BunchKaufman-methods.RdComputes the Bunch-Kaufman factorization of an \(n \times n\) real, symmetric matrix \(A\), which has 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 \(b_{U}\) row-permuted 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 \(b_{L}\) row-permuted unit lower triangular matrices, each having nonzero entries below the diagonal in 1 or 2 columns.
Methods are built on LAPACK routines dsytrf and dsptrf.
BunchKaufman(x, ...)
# S4 method for class 'dsyMatrix'
BunchKaufman(x, warnSing = TRUE, ...)
# S4 method for class 'dspMatrix'
BunchKaufman(x, warnSing = TRUE, ...)
# S4 method for class 'matrix'
BunchKaufman(x, uplo = "U", ...)a finite symmetric matrix or
Matrix to be factorized.
If x is square but not symmetric, then it will be
treated as symmetric; see uplo.
a logical indicating if a warning should
be signaled for singular x.
a string, either "U" or "L",
indicating which triangle of x should be used
to compute the factorization.
further arguments passed to or from methods.
An object representing the factorization, inheriting from
virtual class BunchKaufmanFactorization.
The specific class is BunchKaufman unless
x inherits from virtual class packedMatrix,
in which case it is pBunchKaufman.
Classes BunchKaufman and
pBunchKaufman and their methods.
Classes dsyMatrix and
dspMatrix.
Generic functions expand1 and expand2,
for constructing matrix factors from the result.
Generic functions Cholesky, Schur,
lu, and qr,
for computing other factorizations.
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
showMethods("BunchKaufman", inherited = FALSE)
#> Function: BunchKaufman (package Matrix)
#> x="dspMatrix"
#> x="dsyMatrix"
#> x="matrix"
#>
set.seed(0)
data(CAex, package = "Matrix")
class(CAex) # dgCMatrix
#> [1] "dgCMatrix"
#> attr(,"package")
#> [1] "Matrix"
isSymmetric(CAex) # symmetric, but not formally
#> [1] TRUE
A <- as(CAex, "symmetricMatrix")
class(A) # dsCMatrix
#> [1] "dsCMatrix"
#> attr(,"package")
#> [1] "Matrix"
## Have methods for denseMatrix (unpacked and packed),
## but not yet sparseMatrix ...
if (FALSE) { # \dontrun{
(bk.A <- BunchKaufman(A))
} # }
(bk.A <- BunchKaufman(as(A, "unpackedMatrix")))
#> Bunch-Kaufman factorization of Formal class 'BunchKaufman' [package "Matrix"] with 5 slots
#> ..@ uplo : chr "U"
#> ..@ x : num [1:5184] 4.16e-16 0.00 0.00 0.00 0.00 ...
#> ..@ perm : int [1:72] 1 2 3 4 5 6 7 8 9 10 ...
#> ..@ Dim : int [1:2] 72 72
#> ..@ Dimnames:List of 2
#> .. ..$ : NULL
#> .. ..$ : NULL
## A ~ U DU U' in floating point
str(e.bk.A <- expand2(bk.A), max.level = 2L)
#> List of 3
#> $ U :Formal class 'dgCMatrix' [package "Matrix"] with 6 slots
#> $ DU:Formal class 'dsCMatrix' [package "Matrix"] with 7 slots
#> $ U.:Formal class 'dgCMatrix' [package "Matrix"] with 6 slots
stopifnot(all.equal(as(A, "matrix"), as(Reduce(`%*%`, e.bk.A), "matrix")))