svd
===

singular value decomposition



Calling Sequence
~~~~~~~~~~~~~~~~


::

    s=svd(X)
    [U,S,V]=svd(X)
    [U,S,V]=svd(X,0) (obsolete)
    [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,0)` 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`_ rank
+ `qr`_ QR decomposition
+ `colcomp`_ column compression, kernel, nullspace
+ `rowcomp`_ row compression, range
+ `sva`_ singular value approximation
+ `spec`_ eigenvalues of matrices and pencils




Used Functions
~~~~~~~~~~~~~~

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

.. _rowcomp: rowcomp.html
.. _colcomp: colcomp.html
.. _sva: sva.html
.. _qr: qr.html
.. _rank: rank.html
.. _spec: spec.html