Use this tool to compute the LUP decomposition of a square matrix A, where P·A = L·U. You can also solve a linear system A·x = b once decomposition is complete.
What is LUP decomposition?
LUP decomposition is a matrix factorization technique used in numerical linear algebra. For a square matrix A, it finds three matrices:
- L: a lower triangular matrix with ones on the diagonal,
- U: an upper triangular matrix,
- P: a permutation matrix that records row swaps.
The relationship is P·A = L·U. The permutation part is essential because many matrices require row swapping to avoid division by very small values or zero during elimination.
Why use LUP instead of plain LU?
Standard LU decomposition without pivoting can fail when a pivot element is zero (or nearly zero). LUP decomposition uses partial pivoting, which swaps rows so each pivot is the largest available magnitude in that column. This improves robustness and reduces numerical instability.
In practical computation, LUP is often preferred because it supports:
- Reliable solving of linear systems,
- Fast determinant calculation,
- Reusable factorization for multiple right-hand sides,
- A path toward computing matrix inverses.
How this calculator works
1) Enter a square matrix
Set the matrix size and fill in entries of A. The tool supports dimensions from 2×2 to 8×8.
2) Optional: provide vector b
If you check the solver option, you can enter a vector b and the calculator will compute x in A·x = b using the decomposition.
3) Compute and inspect outputs
You will get:
- Matrices L, U, and P,
- Determinant of A,
- Verification residual max|P·A - L·U|,
- Solution vector x (if b is provided).
Understanding the output
L matrix
The lower triangular matrix stores elimination multipliers. Its diagonal entries are always 1 in this implementation.
U matrix
The upper triangular matrix contains the transformed coefficients after elimination and row swaps.
P matrix
The permutation matrix indicates row reorderings. Multiplying P·A reorders rows of A exactly as used during pivoting.
Common applications
- Solving engineering and physics systems with many equations,
- Econometrics and optimization pipelines,
- Computer graphics and simulation,
- Machine learning preprocessing and numerical kernels.
Numerical tips
- If the matrix is singular (or nearly singular), decomposition may fail or become unstable.
- Very large or very tiny magnitudes can amplify floating-point roundoff.
- Use scaled data when possible for better conditioning.
- For repeated solves with new b, reuse the same decomposition instead of recomputing from scratch.
Quick FAQ
Does this handle non-square matrices?
No. LUP decomposition in this form requires a square matrix.
What if determinant is zero?
That indicates a singular matrix; no unique solution exists for every b.
How accurate is the result?
The calculator uses JavaScript floating-point arithmetic. For most educational and moderate-size numerical tasks, it performs well, and the residual helps you assess quality.