.Modelica_LinearSystems2.Math.Matrices.LAPACK.dgesvd

Information

Lapack documentation:

   Purpose
   =======

   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.

   Arguments
   =========

   JOBU    (input) 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   (input) 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       (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 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     (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)
           (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     (input) INTEGER
           The leading dimension of the array U.  LDU >= 1; if
           JOBU = 'S' or 'A', LDU >= M.
   VT      (output) 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    (input) INTEGER
           The leading dimension of the array VT.  LDVT >= 1; if
           JOBVT = 'A', LDVT >= N; if JOBVT = 'S', LDVT >= min(M,N).
   WORK    (workspace/output) DOUBLE PRECISION array, dimension (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   (input) INTEGER
           The dimension of the array WORK. LWORK >= 1.
           LWORK >= MAX(3*MIN(M,N)+MAX(M,N),5*MIN(M,N)-4).
           For good performance, LWORK should generally be larger.
   INFO    (output) 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.

   =====================================================================  

Interface

function dgesvd
  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 dgesvd;