.Modelica_LinearSystems2.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 VT;
           = '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 (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),6*min(M,N)).
           If JOBZ = 'O',
             LWORK >= 3*min(M,N)*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)*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 < 0 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

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;