Table of Contents
Advertisement
Appendix A
Advanced Use of
Section 8 includes the basic information needed for the effective use of
the
algorithm. This appendix presents more advanced, supple-
mental considerations regarding (SoLve].
Using
With Polynomials
In many practical applications, functions known as polynomials are
useful for representing physical processes or more complex mathemat-
ical functions. Polynomials are easily understood and can be structured
to have a wide range of mathematical characteristics.
A polynomial of degree n can be represented as
apx® t ap x4+ ... +ax +a,
This function has at most n real values for which the function equals
zero. A limit to the number of positive zeros of this function can be
determined by counting the number of times the signs of the coeffi-
cients change as you scan the polynomial from left to right. Similarly, a
limit to the number of negative zeros can be determined by scanning a
new function obtained by substituting —x in place of x in the original
polynomial. If the actual numberof real positive or negative zeros is less
than its limit, it will differ by an even number. (These relationships are
known as Descartes' Rule of Signs.)
As an example, consider the third-degree polynomial function
f(x) =x%—3x2— 6x + 8.
It can have no more than three real zeros. It has at most two positive
real zeros (observe the sign changes from the first to second.and third
to fourth terms) and one negative real zero (obtained fromf(-x) = -x3
-3x2 + 6x + 8).
214
Advertisement
Table of Contents
