How Solve Samples - HP -15C Advanced Functions Handbook

How SOLVE Samples
The [SOLVE I routine uses only five registers of allocatable memory
in the HP-15C. The five registers hold three sample values (a, b,
and c) and two previous function values (/(a) and f(b)) while your
function subroutine calculates/(c).
The key to the effectiveness of | SOLVE] is how the next sample value
c is found.
Normally, [SOLVE | uses the secant method to select the next value.
This method uses the values of a, b,f(a), andf(b) to predict a value
c where /(c) might be close to zero.
f(x)
If c isn't a root, but /(c) is closer to zero than f(b), then b is
relabeled as a, c is relabeled as b, and the prediction process is
repeated. Provided the graph of f(x) is smooth and provided the
initial values of a and b are close to a simple root, the secant
method rapidly converges to a root.
However, under certain conditions the secant method doesn't
suggest a next value that will bound the search or move the search
closer to a root, such as finding a sign change or a smaller function
magnitude. In such cases, [SOLVE[uses a different approach.
If the calculated secant is nearly horizontal, | SOLVE | modifies the
secant method to ensure that | c — 61 ^ 1001 a — 61. This is especially
important because it also reduces the tendency for the secant
method to go astray when rounding error becomes significant near
a root.