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.


 The divide and conquer algorithm makes very mild assumptions about


 floating point arithmetic. It will work on machines with a guard


 digit in add/subtract, or on those binary machines without guard


 digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or


 Cray-2. It could conceivably fail on hexadecimal or decimal machines


 without guard digits, but we know of none.

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:  successful exit.


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


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

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

June 2016

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