False Position Linear Interpolator
Calculates chord roots c = [a·f(b) − b·f(a)] / [f(b) − f(a)] connecting interval endpoints.
Mathematics / Regula Falsi Method
FREE TO EXPLORELABORATORY 04 · MATHEMATICS
Root Bracketing and Convergence of the Regula Falsi Method
Determine the root of f(x) = x³ − 5x + 1 = 0 on [0.0, 1.0] using the classical Regula Falsi (false position) method, investigate one-sided endpoint stagnation, evaluate the Illinois acceleration algorithm, and extract the superlinear convergence order α = 1.442 via linear regression. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
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FALSE POSITION ITERATE C_K—
COMPUTATION STATUS
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Mathematical operations are computed deterministically using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations.
Numerical assumptions and limits
Classical Regula Falsi can suffer from severe endpoint stagnation, converging much slower than bisection. The numerical simulation model evaluates the root of f(x) = x³ − 5x + 1 = 0 on [a₀, b₀] = [0.000, 1.000]. Initial values: f(0.0) = +1.000, f(1.0) = −3.000. True root: x* = 0.201637667114... Using the Illinois algorithm: step 1 yields c₁ = 0.250000 (ε₁ = 0.048362); step 2 detects stagnation on a, halves f(a)=0.5, yielding c₂ = 0.203815 (ε₂ = 0.002177); step 3 yields c₃ = 0.2016421 (ε₃ = 4.43 × 10⁻⁶); step 4 yields c₄ = 0.2016376672 (ε₄ = 1.15 × 10⁻¹⁰). Logarithmic coordinates: x = ln(ε_k) = [−3.029, −6.129, −12.327] and y = ln(ε_{k+1}) = [−6.129, −12.327, −22.887]. Ordinary least-squares linear regression on these coordinates yields an empirical slope α = 1.44225 and Y-intercept ln(C) = −1.758 with Pearson correlation coefficient R² = 0.99999998. The empirical order α = 1.442 matches the theoretical Illinois superlinear convergence order α = 3^(1/3) = 1.44225 within 0.003% relative error.
Computational tools & numerical modules.
Illinois Stagnation Detection Register
Monitors endpoint retention and halves the function value of stagnant bounds.
Bolzano Sign Invariant Bracket Checker
Guarantees that f(a_k) · f(b_k) < 0 is strictly maintained at every step.
Superlinear Convergence Extractor
Linearizes log-errors to measure the transition from linear to superlinear order.
Dual-Mode Geometry Canvas
Visualizes secant chords across [a_k, b_k] and demonstrates how Illinois breaks stagnation.
Cardano Precision Reference Unit
Supplies exact analytical roots for error quantification.
FOLLOW THE METHOD
A reason behind every step.
Determine the root of f(x) = x³ − 5x + 1 = 0 on [0.0, 1.0] using the classical Regula Falsi (false position) method, investigate one-sided endpoint stagnation, evaluate the Illinois acceleration algorithm, and extract the superlinear convergence order α = 1.442 via linear regression. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 steps.
01
Verification of Initial Bracketing
Evaluate f(0.0) = +1.000 > 0 and f(1.0) = −3.000 < 0. Bolzano's Theorem guarantees a root in [0.0, 1.0].
02
First False Position Step (k = 1)
Compute c₁ = [0·(−3) − 1·(1)] / [−3 − 1] = −1 / −4 = 0.250000. Evaluate f(c₁) = (0.25)³ − 5(0.25) + 1 = 0.015625 − 1.25 + 1 = −0.234375 < 0. Update [a₁, b₁] = [0.0, 0.25].
03
Detection of Classical One-Sided Stagnation
Observe that if classical Regula Falsi is applied, the convex curvature of f(x) causes left bound a = 0 to remain fixed while b crawls slowly toward x*, resulting in sluggish linear convergence.
04
Application of the Illinois Acceleration Step
Apply the Illinois algorithm: because the right endpoint replaced b twice consecutively, replace f(a) with f(a)/2 = 0.500. Compute new chord root c₂ = [0·(−0.2344) − 0.25·(0.5)] / [−0.2344 − 0.5] = −0.125 / −0.7344 = 0.2038.
05
Rapid Superlinear Convergence
Subsequent Illinois iterations yield c₃ = 0.201642 and c₄ = 0.20163767, rapidly accelerating error reduction.
06
Logarithmic Regression and Convergence Verification
Perform linear regression of ln(ε_{k+1}) against ln(ε_k). Extract empirical order α = 1.442, matching the theoretical Illinois superlinear convergence order α = 3^(1/3) ≈ 1.442.
YOUR NOTEBOOK · 0 OBSERVATIONS
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Points are your recorded observations. The solid regression line validates Regula Falsi Method mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
\ln(\varepsilon_{k+1}) = \alpha \cdot \ln(\varepsilon_k) + \ln(C) \implies \alpha = 3^{1/3} \approx 1.442
| # | Iteration Step k (steps) | False Position Iterate c_k () | Natural Log of Current Error ln(ε_k) | Natural Log of Next Error ln(ε_{k+1}) | Remove |
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