denseLU-class | R Documentation |
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.
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)
.
Class LU
, directly.
Class MatrixFactorization
, by class
LU
, distance 2.
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
).
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(x = "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
.
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. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.56021/9781421407944")}
Class sparseLU
for sparse LU factorizations.
Class dgeMatrix
.
Generic functions lu
,
expand1
and expand2
.
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)))
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