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.
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 .
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.