Dense Matrix BLAS Level-2 Elimination Engine
Executes row multiplier scalings m_{ik} and rank-1 outer product subtractions A_{i,k:n} ← A_{i,k:n} − m_{ik}·A_{k,k:n} in 64-bit IEEE 754 precision.
Mathematics / Gaussian Elimination
FREE TO EXPLORELABORATORY 11 · MATHEMATICS
Gaussian Elimination with Partial Pivoting and Cubic FLOP Complexity
Solve dense n×n linear algebraic systems Ax = b using Gaussian elimination with row-scaled partial pivoting, eliminate numerical instability from vanishingly small pivot elements, and extract the asymptotic floating-point operational scaling constant C = 2/3 ≈ 0.667 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.
TRIANGULAR ELIMINATION 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
Dense Gaussian elimination scales as O(n³), making it computationally prohibitive for massive systems (n > 20,000) without sparse techniques. The numerical simulation model evaluates dense n×n linear systems with partial pivoting 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 forward elimination FLOP count: FLOPs(n) = (2/3)n³ = [0.667, 5.333, 18.000, 42.667, 83.333, 144.000] kFLOPs. Ordinary least-squares linear regression of elimination FLOPs against x = n³/1000 produces an empirical slope C = 0.66667 and Y-intercept b = 0.000 with coefficient of determination R² = 1.000000. The empirical scaling constant C = 0.667 matches the analytical constant C = 2/3 = 0.66667 within 0.0000% deviation.
Computational tools & numerical modules.
Row Permutation and Pivot Selection Unit
Scans column k for the maximal absolute entry max_{i≥k} |a_{ik}| and exchanges rows to suppress round-off amplification.
Back-Substitution Vector Solver
Solves upper-triangular system Ux = c via backward recurrence x_i = (c_i − ∑_{j>i} u_{ij}x_j)/u_{ii} in O(n²) operations.
Asymptotic FLOP Scaling Counter
Records cumulative multiplication and subtraction operations across the forward elimination phase.
Interactive Matrix Heatmap Canvas
Visualizes non-zero matrix elements, pivot selection highlights, and progressive upper-triangular formation.
Wilkinson Backward Stability Benchmark Unit
Evaluates the pivot growth factor ρ_n = max_{i,j,k} |a_{ij}^{(k)}| / max_{i,j} |a_{ij}| to verify numerical stability.
FOLLOW THE METHOD
A reason behind every step.
Solve dense n×n linear algebraic systems Ax = b using Gaussian elimination with row-scaled partial pivoting, eliminate numerical instability from vanishingly small pivot elements, and extract the asymptotic floating-point operational scaling constant C = 2/3 ≈ 0.667 FLOPs/n³ via ordinary least-squares regression. Standardized input settings: 10.0, 20.0, 30.0, 40.0, 50.0, 60.0 rows.
01
System Dimension Initialization
Initialize an n×n dense linear system Ax = b across dimensions n ∈ {10, 20, 30, 40, 50, 60}. Verify that det(A) ≠ 0.
02
Partial Pivot Selection (Column k)
At step k, scan column elements |a_{i,k}| for i = k, ..., n. Locate row p having the largest absolute magnitude: |a_{p,k}| = max_{i≥k} |a_{i,k}|. If p ≠ k, swap row k and row p in both A and b.
03
Forward Row Elimination
For each row i from k + 1 to n, compute the multiplier m_{i,k} = a_{i,k} / a_{k,k}. Subtract m_{i,k} times row k from row i: a_{i,j} ← a_{i,j} − m_{i,k} · a_{k,j} for j = k+1, ..., n, and b_i ← b_i − m_{i,k} · b_k.
04
Triangular Form Attainment
Repeat steps 2 and 3 for all columns k = 1 to n − 1 until the coefficient matrix is reduced to upper-triangular form U.
05
Back-Substitution Execution
Solve the upper-triangular system Ux = c from row n down to row 1: x_n = c_n / u_{n,n}, and x_i = (c_i − ∑_{j=i+1}^n u_{i,j} x_j) / u_{i,i}.
06
Operational Scaling Regression
Plot elimination FLOPs against normalized cubic dimension x = n³/1000. Perform ordinary least-squares regression to extract empirical slope C = 0.667, verifying theoretical (2/3)n³ FLOP scaling.
YOUR NOTEBOOK · 0 OBSERVATIONS
Turn measurements into mathematical insight.
Points are your recorded observations. The solid regression line validates Gaussian Elimination 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) | Triangular Elimination FLOPs (kFLOPs) | Normalized Dimension Cubic n³/1000 | Triangular Elimination Operations (kFLOPs) | Remove |
|---|
No readings yet. Execute a computation and record it.
Calculate it yourself.
CHECK YOUR UNDERSTANDING