DGESDD(3)      LAPACK routine of NEC Numeric Library Collection      DGESDD(3)



NAME
       DGESDD

SYNOPSIS
       SUBROUTINE DGESDD (JOBZ, M, N, A, LDA, S, U, LDU, VT, LDVT, WORK,
           LWORK, IWORK, INFO)



PURPOSE
            DGESDD 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.




ARGUMENTS
           JOBZ      (input)
                     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         (input)
                     M is INTEGER
                     The number of rows of the input matrix A.  M >= 0.

           N         (input)
                     N is INTEGER
                     The number of columns of the input matrix A.  N >= 0.

           A         (input/output)
                     A is DOUBLE PRECISION 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       (input)
                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,M).

           S         (output)
                     S is DOUBLE PRECISION array, dimension (min(M,N))
                     The singular values of A, sorted so that S(i) >= S(i+1).

           U         (output)
                     U is DOUBLE PRECISION 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       (input)
                     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        (output)
                     VT is DOUBLE PRECISION 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      (input)
                     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      (output)
                     WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK;

           LWORK     (input)
                     LWORK is INTEGER
                     The dimension of the array WORK. LWORK >= 1.
                     If JOBZ = 'N',
                       LWORK >= 3*min(M,N) + max(max(M,N),7*min(M,N)).
                     If JOBZ = 'O',
                       LWORK >= 3*min(M,N) +
                                max(max(M,N),5*min(M,N)*min(M,N)+4*min(M,N)).
                     If JOBZ = 'S' or 'A'
                       LWORK >= 3*min(M,N) +
                                max(max(M,N),4*min(M,N)*min(M,N)+3*min(M,N)+max(M,N)).
                     For good performance, LWORK should generally be larger.
                     If LWORK = -1 but other input arguments are legal, WORK(1)
                     returns the optimal LWORK.

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

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit.
                     < 0:  if INFO = -i, the i-th argument had an illegal value.
                     > 0:  DBDSDC did not converge, updating process failed.



LAPACK routine                  31 October 2017                      DGESDD(3)