BunchKaufman-class function

Dense Bunch-Kaufman Factorizations

Classes BunchKaufman and pBunchKaufman represent Bunch-Kaufman factorizations of $n-by-n$ real, symmetric matrices $A$, having the general form [REMOVE_ME]$A = U D_{U} U' = L D_{L} L'A = U * DU * U' = L * DL * L' [REMOVE_ME_2]$

where $DU$ and $DL$ are symmetric, block diagonal matrices composed of $bU$ and $bL$

$1-by-1$ or $2-by-2$ diagonal blocks; $U = prod(Pk * Uk : k = 1,...,bU)$

is the product of $bU$ row-permuted unit upper triangular matrices, each having nonzero entries above the diagonal in 1 or 2 columns; and $L = prod(Pk * Lk : k = 1,...,bL)$

is the product of $bL$ 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 $2*bU+1$ or $2*bL+1$ factors, which are individually sparse, in a dense format as a vector of length $n*n$ (BunchKaufman) or $n*(n+1)/2$ (pBunchKaufman), the latter giving the packed representation.

class

Description

Classes BunchKaufman and pBunchKaufman represent Bunch-Kaufman factorizations of $n-by-n$ real, symmetric matrices $A$, having the general form

$$A = U D_{U} U' = L D_{L} L'A = U * DU * U' = L * DL * L'$$

where $DU$ and $DL$ are symmetric, block diagonal matrices composed of $bU$ and $bL$

$1-by-1$ or $2-by-2$ diagonal blocks; $U = prod(Pk * Uk : k = 1,...,bU)$

is the product of $bU$ row-permuted unit upper triangular matrices, each having nonzero entries above the diagonal in 1 or 2 columns; and $L = prod(Pk * Lk : k = 1,...,bL)$

is the product of $bL$ 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 $2*bU+1$ or $2*bL+1$ factors, which are individually sparse, in a dense format as a vector of length $n*n$ (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 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.

Note

In Matrix < 1.6-0, 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 $n-by-n$

triangular matrix. Matrix 1.6-0 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.

See Also

Class dsyMatrix and its packed counterpart.

Generic functions BunchKaufman, expand1, and expand2.

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

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)))