1. Regula-Falsi Method

2. 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.

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 ( x rfp ) which is the solution to the linear equation connecting the endpoints. Look at the sign of f(x rfp ): If sign ( f(x rfp ) ) = 0 then end algorithm else If sign ( f ( x rfp )) = sign ( f(a)) then set a = x rfp else set b = x rfp x -axis ab f(b) f(a) actual root f(x) x rfp equation of line: solving for x rfp

4. Example Lets look for a solution to the equation x 3 -2 x -3=0. We consider the function f(x)=x 3 -2 x -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). 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]. 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].

5. 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 ( x rfp ) to not change more than a prescribed amount (usually denoted  ).