## LU LU Decomposition for Matrices

Section: Transforms/Decompositions

### Usage

Computes the LU decomposition for a matrix. The form of the command depends on the type of the argument. For full (non-sparse) matrices, the primary form for `lu` is

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

where `L` is lower triangular, `U` is upper triangular, and `P` is a permutation matrix such that `L*U = P*A`. The second form is

       [V,U] = lu(A),

where `V` is `P'*L` (a row-permuted lower triangular matrix), and `U` is upper triangular. For sparse, square matrices, the LU decomposition has the following form:

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

where `A` is a sparse matrix of either `double` or `dcomplex` type. The matrices are such that `L*U=P*R*A*Q`, where `L` is a lower triangular matrix, `U` is upper triangular, `P` and `Q` are permutation vectors and `R` is a diagonal matrix of row scaling factors. The decomposition is computed using UMFPACK for sparse matrices, and LAPACK for dense matrices.

### Example

First, we compute the LU decomposition of a dense matrix.

    --> a = float([1,2,3;4,5,8;10,12,3])

    a =
      1  2  3
      4  5  8
     10 12  3

    --> [l,u,p] = lu(a)
    l =
        1.0000         0         0
        0.1000    1.0000         0
        0.4000    0.2500    1.0000

    u =
       10.0000   12.0000    3.0000
             0    0.8000    2.7000
             0         0    6.1250

    p =
     0 0 1
     1 0 0
     0 1 0

    --> l*u

    ans =
     10 12  3
      1  2  3
      4  5  8

    --> p*a

    ans =
     10 12  3
      1  2  3
      4  5  8

Now we repeat the exercise with a sparse matrix, and demonstrate the use of the permutation vectors.

    --> a = sparse([1,0,0,4;3,2,0,0;0,0,0,1;4,3,2,4])

    a =
     1 1 1
     2 1 3
     4 1 4
     2 2 2
     4 2 3
     4 3 2
     1 4 4
     3 4 1
     4 4 4
    --> [l,u,p,q,r] = lu(a)
    l =
     1 1 1
     2 2 1
     3 3 1
     4 4 1
    u =
     1 1 0.153846
     1 2 0.230769
     2 2 0.4
     1 3 0.307692
     2 3 0.6
     3 3 0.2
     1 4 0.307692
     3 4 0.8
     4 4 1
    p =
     4
     2
     1
     3

    q =
     3
     2
     1
     4

    r =
     1 1 0.2
     2 2 0.2
     3 3 1
     4 4 0.0769231
    --> full(l*a)

    ans =
     1 0 0 4
     3 2 0 0
     0 0 0 1
     4 3 2 4

    --> b = r*a

    b =
     1 1 0.2
     2 1 0.6
     3 1 0
     4 1 0.307692
     1 2 0
     2 2 0.4
     3 2 0
     4 2 0.230769
     1 3 0
     2 3 0
     3 3 0
     4 3 0.153846
     1 4 0.8
     2 4 0
     3 4 1
     4 4 0.307692
    --> full(b(p,q))

    ans =
        0.1538    0.2308    0.3077    0.3077
             0    0.4000    0.6000         0
             0         0    0.2000    0.8000
             0         0         0    1.0000