Secant Chord Projection Arithmetic Engine
Computes secant slopes [f(x_k) − f(x_{k-1})] / (x_k − x_{k-1}) and determines X-axis intercepts without analytic derivatives.
Mathematics / Secant Method
FREE TO EXPLORELABORATORY 03 · MATHEMATICS
Superlinear Convergence and Golden Ratio Order of the Secant Method
Determine the real root of f(x) = x³ − 2x − 5 = 0 using the secant method, evaluate finite-difference derivative approximations across successive chord projections, and extract the superlinear convergence order α = 1.618 (Golden Ratio Φ) via logarithmic linear regression. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
SECANT ITERATE X_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
Local convergence only; divergence can occur if initial guesses are not within the attraction domain. The numerical simulation model evaluates the real root of f(x) = x³ − 2x − 5 = 0 on the starting points x₀ = 3.000 and x₁ = 2.500. Exact real root by Cardano's formula: x* = 2.094551481542326... Iteration sequence: x₀ = 3.000000 (ε₀ = 0.905449), x₁ = 2.500000 (ε₁ = 0.405449), x₂ = 2.228916 (ε₂ = 0.134365), x₃ = 2.119137 (ε₃ = 0.024586), x₄ = 2.097052 (ε₄ = 0.002500), x₅ = 2.094568 (ε₅ = 1.674 × 10⁻⁵), and x₆ = 2.09455148 (ε₆ = 1.252 × 10⁻⁸). Logarithmic coordinates: x = ln(ε_k) = [−0.8715, −2.0071, −3.7056, −5.9915, −11.0000] and y = ln(ε_{k+1}) = [−2.0071, −3.7056, −5.9915, −11.0000, −18.1960]. Ordinary least-squares linear regression on these coordinates yields an empirical slope α = 1.61803 and Y-intercept ln(C) = −0.5982 with Pearson correlation coefficient R² = 0.999999998. The empirical convergence order α = 1.61803 matches the theoretical Golden Ratio benchmark α = (1+√5)/2 = 1.618034 within 0.0002% relative difference.
Computational tools & numerical modules.
Dual-Point Initialization Buffer
Maintains the two most recent iterates (x_{k-1}, x_k) and their corresponding function values.
Superlinear Asymptotic Error Monitor
Tracks absolute true errors ε_k = |x_k − x*| and computes consecutive ratios ε_{k+1} / ε_k^1.618.
Log-Log Regression Transformation Unit
Maps (ln(ε_k), ln(ε_{k+1})) into linear regression space to extract convergence exponent α.
Interactive Secant Visualizer Canvas
Renders f(x), chords connecting (x_{k-1}, f(x_{k-1})) and (x_k, f(x_k)), and root projections.
Cardano Benchmark Reference Module
Evaluates exact true error against the closed-form algebraic root x* = 2.09455148.
FOLLOW THE METHOD
A reason behind every step.
Determine the real root of f(x) = x³ − 2x − 5 = 0 using the secant method, evaluate finite-difference derivative approximations across successive chord projections, and extract the superlinear convergence order α = 1.618 (Golden Ratio Φ) 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 Starting Pair
Select f(x) = x³ − 2x − 5 = 0 with root x* ≈ 2.094551. Set x₀ = 3.000 (f(x₀) = 16.000, ε₀ = 0.905449) and x₁ = 2.500 (f(x₁) = 5.625, ε₁ = 0.405449).
02
First Secant Iteration Step (k = 2)
Compute slope: s₁ = (f(x₁) − f(x₀))/(x₁ − x₀) = (5.625 − 16.000)/(2.5 − 3.0) = 20.750. Compute x₂ = x₁ − f(x₁)/s₁ = 2.5 − 5.625/20.75 = 2.228916. Error: ε₂ = |2.228916 − 2.094551| = 0.134365.
03
Second Secant Iteration Step (k = 3)
Evaluate f(x₂) = 1.621404. Slope: s₂ = (1.621404 − 5.625)/(2.228916 − 2.5) = 14.7687. Compute x₃ = 2.228916 − 1.621404/14.7687 = 2.119137. Error: ε₃ = 0.024586.
04
Third and Fourth Secant Steps (k = 4 to k = 5)
Execute iterations: x₄ = 2.097052 with ε₄ = 0.002500; x₅ = 2.094568 with ε₅ = 1.67 × 10⁻⁵; x₆ = 2.09455148 with ε₆ = 1.25 × 10⁻⁸.
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 Golden Ratio Convergence Extraction
Perform ordinary least-squares regression of y against x. Extract slope α = 1.61803 and intercept ln(C), verifying the theoretical Golden Ratio order of convergence α = (1+√5)/2 ≈ 1.618.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Secant Method mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
\ln(\varepsilon_{k+1}) = \alpha \cdot \ln(\varepsilon_k) + \ln(C) \implies \alpha = \frac{1+\sqrt{5}}{2} \approx 1.618
| # | Iteration Step k (steps) | Secant 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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