R: Singular Value Decomposition of a Matrix

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 real or complex 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)`.
`LINPACK` | logical. Should LINPACK be used (for compatibility with R < 1.7.0)? In this case `nu` must be `0`, `nrow(x)` or `ncol(x)`.

### Details

The singular value decomposition plays an important role in many statistical techniques. `svd` and `La.svd` provide two slightly different interfaces. The main functions used are the LAPACK routines DGESDD and ZGESVD; `svd(LINPACK = TRUE)` provides an interface to the LINPACK routine DSVDC, purely for backwards compatibility.

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

Unsuccessful results from the underlying LAPACK code will result in an error giving a positive error code: these can only be interpreted by detailed study of the FORTRAN code.

### Value

The SVD decomposition of the matrix as computed by LAPACK/LINPACK,

X = U D V',

where U and V are orthogonal, V' means V transposed, and D is a diagonal matrix with the singular values D[i,i]. Equivalently, D = U' X V, which is verified in the examples, below.
The returned value is a list with components

`d` | a vector containing the singular values of `x`, of length `min(n, p)`.
`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)`.

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

### 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.

### See Also

`eigen`, `qr`.

### Examples

    hilbert <- function(n) { i <- 1:n; 1 / outer(i - 1, i, "+") }
    X <- hilbert(9)[,1:6]
    (s <- svd(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

[Package base version 2.8.1 Index]