Golub-Kahan Bidiagonalization Engine
Applies Householder reflections from left and right to reduce A to upper bidiagonal form B.
Mathematics / Singular Value Decomposition
FREE TO EXPLORELABORATORY 19 · MATHEMATICS
Singular Value Decomposition and Moore-Penrose Pseudoinverse Conditioning
Decompose ill-conditioned and rank-deficient rectangular matrices into A = UΣVᵀ using Golub-Kahan bidiagonalization, construct the Moore-Penrose pseudoinverse A⁺ = VΣ⁺Uᵀ, and extract the 2-norm matrix condition number κ₂(A) = σ_max / σ_min = 25.000 via singular spectrum linear regression. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
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SINGULAR VALUE RATIO Σ_1/Σ_R—
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
SVD is computationally expensive: full SVD requires approximately 4mn² + 8n³ FLOPs. The numerical simulation model executes SVD on a 3×3 matrix with singular spectrum σ = [10.000, 4.000, 0.400]. Analytical 2-norm condition number: κ₂(A) = σ₁ / σ₃ = 10.000 / 0.400 = 25.000000. Across the six standardized iteration steps k = [1, 2, 3, 4, 5, 6], the cumulative condition metric is y = κ · k = [25.000, 50.000, 75.000, 100.000, 125.000, 150.000]. Ordinary least-squares linear regression on these coordinates produces an empirical slope m = 25.00000 and Y-intercept b = 0.000 with coefficient of determination R² = 1.000000. The empirical condition number κ₂(A) = 25.000 matches the theoretical benchmark κ₂(A) = 25.000 within 0.0000% deviation.
Computational tools & numerical modules.
Implicitly Shifted QR Singular Spectrum Solver
Iterates on BᵀB using zero-shift QR to extract singular values σ_i to machine precision.
Moore-Penrose Pseudoinverse Generator
Inverts non-zero singular values σ_i⁺ = 1/σ_i and forms A⁺ = VΣ⁺Uᵀ.
Spectral Condition Number Analyzer
Evaluates κ₂(A) = σ₁ / σ_r and measures ill-conditioning sensitivity.
Interactive Singular Ellipsoid Canvas
Visualizes the unit sphere in ℝⁿ mapping onto the hyperellipsoid in ℝᵐ with semi-axes σ_i u_i.
Eckart-Young-Mirsky Low-Rank Approximator
Computes optimal rank-k approximation A_k = ∑_{i=1}^k σ_i u_i v_iᵀ.
FOLLOW THE METHOD
A reason behind every step.
Decompose ill-conditioned and rank-deficient rectangular matrices into A = UΣVᵀ using Golub-Kahan bidiagonalization, construct the Moore-Penrose pseudoinverse A⁺ = VΣ⁺Uᵀ, and extract the 2-norm matrix condition number κ₂(A) = σ_max / σ_min = 25.000 via singular spectrum linear regression. Standardized input settings: 1.0, 2.0, 3.0, 4.0, 5.0, 6.0 steps.
01
Matrix Initialization
Initialize m×n matrix A with known singular spectrum σ = [10.000, 4.000, 0.400]. Verify that condition number κ₂(A) = 10.0 / 0.4 = 25.000.
02
Golub-Kahan Bidiagonal Reduction
Apply alternating Householder reflections: P₁ A Q₁ ... P_n A Q_{n-2} reduces A to upper-bidiagonal form B in 4mn² − (4/3)n³ operations.
03
Singular Value Extraction
Apply the Golub-Reinsch implicitly shifted QR algorithm to bidiagonal matrix B, driving off-diagonal entries to zero to extract singular values σ_i.
04
Moore-Penrose Pseudoinverse Construction
Form diagonal matrix Σ⁺ where σ_i⁺ = 1/σ_i if σ_i > 10⁻¹², and 0 otherwise. Compute pseudoinverse A⁺ = V Σ⁺ Uᵀ.
05
Minimal-Norm Solution Evaluation
For right-hand side b, evaluate x_{LS} = A⁺ b. Confirm that x_{LS} is the unique solution minimizing both residual ‖Ax − b‖₂ and solution norm ‖x‖₂.
06
Linear Regression and Condition Number Extraction
Plot cumulative condition metric y = 25.000 · k against step k. Perform ordinary least-squares regression to extract empirical slope m = 25.000, confirming matrix condition number κ₂(A) = 25.000.
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Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Singular Value Decomposition mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
\kappa_2(A) = \frac{\sigma_{\max}}{\sigma_{\min}} = \frac{\sigma_1}{\sigma_r} = \frac{10.000}{0.400} = 25.000
| # | Iteration Step k (steps) | Singular Value Ratio σ_1/σ_r () | Iteration Step k | Cumulative Condition Ratio (25.000 · k) | Remove |
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