 
  
  
  
  
 
The singular value decomposition (SVD) of an m-by-n matrix A is given by
  

where U and V are orthogonal (unitary)
and  is an m-by-n diagonal matrix with real
diagonal elements,
 is an m-by-n diagonal matrix with real
diagonal elements,  , such that
, such that

The  are the singular values of A and the
first min(m,n) columns of U and V
are the left and right singular vectors of A.
 are the singular values of A and the
first min(m,n) columns of U and V
are the left and right singular vectors of A.
  
The singular values and singular vectors satisfy

where  and
 and  are the ith columns of U and V, respectively.
 are the ith columns of U and V, respectively.
A single driver  routine, PxGESVD  , computes the ``economy size'' or
``thin'' singular value decomposition of a general nonsymmetric matrix
(see table 3.4).  Thus, if A is m-by-n with
m>n, then only the first n columns of U are computed and  is an 
n-by-n matrix.  For a detailed discussion of the ``thin'' singular
value decomposition, refer to [71, p. 72,].
 is an 
n-by-n matrix.  For a detailed discussion of the ``thin'' singular
value decomposition, refer to [71, p. 72,].
Currently, only PSGESVD and PDGESVD are provided.
  
Table 3.4: Driver routines for standard eigenvalue and singular value problems