HP 15c - Manual (Page 11)

Section 1: Using _ Effectively 11
If _ hasn't found a sign change and a sample value c doesn't yield a function value
with diminished magnitude, then _ fits a parabola through the function values at a, b,
and c. _ finds the value d at which the parabola has its maximum or minimum,
relabels d as a, and then continues the search using the secant method.
_ abandons the search for a root only when three successive parabolic fits yield no
decrease in the function magnitude or when d = b. Under these conditions, the calculator
displays Error 8. Because b represents the point with the smallest sampled function
magnitude, b and f(b) are returned in the X- and Z-registers, respectively. The Y-register
contains the value of a or c. With this information, you can decide what to do next. You
might resume the search where it left off, or direct the search elsewhere, or decide that f(b) is
negligible so that x = b is a root, or transform the equation into another equation easier to
solve, or conclude that no root exists.
Handling Troublesome Situations
The following information is useful for working with problems that could yield misleading
results. Inaccurate roots are caused by calculated function values that differ from the intended
function values. You can frequently avoid trouble by knowing how to diagnose inaccuracy
and reduce it.
Easy Versus Hard Equations
The two equations f(x) = 0 and e
f(x)
− 1 = 0 have the same real roots, yet one is almost always
much easier to solve numerically than the other. For instance, when f(x) = 6x − x
4
− 1, the
first equation is easier. When f(x) = ln(6x − x
4
), the second is easier. The difference lies in
how the function's graph behaves, particularly in the vicinity of a root.