## 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 =
    	Matrix is sparse with 9 nonzeros
    --> [l,u,p,q,r] = lu(a)
    l =
    	Matrix is sparse with 4 nonzeros
    u =
    	Matrix is sparse with 9 nonzeros
    p =

     4 2 1 3

    q =

     3 2 1 4

    r =
    	Matrix is sparse with 4 nonzeros
    --> full(l*a)

    ans =

     1 0 0 4
     3 2 0 0
     0 0 0 1
     4 3 2 4

    --> b = r*a

    b =
    	Matrix is sparse with 9 nonzeros
    --> 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