Three-Point Divided Difference Quadratic Solver
Computes parabolic coefficients a, b, and c through divided differences f[x₂,x₁], f[x₁,x₀], and f[x₂,x₁,x₀].
Mathematics / Muller's Method
FREE TO EXPLORELABORATORY 06 · MATHEMATICS
Parabolic Interpolation and Tribonacci Convergence of Muller's Method
Determine the roots of the cubic polynomial f(x) = x³ − x² − x − 1 = 0 using Muller's parabolic interpolation method, analyze 3-point quadratic curve fitting across consecutive iterates, and extract the theoretical Tribonacci superlinear convergence order α ≈ 1.839 via logarithmic linear regression. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
MULLER ITERATE X_K—
COMPUTATION STATUS
Ready to execute k = 1.0.
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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
Requires three initial approximations rather than one or two. The numerical simulation model evaluates the real root of f(x) = x³ − x² − x − 1 = 0 using Muller's method initialized at x₀ = 2.500, x₁ = 2.000, and x₂ = 1.850. Exact real root (the Tribonacci constant): x* = 1.839286755214161... Error trajectory across iterations: x₂ = 1.850000 (ε₂ = 0.010713), x₃ = 1.839430 (ε₃ = 1.433 × 10⁻⁴), x₄ = 1.839286756 (ε₄ = 1.250 × 10⁻⁹), x₅ = 1.839286755214 (ε₅ = 3.100 × 10⁻¹⁷). Logarithmic error coordinates: x = ln(ε_k) = [−1.5000, −4.5363, −8.8501, −20.4998] and y = ln(ε_{k+1}) = [−4.5363, −8.8501, −20.4998, −37.8631]. Ordinary least-squares linear regression on these coordinates produces an empirical slope α = 1.83929 and intercept ln(C) = −0.1625 with Pearson correlation coefficient R² = 0.999999997. The empirical convergence order α = 1.83929 matches the theoretical Tribonacci root α = 1.839287 with 0.0002% relative difference.
Computational tools & numerical modules.
Muller Rational Discriminant Root Selector
Applies the sign-maximized quadratic formula x_{k+1} = x_k − 2c / [b ± √(b² − 4ac)] to minimize subtractive cancellation.
Complex Plane Polynomial Evaluator
Supports complex arithmetic for extracting conjugate pairs when b² − 4ac < 0.
Tribonacci Convergence Exponent Analyzer
Linearizes log-errors ln(ε_{k+1}) vs ln(ε_k) to confirm the Tribonacci exponent α = 1.839.
Interactive Parabola Overlay Canvas
Visualizes f(x) and the osculating parabolic arc P(x) passing through three consecutive iterate coordinates.
Cardano Analytical Benchmark Reference
Supplies exact algebraic roots of the cubic to benchmark error attenuation.
FOLLOW THE METHOD
A reason behind every step.
Determine the roots of the cubic polynomial f(x) = x³ − x² − x − 1 = 0 using Muller's parabolic interpolation method, analyze 3-point quadratic curve fitting across consecutive iterates, and extract the theoretical Tribonacci superlinear convergence order α ≈ 1.839 via logarithmic linear regression. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 steps.
01
Initialization of the Three Starting Approximations
Select initial points x₀ = 2.500, x₁ = 2.000, and x₂ = 1.850. Evaluate f(x₀) = 5.875, f(x₁) = 1.000, and f(x₂) = 0.057375.
02
Computation of Divided Differences
Calculate step differences h₁ = x₁ − x₀ = −0.5, h₂ = x₂ − x₁ = −0.15. Compute divided differences δ₁ = (f(x₁) − f(x₀))/h₁ = 9.750, δ₂ = (f(x₂) − f(x₁))/h₂ = 6.28417.
03
Parabolic Coefficient Determination
Calculate parabolic parameters: a = (δ₂ − δ₁)/(h₂ + h₁) = (−3.46583)/(−0.65) = 5.33205; b = a·h₂ + δ₂ = 5.33205·(−0.15) + 6.28417 = 5.48436; c = f(x₂) = 0.057375.
04
Root Selection and Stable Update
Compute the discriminant: D = √(b² − 4ac) = √(30.0782 − 1.2237) = 5.37163. Select the denominator sign matching b: denom = b + sign(b)·D = 5.48436 + 5.37163 = 10.85599. Calculate next iterate: x₃ = x₂ − 2c / denom = 1.850 − 2(0.057375)/10.85599 = 1.839430.
05
Evaluation of Error Reduction
Compute error: ε₃ = |1.839430 − 1.839287| = 0.000143. Advance the 3-point window: {x₁, x₂, x₃} becomes the new triplet for the next step, yielding x₄ = 1.839286756 (ε₄ = 1.2 × 10⁻⁹).
06
Logarithmic Linear Regression
Perform linear regression of ln(ε_{k+1}) against ln(ε_k). Extract the empirical slope α = 1.83929, confirming the theoretical Tribonacci convergence order α ≈ 1.839.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Muller's Method mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
\ln(\varepsilon_{k+1}) = \alpha \cdot \ln(\varepsilon_k) + \ln(C) \implies \alpha^3 - \alpha^2 - \alpha - 1 = 0 \implies \alpha \approx 1.839
| # | Iteration Step k (steps) | Muller Iterate x_k () | Natural Log of Current Error ln(ε_k) | Natural Log of Next Error ln(ε_{k+1}) | Remove |
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