rankMatrix {Matrix}  R Documentation 
Compute ‘the’ matrix rank, a welldefined functional in theory(*), somewhat ambigous in practice. We provide several methods, the default corresponding to Matlab's definition.
(*) The rank of a n x m matrix A, rk(A) is the maximal number of linearly independent columns (or rows); hence rk(A) <= min(n,m).
rankMatrix(x, tol = NULL, method = c("tolNorm2", "qr.R", "qrLINPACK", "qr", "useGrad", "maybeGrad"), sval = svd(x, 0, 0)$d, warn.t = TRUE)
x 
numeric matrix, of dimension n x m, say. 
tol 
nonnegative number specifying a (relative,
“scalefree”) tolerance for testing of
“practically zero” with specific meaning depending on

method 
a character string specifying the computational method for the rank, can be abbreviated:

sval 
numeric vector of nonincreasing singular values of

warn.t 
logical indicating if 
If x
is a matrix of all 0
, the rank is zero; otherwise,
a positive integer in 1:min(dim(x))
with attributes detailing
the method used.
For large sparse matrices x
, unless you can specify
sval
yourself, currently method = "qr"
may
be the only feasible one, as the others need sval
and call
svd()
which currently coerces x
to a
denseMatrix
which may be very slow or impossible,
depending on the matrix dimensions.
Note that in the case of sparse x
, method = "qr"
, all
nonstrictly zero diagonal entries d_i where counted, up to
including Matrix version 1.10, i.e., that method implicitly
used tol = 0
, see also the seed(42) example below.
Martin Maechler; for the "*Grad" methods, building on suggestions by Ravi Varadhan.
rankMatrix(cbind(1, 0, 1:3)) # 2 (meths < eval(formals(rankMatrix)$method)) ## a "border" case: H12 < Hilbert(12) rankMatrix(H12, tol = 1e20) # 12; but 11 with default method & tol. sapply(meths, function(.m.) rankMatrix(H12, method = .m.)) ## tolNorm2 qr qr.R qrLINPACK useGrad maybeGrad ## 11 12 11 12 11 11 ## The meaning of 'tol' for method="qrLINPACK" and *dense* x is not entirely "scale free" rMQL < function(ex, M) rankMatrix(M, method="qrLINPACK",tol = 10^ex) rMQR < function(ex, M) rankMatrix(M, method="qr.R", tol = 10^ex) sapply(5:15, rMQL, M = H12) # result is platform dependent ## 7 7 8 10 10 11 11 11 12 12 12 {x86_64} sapply(5:15, rMQL, M = 1000 * H12) # not identical unfortunately ## 7 7 8 10 11 11 12 12 12 12 12 sapply(5:15, rMQR, M = H12) ## 5 6 7 8 8 9 9 10 10 11 11 sapply(5:15, rMQR, M = 1000 * H12) # the *same* ## "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 < 250000; p < 33; nnz < 10000 L < sparseMatrix(i = sample.int(n, nnz, replace=TRUE), j = sample.int(p, nnz, replace=TRUE), x = rnorm(nnz)) (st1 < system.time(r1 < rankMatrix(L))) # warning+ ~1.5 sec (2013) (st2 < system.time(r2 < rankMatrix(L, method = "qr"))) # considerably faster! r1[[1]] == print(r2[[1]]) ## > ( 33 TRUE ) ## another sparse"qr" one, which ``failed'' till 20131123: set.seed(42) f1 < factor(sample(50, 1000, replace=TRUE)) f2 < factor(sample(50, 1000, replace=TRUE)) f3 < factor(sample(50, 1000, replace=TRUE)) rbind. < if(getRversion() < "3.2.0") rBind else rbind D < t(do.call(rbind., lapply(list(f1,f2,f3), as, 'sparseMatrix'))) dim(D); nnzero(D) ## 1000 x 150 // 3000 nonzeros (= 2%) stopifnot(rankMatrix(D, method='qr') == 148, rankMatrix(crossprod(D),method='qr') == 148) ## zero matrix has rank 0 : stopifnot(sapply(meths, function(.m.) rankMatrix(matrix(0, 2, 2), method = .m.)) == 0)