Positive Semi-definite Dense Numeric Matrices

Description

The "dpoMatrix" class is the class of positive-semidefinite symmetric matrices in nonpacked storage. The "dppMatrix" class is the same except in packed storage. Only the upper triangle or the lower triangle is required to be available.

The "corMatrix" class extends "dpoMatrix" with a slot sd.

Objects from the Class

Objects can be created by calls of the form new("dpoMatrix", ...) or from crossprod applied to an "dgeMatrix" object.

Slots

uplo:

Object of class "character". Must be either "U", for upper triangular, and "L", for lower triangular.

x:

Object of class "numeric". The numeric values that constitute the matrix, stored in column-major order.

Dim:

Object of class "integer". The dimensions of the matrix which must be a two-element vector of non-negative integers.

Dimnames:

inherited from class "Matrix"

factors:

Object of class "list". A named list of factorizations that have been computed for the matrix.

sd:

(for "corMatrix") a numeric vector of length n containing the (original) sqrt(var(.)) entries which allow reconstruction of a covariance matrix from the correlation matrix.

Extends

Class "dsyMatrix", directly.
Classes "dgeMatrix", "symmetricMatrix", and many more by class "dsyMatrix".

Methods

chol

signature(x = "dpoMatrix"): Returns (and stores) the Cholesky decomposition of x, see chol.

rcond

signature(x = "dpoMatrix", norm = "character"): Returns (and stores) the reciprocal of the condition number of x. The norm can be "O" for the one-norm (the default) or "I" for the infinity-norm. For symmetric matrices the result does not depend on the norm.

solve

signature(a = "dpoMatrix", b = "missing"): Return the inverse of a.

solve

signature(a = "dpoMatrix", b = "numeric"): Solve the linear system defined by a and b, where b can be a numeric vector, or a matrix, or a dgeMatrix object. The Cholesky decomposition of a is calculated (if needed) while solving the system.

Arith

signature(e1 = "dpoMatrix", e2 = "numeric") (and quite a few other signatures): The result of (“elementwise” defined) arithmetic operations is typically not positive-definite anymore. The only exceptions, currently, are multiplications, divisions or additions with positive length(.) == 1 numbers (or logicals).

See Also

Classes dsyMatrix and dgeMatrix; further, Matrix, rcond, chol, solve, crossprod.

Examples

h6 <- Hilbert(6)
rcond(h6)
str(h6)
h6 * 27720 # is ``integer''
solve(h6)
str(hp6 <- as(h6, "dppMatrix"))

### Note that  as(*, "corMatrix")  *scales* the matrix
(ch6 <- as(h6, "corMatrix"))
stopifnot(all.equal(h6 * 27720, round(27720 * h6), tol = 1e-14),
          all.equal(ch6@sd^(-2), 2*(1:6)-1, tol= 1e-12))
chch <- chol(ch6)
stopifnot(identical(chch, ch6@factors$Cholesky),
          all(abs(crossprod(chch) - ch6) < 1e-10))