denseLU-class {Matrix} R Documentation

## Dense LU Factorizations

### Description

denseLU is the class of dense, row-pivoted LU factorizations of m \times n real matrices A, having the general form

P_{1} A = L U

or (equivalently)

A = P_{1}' L U

where P_{1} is an m \times m permutation matrix, L is an m \times \min(m,n) unit lower trapezoidal matrix, and U is a \min(m,n) \times n upper trapezoidal matrix. If m = n, then the factors L and U are triangular.

### Slots

Dim, Dimnames

inherited from virtual class MatrixFactorization.

x

a numeric vector of length prod(Dim) storing the triangular L and U factors together in a packed format. The details of the representation are specified by the manual for LAPACK routine dgetrf.

perm

an integer vector of length min(Dim) specifying the permutation P_{1} as a product of transpositions. The corresponding permutation vector can be obtained as asPerm(perm).

### Extends

Class LU, directly. Class MatrixFactorization, by class LU, distance 2.

### Instantiation

Objects can be generated directly by calls of the form new("denseLU", ...), but they are more typically obtained as the value of lu(x) for x inheriting from denseMatrix (often dgeMatrix).

### Methods

coerce

signature(from = "denseLU", to = "dgeMatrix"): returns a dgeMatrix with the dimensions of the factorized matrix A, equal to L below the diagonal and equal to U on and above the diagonal.

determinant

signature(from = "denseLU", logarithm = "logical"): computes the determinant of the factorized matrix A or its logarithm.

expand

signature(x = "denseLU"): see expand-methods.

expand1

signature(x = "denseLU"): see expand1-methods.

expand2

signature(x = "denseLU"): see expand2-methods.

solve

signature(a = "denseLU", b = "missing"): see solve-methods.

### References

The LAPACK source code, including documentation; see https://netlib.org/lapack/double/dgetrf.f.

Golub, G. H., & Van Loan, C. F. (2013). Matrix computations (4th ed.). Johns Hopkins University Press. doi:10.56021/9781421407944

Class sparseLU for sparse LU factorizations.

Class dgeMatrix.

Generic functions lu, expand1 and expand2.

### Examples


showClass("denseLU")
set.seed(1)

n <- 3L
(A <- Matrix(round(rnorm(n * n), 2L), n, n))

## With dimnames, to see that they are propagated :
dimnames(A) <- dn <- list(paste0("r", seq_len(n)),
paste0("c", seq_len(n)))

(lu.A <- lu(A))
str(e.lu.A <- expand2(lu.A), max.level = 2L)

## Underlying LAPACK representation
(m.lu.A <- as(lu.A, "dgeMatrix")) # which is L and U interlaced
stopifnot(identical(as(m.lu.A, "matrix"), dim<-(lu.A@x, lu.A@Dim)))

ae1 <- function(a, b, ...) all.equal(as(a, "matrix"), as(b, "matrix"), ...)
ae2 <- function(a, b, ...) ae1(unname(a), unname(b), ...)

## A ~ P1' L U in floating point
stopifnot(exprs = {
identical(names(e.lu.A), c("P1.", "L", "U"))
identical(e.lu.A[["P1."]],
new(  "pMatrix", Dim = c(n, n), Dimnames = c(dn[1L], list(NULL)),
margin = 1L, perm = invertPerm(asPerm(lu.A@perm))))
identical(e.lu.A[["L"]],
new("dtrMatrix", Dim = c(n, n), Dimnames = list(NULL, NULL),
uplo = "L", diag = "U", x = lu.A@x))
identical(e.lu.A[["U"]],
new("dtrMatrix", Dim = c(n, n), Dimnames = c(list(NULL), dn[2L]),
uplo = "U", diag = "N", x = lu.A@x))
ae1(A, with(e.lu.A, P1. %*% L %*% U))
ae2(A[asPerm(lu.A@perm), ], with(e.lu.A, L %*% U))
})

## Factorization handled as factorized matrix
b <- rnorm(n)
stopifnot(identical(det(A), det(lu.A)),
identical(solve(A, b), solve(lu.A, b)))


[Package Matrix version 1.6-5 Index]