LU Decomposition

Updated 01 Oct 2026

•
What it is
- •
  factors a matrix into lower- and upper-triangular factors for efficient solution of linear systems
•
Formula
- •
  with the convention $PA=LU$ , solve $Ly=Pb$ , then $Ux=y$ .
- •
  state the permutation convention explicitly: some software uses an equivalent arrangement with a differently defined $P$ .
•
Variants
- •
  doolittle uses a unit diagonal in $L$ ; Crout uses a unit diagonal in $U$
•
Use
- •
  factor once, then reuse the factors for multiple right-hand sides
•
Related
- •
  Gaussian Elimination
- •
  Cholesky Decomposition

Linked references 3