Regula falsi

In numerical analysis, the false position method or regula falsi method is a root-finding algorithm that combines features from the bisection method and the secant method.

The Method

Let r be a x-intercept of a secant line with points

$(s,f(s))\,\!$

$(t,f(t))\,\!$

and slope of

${\frac {f(s)-f(t)}{s-t}}$

In point-slope form, the secant line can be defined as

$f(x)-f(t)={\frac {f(s)-f(t)}{s-t}}(x-t)$

For f(r), we have

${\frac {f(s)-f(t)}{s-t}}(r-t)+f(t)=0$

which, when solved for the x-intercept r, gives the false position method formula

$r=t-f(t){\frac {s-t}{f(s)-f(t)}}$

To find a root in interval [a, b], the false position method would initially have s = a and t = b. Then, r would be calculated and if the root is bracketed by s and r, the new interval to check would be [s, r] so t would be replaced by r. Otherwise, if the root is bracketed by r and t, the new interval to check would be [r, t] so s would be replaced by r.

Example code

The following C code was wrote for clarity instead of efficiency. It was designed to solve the same problem as solved by the Newton's method and Secant method code: to find the positive number x where cos(x) = x³. This problem is transformed into a root-finding problem of the form f(x) = cos(x) - x³ = 0.

#include <stdio.h>
#include <math.h>

double f(double x)
{
    return cos(x) - x*x*x;
}

double FalsiMethod(double s, double t, double e, double m)
{
    int n;
    double r;
    double d;
    for (n = 1; n <= m; n++)
    {
        r = t - f(t) * (s - t) / (f(s) - f(t));
        if (f(r) < e)
            return r;
        if (f(s) * f(r) < 0)
            t = r;
        else if (f(r) * f(t) < 0)
            s = r;
   }
   return r;
}

int main(void)
{
    printf("%0.15f\n", FalsiMethod(0, 1, 5E-11, 100));
    return 0;
}