The False-Position Method

Introduction

The poor convergence of the bisection method as well as its poor adaptability to higher dimensions motivate the use of better techniques. One such method is the Method of False Position.

Methodology we start with an initial interval [x1,x2], and we assume that the function changes sign only once in this interval. Now we find an x3 in this interval, which is given by the intersection of the x axis and the straight line passing through (x1,f(x1)) and (x2,f(x2)). It is easy to verify that x3 is given by

Methodology

Now, we choose the new interval from the two choices [x1,x3] or [x3,x2] depending on in which interval the function changes sign.

NoteThe False-Position and Bisection algorithms are quite similar. The only difference is the formula used to calculate the new estimate of the root x3

Graphical look

Numerical Example Find a root of 3x + sin(x) - exp(x) =0. The graph of this equation is given in the figure. From this it's clear that there is a root between 0 and 0.5 and also another root between 1.5 and 2.0. Now let us consider the function f (x) in the interval [0, 0.5] where f (0) * f (0.5) is less than zero and use the regula-falsi scheme to obtain the zero of f (x) = 0.

So one of the roots of 3x + sin(x) - exp(x) = 0 is approximately 0.36. Note : Although the length of the interval is getting smaller in each iteration, it is possible that it may not go to zero. If the graph y = f(x) is concave near the root 's', one of the endpoints becomes fixed and the other end marches towards the root.

Matlab implementation

Thank You…!

Presented by

Tayyaba AbbasRoll no 5