Secant method
The secant method uses secant lines of the graph of a locally linear function to approximate its real or complex roots. For a function , the approximations are defined recursively by
The values
and
used initially in the recursion are guesses.
Derivation
Since the secant line between points and
is a linear function, it has exactly one root (unless
, in which case the method fails). The root of the secant line serves as
, an improved approximation to the root of
.
The slope of the secant line is The point
is on the secant line, so
Rearranging,
The reformulation
more clearly shows the symmetry between
and
in the calculation of
.
Failure cases
The secant method fails when there are two adjacent estimates and
for which
(analogous to reaching a zero derivative in Newton's method), and like Newton's method may converge to an unexpected root, cycle periodically, or diverge.