# LU-class {Matrix}

### Description

The `"LU"`

class is the *virtual* class of LU decompositions of real matrices. `"denseLU"`

the class of LU decompositions of dense real matrices.

### Details

The decomposition is of the form A = P L U where typically all matrices are of size n by n, and the matrix P is a permutation matrix, L is lower triangular and U is upper triangular (both of class `dtrMatrix`

).

Note that the *dense* decomposition is also implemented for a m by n matrix A, when m != n.

If m < n (“wide case”), U is m by n, and hence not triangular.

If m > n (“long case”), L is m by n, and hence not triangular.

### Objects from the Class

Objects can be created by calls of the form `new("denseLU", ...)`

. More commonly the objects are created explicitly from calls of the form `lu(mm)`

where `mm`

is an object that inherits from the `"dgeMatrix"`

class or as a side-effect of other functions applied to `"dgeMatrix"`

objects.

### Extends

`"LU"`

directly extends the virtual class `"MatrixFactorization"`

.

`"denseLU"`

directly extends `"LU"`

.

### Slots

`x`

:- object of class
`"numeric"`

. The`"L"`

(unit lower triangular) and`"U"`

(upper triangular) factors of the original matrix. These are stored in a packed format described in the Lapack manual, and can retrieved by the`expand()`

method, see below. `perm`

:- Object of class
`"integer"`

- a vector of length`min(Dim)`

that describes the permutation applied to the rows of the original matrix. The contents of this vector are described in the Lapack manual. `Dim`

:- the dimension of the original matrix; inherited from class
`MatrixFactorization`

.

### Methods

### See Also

class `sparseLU`

for LU decompositions of *sparse* matrices; further, class `dgeMatrix`

and functions `lu`

, `expand`

.

### Examples

Documentation reproduced from package Matrix, version 1.2-3. License: GPL (>= 2)