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 regula-falsi point (xrfp) which is the solution to the linear equation connecting the endpoints.

Look at the sign of f(xrfp):

If sign(f(xrfp)) = 0 then end algorithm

else If sign(f(xrfp)) = sign(f(a)) then set a = xrfp

else set b = xrfp

x-axisa b

f(b)

f(a) actual root

f(x)xrfp

equation of line:

axab

afbfafy

)()(

)(

solving for xrfp

)()(

)(

)()(

)(

)()()(0

afbf

abafax

axafbf

abaf

axab

afbfaf

rfp

rfp

rfp

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).

2

3

4

6

31

)2(3

)0()2(

02)0(0

ff

fxrfp

8

21

2

3)(

fxf rfp

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].

29

54

58

21

2

3

12

3

)2(

2

2

3

821

21

821

23

23

23

ff

fxrfp

267785.029

54)(

fxf rfp

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 ).