MATLAB Mathematics - SERC - Index of

Cholesky, LU, and QR Factorizations
LU Factorization
LU factorization, or Gaussian elimination, expresses any square matrix A as
the product of a permutation of a lower triangular matrix and an upper
triangular matrix
A
=
LU
where L is a permutation of a lower triangular matrix with ones on its diagonal
and U is an upper triangular matrix.
The permutations are necessary for both theoretical and computational
reasons. The matrix
0 1
1 0
cannot be expressed as the product of triangular matrices without
interchanging its two rows. Although the matrix
ε 1
1 0
can be expressed as the product of triangular matrices, when ε is small the
elements in the factors are large and magnify errors, so even though the
permutations are not strictly necessary, they are desirable. Partial pivoting
ensures that the elements of L are bounded by one in magnitude and that the
elements of U are not much larger than those of A.
For example
[L,U] = lu(B)
L =
1.0000 0 0
0.3750 0.5441 1.0000
0.5000 1.0000 0
U =
8.0000 1.0000 6.0000
0 8.5000 -1.0000
0 0 5.2941
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