Jacobi Component-Wise Vector Update Unit
Evaluates x_i^{(k+1)} = (b_i − ∑_{j≠i} a_{ij}x_j^{(k)}) / a_{ii} simultaneously in O(n²) operations.
Mathematics / Jacobi Iteration
FREE TO EXPLORELABORATORY 14 · MATHEMATICS
Spectral Radius and Asymptotic Convergence of the Jacobi Iteration Method
Solve strictly diagonally dominant linear systems Ax = b using the stationary Jacobi splitting method A = D − (L + U), analyze the spectral radius ρ(T_J) = max |λ_i| of the Jacobi iteration matrix T_J = D⁻¹(L + U), and extract the asymptotic linear convergence factor ρ = 0.500 via linear regression of logarithmic error norms. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
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EUCLIDEAN ERROR NORM ‖E_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
Convergence is strictly linear and can be excruciatingly slow if the spectral radius is close to 1. The numerical simulation model evaluates a strictly diagonally dominant 4×4 linear system Ax = b with analytical solution x* = [1.0, 2.0, 3.0, 4.0]ᵀ and Jacobi iteration matrix T_J = D⁻¹(L + U) having eigenvalues λ = [0.500, −0.500, 0.250, −0.250], giving spectral radius ρ(T_J) = 0.500000. Initial error norm: ‖e_0‖ = 0.367879 (ln(‖e_0‖) = −1.0000). Across six iterations, the Euclidean error norms are: ‖e_1‖ = 0.183940 (ln = −1.6931); ‖e_2‖ = 0.091970 (ln = −2.3863); ‖e_3‖ = 0.045985 (ln = −3.0794); ‖e_4‖ = 0.022992 (ln = −3.7726); ‖e_5‖ = 0.011496 (ln = −4.4657); ‖e_6‖ = 0.005748 (ln = −5.1589). Linear regression of y = ln(‖e_{k+1}‖) against x = ln(‖e_k‖) yields slope m = 1.00000 and intercept ln(ρ) = −0.693147 with coefficient of determination R² = 1.000000. The empirical spectral radius is extracted as ρ = e^(−0.693147) = 0.50000, matching the theoretical spectral radius ρ = 0.50000 within 0.0000% deviation.
Computational tools & numerical modules.
Gerschgorin Circle Invariant Checker
Calculates row disk radii R_i = ∑_{j≠i} |a_{ij}| to confirm strict diagonal dominance.
Spectral Radius Matrix Analyzer
Computes eigenvalues of T_J = D⁻¹(L + U) and verifies ρ(T_J) = 0.500.
Euclidean Residual Norm Monitor
Tracks ‖r^{(k)}‖₂ = ‖b − Ax^{(k)}‖₂ across iterations.
Interactive Vector Convergence Canvas
Renders coordinate trajectories (x₁, x₂) in state space converging to the exact solution.
Asymptotic Convergence Rate Extractor
Linearizes ln(‖e_{k+1}‖) vs ln(‖e_k‖) to extract the spectral radius ρ.
FOLLOW THE METHOD
A reason behind every step.
Solve strictly diagonally dominant linear systems Ax = b using the stationary Jacobi splitting method A = D − (L + U), analyze the spectral radius ρ(T_J) = max |λ_i| of the Jacobi iteration matrix T_J = D⁻¹(L + U), and extract the asymptotic linear convergence factor ρ = 0.500 via linear regression of logarithmic error norms. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 steps.
01
Verification of Strict Diagonal Dominance
For each row i, compute the diagonal magnitude |a_{ii}| and sum of off-diagonals ∑_{j≠i} |a_{ij}|. Confirm |a_{ii}| > ∑_{j≠i} |a_{ij}| to guarantee global convergence.
02
Initialization of the Solution Vector
Set initial approximation x^{(0)} = [0, 0, ..., 0]ᵀ and compute initial error norm ‖e_0‖ = ‖x^{(0)} − x*‖ = 0.367879 (ln(‖e_0‖) = −1.000).
03
First Jacobi Step (k = 1)
Compute all components simultaneously: x_i^{(1)} = (b_i − ∑_{j≠i} a_{ij} x_j^{(0)}) / a_{ii}. Error drops to ‖e_1‖ = 0.183940 (ln(‖e_1‖) = −1.6931).
04
Second and Third Jacobi Steps (k = 2 to k = 3)
Execute simultaneous updates: step 2 yields ‖e_2‖ = 0.091970 (ln = −2.3863); step 3 yields ‖e_3‖ = 0.045985 (ln = −3.0794).
05
Subsequent Steps to k = 6
Execute iterations up to k = 6: step 4 yields ‖e_4‖ = 0.022992; step 5 yields ‖e_5‖ = 0.011496; step 6 yields ‖e_6‖ = 0.005748.
06
Logarithmic Linear Regression
Plot ln(‖e_{k+1}‖) against ln(‖e_k‖). Perform ordinary least-squares regression: y = m·x + b. Verify slope m = 1.000 and extract spectral radius ρ = e^b = e^(−0.6931) = 0.500.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Jacobi Iteration mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
\ln(\Vert e_{k+1}\Vert) = 1.000 \cdot \ln(\Vert e_k\Vert) + \ln(\rho) \implies \rho = e^{\ln(\rho)} = 0.500
| # | Iteration Step k (steps) | Euclidean Error Norm ‖e_k‖ () | Current Log Error Norm ln(‖e_k‖) | Next Log Error Norm ln(‖e_{k+1}‖) | Remove |
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