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40 Some Tools and Examples of Filter Synthesis
imaginary parts with (3.3); this straightforward process yields
n
u= L:(a,X,-b,Y,),
o
(3.29)
n
v= L: (a,Y,+b,Xd.
(3.30)
o
Furthermore, differentiating (3.1) with respect to z and following the same
procedure yields
(3.31 )
(3.32)
Clearly, (3.29)-(3.32) can be programmed easily in the BASIC language,
especially since complex-variable calculations have been avoided. For the
common situation where the given polynomials have only real coefficients,
half the work in (3.29)-(3.32) can be eliminated, because all b, are zero. This
and savings in synthetic division by quadratic factors make it worthwhile to
have a separate real-coefficient, root-finding program.
3.1.4. Root-Finder Program BASIC language Program B3-I is documented
in Appendix B, including a flowchart and listing. This is similar to the
Hewlett-Packard Co. (1976a) program in the RPN language. Given a polynomial
as in (3.1), the program always starts at the point z=O.1 +jl (see Figure
3.1). Subroutine 3000 calculates (3.28), (3.29), and (3.30). Only program lines
2040-2070 are required to obtain the derivatives in (3.31) and (3.32), so that
the adjustments in x and y can be calculated for the Newton-Raphson step in
(3.9) and (3.10). If taking that step increases the objective function (goes too
far up an opposite hill in Figure 3.1), then the steps are reduced by a factor of
4 in the flowchart loop to reentry point 2190 until a lower objective value is
obtained. Note that while in that cutback loop, new derivatives are not
required, because the search direction is unchanged. It is interesting to observe
how seldom cutback is required by temporarily adding the lines in Table 3.1.
Table 3.1.
Temporary Code to Print Search
Cutback in Program 83-1
4005 PRINT ""'''''''CUTBACK''''''*ON ITER#"; L
5035 PRINT" ITERS -"; L •
Example 3.3. Input the coefficient real and imaginary parts for the polynomial
f(z) = I - Z8. The roots are on the unit circle; they are located at the four