HP 15c User Manual
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Section 1: Using
_ Effectively
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If c isn't a root, but f(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 the root, such as finding a sign change or a
smaller function magnitude. In such cases, _ uses a different approach.
If the calculated secant is nearly horizontal, _ modifies the secant method to ensure
that |c − b| ≤ 100 |a − b|. 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.
If _
has already found values a and b such that f(a) and f(b) have opposite signs, it
modifies the secant method to ensure that c always lies within the interval containing the sign
change. This guarantees that the search interval decreases with each iteration, eventually
finding a root.