False Position Method (Regula Falsi)

Summary

False position keeps a sign-changing bracket like bisection, but places the next sample at the intercept of the secant through $(a,f(a))$ and $(b,f(b))$ . It often reduces the residual faster than pure bisection while preserving a bracket.

Prerequisites

• Bolzano's Theorem
• Bisection Method (bracketing idea)

Problem Type

Solve $f(x)=0$ on a continuous bracket with opposite signs.

Method Definition

Given continuous $f$ on $[a,b]$ with $f(a)f(b)<0$ , set

and replace the endpoint whose function value has the same sign as $f(c)$ .^[1]

Assumptions / Requirements

• Continuity on $[a,b]$
• $f(a)f(b)<0$
• $f(b)\ne f(a)$ (secant is defined)

Algorithm

1. Evaluate $f(a),f(b)$ with opposite signs.
2. Compute $c$ from the formula above.
3. If $|f(c)|$ or $|b-a|$ is small enough, stop.
4. If $f(a)f(c)<0$ then $b\leftarrow c$ , else $a\leftarrow c$ .
5. Repeat.

Formula / Iteration Rule

Equivalent form:

Convergence

Typically faster residual reduction than bisection when $f$ is roughly linear near the root. Convergence is still linear in general. One endpoint can stagnate for convex/concave $f$ ; modified regula falsi variants fix this.

Error / Accuracy

Common stops: $|f(c)|<\epsilon$ or $|c_{\text{new}}-c_{\text{old}}|<\epsilon$ . The interval length need not shrink as aggressively as in bisection.

Worked Example

Again use $f(x)=x^3-2x-5$ . Valid bracket: $[2,3]$ because $f(2)=-1$ and $f(3)=16$ .

Iteration 1:

Iteration 2:

Subsequent iterates approach $\approx 2.09455$ from below while the right endpoint stays at $3$ for several steps (classic regula falsi stagnation pattern).

Common Failure Modes

• Invalid initial bracket
• One endpoint stuck for many iterations
• Near-zero denominator if $f(a)\approx f(b)$ (should not happen under opposite signs unless values are tiny)

Connections

• Secant Method drops the bracketing constraint
• Bisection Method for guaranteed interval halving
• Root Finding

References