Analysis
If the initial end-points a0 and b0 are chosen such that f(a0) and f(b0) are of opposite signs, then at each step, one of the end-points will get closer to a root of f. If the second derivative of f is of constant sign (so there is no inflection point) in the interval, then one endpoint (the one where f also has the same sign) will remain fixed for all subsequent iterations while the converging endpoint becomes updated. As a result, unlike the bisection method, the width of the bracket does not tend to zero (unless the zero is at an inflection point around which sign(f)=-sign(f″)). As a consequence, the linear approximation to f(x), which is used to pick the false position, does not improve in its quality.
One example of this phenomenon is the function
on the initial bracket . The left end, −1, is never replaced (after the first three iterations, f″ is negative on the interval) and thus the width of the bracket never falls below 1. Hence, the right endpoint approaches 0 at a linear rate (the number of accurate digits grows linearly, with a rate of convergence of 2/3).
For discontinuous functions, this method can only be expected to find a point where the function changes sign (for example at x=0 for 1/x or the sign function). In addition to sign changes, it is also possible for the method to converge to a point where the limit of the function is zero, even if the function is undefined (or has another value) at that point (for example at x=0 for the function given by f(x)=abs(x)-x² when x≠0 and by f(0)=5, starting with the interval ). It is mathematically possible with discontinuous functions for the method to fail to converge to a zero limit or sign change, but this is not a problem in practice since it would require an infinite sequence of coincidences for both endpoints to get stuck converging to discontinuities where the sign does not change (for example at x=±1 in f(x)=1/(x-1)²+1/(x+1)²). The method of bisection avoids this hypothetical convergence problem.
Read more about this topic: False Position Method
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