.Modelica_LinearSystems2.Math.Matrices.LAPACK.dgees

Information

Lapack documentation:

   Purpose
   =======

   DGEES computes for an N-by-N real nonsymmetric matrix A, the
   eigenvalues, the real Schur form T, and, optionally, the matrix of
   Schur vectors Z.  This gives the Schur factorization A = Z*T*(Z**T).

   Optionally, it also orders the eigenvalues on the diagonal of the
   real Schur form so that selected eigenvalues are at the top left.
   The leading columns of Z then form an orthonormal basis for the
   invariant subspace corresponding to the selected eigenvalues.

   A matrix is in real Schur form if it is upper quasi-triangular with
   1-by-1 and 2-by-2 blocks. 2-by-2 blocks will be standardized in the
   form
           [  a  b  ]
           [  c  a  ]

   where b*c < 0. The eigenvalues of such a block are a +- sqrt(bc).

   Arguments
   =========

   JOBVS   (input) CHARACTER*1
           = 'N': Schur vectors are not computed;
           = 'V': Schur vectors are computed.

   SORT    (input) CHARACTER*1
           Specifies whether or not to order the eigenvalues on the
           diagonal of the Schur form.
           = 'N': Eigenvalues are not ordered;
           = 'S': Eigenvalues are ordered (see SELECT).

   SELECT  (external procedure) LOGICAL FUNCTION of two DOUBLE PRECISION arguments
           SELECT must be declared EXTERNAL in the calling subroutine.
           If SORT = 'S', SELECT is used to select eigenvalues to sort
           to the top left of the Schur form.
           If SORT = 'N', SELECT is not referenced.
           An eigenvalue WR(j)+sqrt(-1)*WI(j) is selected if
           SELECT(WR(j),WI(j)) is true; i.e., if either one of a complex
           conjugate pair of eigenvalues is selected, then both complex
           eigenvalues are selected.
           Note that a selected complex eigenvalue may no longer
           satisfy SELECT(WR(j),WI(j)) = .TRUE. after ordering, since
           ordering may change the value of complex eigenvalues
           (especially if the eigenvalue is ill-conditioned); in this
           case INFO is set to N+2 (see INFO below).

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

   A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
           On entry, the N-by-N matrix A.
           On exit, A has been overwritten by its real Schur form T.

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

   SDIM    (output) INTEGER
           If SORT = 'N', SDIM = 0.
           If SORT = 'S', SDIM = number of eigenvalues (after sorting)
                          for which SELECT is true. (Complex conjugate
                          pairs for which SELECT is true for either
                          eigenvalue count as 2.)

   WR      (output) DOUBLE PRECISION array, dimension (N)
   WI      (output) DOUBLE PRECISION array, dimension (N)
           WR and WI contain the real and imaginary parts,
           respectively, of the computed eigenvalues in the same order
           that they appear on the diagonal of the output Schur form T.
           Complex conjugate pairs of eigenvalues will appear
           consecutively with the eigenvalue having the positive
           imaginary part first.

   VS      (output) DOUBLE PRECISION array, dimension (LDVS,N)
           If JOBVS = 'V', VS contains the orthogonal matrix Z of Schur
           vectors.
           If JOBVS = 'N', VS is not referenced.

   LDVS    (input) INTEGER
           The leading dimension of the array VS.  LDVS >= 1; if
           JOBVS = 'V', LDVS >= N.

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

   LWORK   (input) INTEGER
           The dimension of the array WORK.  LWORK >= max(1,3*N).
           For good performance, LWORK must 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.

   BWORK   (workspace) LOGICAL array, dimension (N)
           Not referenced if SORT = 'N'.

   INFO    (output) INTEGER
           = 0: successful exit
           < 0: if INFO = -i, the i-th argument had an illegal value.
           > 0: if INFO = i, and i is
              <= N: the QR algorithm failed to compute all the
                    eigenvalues; elements 1:ILO-1 and i+1:N of WR and WI
                    contain those eigenvalues which have converged; if
                    JOBVS = 'V', VS contains the matrix which reduces A
                    to its partially converged Schur form.
              = N+1: the eigenvalues could not be reordered because some
                    eigenvalues were too close to separate (the problem
                    is very ill-conditioned);
              = N+2: after reordering, roundoff changed values of some
                    complex eigenvalues so that leading eigenvalues in
                    the Schur form no longer satisfy SELECT=.TRUE.  This
                    could also be caused by underflow due to scaling.

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

Interface

function dgees
  input Real A[:, size(A, 1)] "Square matrix";
  output Real T[size(A, 1), size(A, 2)] = A "Real Schur form with A = Z*T*Z'";
  output Real Z[size(A, 1), size(A, 1)] "orthogonal matrix Z of Schur vectors";
  output Real eval_real[size(A, 1)] "real part of the eigenvectors of A";
  output Real eval_imag[size(A, 1)] "imaginary part of the eigenvectors of A";
  output Integer info;
end dgees;