Section 1: Using; Finding Roots - HP -15C Advanced Functions Handbook

Section 1
Using | SOLVE | Effectively
The [SOLVE| algorithm provides an effective method for finding a
root of an equation. This section describes the numerical method
used by [SOLVE | and gives practical information about using
| SOLVE | in various situations.
Finding Roots
In general, no numerical technique can be guaranteed to find a root
of every equation that has one. Because a finite number of digits
are used, the calculated function may differ from the theoretical
function in certain intervals of x, it may not be possible to
represent the roots exactly, or it may be impossible to distinguish
between zeros and discontinuities of the function being used.
Because the function can be sampled at only a finite number of
places, it's also possible to conclude falsely that the equation has
no roots.
Despite these inherent limitations on any numerical method for
finding roots, an effective method—like that used by [ SOLVE | —
should strive to meet each of the following objectives:
• If a real root exists and can be exactly represented by the
calculator, it should be returned. Note that the calculated
function may underflow (and be set to zero) for some values of
x other than the true roots.
• If a real root exists, but it can't be exactly represented by the
calculator, the value returned should differ from the true root
only in the last significant digit.
• If no real root exists, an error message should be displayed.
The | SOLVE | algorithm was designed with these objectives in mind.
It is also easy to use and requires little of the calculator's memory.
And because I SOLVE | in a program can detect the situation of not
finding a root, your programs can remain entirely automatic
regardless of whether | SOLVE | finds a root.
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