SGESVDX computes the singular value decomposition (SVD) of a real


  M-by-N matrix A, optionally computing the left and/or right singular


  vectors. The SVD is written


      A = U * SIGMA * transpose(V)


  where SIGMA is an M-by-N matrix which is zero except for its


  min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and


  V is an N-by-N orthogonal matrix.  The diagonal elements of SIGMA


  are the singular values of A; they are real and non-negative, and


  are returned in descending order.  The first min(m,n) columns of


  U and V are the left and right singular vectors of A.


  SGESVDX uses an eigenvalue problem for obtaining the SVD, which


  allows for the computation of a subset of singular values and


  vectors. See SBDSVDX for details.


  Note that the routine returns V**T, not V.

JOBU

          JOBU is CHARACTER*1


          Specifies options for computing all or part of the matrix U:


          = 'V':  the first min(m,n) columns of U (the left singular


                  vectors) or as specified by RANGE are returned in


                  the array U;


          = 'N':  no columns of U (no left singular vectors) are


                  computed.

JOBVT

          JOBVT is CHARACTER*1


           Specifies options for computing all or part of the matrix


           V**T:


           = 'V':  the first min(m,n) rows of V**T (the right singular


                   vectors) or as specified by RANGE are returned in


                   the array VT;


           = 'N':  no rows of V**T (no right singular vectors) are


                   computed.

RANGE

          RANGE is CHARACTER*1


          = 'A': all singular values will be found.


          = 'V': all singular values in the half-open interval (VL,VU]


                 will be found.


          = 'I': the IL-th through IU-th singular values will be found.

M

          M is INTEGER


          The number of rows of the input matrix A.  M >= 0.

N

          N is INTEGER


          The number of columns of the input matrix A.  N >= 0.

A

          A is REAL array, dimension (LDA,N)


          On entry, the M-by-N matrix A.


          On exit, the contents of A are destroyed.

LDA

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,M).

VL

          VL is REAL


          If RANGE='V', the lower bound of the interval to


          be searched for singular values. VU > VL.


          Not referenced if RANGE = 'A' or 'I'.

VU

          VU is REAL


          If RANGE='V', the upper bound of the interval to


          be searched for singular values. VU > VL.


          Not referenced if RANGE = 'A' or 'I'.

IL

          IL is INTEGER


          If RANGE='I', the index of the


          smallest singular value to be returned.


          1 <= IL <= IU <= min(M,N), if min(M,N) > 0.


          Not referenced if RANGE = 'A' or 'V'.

IU

          IU is INTEGER


          If RANGE='I', the index of the


          largest singular value to be returned.


          1 <= IL <= IU <= min(M,N), if min(M,N) > 0.


          Not referenced if RANGE = 'A' or 'V'.

NS

          NS is INTEGER


          The total number of singular values found,


          0 <= NS <= min(M,N).


          If RANGE = 'A', NS = min(M,N); if RANGE = 'I', NS = IU-IL+1.

S

          S is REAL array, dimension (min(M,N))


          The singular values of A, sorted so that S(i) >= S(i+1).

U

          U is REAL array, dimension (LDU,UCOL)


          If JOBU = 'V', U contains columns of U (the left singular


          vectors, stored columnwise) as specified by RANGE; if


          JOBU = 'N', U is not referenced.


          Note: The user must ensure that UCOL >= NS; if RANGE = 'V',


          the exact value of NS is not known in advance and an upper


          bound must be used.

LDU

          LDU is INTEGER


          The leading dimension of the array U.  LDU >= 1; if


          JOBU = 'V', LDU >= M.

VT

          VT is REAL array, dimension (LDVT,N)


          If JOBVT = 'V', VT contains the rows of V**T (the right singular


          vectors, stored rowwise) as specified by RANGE; if JOBVT = 'N',


          VT is not referenced.


          Note: The user must ensure that LDVT >= NS; if RANGE = 'V',


          the exact value of NS is not known in advance and an upper


          bound must be used.

LDVT

          LDVT is INTEGER


          The leading dimension of the array VT.  LDVT >= 1; if


          JOBVT = 'V', LDVT >= NS (see above).

WORK

          WORK is REAL array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) returns the optimal LWORK;

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.


          LWORK >= MAX(1,MIN(M,N)*(MIN(M,N)+4)) for the paths (see


          comments inside the code):


             - PATH 1  (M much larger than N)


             - PATH 1t (N much larger than M)


          LWORK >= MAX(1,MIN(M,N)*2+MAX(M,N)) for the other paths.


          For good performance, LWORK should generally be larger.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates the optimal size of the WORK array, returns


          this value as the first entry of the WORK array, and no error


          message related to LWORK is issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (12*MIN(M,N))


          If INFO = 0, the first NS elements of IWORK are zero. If INFO > 0,


          then IWORK contains the indices of the eigenvectors that failed


          to converge in SBDSVDX/SSTEVX.

INFO

     INFO is INTEGER


           = 0:  successful exit


           < 0:  if INFO = -i, the i-th argument had an illegal value


           > 0:  if INFO = i, then i eigenvectors failed to converge


                 in SBDSVDX/SSTEVX.


                 if INFO = N*2 + 1, an internal error occurred in


                 SBDSVDX

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