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Wed Jul 18 14:46:20 2007 UTC (12 years, 1 month ago) by maechler
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norm() for sparse; systematic checking of norm() and all "Summary" methods
\title{Positive Semi-definite Dense Numeric Matrices}
  The \code{"dpoMatrix"} class is the class of
  positive-semidefinite symmetric matrices in nonpacked storage.
  The \code{"dppMatrix"} class is the same except in packed storage.
  Only the upper triangle or the lower triangle is required to be

  The \code{"corMatrix"} class extends \code{"dpoMatrix"} with a slot \code{sd}.

\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.}
    \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
    \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.}
  Class \code{"dsyMatrix"}, directly.\cr
  Classes \code{"dgeMatrix"}, \code{"symmetricMatrix"}, and many more
  by class \code{"dsyMatrix"}.
    \item{chol}{\code{signature(x = "dpoMatrix")}:
      Returns (and stores) the Cholesky decomposition of the matrix
    \item{rcond}{\code{signature(x = "dpoMatrix", type = "character")}:
      Returns (and stores) the reciprocal of the condition number of
      \code{x}.  The \code{type} 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 type.}
    \item{solve}{\code{signature(a = "dpoMatrix", b = "missing")}:
      Return the inverse of \code{a}.}
    \item{solve}{\code{signature(a = "dpoMatrix", b = "numeric")}:
      Solve the linear system defined by \code{a} and \code{b}, where
      \code{b} can be a numeric vector, or a matrix, or a dgeMatrix
      object.  The Cholesky decomposition of \code{a} is calculated (if
      needed) while solving the system.}
  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}}.
h6 <- Hilbert(6)
str(hp6 <- as(h6, "dppMatrix"))

### Note that  as(*, "corMatrix")  *scales* the matrix
(ch6 <- as(h6, "corMatrix"))
stopifnot(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))
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