method of false position

A bracketing method that, like the secant method, replaces the midpoint with the chord's x-intercept -- but always keeps a valid bracket the way bisection does.

The method of false position (regula falsi) narrows a bracket $[a,b]$ by replacing the plain midpoint bisection uses with $c$ , the $x$ -intercept of the secant line through $(a,f(a))$ and $(b,f(b))$ , then discards whichever endpoint shares $f(c)$ 's sign -- the same bracket-preserving rule bisection uses, applied to a better-chosen point.

Examples

One classic failure mode: if $f$ is strongly curved, one endpoint of the bracket can stay fixed for many iterations while only the other side inches in -- much slower than its per-step accuracy would suggest.