svd {base} R Documentation

## Singular Value Decomposition of a Matrix

### Description

Compute the singular-value decomposition of a rectangular matrix.

### Usage

svd(x, nu = min(n, p), nv = min(n, p), LINPACK = FALSE)

La.svd(x, nu = min(n, p), nv = min(n, p))


### Arguments

 x a numeric or complex matrix whose SVD decomposition is to be computed. Logical matrices are coerced to numeric. 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). LINPACK logical. Defunct and an error.

### Details

The singular value decomposition plays an important role in many statistical techniques. svd and La.svd provide two interfaces which differ in their return values.

Computing the singular vectors is the slow part for large matrices. The computation will be more efficient if both nu <= min(n, p) and nv <= min(n, p), and even more so if both are zero.

Unsuccessful results from the underlying LAPACK code will result in an error giving a positive error code (most often 1): these can only be interpreted by detailed study of the FORTRAN code but mean that the algorithm failed to converge.

### Value

The SVD decomposition of the matrix as computed by LAPACK,

 \bold{X = U D V'},

where \bold{U} and \bold{V} are orthogonal, \bold{V'} means V transposed (and conjugated for complex input), and \bold{D} is a diagonal matrix with the (non-negative) singular values D_{ii} in decreasing order. Equivalently, \bold{D = U' X V}, which is verified in the examples.

The returned value is a list with components

 d a vector containing the singular values of x, of length min(n, p), sorted decreasingly. u a matrix whose columns contain the left singular vectors of x, present if nu > 0. Dimension c(n, nu). v a matrix whose columns contain the right singular vectors of x, present if nv > 0. Dimension c(p, nv).

Recall that the singular vectors are only defined up to sign (a constant of modulus one in the complex case). If a left singular vector has its sign changed, changing the sign of the corresponding right vector gives an equivalent decomposition.

For La.svd the return value replaces v by vt, the (conjugated if complex) transpose of v.

### Source

The main functions used are the LAPACK routines DGESDD and ZGESDD.

LAPACK is from https://www.netlib.org/lapack/ and its guide is listed in the references.

### References

Anderson. E. and ten others (1999) LAPACK Users' Guide. Third Edition. SIAM.
Available on-line at https://www.netlib.org/lapack/lug/lapack_lug.html.

The ‘Singular-value decomposition’ Wikipedia article.

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

eigen, qr.
hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, +) }
D <- diag(s$d) s$u %*% D %*% t(s$v) # X = U D V' t(s$u) %*% X %*% s\$v #  D = U' X V