gpuSvd: Singular Value Decomposition of a Matrix with a GPU
Description
Compute the singular-value decomposition of a rectangular matrix
using the Cula library to compute the decomposition using a GPU.
Usage
gpuSvd(x, nu = min(n, p), nv = min(n, p))
Arguments
x
a real matrix whose SVD decomposition is to be computed.
nu
the number of left singular vectors to be computed.
This must between 0 and n = nrow(x).
nv
the number of right singular vectors to be computed.
This must be between 0 and p = ncol(x).
Value
The SVD decomposition of the matrix,
$$\bold{X = U D V'},$$ where $\bold{U}$ and $\bold{V}$ are
orthogonal, $\bold{V'}$ means V transposed, and
$\bold{D}$ is a diagonal matrix with the singular
values $D_{ii}$. Equivalently, $\bold{D = U' X V}$,
which is verified in the examples, below.
The returned value is a list with components
da vector containing the singular values of x, of
length min(n, p).
ua matrix whose columns contain the left singular vectors of
x, present if nu > 0. Dimension c(n, nu).
va matrix whose columns contain the right singular vectors of
x, present if nv > 0. Dimension c(p, nv).
Details
The computation will be more efficient if nu <= min(n,="" p)<="" code=""> and
nv <= min(n,="" p)<="" code="">, and even more efficient if one or both are zero.
References
Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988)
The New S Language.
Wadsworth & Brooks/Cole.
Dongarra, J. J., Bunch, J. R., Moler, C. B. and Stewart, G. W. (1978)
LINPACK Users Guide. Philadelphia: SIAM Publications.
Anderson. E. and ten others (1999)
LAPACK Users' Guide. Third Edition. SIAM.
Available on-line at
http://www.netlib.org/lapack/lug/lapack_lug.html.
hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, "+") }
X <- hilbert(9)[,1:6]
(s <- gpuSvd(X))
D <- diag(s$d)
s$u %*% D %*% t(s$v) # X = U D V't(s$u) %*% X %*% s$v # D = U' X V