DGESVD 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.


 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:


          = 'A':  all M columns of U are returned in array U:


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


                  vectors) are returned in the array U;


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


                  vectors) are overwritten on the array A;


          = '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:


          = 'A':  all N rows of V**T are returned in the array VT;


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


                  vectors) are returned in the array VT;


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


                  vectors) are overwritten on the array A;


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


                  computed.


          JOBVT and JOBU cannot both be 'O'.

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 DOUBLE PRECISION array, dimension (LDA,N)


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


          On exit,


          if JOBU = 'O',  A is overwritten with the first min(m,n)


                          columns of U (the left singular vectors,


                          stored columnwise);


          if JOBVT = 'O', A is overwritten with the first min(m,n)


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


                          stored rowwise);


          if JOBU .ne. 'O' and JOBVT .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 DOUBLE PRECISION array, dimension (min(M,N))


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

U

          U is DOUBLE PRECISION array, dimension (LDU,UCOL)


          (LDU,M) if JOBU = 'A' or (LDU,min(M,N)) if JOBU = 'S'.


          If JOBU = 'A', U contains the M-by-M orthogonal matrix U;


          if JOBU = 'S', U contains the first min(m,n) columns of U


          (the left singular vectors, stored columnwise);


          if JOBU = 'N' or 'O', U is not referenced.

LDU

          LDU is INTEGER


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


          JOBU = 'S' or 'A', LDU >= M.

VT

          VT is DOUBLE PRECISION array, dimension (LDVT,N)


          If JOBVT = 'A', VT contains the N-by-N orthogonal matrix


          V**T;


          if JOBVT = 'S', VT contains the first min(m,n) rows of


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


          if JOBVT = 'N' or 'O', VT is not referenced.

LDVT

          LDVT is INTEGER


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


          JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))


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


          if INFO > 0, WORK(2:MIN(M,N)) contains the unconverged


          superdiagonal elements of an upper bidiagonal matrix B


          whose diagonal is in S (not necessarily sorted). B


          satisfies A = U * B * VT, so it has the same singular values


          as A, and singular vectors related by U and VT.

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.


          LWORK >= MAX(1,5*MIN(M,N)) for the paths (see comments inside code):


             - PATH 1  (M much larger than N, JOBU='N') 


             - PATH 1t (N much larger than M, JOBVT='N')


          LWORK >= MAX(1,3*MIN(M,N)+MAX(M,N),5*MIN(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.

INFO

          INFO is INTEGER


          = 0:  successful exit.


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


          > 0:  if DBDSQR did not converge, INFO specifies how many


                superdiagonals of an intermediate bidiagonal form B


                did not converge to zero. See the description of WORK


                above for details.

Univ. of Tennessee

Univ. of California Berkeley

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NAG Ltd.

April 2012