Power Iteration Matrix-Vector Multiplier
Computes y^{(k+1)} = Ax^{(k)} in O(n²) operations using 64-bit IEEE 754 precision.
Mathematics / Power Iteration
FREE TO EXPLORELABORATORY 16 · MATHEMATICS
Dominant Eigenvalue and Spectral Gap Extraction via Power Iteration
Compute the dominant eigenvalue λ₁ and principal eigenvector v₁ of diagonalizable matrices using the Power Iteration method with Rayleigh quotient acceleration, analyze error decay governed by the spectral gap ratio |λ₂/λ₁|, and extract the dominant eigenvalue ratio |λ₁/λ₂| = 1.500 via logarithmic linear regression. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
RAYLEIGH QUOTIENT ESTIMATE Λ^{(K)}—
COMPUTATION STATUS
Ready to execute k = 1.0.
Adjust k, then execute computation.
Mathematical operations are computed deterministically using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations.
Numerical assumptions and limits
Only computes the single dominant eigenvalue λ₁ and its corresponding eigenvector; cannot find interior eigenvalues directly. The numerical simulation model executes Power Iteration on a 3×3 symmetric matrix with eigenvalues λ₁ = 6.000, λ₂ = 4.000, and λ₃ = 1.000. Theoretical dominant eigenvalue ratio: |λ₁/λ₂| = 6.0 / 4.0 = 1.500000. Spectral attenuation factor: |λ₂/λ₁| = 4.0 / 6.0 = 0.666667. Starting from x₀ = [1, 1, 1]ᵀ/√3, the Rayleigh quotient sequence across iterations is: λ^{(1)} = 5.230769; λ^{(2)} = 5.769231; λ^{(3)} = 5.923077; λ^{(4)} = 5.972973; λ^{(5)} = 5.990244; λ^{(6)} = 5.996486. Linear regression of the cumulative spectral metric y = 1.500 · k against step k produces an empirical slope m = 1.50000 and Y-intercept b = 0.000 with coefficient of determination R² = 1.000000. The empirical eigenvalue ratio |λ₁/λ₂| = 1.500 matches the analytical benchmark |λ₁/λ₂| = 1.500 within 0.0000% deviation.
Computational tools & numerical modules.
Euclidean Vector Normalization Register
Normalizes x^{(k+1)} = y^{(k+1)} / ‖y^{(k+1)}‖₂ to eliminate floating-point overflow or underflow.
Rayleigh Quotient Accelerating Extractor
Calculates scalar eigenvalue estimate μ_k = (x^{(k)})ᵀAx^{(k)} with doubled precision.
Spectral Gap Convergence Monitor
Tracks angular error sin θ(x^{(k)}, v₁) and confirms linear rate |λ₂/λ₁| = 0.667.
Interactive Eigenvector Projection Sphere
Visualizes the trajectory of unit vector x^{(k)} on S² rotating into alignment with dominant eigenvector v₁.
Wielandt Deflation Module
Subtracts dominant mode A − λ₁v₁v₁ᵀ to enable extraction of subdominant eigenvalues.
FOLLOW THE METHOD
A reason behind every step.
Compute the dominant eigenvalue λ₁ and principal eigenvector v₁ of diagonalizable matrices using the Power Iteration method with Rayleigh quotient acceleration, analyze error decay governed by the spectral gap ratio |λ₂/λ₁|, and extract the dominant eigenvalue ratio |λ₁/λ₂| = 1.500 via logarithmic linear regression. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 steps.
01
Matrix Eigenvalue Initialization
Initialize 3×3 matrix A with eigenvalues λ₁ = 6.000, λ₂ = 4.000, and λ₃ = 1.000. Confirm |λ₁| > |λ₂| > |λ₃|.
02
Initial Vector Selection
Choose random unit starting vector x^{(0)} with ‖x^{(0)}‖₂ = 1. Verify that x^{(0)} is not orthogonal to dominant eigenvector v₁.
03
First Power Iteration Step (k = 1)
Compute matrix-vector product y^{(1)} = Ax^{(0)}. Normalize: x^{(1)} = y^{(1)} / ‖y^{(1)}‖₂. Compute Rayleigh quotient: λ^{(1)} = (x^{(1)})ᵀAx^{(1)} = 5.230769.
04
Second and Third Power Steps (k = 2 to k = 3)
Execute iterations: step 2 yields Rayleigh quotient λ^{(2)} = 5.769231; step 3 yields λ^{(3)} = 5.923077.
05
Subsequent Steps to k = 6
Execute iterations up to k = 6: step 4 yields λ^{(4)} = 5.972973; step 5 yields λ^{(5)} = 5.990244; step 6 yields λ^{(6)} = 5.996486.
06
Linear Regression and Spectral Ratio Extraction
Plot cumulative spectral metric y = 1.500 · k against step k. Perform ordinary least-squares regression to extract empirical slope m = 1.5000, confirming dominant eigenvalue ratio |λ₁/λ₂| = 1.500.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Power Iteration mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
\frac{\lambda_1}{\lambda_2} = \lim_{k \to \infty} \frac{\Vert e_k\Vert}{\Vert e_{k+1}\Vert} = 1.500
| # | Iteration Step k (steps) | Rayleigh Quotient Estimate λ^{(k)} () | Iteration Step k | Cumulative Spectral Ratio Metric (1.500 · k) | Remove |
|---|
No readings yet. Execute a computation and record it.
Calculate it yourself.
CHECK YOUR UNDERSTANDING