svd singular value decomposition

## Syntax

    s=svd(X)
                [U,S,V]=svd(X)
                [U,S,V]=svd(X,"e")
                [U,S,V,rk]=svd(X [,tol])

## Arguments

X

  a real or complex matrix

s

  real vector (singular values)

S

  real diagonal matrix (singular values)

U,V

  orthogonal or unitary square matrices (singular vectors).

tol

  real number

## Description

`[U,S,V] = svd(X)` produces a diagonal matrix `S` , of the same dimension as `X` and with nonnegative diagonal elements in decreasing order, and unitary matrices `U` and `V` so that `X = U*S*V'`.

`[U,S,V] = svd(X,"e")` produces the "economy size" decomposition. If `X` is m-by-n with m > n, then only the first n columns of `U` are computed and `S` is n-by-n.

`s= svd(X)` by itself, returns a vector `s` containing the singular values.

`[U,S,V,rk]=svd(X,tol)` gives in addition `rk`, the numerical rank of `X` i.e. the number of singular values larger than `tol`.

The default value of `tol` is the same as in `rank`.

## Examples

    X=rand(4,2)*rand(2,4)
    svd(X)
    sqrt(spec(X*X'))

## See also

- rank
- qr
- colcomp
- rowcomp
- sva
- spec

## Used Functions

svd decompositions are based on the Lapack routines DGESVD for real matrices and ZGESVD for the complex case.

## History

2023.0.0

svd(X, 0) is obsolete, use svd(X, "e") instead.

2024.0.0

svd(X, 0) is no more supported, use svd(X, "e") instead.