LU Factorization Calculator
Decompose matrices into Lower and Upper triangular matrices using advanced factorization methods. Our comprehensive linear algebra calculator supports Doolittle, Crout, and partial pivoting techniques with step-by-step solutions, determinant calculation, and complete numerical analysis for educational and professional applications.
Last updated: December 15, 2024
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Quick Example Result
3×3 matrix decomposed into L and U triangular matrices
Determinant
4
Method
Doolittle
Status
Invertible
How This Calculator Works
Our LU Factorization calculator implements three fundamental matrix decomposition algorithms: Doolittle, Crout, and partial pivoting methods. Each technique systematically transforms the input matrix into the product of lower and upper triangular matrices, enabling efficient linear system solutions, determinant calculations, and matrix inversions with optimal computational complexity.
LU Factorization Methods
Doolittle Method (L unit lower)
L has ones on diagonal, U has computed diagonal elements
L[i][i] = 1, U[i][j] = A[i][j] - Σ L[i][k]U[k][j]Crout Method (U unit upper)
U has ones on diagonal, L has computed diagonal elements
U[i][i] = 1, L[i][j] = A[i][j] - Σ L[i][k]U[k][j]Partial Pivoting Method
Row swaps for numerical stability, creates PA = LU
Choose max |A[i][k]| as pivot, swap rows as neededApplications & Analysis
Solve Ax = b via Ly = b, then Ux = y
det(A) = ±∏ U[i][i], efficient for multiple RHSEach method provides the same mathematical result but with different computational characteristics and numerical stability properties, making them suitable for different applications and matrix types.
Interactive display showing how A matrix splits into L and U triangular components
Computational Advantages and Applications
LU factorization provides significant computational benefits over direct methods. Once computed, the L and U matrices can be reused to solve multiple linear systems with different right-hand sides in O(n²) time instead of O(n³). This makes LU factorization essential for iterative algorithms, optimization problems, and numerical simulations requiring repeated system solutions.
- Linear system solving: Efficient solution of Ax = b through forward and backward substitution
- Matrix inversion: Compute A⁻¹ by solving AX = I using LU factors
- Determinant calculation: det(A) = product of U diagonal elements (±sign from pivoting)
- Numerical stability: Partial pivoting prevents division by small numbers
Educational Standards & Applications
- Society for Industrial and Applied Mathematics (SIAM) - Numerical Linear Algebra StandardsProfessional standards for matrix factorization and computational methods
- International Linear Algebra Society (ILAS) - Matrix Decomposition ResearchAdvanced research and applications in matrix factorization techniques
- Institute of Electrical and Electronics Engineers (IEEE) - Computational Mathematics StandardsIndustry standards for numerical algorithms and computational efficiency
Explore more matrix operations with our matrix determinant calculator and row reduction calculator for comprehensive linear algebra analysis.
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Original Matrix A:
Key Steps:
- Initialize L as identity, U as original matrix
- L[2,1] = U[2,1]/U[1,1] = 4/2 = 2
- L[3,1] = U[3,1]/U[1,1] = 8/2 = 4
- Update U matrix using elimination
- L[3,2] = U[3,2]/U[2,2] = 3/1 = 3
- Complete factorization and verify A = LU
Result Verification:
Matrix multiplication: L × U = A
Determinant: det(A) = 2 × 1 × 2 = 4
Status: Matrix is invertible (det ≠ 0)
L Matrix (Lower Triangular):
U Matrix (Upper Triangular):
Success: Matrix successfully factorized using Doolittle method
The factorization A = LU is complete with L containing unit diagonal elements and U containing the computed upper triangular elements. This decomposition enables efficient solution of linear systems and matrix operations.
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