Transcript Regula-Falsi Method - Muskingum University

Regula-Falsi Method
Regula-Falsi Method
Type of Algorithm (Equation Solver)
The Regula-Falsi Method (sometimes called the False Position Method) is a
method used to find a numerical estimate of an equation.
This method attempts to solve an equation of the form f(x)=0. (This is very
common in most numerical analysis applications.) Any equation can be written in
this form.
Algorithm Requirements
This algorithm requires a function f(x) and two points a and b for which f(x) is
positive for one of the values and negative for the other. We can write this
condition as f(a)f(b)<0.
If the function f(x) is continuous on the interval [a,b] with f(a)f(b)<0, the algorithm
will eventually converge to a solution.
This algorithm can not be implemented to find a tangential root. That is a root
that is tangent to the x-axis and either positive or negative on both side of the root.
For example f(x)=(x-3)2, has a tangential root at x=3.
Regula-Falsi Algorithm
The idea for the Regula-Falsi method is to connect the points (a,f(a)) and (b,f(b))
with a straight line.
Since linear equations are the simplest equations to solve for find the regulafalsi point (xrfp) which is the solution to the linear equation connecting the
endpoints.
Look at the sign of f(xrfp):
equation of line:
If sign(f(xrfp)) = 0 then end algorithm
else If sign(f(xrfp)) = sign(f(a)) then set a = xrfp
y  f (a ) 
f (b )  f ( a )
ba
else set b = xrfp
x  a 
solving for xrfp
f(b)
0  f (a ) 
f(x)
xrfp
a
f(a)
x-axis
b
actual root
f (b )  f ( a )
ba
 f ( a ) b  a 
f (b )  f ( a )
x rfp  a 
x
rfp
 x rfp  a
f ( a ) b  a 
f (b )  f ( a )
 a
Example
Lets look for a solution to the equation x3-2x-3=0.
We consider the function f(x)=x3-2x-3
On the interval [0,2] the function is negative at 0 and positive at 2. This means
that a=0 and b=2 (i.e. f(0)f(2)=(-3)(1)=-3<0, this means we can apply the
algorithm).
f ( 0 ) 2  0 
x rfp  0 
f (2)  f (0)

 3   21
f ( x rfp )  f   
8
2
x rfp 
3
2

 32 2  32 

3
f (2)  f  2 
f
3
2
 54 
f ( x rfp )  f 
   0 . 267785
 29 
 3( 2 )
1  3

6

4
3
2
This is negative and we will make the a =3/2
and b is the same and apply the same thing
to the interval [3/2,2].

 21
8
 12 
1
 21
8

3
2

21
58

54
29
This is negative and we will make the a =54/29
and b is the same and apply the same thing to
the interval [54/29,2].
Stopping Conditions
Aside from lucking out and actually hitting the root, the stopping condition is
usually fixed to be a certain number of iterations or for the Standard Cauchy Error
in computing the Regula-Falsi Point (xrfp) to not change more than a prescribed
amount (usually denoted ).