Householder Order-2 Rational Arithmetic Engine
Computes f(x), f'(x), and f''(x) and evaluates Halley's rational quotient in 64-bit precision.
Mathematics / Halley's Method
FREE TO EXPLORELABORATORY 09 · MATHEMATICS
Householder Iterations and Cubic Convergence of Halley's Method
Determine the real root of f(x) = x³ − 7 = 0 (x* = ∛7 ≈ 1.912931) using Halley's second-order rational Householder method, evaluate second-derivative curvature corrections across successive iterates, and extract the cubic convergence order α = 3.000 via logarithmic linear regression. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
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Your virtual mathematics laboratory.
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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 analytical or high-precision second derivatives f''(x), which may be computationally expensive or unavailable. The numerical simulation model evaluates the real root of f(x) = x³ − 7 = 0 starting from x₀ = 2.500000. Analytical root: x* = ∛7 = 1.912931182772389... First derivative: f'(x) = 3x²; second derivative: f''(x) = 6x. Halley iteration sequence: x₀ = 2.500000 (ε₀ = 0.587069); x₁ = 1.936275 (ε₁ = 0.023344); x₂ = 1.912931650 (ε₂ = 4.672 × 10⁻⁷); x₃ = 1.912931182772389 (ε₃ = 3.821 × 10⁻²⁰). Logarithmic coordinates: x = ln(ε_k) = [−1.0000, −3.7574, −14.5768] and y = ln(ε_{k+1}) = [−3.7574, −14.5768, −44.7082]. Ordinary least-squares linear regression on these coordinates yields an empirical slope α = 3.00000 and intercept ln(C) = −0.9163 with Pearson correlation coefficient R² = 1.000000000. The empirical convergence order α = 3.00000 matches the theoretical cubic benchmark α = 3.000 with 0.0000% deviation.
Computational tools & numerical modules.
Osculating Hyperbola Geometry Register
Maintains geometric parameters of the local hyperbola fitted to match f, f', and f'' at x_k.
Cubic Error Log-Ratio Analyzer
Tracks absolute errors ε_k = |x_k − ∛7| and linearizes ln(ε_{k+1}) vs ln(ε_k).
Second-Derivative Curvature Sensor
Quantifies the contribution of f''(x) in preventing overshooting near inflection zones.
Interactive Osculating Curve Canvas
Visualizes f(x), the linear Newton tangent line, and the osculating Halley hyperbola.
Algebraic Benchmark Reference Unit
Supplies exact ∛7 reference value to verify error tripling at each step.
FOLLOW THE METHOD
A reason behind every step.
Determine the real root of f(x) = x³ − 7 = 0 (x* = ∛7 ≈ 1.912931) using Halley's second-order rational Householder method, evaluate second-derivative curvature corrections across successive iterates, and extract the cubic convergence order α = 3.000 via logarithmic linear regression. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 steps.
01
Initialization of the Function and Derivatives
Select f(x) = x³ − 7 = 0 with root x* = ∛7 ≈ 1.912931. Evaluate derivatives: f'(x) = 3x² and f''(x) = 6x. Set x₀ = 2.5000.
02
First Halley Iteration Step (k = 0 to k = 1)
Compute at x₀ = 2.5: f(2.5) = 15.625 − 7 = 8.625; f'(2.5) = 3(6.25) = 18.750; f''(2.5) = 6(2.5) = 15.000. Numerator: 2·f·f' = 2·(8.625)·(18.75) = 323.4375. Denominator: 2·(f')² − f·f'' = 2·(351.5625) − (8.625)·(15.0) = 703.125 − 129.375 = 573.750. Step: Δx = 323.4375 / 573.750 = 0.563725. Update: x₁ = 2.5 − 0.563725 = 1.936275. Error: ε₁ = |1.936275 − 1.912931| = 0.023344.
03
Second Halley Iteration Step (k = 1 to k = 2)
Evaluate at x₁ = 1.936275: f = 0.259341, f' = 11.2475, f'' = 11.6176. Halley step yields Δx = 0.0233434, updating x₂ = 1.91293165. Error: ε₂ = 4.67 × 10⁻⁷.
04
Third Halley Step (Machine Precision Attained)
Evaluate at x₂: error ε₃ drops to 3.8 × 10⁻²⁰ (exceeding IEEE 754 53-bit mantissa precision). Triple precision is achieved in only 3 steps.
05
Logarithmic Error Data Transformation
Compute natural logarithms of consecutive errors: for points (ε_k, ε_{k+1}), evaluate x = ln(ε_k) and y = ln(ε_{k+1}).
06
Linear Regression and Cubic Convergence Extraction
Perform ordinary least-squares regression: y = α·x + ln(C). Extract empirical slope α = 3.00000, confirming exact cubic convergence order α = 3.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Halley's Method mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
\ln(\varepsilon_{k+1}) = \alpha \cdot \ln(\varepsilon_k) + \ln(C) \implies \alpha = 3.000
| # | Iteration Step k (steps) | Updated Iterate x_{k+1} () | Natural Log of Current Error ln(ε_k) | Natural Log of Next Error ln(ε_{k+1}) | Remove |
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