Matrix (version 1.5-1)

LU-class: LU (dense) Matrix Decompositions

Description

The "LU" class is the virtual class of LU decompositions of real matrices. "denseLU" the class of LU decompositions of dense real matrices.

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

expand

signature(x = "denseLU"): Produce the "L" and "U" (and "P") factors as a named list of matrices, see also the example below.

solve

signature(a = "denseLU", b = "missing"): Compute the inverse of A, \(A^{-1}\), solve(A) using the LU decomposition, see also solve-methods.

Details

The decomposition is of the form $$A = P L U$$ where typically all matrices are of size \(n\times n\), 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 \(m\times n\) matrix \(A\), when \(m \ne n\).

If \(m < n\) (“wide case”), \(U\) is \(m\times n\), and hence not triangular.
If \(m > n\) (“long case”), \(L\) is \(m\times n\), and hence not triangular.

See Also

class sparseLU for LU decompositions of sparse matrices; further, class dgeMatrix and functions lu, expand.

Examples

set.seed(1)
mm <- Matrix(round(rnorm(9),2), nrow = 3)
mm
str(lum <- lu(mm))
elu <- expand(lum)
elu # three components: "L", "U", and "P", the permutation
elu$L %*% elu$U
(m2 <- with(elu, P %*% L %*% U)) # the same as 'mm'
stopifnot(all.equal(as(mm, "matrix"),
                    as(m2, "matrix")))