lu

Scilab Function
Last update : April 1993

lu - LU factors of Gaussian elimination

### Calling Sequence

  `[L,U]= lu(A)`
  `[L,U,E]= lu(A)`

### Parameters

- `A`: real or complex matrix (m x n).
- `L`: real or complex matrices (m x min(m,n)).
- `U`: real or complex matrices (min(m,n) x n ).
- `E`: a (n x n) permutation matrix.

### Description

`[L,U]= lu(A)` produces two matrices `L` and `U` such that `A = L*U` with `U` upper triangular and `E*L` lower triangular for a permutation matrix `E`.

If `A` has rank `k`, rows `k+1` to `n` of `U` are zero.

`[L,U,E]= lu(A)` produces three matrices `L`, `U` and `E` such that `E*A = L*U` with `U` upper triangular and `E*L` lower triangular for a permutation matrix `E`.

If `A` is a real matrix, using the function `lufact` and `luget` it is possible to obtain the permutation matrices and also when `A` is not full rank the column compression of the matrix `L`.

### Examples

    a=rand(4,4);
    [l,u]=lu(a)
    norm(l*u-a)

    [h,rk]=lufact(sparse(a))  // lufact fonctionne avec des matrices creuses
    [P,L,U,Q]=luget(h);
    ludel(h)
    P=full(P);L=full(L);U=full(U);Q=full(Q);
    norm(P*L*U*Q-a) // P,Q sont des matrices de permutation

### See Also

`lufact`, `luget`, `lusolve`, `qr`, `svd`,

### Used Function

lu decompositions are based on the Lapack routines DGETRF for real matrices and ZGETRF for the complex case.