HP 15c - Manual (Page 238)

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 one root (with a
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.