# R/rankMatrix.R In Matrix: Sparse and Dense Matrix Classes and Methods

#### Documented in qr2rankMatrixrankMatrix

```#### Determine *the* rank of a matrix
#### --------------------------------
##
## As this is not such a well-defined problem as people think,
## we provide *some* possibilities here, including the Matlab one.
##
## Ideas by Martin Maechler (April 2007) and Ravi Varadhan (October 2007)

qr2rankMatrix <- function(qr, tol = NULL, isBqr = is.qr(qr), do.warn=TRUE) {
## NB: 1) base::qr(*, LAPACK = TRUE/FALSE)  differ via attr(.,"useLAPACK")
##     2) if LAPACK=TRUE, .\$rank is useless (always = full rank)
##
## return ( . ) :
if(isBqr && !isTRUE(attr(qr, "useLAPACK")))
qr\$rank
else {
diagR <- if(isBqr) # hence "useLAPACK" here
diag(qr\$qr) # faster than, but equivalent to   diag(qr.R(q.r))
else ## ==> assume Matrix::qr() i.e., currently "sparseQR"
## FIXME: Here, we could be quite a bit faster,
## by not returning the full sparseQR, but just
## doing the following in C, and return the rank.
diag(qr@R)

if(anyNA(diagR) || !all(is.finite(diagR))) {
if(do.warn) {
ifi <- is.finite(diagR)
warning(gettextf(
"qr2rankMatrix(.): QR with only %d out of %d finite diag(R) entries",
sum(ifi), length(ifi)))
}
## return
NA_integer_
## alternative: gives *too* small rank in typical cases
## reduce the maximal rank by omitting all non-finite entries:
## diagR <- diagR[is.finite(diagR)]
## if(length(diagR) == 0)
##     return(NA_integer_)
} else {
if(isBqr) diagR <- abs(diagR) # in base qr(), sign( diag(R) ) are *not* coerced to positive
else if(do.warn && any(diagR < 0))
warning(gettextf("qr2rankMatrix(.): QR has negative diag(R) entries"))
## declare those entries to be zero that are < tol*max(.)
if((mdi <- max(diagR, na.rm=TRUE)) > 0) {
if(!is.numeric(tol)) {
## d := dim(x) extracted from qr, in both (dense and sparse) qr() cases
d <- dim(if(isBqr) qr\$qr else qr)
tol <- max(d) * .Machine\$double.eps
}
sum(diagR >= tol * mdi)
## was sum(diag(q.r@R) != 0)
}
else 0L # for 0-matrix or all NaN or negative diagR[]
}
} ## else {Lapack or sparseQR}
}

rankMatrix <- function(x, tol = NULL,
method = c("tolNorm2", "qr.R", "qrLINPACK", "qr",
sval = svd(x, 0,0)\$d, warn.t = TRUE, warn.qr = TRUE)
{
## Purpose: rank of a matrix ``as Matlab'' or "according to Ravi V"
## ----------------------------------------------------------------------
## Arguments: x: a numerical matrix, maybe non-square
##          tol: numerical tolerance (compared to singular values)
##         sval: vector of non-increasing singular values of  x
##               (pass as argument if already known)
## ----------------------------------------------------------------------
## Author: Martin Maechler, Date: 7 Apr 2007, 16:16
## ----------------------------------------------------------------------
##
## maybeGrad (Ravi V.): This algorithm determines the rank based on the
## absolute, singular values, rather than enforcing a rigid
## tolerance criterion,
##
## Author: Ravi Varadhan, Date: 22 October 2007 // Tweaks: MM, Oct.23
## ----------------------------------------------------------------------

stopifnot(length(d <- dim(x)) == 2)
p <- min(d)
## miss.meth <- missing(method)
method <- match.arg(method)

stopifnot(length(sval) == p)
if(p > 1) stopifnot(diff(sval) <= 0) # must be sorted non-increasingly: max = s..[1]
if(sval[1] == 0) { ## <==> all singular values are zero  <==> Matrix = 0  <==> rank = 0
method <- eval(formals()[["method"]])[[1]]
} else {
ln.av <- log(abs(sval))
if(any(s0 <- sval == 0)) ln.av[s0] <- - .Machine\$double.xmax # so we get diff() == 0
diff1 <- diff(ln.av)
grad <- (min(ln.av) - max(ln.av)) / p
}#  -------
}
}
x.dense <- is.numeric(x) || is(x,"denseMatrix")
## "qr" is allowed for backcompatibility [change @ 2013-11-24]
if((Meth <- method) == "qr")
method <- if(x.dense) "qrLINPACK" else "qr.R"
else Meth <- substr(method, 1,2)

if(Meth == "qr") {
if(is.null(tol)) tol <- max(d) * .Machine\$double.eps
} else { ## (Meth != "qr"), i.e. "tolNorm2"
if(is.null(tol)) {
if(!x.dense && missing(sval) && prod(d) >= 100000L)
warning(gettextf(
"rankMatrix(<large sparse Matrix>, method = '%s') coerces to dense matrix.
Probably should rather use method = 'qr' !?",
method),
immediate.=TRUE, domain=NA)
## the "Matlab" default:
if(p > 1) stopifnot(diff(sval) <= 0) #=> sval[1]= max(sval)
tol <- max(d) * .Machine\$double.eps
} else stopifnot((tol <- as.numeric(tol)[[1]]) >= 0)
}
}

structure(## rank :
else if(Meth == "qr") {
if((do.t <- (d[1L] < d[2L])) && warn.t)
warning(gettextf(
"rankMatrix(x, method='qr'): computing t(x) as nrow(x) < ncol(x)"))
q.r <- qr(if(do.t) t(x) else x, tol=tol, LAPACK = method != "qrLINPACK")
qr2rankMatrix(q.r, tol=tol, isBqr = x.dense, do.warn = warn.qr)
}
else if(sval[1] > 0) sum(sval >= tol * sval[1]) else 0L, ## "tolNorm2"
"method" = method,
"tol" = if(useGrad) NA else tol)
}

## Ravi's plot of the absolute singular values:
if(FALSE) {
## if (plot.eigen) {
plot(abs(sval), type = "b", xlab = "Index", xaxt = "n",
log = "y", ylab = "|singular value|   [log scaled]")
axis(1, at = unique(c(axTicks(1), rank, p)))
abline(v = rank, lty = 3)
mtext(sprintf("rank = %d (used %s (%g))", rank,