BunchKaufman-class: Dense Bunch-Kaufman Factorizations

Description

Classes BunchKaufman and pBunchKaufman represent Bunch-Kaufman 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 $b_{U}$ 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.

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.

Arguments

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 the x slot is partitioned.
x: a numeric vector of length n*n (BunchKaufman) or n*(n+1)/2 (pBunchKaufman), where n=Dim[1]. The details of the representation are specified by the manual for LAPACK routines dsytrf and dsptrf.
perm: an integer vector of length n=Dim[1] specifying row and column interchanges as described in the manual for LAPACK routines dsytrf and dsptrf.

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 a dtrMatrix, useful for inspecting the internal representation of the factorization; see ‘Note’.
coerce: signature(from = "pBunchKaufman", to = "dtpMatrix"): returns a dtpMatrix, 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"): see expand1-methods.
expand2: signature(x = "p?BunchKaufman"): see expand2-methods.
solve: signature(a = "p?BunchKaufman", b = .): see solve-methods.

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

Examples

Run this code

 
library(stats, pos = "package:base", verbose = FALSE)
library(utils, pos = "package:base", verbose = FALSE)

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)))

Run the code above in your browser using DataLab