Dense BunchKaufman Factorizations
Description
Classes BunchKaufman
and pBunchKaufman
represent
BunchKaufman factorizations of $n \times n$
real,
symmetric matrices $A$
, having the general form
$A = U D_{U} U' = L D_{L} L'$
where
$D_{U}$
and $D_{L}$
are symmetric, block diagonal
matrices composed of $b_{U}$
and $b_{L}$
$1 \times 1$
or $2 \times 2$
diagonal blocks;
$U = \prod_{k = 1}^{b_{U}} P_{k} U_{k}$
is the product of $b_{U}$
rowpermuted unit upper triangular
matrices, each having nonzero entries above the diagonal in 1 or 2 columns;
and
$L = \prod_{k = 1}^{b_{L}} P_{k} L_{k}$
is the product of $b_{L}$
rowpermuted unit lower triangular
matrices, each having nonzero entries below the diagonal in 1 or 2 columns.
These classes store the nonzero entries of the
$2 b_{U} + 1$
or $2 b_{L} + 1$
factors,
which are individually sparse,
in a dense format as a vector of length
$nn$
(BunchKaufman
) or
$n(n+1)/2$
(pBunchKaufman
),
the latter giving the “packed” representation.
Slots

Dim
,Dimnames

inherited from virtual class
MatrixFactorization
. uplo

a string, either
"U"
or"L"
, indicating which triangle (upper or lower) of the factorized symmetric matrix was used to compute the factorization and in turn how thex
slot is partitioned. x

a numeric vector of length
n*n
(BunchKaufman
) orn*(n+1)/2
(pBunchKaufman
), wheren=Dim[1]
. The details of the representation are specified by the manual for LAPACK routinesdsytrf
anddsptrf
. perm

an integer vector of length
n=Dim[1]
specifying row and column interchanges as described in the manual for LAPACK routinesdsytrf
anddsptrf
.
Extends
Class BunchKaufmanFactorization
, directly.
Class MatrixFactorization
, by class
BunchKaufmanFactorization
, distance 2.
Instantiation
Objects can be generated directly by calls of the form
new("BunchKaufman", ...)
or new("pBunchKaufman", ...)
,
but they are more typically obtained as the value of
BunchKaufman(x)
for x
inheriting from
dsyMatrix
or dspMatrix
.
Methods
coerce

signature(from = "BunchKaufman", to = "dtrMatrix")
: returns adtrMatrix
, useful for inspecting the internal representation of the factorization; see ‘Note’. coerce

signature(from = "pBunchKaufman", to = "dtpMatrix")
: returns adtpMatrix
, useful for inspecting the internal representation of the factorization; see ‘Note’. determinant

signature(from = "p?BunchKaufman", logarithm = "logical")
: computes the determinant of the factorized matrix$A$
or its logarithm. expand1

signature(x = "p?BunchKaufman")
: seeexpand1methods
. expand2

signature(x = "p?BunchKaufman")
: seeexpand2methods
. solve

signature(a = "p?BunchKaufman", b = .)
: seesolvemethods
.
Note
In Matrix < 1.60
, class BunchKaufman
extended
dtrMatrix
and class pBunchKaufman
extended
dtpMatrix
, reflecting the fact that the internal
representation of the factorization is fundamentally triangular:
there are $n(n+1)/2$
“parameters”, and these
can be arranged systematically to form an $n \times n$
triangular matrix.
Matrix 1.60
removed these extensions so that methods
would no longer be inherited from dtrMatrix
and dtpMatrix
.
The availability of such methods gave the wrong impression that
BunchKaufman
and pBunchKaufman
represent a (singular)
matrix, when in fact they represent an ordered set of matrix factors.
The coercions as(., "dtrMatrix")
and as(., "dtpMatrix")
are provided for users who understand the caveats.
References
The LAPACK source code, including documentation; see https://netlib.org/lapack/double/dsytrf.f and https://netlib.org/lapack/double/dsptrf.f.
Golub, G. H., & Van Loan, C. F. (2013). Matrix computations (4th ed.). Johns Hopkins University Press. doi:10.56021/9781421407944
See Also
Class dsyMatrix
and its packed counterpart.
Generic functions BunchKaufman
,
expand1
, and expand2
.
Examples
showClass("BunchKaufman")
set.seed(1)
n < 6L
(A < forceSymmetric(Matrix(rnorm(n * n), n, n)))
## With dimnames, to see that they are propagated :
dimnames(A) < rep.int(list(paste0("x", seq_len(n))), 2L)
(bk.A < BunchKaufman(A))
str(e.bk.A < expand2(bk.A, complete = FALSE), max.level = 2L)
str(E.bk.A < expand2(bk.A, complete = TRUE), max.level = 2L)
## Underlying LAPACK representation
(m.bk.A < as(bk.A, "dtrMatrix"))
stopifnot(identical(as(m.bk.A, "matrix"), `dim<`(bk.A@x, bk.A@Dim)))
## Number of factors is 2*b+1, b <= n, which can be nontrivial ...
(b < (length(E.bk.A)  1L) %/% 2L)
ae1 < function(a, b, ...) all.equal(as(a, "matrix"), as(b, "matrix"), ...)
ae2 < function(a, b, ...) ae1(unname(a), unname(b), ...)
## A ~ U DU U', U := prod(Pk Uk) in floating point
stopifnot(exprs = {
identical(names(e.bk.A), c("U", "DU", "U."))
identical(e.bk.A[["U" ]], Reduce(`%*%`, E.bk.A[seq_len(b)]))
identical(e.bk.A[["U."]], t(e.bk.A[["U"]]))
ae1(A, with(e.bk.A, U %*% DU %*% U.))
})
## Factorization handled as factorized matrix
b < rnorm(n)
stopifnot(identical(det(A), det(bk.A)),
identical(solve(A, b), solve(bk.A, b)))