REGULAFALSI METHOD
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 midpoint 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
xaxis. 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) 
(ba) 
(cb) 
=> (cb) * (f(b)f(a))
= (ba) * f(b)
c = b  
f(b) * (ba) 
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 RegulaFalsi
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
Do
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 :
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 regulafalsi scheme to obtain the
zero of f (x) = 0.
Iteration
No.

a

b

c

f(a) * f(c)

1

0

0.5

0.376

1.38 (+ve)

2

0.376

0.5

0.36

0.102 (ve)

3

0.376

0.36

0.36

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)/ (x2)]=0 
Solution 
Exapmple 2 
Find a root of x^{2 }= (exp(2x)
 1) / x 
Solution 
Exapmple 3 
Find a root of exp(x^{2}1)+10sin(2x)5
= 0 
Solution 
Exapmple 4 
Find a root of exp(x)3x^{2}=0 
Solution 
Exapmple 5 
Find a root of tan(x)x1 = 0 
Solution 
Exapmple 6 
Find a root of sin(2x)exp(x1) = 0 
Solution 
Problems
to workout

Highlights
of the scheme
Work out with the 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^4x10
(iii) xsin[x](1/2) (iv)
exp[(x+212+1)]*(x^2+5x+2)+1 
