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[matrix] View of /pkg/Matrix/man/rankMatrix.Rd
 [matrix] / pkg / Matrix / man / rankMatrix.Rd # View of /pkg/Matrix/man/rankMatrix.Rd

Revision 2854 - (download) (as text) (annotate)
Thu Nov 22 07:54:41 2012 UTC (6 years, 10 months ago) by mmaechler
File size: 2734 byte(s)
qr(<sparse>) --> <sparseQR> : finally address the structurally rank-deficient case where cs_sqr() produces "spurious" extra rows in Q i.e. V, and R; add checking code for that case
\name{rankMatrix}
\Rdversion{1.1}
\alias{rankMatrix}
\title{Rank of a Matrix}
\description{
Compute \sQuote{the} matrix rank, a well-defined functional in theory,
somewhat ambigous in practice.  We provide several methods, the
default corresponding to Matlab's definition.
}
\usage{
rankMatrix(x, tol = NULL,
sval = svd(x, 0, 0)$d) } \arguments{ \item{x}{numeric matrix, of dimension \eqn{n \times m}{n x m}, say.} \item{tol}{nonnegative number specifying a tolerance for \dQuote{practically zero} with specific meaning depending on \code{method}; by default, \code{max(dim(x)) * \link{.Machine}$double.eps * abs(max(sval))} is according to
Matlab's default (for its only \code{method} "tolNorm2").}
\item{method}{a character string specifying the computational method,
can be abbreviated:
\describe{
\item{tolNorm2}{the number of singular values \code{>= tol};}
\item{qrLINPACK}{this is the rank of \code{\link{qr}(x, tol,
LAPACK=FALSE)}, which is \code{qr(...)$rank} for a dense matrix, and the rank of \eqn{R} for sparse \code{x} (where \code{qr} uses a "sparseQR" method, see \code{\link{qr-methods}}, and not LINPACK). This used to be \emph{the} recommended way to compute a matrix rank for a while in the past. For this method, \code{sval} are not used (nor computed).} \item{useGrad}{considering the \dQuote{gradient} of the (decreasing) singular values, the index of the \emph{smallest} gap.} \item{maybeGrad}{choosing method \code{"useGrad"} only when that seems \emph{reasonable}; otherwise using \code{"tolNorm2"}.} %% FIXME say more } } \item{sval}{numeric vector of non-increasing singular values of \code{x}; typically unspecified and computed from \code{x}.} } % \details{ % FIXME % } \value{ positive integer in \code{1:min(dim(x))}, with attributes detailing the method used. } % \references{ % %% ~put references to the literature/web site here ~ % } \author{Martin Maechler; for the "*Grad" methods, building on suggestions by Ravi Varadhan. } \seealso{ \code{\link{qr}}, \code{\link{svd}}. } \examples{ rankMatrix(cbind(1, 0, 1:3)) # 2 (meths <- eval(formals(rankMatrix)$method))

## a "border" case:
H12 <- Hilbert(12)
rankMatrix(H12, tol = 1e-20) # 12;  but  11  with default method & tol.
sapply(meths, function(.m.) rankMatrix(H12, method = .m.))
##       11        12        11        11

## "sparse" case:
M15 <- kronecker(diag(x=c(100,1,10)), Hilbert(5))
sapply(meths, function(.m.) rankMatrix(M15, method = .m.))
#--> all 15, but 'useGrad' has 14.
}
\keyword{algebra}
\keyword{array}


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