False Position Method

An Algorithm for finding Roots which uses the point where the linear approximation crosses the axis as the next iteration and keeps the same initial point for each iteration. Using the two-point form of the line

$$y-y_1={f(x_{n-1})-f(x_1)\over x_{n-1}-x_1}(x_n-x_1)$$

with $y=0$, using $y_1=f(x_1)$, and solving for $x_n$ therefore gives the iteration

$$x_n=x_1-{x_{n-1}-x_1\over f(x_{n-1})-f(x_1)} f(x_1).$$

See also Brent's Method, Ridders' Method, Secant Method

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 18, 1972.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Secant Method, False Position Method, and Ridders' Method.'' §9.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 347-352, 1992.