rankMatrix {Matrix}R Documentation

Rank of a Matrix

Description

Compute the rank of matrix, 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,
           method = c("tolNorm2", "qrLINPACK", "useGrad", "maybeGrad"),
           sval = svd(x, 0, 0)$d)

Arguments

x

numeric matrix, of dimension n x m, say.

tol

nonnegative number specifying a tolerance for “practically zero” with specific meaning depending on method; by default, max(dim(x)) * .Machine$double.eps * abs(max(sval)) is according to Matlab's default (for its only method "tolNorm2").

method

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

tolNorm2

the number of singular values >= tol;

qrLINPACK

this is the rank of qr(x, tol, LAPACK=FALSE), which is qr(...)$rank for a dense matrix, and the rank of R for sparse x (where qr uses a "sparseQR" method, see qr-methods, and not LINPACK).

This used to be the recommended way to compute a matrix rank for a while in the past. For this method, sval are not used (nor computed).

useGrad

considering the “gradient” of the (decreasing) singular values, the index of the smallest gap.

maybeGrad

choosing method "useGrad" only when that seems reasonable; otherwise using "tolNorm2".

sval

numeric vector of non-increasing singular values of x; typically unspecified and computed from x.

Value

positive integer in 1:min(dim(x)), with attributes detailing the method used.

Author(s)

Martin Maechler; for the "*Grad" methods, building on suggestions by Ravi Varadhan.

See Also

qr, 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.))
## 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.
[Package Matrix version 1.0-6 Index]