error function to be minimized over the (x, y) space is
F= ifI'=u'+v', (3.5)
as illustrated in Figure 3.1. The positive, real function F in (3.5) must have
exactly n zeros, as does the given complex function f in (3.1) or (3.4). It is well
known that u and v are well-behaved functions of x and y; i.e., they are
continuous, and their derivatives exist. In such cases, the Cauchy-Riemann
condition defines I'(z), tbe derivative of f with respect to z:
(3.6)
Furthermore, (3.6) defines a relationship between real parts and between
imaginary parts; consequently, knowledge of partial derivatives with respect to
x will furnish partial derivatives with respect to y without further work.
Proceeding, the partial derivative of F with respect to x is written by inspection
of (3.5):
(3.7)
The partial derivative of F with respect to y is similarly written, but the
Flzl
y
,
F=O
Figure 3.1.
Polynomial error surface near a root.