HP 15c Scientific manual Choosing Initial Estimates

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.