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Complex Zeros of Complex Polynomials 41
axis intersections and spaced between these at 45 degrees. Running the
program shows that these roots have coordinates equal to either unity or
I/.fi, with agreement through eight significant figures on most computers.
Adding the temporary statements in Table 3.1 and running the program again
show how few times the algorithm needs to reduce the step length in a chosen
search direction. Such reductions usually occur early in the search at some
distance from the root (minimum) location.
The roots are printed whenever changes in x and yare less than I.E - 5 or,
following 10 step-size reductions, when F is no greater than I.E-8. The
algorithm is aborted if the latter condition fails or when there have been more
than 50 iterations (search directions). Little memory is required; there are two
vectors (single-subscript arrays) for the coefficient's rectangular components
a k and b k
and two more vectors for X k and Yk in (3.26). These are dimensioned
to hold N elements, where N is the maximum polynomial degree.
However, on computers with exponent ranges of about 1Oexp( + / - 37),
numerical overflow occurs for polynomials of degree greater than 20. Exponent
ranges to lOexp( + / -99) usually solve polynomials up to degree 35. The
difficulty occurs in the large polynomial value because of the poor initial root
guess of z=O.1 +jl.
Gradient root finders such as Moore's suffer from a chronic problem with
multiple roots. Consideration of a function such as y=(x-I)' and its derivative
shows that repeated (multiple) roots cause gradients (coordinate derivatives)
that tend to zero in the neighborhood of the root. This causes some
inaccuracy in repeated root values, because Moore's method depends on
gradient scaling in the step length formulas (3.9)-(3.11). The code in Table 3.2
can be added to print the value of (3.11).
Table 3.2.
Temporary Code to Print the
Squared Length of a Gradieut
2085 PRINT'GRAD MAG SQD-"; PM
Example 3.4. Add the program code m Table 3.2 to root-finder Program
B3-1 and solve the polynomial
1080 + 2466z+ 2025z' + 765z 3 + 135z 4 + 9z' = 9(z+ 1)(z+ 2)(z+ 3)(z+4)(z+ 5).
Note that the "GRAD MAG SQD" value (3.11) is well scaled. Then solve the
polynomial
54+ 135z+ 126z' + 56z 3 + 12z 4 +z'=(z+ 1)(z+2)(z+3)3.
Note that the squared gradient length used as a divisor in the search step
adjustment is well behaved until the z= - 3 repeated root is encountered.
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