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238
Appendix D: A Detailed Look at _
Using the same initial estimates each
time, you have found four roots for this
equation involving a fourth-degree
polynomial. However, the last two
roots are quite close to each other and
are
actually
multiplicity of 2). That is why the root
was not eliminated when you tried
deflation once at this root. (Round-off
error causes the original function to
have small positive and negative values
for values of x between 8.4999 and
8.5001; for x = 8.5 the function is
exactly zero.)
In general, you will not know in advance the multiplicity of the root you are
trying to eliminate. If, after you have attempted to eliminate a root, _
finds that same root again, you can proceed in a number of ways:
Use different initial estimates with the deflated function in an
attempt to search for a different root.
Use deflation again in an attempt to eliminate a multiple root. If
you do not know the multiplicity of the root, you may need to
repeat this a number of times.
Examine the behavior of the deflated function at x-values near the
known root. If the function's calculated values cross the x-axis
smoothly, either another root or a greater multiplicity is indicated.
Analyze the original function and its derivatives algebraically. It
may be possible to determine its behavior for x-values near the
known root. (A Taylor series representation, for example, may
indicate the multiplicity of a root.)
Limiting the Estimation Time
Occasionally, you may desire to limit the time used by _ to find a
root. You can use two possible techniques to do this – counting iterations
and specifying a tolerance.
one
root
(with
a
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