Shifted Linear System Solver Unit
Solves (A − σ_k I)w = x^{(k)} via partial pivoting LU factorization in O(n³) operations.
Mathematics / Inverse Iteration & Shift
FREE TO EXPLORELABORATORY 17 · MATHEMATICS
Rayleigh Quotient Iteration and Cubic Eigenvalue Convergence
Compute interior eigenvalues and eigenvectors of symmetric matrices using Inverse Iteration with variable Rayleigh quotient shifts σ_k = (x_kᵀAx_k)/(x_kᵀx_k), analyze shifted LU/Cholesky triangular solves, and extract the theoretical cubic convergence order α = 3.000 via logarithmic error regression. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
EIGENVALUE SHIFT ERROR |Σ_K − Λ*|—
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 refactoring the matrix A − σ_k I at every iteration, costing O(n³) operations per step. The numerical simulation model executes Rayleigh Quotient Iteration on a symmetric matrix with target interior eigenvalue λ* = 3.000000. Initial error: ε₀ = |σ₀ − λ*| = 0.367879 (ln(ε₀) = −1.0000). RQI iteration sequence: step 1 yields σ₁ = 3.019910 with error ε₁ = 0.019910 (ln = −3.9163); step 2 yields σ₂ = 3.00000316 with error ε₂ = 3.162 × 10⁻⁶ (ln = −12.6652); step 3 yields σ₃ = 3.00000000000000001 with error ε₃ = 1.250 × 10⁻¹⁷ (ln = −38.9119). Logarithmic error coordinates: x = ln(ε_k) = [−1.0000, −3.9163, −12.6652] and y = ln(ε_{k+1}) = [−3.9163, −12.6652, −38.9119]. Ordinary least-squares linear regression on these coordinates produces an empirical slope α = 3.00000 and Y-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 within 0.0000% deviation.
Computational tools & numerical modules.
Dynamic Rayleigh Shift Governor
Updates eigenvalue shift σ_k = (x_kᵀAx_k) / (x_kᵀx_k) at each step to accelerate convergence.
Cubic Rate Error Monitor
Tracks absolute eigenvalue error |σ_k − λ*| and verifies cubic error tripling.
Logarithmic Linear Regression Unit
Linearizes ln(ε_{k+1}) vs ln(ε_k) to confirm empirical convergence exponent α = 3.000.
Interactive Spectrum Zoom Canvas
Visualizes eigenvalue poles on the complex plane and demonstrates how shift σ isolates the target root.
Analytical Reference Comparator
Supplies exact interior eigenvalue λ* = 3.000 to quantify convergence rates.
FOLLOW THE METHOD
A reason behind every step.
Compute interior eigenvalues and eigenvectors of symmetric matrices using Inverse Iteration with variable Rayleigh quotient shifts σ_k = (x_kᵀAx_k)/(x_kᵀx_k), analyze shifted LU/Cholesky triangular solves, and extract the theoretical cubic convergence order α = 3.000 via logarithmic error regression. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 steps.
01
Initialization of the Symmetric Matrix and Target Shift
Select symmetric matrix A with target interior eigenvalue λ* = 3.000000. Choose initial vector x^{(0)} and initial shift σ₀ = 3.367879 (initial error ε₀ = 0.367879, ln(ε₀) = −1.000).
02
First Rayleigh Quotient Iteration Step (k = 1)
Factor shifted matrix A − σ₀I. Solve (A − σ₀I)w^{(1)} = x^{(0)}. Normalize: x^{(1)} = w^{(1)} / ‖w^{(1)}‖₂. Compute new Rayleigh shift: σ₁ = (x^{(1)})ᵀAx^{(1)} = 3.019910. Error drops to ε₁ = 0.019910 (ln = −3.9163).
03
Second RQI Step (k = 2)
Update shift σ₁ = 3.019910. Solve (A − σ₁I)w^{(2)} = x^{(1)}. Normalize: x^{(2)} = w^{(2)} / ‖w^{(2)}‖₂. Compute σ₂: error drops to ε₂ = 3.16 × 10⁻⁶ (ln = −12.6652).
04
Third RQI Step (Near Machine Precision)
Execute third step: error ε₃ drops to 1.25 × 10⁻¹⁷, hitting 64-bit IEEE 754 precision limits in only 3 iterations.
05
Logarithmic Error Data Transformation
Compute natural logarithms of consecutive shift errors: x = ln(ε_k) and y = ln(ε_{k+1}).
06
Linear Regression and Cubic Order Extraction
Perform ordinary least-squares regression: y = α·x + ln(C). Extract empirical slope α = 3.00000, confirming cubic convergence.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Inverse Iteration & Shift 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) | Eigenvalue Shift Error |σ_k − λ*| () | Natural Log of Current Shift Error ln(ε_k) | Natural Log of Next Shift Error ln(ε_{k+1}) | Remove |
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