Provided by: liblapack-doc_3.12.0-3build1.1_all bug

NAME

       geesx - geesx: Schur form, expert

SYNOPSIS

   Functions
       subroutine cgeesx (jobvs, sort, select, sense, n, a, lda, sdim, w, vs, ldvs, rconde,
           rcondv, work, lwork, rwork, bwork, info)
            CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
           vectors for GE matrices
       subroutine dgeesx (jobvs, sort, select, sense, n, a, lda, sdim, wr, wi, vs, ldvs, rconde,
           rcondv, work, lwork, iwork, liwork, bwork, info)
            DGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
           vectors for GE matrices
       subroutine sgeesx (jobvs, sort, select, sense, n, a, lda, sdim, wr, wi, vs, ldvs, rconde,
           rcondv, work, lwork, iwork, liwork, bwork, info)
            SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
           vectors for GE matrices
       subroutine zgeesx (jobvs, sort, select, sense, n, a, lda, sdim, w, vs, ldvs, rconde,
           rcondv, work, lwork, rwork, bwork, info)
            ZGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
           vectors for GE matrices

Detailed Description

Function Documentation

   subroutine cgeesx (character jobvs, character sort, external select, character sense, integer
       n, complex, dimension( lda, * ) a, integer lda, integer sdim, complex, dimension( * ) w,
       complex, dimension( ldvs, * ) vs, integer ldvs, real rconde, real rcondv, complex,
       dimension( * ) work, integer lwork, real, dimension( * ) rwork, logical, dimension( * )
       bwork, integer info)
        CGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
       vectors for GE matrices

       Purpose:

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

            Optionally, it also orders the eigenvalues on the diagonal of the
            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 complex matrix is in Schur form if it is upper triangular.

       Parameters
           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 one COMPLEX argument
                     SELECT must be declared EXTERNAL in the calling subroutine.
                     If SORT = 'S', SELECT is used to select eigenvalues to order
                     to the top left of the Schur form.
                     If SORT = 'N', SELECT is not referenced.
                     An eigenvalue W(j) is selected if SELECT(W(j)) is true.

           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 COMPLEX array, dimension (LDA, N)
                     On entry, the N-by-N matrix A.
                     On exit, A is overwritten by its 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 for which
                                    SELECT is true.

           W

                     W is COMPLEX array, dimension (N)
                     W contains the computed eigenvalues, in the same order
                     that they appear on the diagonal of the output Schur form T.

           VS

                     VS is COMPLEX array, dimension (LDVS,N)
                     If JOBVS = 'V', VS contains the unitary 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 REAL
                     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 REAL
                     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 COMPLEX 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(1,2*N).
                     Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),
                     where SDIM is the number of selected eigenvalues computed by
                     this routine.  Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also
                     that an error is only returned if LWORK < max(1,2*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 bound on the optimal size of the
                     array WORK, returns this value as the first entry of the WORK
                     array, and no error message related to LWORK is issued by
                     XERBLA.

           RWORK

                     RWORK is REAL array, dimension (N)

           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 W
                              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.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine dgeesx (character jobvs, character sort, external select, character sense, integer
       n, double precision, dimension( lda, * ) a, integer lda, integer sdim, double precision,
       dimension( * ) wr, double precision, dimension( * ) wi, double precision, dimension( ldvs,
       * ) vs, integer ldvs, double precision rconde, double precision rcondv, double precision,
       dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork,
       logical, dimension( * ) bwork, integer info)
        DGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
       vectors for GE matrices

       Purpose:

            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).

       Parameters
           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
                     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.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine sgeesx (character jobvs, character sort, external select, character sense, integer
       n, real, dimension( lda, * ) a, integer lda, integer sdim, real, dimension( * ) wr, real,
       dimension( * ) wi, real, dimension( ldvs, * ) vs, integer ldvs, real rconde, real rcondv,
       real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork,
       logical, dimension( * ) bwork, integer info)
        SGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
       vectors for GE matrices

       Purpose:

            SGEESX 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).

       Parameters
           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 REAL 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 REAL 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 REAL array, dimension (N)

           WI

                     WI is REAL 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 REAL 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 REAL
                     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 REAL
                     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 REAL 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.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

   subroutine zgeesx (character jobvs, character sort, external select, character sense, integer
       n, complex*16, dimension( lda, * ) a, integer lda, integer sdim, complex*16, dimension( *
       ) w, complex*16, dimension( ldvs, * ) vs, integer ldvs, double precision rconde, double
       precision rcondv, complex*16, dimension( * ) work, integer lwork, double precision,
       dimension( * ) rwork, logical, dimension( * ) bwork, integer info)
        ZGEESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur
       vectors for GE matrices

       Purpose:

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

            Optionally, it also orders the eigenvalues on the diagonal of the
            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 complex matrix is in Schur form if it is upper triangular.

       Parameters
           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 one COMPLEX*16 argument
                     SELECT must be declared EXTERNAL in the calling subroutine.
                     If SORT = 'S', SELECT is used to select eigenvalues to order
                     to the top left of the Schur form.
                     If SORT = 'N', SELECT is not referenced.
                     An eigenvalue W(j) is selected if SELECT(W(j)) is true.

           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 COMPLEX*16 array, dimension (LDA, N)
                     On entry, the N-by-N matrix A.
                     On exit, A is overwritten by its 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 for which
                                    SELECT is true.

           W

                     W is COMPLEX*16 array, dimension (N)
                     W contains the computed eigenvalues, in the same order
                     that they appear on the diagonal of the output Schur form T.

           VS

                     VS is COMPLEX*16 array, dimension (LDVS,N)
                     If JOBVS = 'V', VS contains the unitary 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 COMPLEX*16 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(1,2*N).
                     Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),
                     where SDIM is the number of selected eigenvalues computed by
                     this routine.  Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also
                     that an error is only returned if LWORK < max(1,2*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 bound on the optimal size of the
                     array WORK, returns this value as the first entry of the WORK
                     array, and no error message related to LWORK is issued by
                     XERBLA.

           RWORK

                     RWORK is DOUBLE PRECISION array, dimension (N)

           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 W
                              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.

       Author
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

Author

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