HP 15c - Manual (Page 188)

188 Section 13: Finding the Roots of an Equation
The final case points out a potential deficiency in the subroutine rather than
a limitation of the root-finding routine. Improper operations may sometimes
be avoided by specifying initial estimates that focus the search in a region
where such an outcome will not occur. However, the _
routine is very
aggressive and may sample the function over a wide range. It is a good
practice to have your subroutine test or adjust potentially improper
arguments prior to performing an operation (for instance, use a prior to
¤). Rescaling variables to avoid large numbers can also be helpful.
The success of the _ routine in locating a root depends primarily
upon the nature of the function it is analyzing and the initial estimates at
which it begins searching. The mere existence of a root does not ensure that
the casual use of the _ key will find it. If the function f(x) has a
nonzero horizontal asymptote or a local minimum of its magnitude, the
routine can be expected to find a root of f(x) = 0 only if the initial estimates
do not concentrate the search in one of these unproductive regions—and, of
course, if a root actually exists.
Choosing Initial Estimates
When you use _ to find the root of an equation, the two initial
estimates that you provide determine the values of the variable x at which
the routine begins its search. In general, the likelihood that you will find the
particular root you are seeking increases with the level of understanding that
you have about the function you are analyzing. Realistic, intelligent
estimates greatly facilitate the determination of a root.
The initial estimates that you use may be chosen in a number of ways:
If the variable x has a limited range in which it is conceptually meaningful
as a solution, it is reasonable to choose initial estimates within this range.
Frequently an equation that is applicable to a real problem has, in addition
to the desired solution, other roots that are physically meaningless. These
usually occur because the equation being analyzed is appropriate only
between certain limits of the variable. You should recognize this restriction
and interpret the results accordingly.