DGEESX 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;


 computes a reciprocal condition number for the average of the


 selected eigenvalues (RCONDE); and computes a reciprocal condition


 number for the right invariant subspace corresponding to the


 selected eigenvalues (RCONDV).  The leading columns of Z form an


 orthonormal basis for this invariant subspace.


 For further explanation of the reciprocal condition numbers RCONDE


 and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where


 these quantities are called s and sep respectively).


 A real 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 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


          are.  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 may be set to N+3 (see INFO below).

SENSE

          SENSE is CHARACTER*1


          Determines which reciprocal condition numbers are computed.


          = 'N': None are computed;


          = 'E': Computed for average of selected eigenvalues only;


          = 'V': Computed for selected right invariant subspace only;


          = 'B': Computed for both.


          If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.

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 is 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 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, and if


          JOBVS = 'V', LDVS >= N.

RCONDE

          RCONDE is DOUBLE PRECISION


          If SENSE = 'E' or 'B', RCONDE contains the reciprocal


          condition number for the average of the selected eigenvalues.


          Not referenced if SENSE = 'N' or 'V'.

RCONDV

          RCONDV is DOUBLE PRECISION


          If SENSE = 'V' or 'B', RCONDV contains the reciprocal


          condition number for the selected right invariant subspace.


          Not referenced if SENSE = 'N' or 'E'.

WORK

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


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

LWORK

          LWORK is INTEGER


          The dimension of the array WORK.  LWORK >= max(1,3*N).


          Also, if SENSE = 'E' or 'V' or 'B',


          LWORK >= N+2*SDIM*(N-SDIM), where SDIM is the number of


          selected eigenvalues computed by this routine.  Note that


          N+2*SDIM*(N-SDIM) <= N+N*N/2. Note also that an error is only


          returned if LWORK < max(1,3*N), but if SENSE = 'E' or 'V' or


          'B' this may not be large enough.


          For good performance, LWORK must generally be larger.


          If LWORK = -1, then a workspace query is assumed; the routine


          only calculates upper bounds on the optimal sizes of the


          arrays WORK and IWORK, returns these values as the first


          entries of the WORK and IWORK arrays, and no error messages


          related to LWORK or LIWORK are issued by XERBLA.

IWORK

          IWORK is INTEGER array, dimension (MAX(1,LIWORK))


          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.

LIWORK

          LIWORK is INTEGER


          The dimension of the array IWORK.


          LIWORK >= 1; if SENSE = 'V' or 'B', LIWORK >= SDIM*(N-SDIM).


          Note that SDIM*(N-SDIM) <= N*N/4. Note also that an error is


          only returned if LIWORK < 1, but if SENSE = 'V' or 'B' this


          may not be large enough.


          If LIWORK = -1, then a workspace query is assumed; the


          routine only calculates upper bounds on the optimal sizes of


          the arrays WORK and IWORK, returns these values as the first


          entries of the WORK and IWORK arrays, and no error messages


          related to LWORK or LIWORK are 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 transformation 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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November 2011