Method of False Position

Numerical Analysis · Axiom Academy

The False Position Method - A smarter bracket refinement From Bisection to False Position The bisection method always picks the midpoint of the interval. But what if we could be smarter? Key Insight: Why not use information about the function values to make a better guess? If f(a) is much smaller than f(b), the root is probably closer to a! Ignores the actual function values Uses f(a) and f(b) to estimate root The name "regula falsi" is Latin for "false position" - we're using a "false" linear approximation to find the root position. We draw a secant line through the points (a, f(a)) and (b, f(b)), then find where this line crosses zero. The formula finds the x-intercept of the line connecting (a, f(a)) and (b, f(b)). Solving for c gives us the regula falsi formula! Like bisection, regula falsi maintains a bracket [a, b] where the root lives. Start with interval [a, b] where f(a) · f(b) If |f(c)| stop - c is our root Always converges (guaranteed!) Uses function value information Watch regula falsi find the root of f(x) = x³ - x - 2 on [1, 2]. Regula falsi has a notorious weakness: one endpoint can get "stuck" and never move! If the function is highly curved, the secant line may consistently underestimate (or overestimate) where the root is. The same endpoint keeps getting replaced while the other stays fixed. Result: Convergence can be much slower than bisection! Illinois Method: Halve f(a) or f(b) if they're "stuck" Pegasus Method: More sophisticated endpoint adjustment

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