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Wed Jul 3 10:13:08 2013 UTC (6 years, 2 months ago) by mmaechler
File size: 3812 byte(s)
rankMatrix(): method="qr.." no longer needs sval --> even more useful for large sparse
\title{Rank of a Matrix}
  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.
rankMatrix(x, tol = NULL,
           method = c("tolNorm2", "qrLINPACK", "useGrad", "maybeGrad"),
           sval = svd(x, 0, 0)$d, warn.t = TRUE)
  \item{x}{numeric matrix, of dimension \eqn{n \times m}{n x m}, say.}
  \item{tol}{nonnegative number specifying a tolerance for
    \dQuote{practically zero} with specific meaning depending on
    \code{method}; by default, \code{max(dim(x)) *
      \link{.Machine}$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:
      \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), which may be crucially important for a
	large sparse matrix \code{x}.}
      \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} when
    needed, i.e., unless \code{method = "qrLINPACK"}.}
  \item{warn.t}{logical indicating if \code{rankMatrix()} should warn
    when it needs \code{\link{t}(x)} instead of \code{x}. Currently, for
    \code{method = "qrLINPACK"} only.}
% \details{
% }
\note{For large sparse matrices \code{x}, unless you can specify
  \code{sval} yourself, currently \code{method = "qrLINPACK"} may
  be the only feasible one, as the others need \code{sval} and call
  \code{\link{svd}()} which currently coerces \code{x} to a
  \code{\linkS4class{denseMatrix}} which may be very slow or impossible,
  depending on the matrix dimensions.
  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.
 \code{\link{qr}}, \code{\link{svd}}.
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.))
## tolNorm2 qrLINPACK   useGrad maybeGrad
##       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.

## "large" sparse
n <- 200000; p <- 33; nnz <- 10000
L <- sparseMatrix(i =, nnz, replace=TRUE),
                  j =, nnz, replace=TRUE), x = rnorm(nnz))
(st1 <- system.time(r1 <- rankMatrix(L)))                # almost 1 sec
(st2 <- system.time(r2 <- rankMatrix(L, method = "qr"))) # considerably faster!
r1[[1]] == print(r2[[1]]) ## -->  ( 33  TRUE )
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