THE FALSE-POSITION METHOD

Although bisection is a perfectly valid technique for determining roots, its “brute-force” approach is relatively inefficient. False position is an alternative based on a graphical insight. 

A shortcoming of the bisection method is that, in dividing the interval from Xl to Xu into equal halves, no account is taken of the magnitudes of f(Xl) and f(xu). For example, if f(Xl) is much closer to zero than f(xu), it is likely that the root is closer to xl than to Xu. An alternative method that exploits this graphical insight is to join f(Xl) and f(Xu) by a straight line. 

The intersection of this line with the x axis represents an improved estimate of the root. The fact that the replacement of the curve by a straight line gives a “false position” of the root is the origin of the name, method of false position, or in Latin. Regula falsi. It is also called the linear interpolation method.