`lu`

LU factorization of a matrix.

Syntax

[L, U] = lu(A)

Returns a permuted lower triangular L and upper triangular U with L*U = A.

[L, U, P] = lu(A)

Returns L truly lower triangular and a permutation matrix P with P*A = L*U.

[L, U, p] = lu(A, 'vector')

Returns the permutation as a row vector p, with A(p,:) = L*U.

Y = lu(A)

Returns the packed factors in one matrix.

Description

Gaussian elimination written as a matrix product. The rows are swapped so that each pivot is the largest available in its column (partial pivoting), which keeps the multipliers at most 1 in size and the method stable.

Once factored, a system A*x = b is solved with two triangular solves, cheap for repeated right-hand sides. det(A) follows from the diagonal of U and the sign of the permutation.

Input arguments

A: Matrix, usually square.

Output arguments

L, U: Lower and upper triangular factors; L has unit diagonal.
P, p: Row permutation as a matrix or as an index vector.

Example

With a permutation matrix

A = [4 3; 6 3];
[L, U, P] = lu(A)
disp(norm(P*A - L*U) < 1e-12)

Output

L = 2×2

    1.0000         0
    0.6667    1.0000
U = 2×2

     6     3
     0     1
P = 2×2

     0     1
     1     0
   1

lu