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).

JOBVS

          JOBVS is CHARACTER*1


          = 'N': Schur vectors are not computed;


          = 'V': Schur vectors are computed.

SORT

          SORT is 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

          SELECT is a 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

          N is INTEGER


          The order of the matrix A. N >= 0.

A

          A is 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

          LDA is INTEGER


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

SDIM

          SDIM is 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

          WR is DOUBLE PRECISION array, dimension (N)

WI

          WI is 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

          VS is 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

          LDVS is INTEGER


          The leading dimension of the array VS.  LDVS >= 1; if


          JOBVS = 'V', LDVS >= N.

WORK

          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))


          On exit, if INFO = 0, WORK(1) contains the optimal LWORK.

LWORK

          LWORK is 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

          BWORK is LOGICAL array, dimension (N)


          Not referenced if SORT = 'N'.

INFO

          INFO is 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.

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NAG Ltd.

December 2016