Compact In-Place LU Factorization Unit
Overwrites matrix A with L (strictly lower) and U (upper triangular) to eliminate auxiliary memory allocation.
Mathematics / LU Decomposition
FREE TO EXPLORELABORATORY 12 · MATHEMATICS
LU Triangular Matrix Factorization and Forward-Backward Substitution
Factorize square non-singular matrices A into unit lower-triangular matrix L and upper-triangular matrix U (PA = LU) using Doolittle and Crout algorithms, solve multiple right-hand-side vectors in O(n²) time via forward and back substitution, and extract the asymptotic factorization constant C = 2/3 ≈ 0.667 via linear regression. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
LU FACTORIZATION OPERATIONS—
COMPUTATION STATUS
Ready to execute n = 20.0.
Adjust n, 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
Dense LU factorization requires (2/3)n³ operations and O(n²) memory, becoming impractical for massive systems without sparsity. The numerical simulation model executes in-place Doolittle LU factorization across dense systems from n = [10, 20, 30, 40, 50, 60]. Normalized cubic coordinate: x = n³/1000 = [1.00, 8.00, 27.00, 64.00, 125.00, 216.00]. Theoretical factorization operations: FLOPs(n) = (2/3)n³ = [0.667, 5.333, 18.000, 42.667, 83.333, 144.000] kFLOPs. Ordinary least-squares linear regression on these coordinates produces an empirical slope C = 0.66667 and Y-intercept b = 0.000 with coefficient of determination R² = 1.000000. The empirical factorization multiplier C = 0.667 matches the theoretical benchmark C = 2/3 = 0.66667 within 0.0000% deviation.
Computational tools & numerical modules.
Permutation Vector Register P
Maintains integer pivot indices P = [p₁, p₂, ..., p_n] to apply row swaps to subsequent right-hand side vectors.
Dual-Phase Triangular Solver
Executes forward substitution Ly = Pb followed by backward substitution Ux = y in 2n² FLOPs.
Factorization FLOP Accumulator
Tracks the (2/3)n³ operations required to compute outer products and Schur complements.
Interactive LU Heatmap Canvas
Displays L and U triangular factors with color-coded entry magnitudes and sparsity patterns.
Matrix Determinant Extractor
Computes det(A) = det(P) · ∏_{i=1}^n u_{ii} directly from the diagonal of U in O(n) time.
FOLLOW THE METHOD
A reason behind every step.
Factorize square non-singular matrices A into unit lower-triangular matrix L and upper-triangular matrix U (PA = LU) using Doolittle and Crout algorithms, solve multiple right-hand-side vectors in O(n²) time via forward and back substitution, and extract the asymptotic factorization constant C = 2/3 ≈ 0.667 via linear regression. Standardized input settings: 10.0, 20.0, 30.0, 40.0, 50.0, 60.0 rows.
01
System Dimension Initialization
Select matrix dimension n ∈ {10, 20, 30, 40, 50, 60}. Initialize n×n dense matrix A and permutation vector P = [1, 2, ..., n].
02
Pivot Selection and Row Exchange
At stage k, locate pivot row p maximizing |a_{p,k}| for p ≥ k. Swap row k and row p in both A and P.
03
Doolittle Factorization Recurrence
For i = k+1 to n, compute multiplier l_{i,k} = a_{i,k} / a_{k,k} and store it directly in a_{i,k}. For j = k+1 to n, update u_{i,j} = a_{i,j} ← a_{i,j} − l_{i,k} · a_{k,j}.
04
Repeat Across All Columns
Continue the recurrence for k = 1 to n − 1. Upon completion, the upper triangle contains U and the strictly lower triangle contains L.
05
Dual-Phase Substitution for Solutions
Given vector b, permute Pb. Solve Ly = Pb via forward substitution in n² operations. Solve Ux = y via back substitution in n² operations.
06
Linear Regression and Scaling Extraction
Plot factorization operations against normalized cubic coordinate n³/1000. Perform least-squares regression to extract empirical slope C = 0.667, confirming (2/3)n³ factorization complexity.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates LU Decomposition mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
\text{FLOPs}(n) = C \cdot \frac{n^3}{1000} \implies C = \frac{2}{3} \approx 0.667
| # | Matrix Dimension n (rows) | LU Factorization Operations (kFLOPs) | Normalized Dimension Cubic n³/1000 | Factorization Operations (kFLOPs) | Remove |
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