.Modelica.Math.Matrices.LAPACK.dgesdd

Information

Lapack documentation
    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) 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) INTEGER
            The number of rows of the input matrix A.  M >= 0.

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

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

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

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

    LWORK   (input) 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)+4*min(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   (workspace) INTEGER array, dimension (8*min(M,N))

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

    Further Details
    ===============

    Based on contributions by
       Ming Gu and Huan Ren, Computer Science Division, University of
       California at Berkeley, USA

Interface

pure function dgesdd
  extends Modelica.Icons.Function;
  input Real A[:, :];
  output Real sigma[min(size(A, 1), size(A, 2))];
  output Real U[size(A, 1), size(A, 1)] = zeros(size(A, 1), size(A, 1));
  output Real VT[size(A, 2), size(A, 2)] = zeros(size(A, 2), size(A, 2));
  output Integer info;
end dgesdd;