The convergce process in the bisection method is very slow. It depends only on the choice of end points of the interval [a,b]. The function f(x) does not have any role in finding the point c (which is just  the mid-point of a and b). It is used only to decide the next smaller interval [a,c] or [c,b]. A better approximation to c can be obtained by taking the straight line L joining the points (a,f(a)) and (b,f(b)) intersecting the x-axis. To obtain the value of c we can equate the two expressions of the slope m of the line L. 

m =  f(b) - f(a)  =  0 - f(b)
   (b-a)  (c-b)

      => (c-b) * (f(b)-f(a)) = -(b-a) * f(b)


c = b - f(b) * (b-a)
 f(b) - f(a)

Now the next smaller interval which brackets the root can be obtained by checking

f(a) * f(b) < 0 then b = c
                > 0 then a = c
                           = 0 then c is the root.

Selecting c by the above expression is called Regula-Falsi method or False position method.

Algorithm - False Position Scheme

Given a function f (x) continuos on an interval [a,b] such that f (a) * f (b) < 0 

c =  a*f(b) - b*f(a)
    f(b) - f(a)

       if f (a) * f (c) < 0 then  b = c 
                                  else  a = c 
while (none of the convergence criterion C1, C2 or C3 is satisfied)

The false position method is again bound to converge because it brackets the root in the whole of its convergence process.

Numerical Example :
exp0.jpg for 3x+sin[x]-exp[x]
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.

f(a) * f(c)
 1.38 (+ve)
-0.102 (-ve)
-0.085 (-ve)

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. 

Worked out problems
 Exapmple 1  Find a root of x * cos[(x)/ (x-2)]=0  Solution
 Exapmple 2  Find a root of x2 = (exp(-2x) - 1) / x  Solution
 Exapmple 3  Find a root of exp(x2-1)+10sin(2x)-5 = 0  Solution
 Exapmple 4  Find a root of exp(x)-3x2=0  Solution
 Exapmple 5  Find a root of tan(x)-x-1 = 0  Solution
 Exapmple 6  Find a root of sin(2x)-exp(x-1) = 0  Solution
Problems to workout


Highlights of the scheme

Work out with the Regula-Falsi method here

Note : Please enter equation like 3x+sin[x]-exp[x]. Use "[ ]" brackets for transcendentals and "( )" for others eg., 3x+sin[(x+2)]+(3/4). 'a' and 'b' are the limits within which you are going to find the root. Few examples of how to enter equations are given below . . . (i) exp[-x]*(x^2+5x+2)+1  (ii) x^4-x-10  (iii) x-sin[x]-(1/2)  (iv) exp[(-x+2-1-2+1)]*(x^2+5x+2)+1

Solution of Transcendental Equations | Solution of Linear System of Algebraic Equations | Interpolation & Curve Fitting
Numerical Differentiation & Integration | Numerical Solution of Ordinary Differential Equations
Numerical Solution of Partial Differential Equations