Cholesky and Bunch-Kaufman Decompositions

Description

The "Cholesky" class is the class of Cholesky decompositions of positive-semidefinite, real dense matrices. The "BunchKaufman" class is the class of Bunch-Kaufman decompositions of symmetric, real matrices. The "pCholesky" and "pBunchKaufman" classes are their packed storage versions.

Objects from the Class

Objects can be created by calls of the form new("Cholesky", ...) or new("BunchKaufman", ...), etc, or rather by calls of the form chol(pm) or BunchKaufman(pm) where pm inherits from the "dpoMatrix" or "dsyMatrix" class or as a side-effect of other functions applied to "dpoMatrix" objects (see dpoMatrix).

Slots

A Cholesky decomposition extends class MatrixFactorization but is basically a triangular matrix extending the "dtrMatrix" class.

uplo:

inherited from the "dtrMatrix" class.

diag:

inherited from the "dtrMatrix" class.

x:

inherited from the "dtrMatrix" class.

Dim:

inherited from the "dtrMatrix" class.

Dimnames:

inherited from the "dtrMatrix" class.

A Bunch-Kaufman decomposition also extends the "dtrMatrix" class and has a perm slot representing a permutation matrix. The packed versions extend the "dtpMatrix" class.

Extends

Class "MatrixFactorization" and "dtrMatrix", directly. Class "dgeMatrix", by class "dtrMatrix". Class "Matrix", by class "dtrMatrix".

Methods

Both these factorizations can directly be treated as (triangular) matrices, as they extend "dtrMatrix", see above. There are currently no further explicit methods defined with class "Cholesky" or "BunchKaufman" in the signature.

Note

1. Objects of class "Cholesky" typically stem from chol(D), applied to a dense matrix D.

   On the other hand, the function Cholesky(S) applies to a sparse matrix S, and results in objects inheriting from class CHMfactor.

2. For traditional matrices m, chol(m) is a traditional matrix as well, triangular, but simply an n * n numeric matrix. Hence, for compatibility, the "Cholesky" and "BunchKaufman" classes (and their "p*" packed versions) also extend triangular Matrix classes (such as "dtrMatrix").

   Consequently, determinant(R) for R <- chol(A) returns the determinant of R, not of A. This is in contrast to class CHMfactor objects C, where determinant(C) gives the determinant of the original matrix A, for C <- Cholesky(A), see also the determinant method documentation on the class CHMfactor page.

See Also

Classes dtrMatrix, dpoMatrix; function chol.

Function Cholesky resulting in class CHMfactor objects, not class "Cholesky" ones, see the section ‘Note’.

Examples

(sm <- as(as(Matrix(diag(5) + 1), "dsyMatrix"), "dspMatrix"))
signif(csm <- chol(sm), 4)

(pm <- crossprod(Matrix(rnorm(18), nrow = 6, ncol = 3)))
(ch <- chol(pm))
if (toupper(ch@uplo) == "U") # which is TRUE
   crossprod(ch)
stopifnot(all.equal(as(crossprod(ch), "matrix"),
                    as(pm, "matrix"), tolerance=1e-14))