Picard Successive Substitution Arithmetic Engine
Evaluates the nonlinear map g(x) = (x² + 15)/16 and updates x_{k+1} ← g(x_k) in 64-bit IEEE 754 precision.
Mathematics / Fixed-Point Iteration
FREE TO EXPLORELABORATORY 05 · MATHEMATICS
Banach Fixed-Point Contraction Mapping and Linear Error Attenuation
Determine the unique real fixed point x* = 1.000 of the contraction map g(x) = (x² + 15)/16 using Picard successive approximations, evaluate the Banach contraction mapping theorem on [0.0, 2.0], and extract the Lipschitz contraction constant L = 0.125 via linear regression of consecutive error distances. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
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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
Convergence is strictly linear unless g'(x*) = 0; if L is close to 1 (e.g. L = 0.99), convergence is painfully slow. The numerical simulation model evaluates the fixed point of g(x) = (x² + 15)/16 starting from x₀ = 1.500000. True attracting fixed point: x* = 1.000000 (positive root of x² − 16x + 15 = 0; the second root x = 15 is repelling with g'(15) = 30/16 = 1.875 > 1). Analytical derivative: g'(x) = 2x/16 = x/8. At the attracting fixed point, the theoretical Lipschitz constant is L = g'(1.0) = 1/8 = 0.125000. Standardized iteration sequence: x₀ = 1.500000 (ε₀ = 0.500000), x₁ = 1.078125 (ε₁ = 0.078125), x₂ = 1.010147 (ε₂ = 0.010147), x₃ = 1.001275 (ε₃ = 0.001275), x₄ = 1.000160 (ε₄ = 0.000160), x₅ = 1.000020 (ε₅ = 0.000020), and x₆ = 1.000003 (ε₆ = 0.000003). Linear regression coordinates: x = ε_k = [0.500000, 0.078125, 0.010147, 0.001275, 0.000160] and y = ε_{k+1} = [0.078125, 0.010147, 0.001275, 0.000160, 0.000020]. Ordinary least-squares regression yields an empirical slope L = 0.12500 and intercept b = 0.00000 with coefficient of determination R² = 1.000000. The empirical contraction constant L = 0.12500 matches the analytical Lipschitz constant L = 1/8 = 0.12500 with 0.0000% deviation.
Computational tools & numerical modules.
Cobweb Trajectory Coordinate Register
Maintains orthogonal path vertices (x_k, x_k) → (x_k, g(x_k)) → (x_{k+1}, x_{k+1}) along the identity line y = x.
Lipschitz Gradient Monitor
Evaluates the analytical derivative g'(x) = x/8 and verifies the contraction condition |g'(x)| < 1 across the iteration sequence.
Metric Space Error Attenuation Log
Records absolute errors ε_k = |x_k − 1.0| and computes empirical contraction ratios ε_{k+1}/ε_k.
Interactive Cobweb Dynamics Canvas
Renders high-resolution SVG curves of y = g(x) and the diagonal y = x alongside animated cobweb step lines.
Banach A Priori Error Estimator
Evaluates the theoretical bound |x_k − x*| ≤ [L^k / (1 − L)] · |x₁ − x₀| to verify uniform contractivity.
FOLLOW THE METHOD
A reason behind every step.
Determine the unique real fixed point x* = 1.000 of the contraction map g(x) = (x² + 15)/16 using Picard successive approximations, evaluate the Banach contraction mapping theorem on [0.0, 2.0], and extract the Lipschitz contraction constant L = 0.125 via linear regression of consecutive error distances. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 steps.
01
Verification of the Contraction Condition
Verify that g(x) = (x² + 15)/16 maps [0, 2] into [0, 2]: g(0) = 15/16 = 0.9375 and g(2) = 19/16 = 1.1875. Compute g'(x) = x/8. For all x ∈ [0, 2], |g'(x)| ≤ 2/8 = 0.250 < 1. Hence, g is a strict contraction mapping.
02
First Picard Iteration (k = 0 to k = 1)
Set x₀ = 1.500000. Compute error ε₀ = |1.500000 − 1.000000| = 0.500000. Evaluate x₁ = g(x₀) = (1.5² + 15)/16 = (2.25 + 15)/16 = 1.078125. Next error ε₁ = |1.078125 − 1.000000| = 0.078125.
03
Second Picard Iteration (k = 1 to k = 2)
Evaluate x₂ = g(x₁) = (1.078125² + 15)/16 = (1.162354 + 15)/16 = 1.010147. Error: ε₂ = 0.010147.
04
Third Picard Iteration (k = 2 to k = 3)
Evaluate x₃ = g(x₂) = (1.010147² + 15)/16 = 1.001275. Error: ε₃ = 0.001275.
05
Fourth and Fifth Picard Iterations (k = 3 to k = 5)
Evaluate x₄ = 1.000160 (ε₄ = 0.000160) and x₅ = 1.000020 (ε₅ = 0.000020). Observe rapid geometric decay of the error.
06
Linear Regression and Contraction Constant Extraction
Plot the next error ε_{k+1} against the current error ε_k. Perform ordinary least-squares linear regression: ε_{k+1} = L · ε_k + b. Extract the empirical slope L = 0.12500, verifying the theoretical Lipschitz constant L = |g'(1.0)| = 1/8 = 0.125.
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Points are your recorded observations. The solid regression line validates Fixed-Point Iteration mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
\varepsilon_{k+1} = L \cdot \varepsilon_k \implies L = |g'(x^*)| = \frac{2(1.0)}{16} = 0.125
| # | Iteration Step k (steps) | Next Iterate x_{k+1} () | Current Error Distance ε_k = |x_k − x*| | Next Error Distance ε_{k+1} = |x_{k+1} − x*| | Remove |
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