\name{dpoMatrix-class}
\title{Positive Semi-definite Dense (Packed | Non-packed) Numeric Matrices}
\docType{class}
\alias{dpoMatrix-class}
\alias{dppMatrix-class}
\alias{corMatrix-class}
%      ---
\alias{coerce,dpoMatrix,corMatrix-method}
\alias{coerce,dpoMatrix,dppMatrix-method}
\alias{coerce,dpoMatrix,lMatrix-method}
\alias{coerce,dpoMatrix,nMatrix-method}
\alias{coerce,dppMatrix,lMatrix-method}
\alias{coerce,dppMatrix,nMatrix-method}
\alias{coerce,dppMatrix,sparseMatrix-method}
\alias{coerce,dppMatrix,CsparseMatrix-method}
\alias{coerce,dppMatrix,dpoMatrix-method}
\alias{coerce,dspMatrix,dpoMatrix-method}
\alias{coerce,dspMatrix,dppMatrix-method}
\alias{coerce,matrix,corMatrix-method}
\alias{coerce,Matrix,corMatrix-method}
\alias{coerce,matrix,dpoMatrix-method}
\alias{coerce,Matrix,dpoMatrix-method}
\alias{coerce,matrix,dppMatrix-method}
\alias{coerce,Matrix,dppMatrix-method}
%\alias{coerce,corMatrix,lMatrix-method}
\alias{Ops,dpoMatrix,logical-method}
\alias{Ops,dpoMatrix,numeric-method}
\alias{Ops,dppMatrix,logical-method}
\alias{Ops,dppMatrix,numeric-method}
\alias{Ops,logical,dpoMatrix-method}
\alias{Ops,logical,dppMatrix-method}
\alias{Ops,numeric,dpoMatrix-method}
\alias{Ops,numeric,dppMatrix-method}
\alias{Arith,dpoMatrix,logical-method}
\alias{Arith,dpoMatrix,numeric-method}
\alias{Arith,dppMatrix,logical-method}
\alias{Arith,dppMatrix,numeric-method}
\alias{Arith,logical,dpoMatrix-method}
\alias{Arith,logical,dppMatrix-method}
\alias{Arith,numeric,dpoMatrix-method}
\alias{Arith,numeric,dppMatrix-method}
%
\alias{rcond,dpoMatrix,character-method}
\alias{rcond,dppMatrix,character-method}
\alias{rcond,dpoMatrix,missing-method}
\alias{rcond,dppMatrix,missing-method}
\alias{determinant,dpoMatrix,logical-method}
\alias{determinant,dppMatrix,logical-method}
\alias{determinant,dpoMatrix,missing-method}
\alias{determinant,dppMatrix,missing-method}
%\alias{solve,dpoMatrix,...-method}--> solve-methods.Rd
\alias{t,dppMatrix-method}
\description{
  \itemize{
    \item{The \code{"dpoMatrix"} class is the class of
      positive-semidefinite symmetric matrices in nonpacked storage.}
    \item{The \code{"dppMatrix"} class is the same except in packed
      storage.  Only the upper triangle or the lower triangle is
      required to be available.}
    \item{The \code{"corMatrix"} class of correlation matrices extends
      \code{"dpoMatrix"} with a slot \code{sd}, which allows to restore
      the original covariance matrix.}
  }
}
\section{Objects from the Class}{Objects can be created by calls of the
  form \code{new("dpoMatrix", ...)} or from \code{crossprod} applied to
  an \code{"dgeMatrix"} object.}
\section{Slots}{
  \describe{
    \item{\code{uplo}:}{Object of class \code{"character"}. Must be
      either "U", for upper triangular, and "L", for lower triangular.}
    \item{\code{x}:}{Object of class \code{"numeric"}. The numeric
      values that constitute the matrix, stored in column-major order.}
    \item{\code{Dim}:}{Object of class \code{"integer"}. The dimensions
      of the matrix which must be a two-element vector of non-negative
      integers.}
    \item{\code{Dimnames}:}{inherited from class \code{"Matrix"}}
    \item{\code{factors}:}{Object of class \code{"list"}.  A named
      list of factorizations that have been computed for the matrix.}

    \item{\code{sd}:}{(for \code{"corMatrix"}) a \code{\link{numeric}}
      vector of length \code{n} containing the (original)
      \eqn{\sqrt{var(.)}}{sqrt(var(.))} entries which allow
	reconstruction of a covariance matrix from the correlation matrix.}
  }
}
\section{Extends}{
  Class \code{"dsyMatrix"}, directly.\cr
  Classes \code{"dgeMatrix"}, \code{"symmetricMatrix"}, and many more
  by class \code{"dsyMatrix"}.
}
\section{Methods}{
  \describe{
    \item{chol}{\code{signature(x = "dpoMatrix")}:
      Returns (and stores) the Cholesky decomposition of \code{x}, see
      \code{\link{chol}}.}
    \item{determinant}{\code{signature(x = "dpoMatrix")}:
      Returns the \code{\link{determinant}} of \code{x}, via
      \code{chol(x)}, see above.}
    \item{rcond}{\code{signature(x = "dpoMatrix", norm = "character")}:
      Returns (and stores) the reciprocal of the condition number of
      \code{x}.  The \code{norm} can be \code{"O"} for the
      one-norm (the default) or \code{"I"} for the infinity-norm.  For
      symmetric matrices the result does not depend on the norm.}
    \item{solve}{\code{signature(a = "dpoMatrix", b = "....")}}{, and}
    \item{solve}{\code{signature(a = "dppMatrix", b = "....")}}{work
      via the Cholesky composition, see also the Matrix \code{\link{solve-methods}}.}
    \item{Arith}{\code{signature(e1 = "dpoMatrix", e2 = "numeric")} (and
      quite a few other signatures): The result of (\dQuote{elementwise}
      defined) arithmetic operations is typically \emph{not}
      positive-definite anymore.  The only exceptions, currently, are
      multiplications, divisions or additions with \emph{positive}
      \code{length(.) == 1} numbers (or \code{\link{logical}}s).}
  }
}
%\references{}
%\author{}
\seealso{
  Classes \code{\linkS4class{dsyMatrix}} and \code{\linkS4class{dgeMatrix}};
  further, \code{\link{Matrix}}, \code{\link{rcond}},
  \code{\link[base]{chol}}, \code{\link[base]{solve}}, \code{\link{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), tolerance = 1e-14),
          all.equal(ch6@sd^(-2), 2*(1:6)-1, tolerance= 1e-12))
chch <- chol(ch6)
stopifnot(identical(chch, ch6@factors$Cholesky),
          all(abs(crossprod(chch) - ch6) < 1e-10))
}
\keyword{classes}
\keyword{algebra}