The false position method is a type of bracketing method, which means that it always generates a
sequence of intervals that contains the root. It is guaranteed to converge to a root if the function is
continuous and changes sign within the interval [a,b]. However, the method may converge very slowly or
not at all if the function is highly non-linear or has multiple roots within the interval.
In bisection method, the interval between x1 and x2 is divided into two equal halves, irrespective of location
of the root. It may be possible that the root is closer to one end than the other. In the figure below, the
root is closer to x1. When the two points, x1 and x2 are joined by the straight line, the point of intersection
of this line with the x-axis (at x0) gives an improved estimate of the root and is called the false position of
the root. This point then replaces one of the initial guesses that has a function value of the same sign as
f(x0). The process is repeated with the new values of x1 and x2. This method is also known as Falsi method
and often known as linear interpolation method (because an approximate root is determined by linear
interpolation).
