# svd {base}

Singular Value Decomposition of a Matrix
Package:
base
Version:
R 3.0.2

### 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 ` ` and `n = nrow(x)`.
nv
the number of right singular vectors to be computed. This must be between ` ` and `p = ncol(x)`.
LINPACK
logical. Defunct and ignored (with a warning for true values).

### Details

The singular value decomposition plays an important role in many statistical techniques. `svd` and `La.svd` provide two slightly different interfaces.

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 (most often `1`): these can only be interpreted by detailed study of the FORTRAN code but mean that the algorithm failed to converge.

### Values

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 \bold{D} is a diagonal matrix with the singular values D[i,i]. Equivalently, \bold{D = U' X V}, which is verified in the examples, below.

The returned value is a list with components

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

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)`.

### References

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.

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, "+") }