Cholesky-Banachiewicz Square-Root Engine
Computes diagonal l_{jj} = √(a_{jj} − ∑_{k=1}^{j-1} l_{jk}²) and off-diagonal l_{ij} = (a_{ij} − ∑_{k=1}^{j-1} l_{ik}l_{jk})/l_{jj}.
Mathematics / Cholesky Decomposition
FREE TO EXPLORELABORATORY 13 · MATHEMATICS
Cholesky LLᵀ Factorization of Symmetric Positive-Definite Matrices
Factorize symmetric positive-definite covariance and stiffness matrices A into lower-triangular factor L (A = LLᵀ) using the Cholesky-Banachiewicz algorithm, evaluate Sylvester's criterion on leading principal minors, and extract the half-cost operational constant C = 1/3 ≈ 0.333 FLOPs/n³ via ordinary least-squares regression. Explore interactive numerical methods, evaluate error attenuation and verify convergence orders.
COMPUTE · CONVERGE · ANALYSE
Your virtual mathematics laboratory.
CHOLESKY FACTORIZATION FLOPS—
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
Strictly restricted to symmetric positive-definite matrices; cannot be used for general unsymmetric or indefinite systems. The numerical simulation model executes Cholesky-Banachiewicz factorization on symmetric positive-definite matrices across dimensions 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 Cholesky FLOP count: FLOPs(n) = (1/3)n³ = [0.333, 2.667, 9.000, 21.333, 41.667, 72.000] kFLOPs. Ordinary least-squares linear regression on these coordinates produces an empirical slope C = 0.33333 and Y-intercept b = 0.000 with coefficient of determination R² = 1.000000. The empirical efficiency constant C = 0.333 matches the analytical benchmark C = 1/3 = 0.33333 within 0.0000% deviation.
Computational tools & numerical modules.
Sylvester Positive-Definiteness Detector
Monitors the discriminant under the square root; flags non-positive definite matrices if argument ≤ 0.
Half-Matrix Storage Manager
Stores only the lower triangle of A, cutting memory requirements by 50% compared to general LU.
Operational Halving FLOP Counter
Verifies that the operation count scales strictly as (1/3)n³ rather than (2/3)n³.
Interactive Cholesky Triangle Canvas
Visualizes lower-triangular factor L and demonstrates how LLᵀ reconstructs symmetric matrix A.
Condition Number Reciprocal Sensor
Evaluates ‖L‖·‖L⁻¹‖ to assess matrix conditioning and forward error bounds.
FOLLOW THE METHOD
A reason behind every step.
Factorize symmetric positive-definite covariance and stiffness matrices A into lower-triangular factor L (A = LLᵀ) using the Cholesky-Banachiewicz algorithm, evaluate Sylvester's criterion on leading principal minors, and extract the half-cost operational constant C = 1/3 ≈ 0.333 FLOPs/n³ via ordinary least-squares regression. Standardized input settings: 10.0, 20.0, 30.0, 40.0, 50.0, 60.0 rows.
01
Verification of Matrix Symmetry and Positivity
Initialize symmetric n×n matrix A (A = Aᵀ) across dimensions n ∈ {10, 20, 30, 40, 50, 60}. Verify that all diagonal entries a_{ii} > 0.
02
First Column Factorization
Compute top-left diagonal entry: l_{1,1} = √a_{1,1}. For i = 2 to n, compute column entries: l_{i,1} = a_{i,1} / l_{1,1}.
03
General Column Step (j = 2 to n)
Compute diagonal entry: l_{j,j} = √(a_{j,j} − ∑_{k=1}^{j-1} l_{j,k}²). If the argument under the square root is ≤ 0, terminate: matrix is not positive-definite.
04
Subdiagonal Column Completion
For i = j + 1 to n, compute off-diagonal elements: l_{i,j} = (a_{i,j} − ∑_{k=1}^{j-1} l_{i,k} · l_{j,k}) / l_{j,j}.
05
Solution of Linear System Ax = b
Solve Ly = b via forward substitution in n²/2 operations. Solve Lᵀx = y via back substitution in n²/2 operations.
06
Linear Regression and Half-Cost Verification
Plot factorization FLOPs against n³/1000. Perform ordinary least-squares regression to extract empirical slope C = 0.333, confirming that Cholesky requires exactly half the operations of general LU.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Cholesky Decomposition mathematical scaling.
LINEAR ASYMPTOTIC FIT · y = mx + b
\text{FLOPs}(n) = C \cdot \frac{n^3}{1000} \implies C = \frac{1}{3} \approx 0.333
| # | Matrix Dimension n (rows) | Cholesky Factorization FLOPs (kFLOPs) | Normalized Dimension Cubic n³/1000 | Factorization Operations (kFLOPs) | Remove |
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