denseLU-class function

Dense LU Factorizations

denseLU is the class of dense, row-pivoted LU factorizations of $m-by-n$ real matrices $A$, having the general form [REMOVE_ME]$P_{1} A = L UP1 * A = L * U [REMOVE_ME_2]$

or (equivalently) [REMOVE_ME]$A = P_{1}' L UA = P1' * L * U [REMOVE_ME_2]$

where $P1$ is an $m-by-m$ permutation matrix, $L$ is an $m-by-min(m,n)$

unit lower trapezoidal matrix, and $U$ is a $min(m,n)-by-n$

upper trapezoidal matrix. If $m = n$, then the factors $L$ and $U$ are triangular.

class

Description

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

or (equivalently)

where $P1$ is an $m-by-m$ permutation matrix, $L$ is an $m-by-min(m,n)$

unit lower trapezoidal matrix, and $U$ is a $min(m,n)-by-n$

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 $P1$ 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.

See Also

Class sparseLU for sparse LU factorizations.

Class dgeMatrix.

Generic functions lu, expand1 and expand2.

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. tools:::Rd_expr_doi("10.56021/9781421407944")

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)))