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


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


 vectors.  If singular vectors are desired, it uses a


 divide-and-conquer algorithm.


 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.


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

JOBZ

          JOBZ is CHARACTER*1


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


          = 'A':  all M columns of U and all N rows of V**T are


                  returned in the arrays U and VT;


          = 'S':  the first min(M,N) columns of U and the first


                  min(M,N) rows of V**T are returned in the arrays U


                  and VT;


          = 'O':  If M >= N, the first N columns of U are overwritten


                  on the array A and all rows of V**T are returned in


                  the array VT;


                  otherwise, all columns of U are returned in the


                  array U and the first M rows of V**T are overwritten


                  in the array A;


          = 'N':  no columns of U or rows of V**T are computed.

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,


          if JOBZ = 'O',  A is overwritten with the first N columns


                          of U (the left singular vectors, stored


                          columnwise) if M >= N;


                          A is overwritten with the first M rows


                          of V**T (the right singular vectors, stored


                          rowwise) otherwise.


          if JOBZ .ne. 'O', the contents of A are destroyed.

LDA

          LDA is INTEGER


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

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)


          UCOL = M if JOBZ = 'A' or JOBZ = 'O' and M < N;


          UCOL = min(M,N) if JOBZ = 'S'.


          If JOBZ = 'A' or JOBZ = 'O' and M < N, U contains the M-by-M


          orthogonal matrix U;


          if JOBZ = 'S', U contains the first min(M,N) columns of U


          (the left singular vectors, stored columnwise);


          if JOBZ = 'O' and M >= N, or JOBZ = 'N', U is not referenced.

LDU

          LDU is INTEGER


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


          JOBZ = 'S' or 'A' or JOBZ = 'O' and M < N, LDU >= M.

VT

          VT is REAL array, dimension (LDVT,N)


          If JOBZ = 'A' or JOBZ = 'O' and M >= N, VT contains the


          N-by-N orthogonal matrix V**T;


          if JOBZ = 'S', VT contains the first min(M,N) rows of


          V**T (the right singular vectors, stored rowwise);


          if JOBZ = 'O' and M < N, or JOBZ = 'N', VT is not referenced.

LDVT

          LDVT is INTEGER


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


          if JOBZ = 'A' or JOBZ = 'O' and M >= N, LDVT >= N;


          if JOBZ = 'S', LDVT >= min(M,N).

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 >= 1.


          If LWORK = -1, a workspace query is assumed.  The optimal


          size for the WORK array is calculated and stored in WORK(1),


          and no other work except argument checking is performed.


          Let mx = max(M,N) and mn = min(M,N).


          If JOBZ = 'N', LWORK >= 3*mn + max( mx, 7*mn ).


          If JOBZ = 'O', LWORK >= 3*mn + max( mx, 5*mn*mn + 4*mn ).


          If JOBZ = 'S', LWORK >= 4*mn*mn + 7*mn.


          If JOBZ = 'A', LWORK >= 4*mn*mn + 6*mn + mx.


          These are not tight minimums in all cases; see comments inside code.


          For good performance, LWORK should generally be larger;


          a query is recommended.

IWORK

          IWORK is INTEGER array, dimension (8*min(M,N))

INFO

          INFO is INTEGER


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


          = -4:  if A had a NAN entry.


          >  0:  SBDSDC did not converge, updating process failed.


          =  0:  successful exit.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA