dggesx.f

Go to the documentation of this file.

*> \brief <b> DGGESX computes the eigenvalues, the Schur form, and, optionally, the matrix of Schur vectors for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DGGESX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dggesx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dggesx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dggesx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGESX( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA,
*                          B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,
*                          VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK,
*                          LIWORK, BWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBVSL, JOBVSR, SENSE, SORT
*       INTEGER            INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
*      $                   SDIM
*       ..
*       .. Array Arguments ..
*       LOGICAL            BWORK( * )
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
*      $                   B( LDB, * ), BETA( * ), RCONDE( 2 ),
*      $                   RCONDV( 2 ), VSL( LDVSL, * ), VSR( LDVSR, * ),
*      $                   WORK( * )
*       ..
*       .. Function Arguments ..
*       LOGICAL            SELCTG
*       EXTERNAL           SELCTG
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGGESX computes for a pair of N-by-N real nonsymmetric matrices
*> (A,B), the generalized eigenvalues, the real Schur form (S,T), and,
*> optionally, the left and/or right matrices of Schur vectors (VSL and
*> VSR).  This gives the generalized Schur factorization
*>
*>      (A,B) = ( (VSL) S (VSR)**T, (VSL) T (VSR)**T )
*>
*> Optionally, it also orders the eigenvalues so that a selected cluster
*> of eigenvalues appears in the leading diagonal blocks of the upper
*> quasi-triangular matrix S and the upper triangular matrix T; computes
*> a reciprocal condition number for the average of the selected
*> eigenvalues (RCONDE); and computes a reciprocal condition number for
*> the right and left deflating subspaces corresponding to the selected
*> eigenvalues (RCONDV). The leading columns of VSL and VSR then form
*> an orthonormal basis for the corresponding left and right eigenspaces
*> (deflating subspaces).
*>
*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar w
*> or a ratio alpha/beta = w, such that  A - w*B is singular.  It is
*> usually represented as the pair (alpha,beta), as there is a
*> reasonable interpretation for beta=0 or for both being zero.
*>
*> A pair of matrices (S,T) is in generalized real Schur form if T is
*> upper triangular with non-negative diagonal and S is block upper
*> triangular with 1-by-1 and 2-by-2 blocks.  1-by-1 blocks correspond
*> to real generalized eigenvalues, while 2-by-2 blocks of S will be
*> "standardized" by making the corresponding elements of T have the
*> form:
*>         [  a  0  ]
*>         [  0  b  ]
*>
*> and the pair of corresponding 2-by-2 blocks in S and T will have a
*> complex conjugate pair of generalized eigenvalues.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBVSL
*> \verbatim
*>          JOBVSL is CHARACTER*1
*>          = 'N':  do not compute the left Schur vectors;
*>          = 'V':  compute the left Schur vectors.
*> \endverbatim
*>
*> \param[in] JOBVSR
*> \verbatim
*>          JOBVSR is CHARACTER*1
*>          = 'N':  do not compute the right Schur vectors;
*>          = 'V':  compute the right Schur vectors.
*> \endverbatim
*>
*> \param[in] SORT
*> \verbatim
*>          SORT is CHARACTER*1
*>          Specifies whether or not to order the eigenvalues on the
*>          diagonal of the generalized Schur form.
*>          = 'N':  Eigenvalues are not ordered;
*>          = 'S':  Eigenvalues are ordered (see SELCTG).
*> \endverbatim
*>
*> \param[in] SELCTG
*> \verbatim
*>          SELCTG is a LOGICAL FUNCTION of three DOUBLE PRECISION arguments
*>          SELCTG must be declared EXTERNAL in the calling subroutine.
*>          If SORT = 'N', SELCTG is not referenced.
*>          If SORT = 'S', SELCTG is used to select eigenvalues to sort
*>          to the top left of the Schur form.
*>          An eigenvalue (ALPHAR(j)+ALPHAI(j))/BETA(j) is selected if
*>          SELCTG(ALPHAR(j),ALPHAI(j),BETA(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
*>          SELCTG(ALPHAR(j),ALPHAI(j),BETA(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+3.
*> \endverbatim
*>
*> \param[in] SENSE
*> \verbatim
*>          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 deflating subspaces only;
*>          = 'B':  Computed for both.
*>          If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A, B, VSL, and VSR.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the first of the pair of matrices.
*>          On exit, A has been overwritten by its generalized Schur
*>          form S.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is DOUBLE PRECISION array, dimension (LDB, N)
*>          On entry, the second of the pair of matrices.
*>          On exit, B has been overwritten by its generalized Schur
*>          form T.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] SDIM
*> \verbatim
*>          SDIM is INTEGER
*>          If SORT = 'N', SDIM = 0.
*>          If SORT = 'S', SDIM = number of eigenvalues (after sorting)
*>          for which SELCTG is true.  (Complex conjugate pairs for which
*>          SELCTG is true for either eigenvalue count as 2.)
*> \endverbatim
*>
*> \param[out] ALPHAR
*> \verbatim
*>          ALPHAR is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] ALPHAI
*> \verbatim
*>          ALPHAI is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is DOUBLE PRECISION array, dimension (N)
*>          On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
*>          be the generalized eigenvalues.  ALPHAR(j) + ALPHAI(j)*i
*>          and BETA(j),j=1,...,N  are the diagonals of the complex Schur
*>          form (S,T) that would result if the 2-by-2 diagonal blocks of
*>          the real Schur form of (A,B) were further reduced to
*>          triangular form using 2-by-2 complex unitary transformations.
*>          If ALPHAI(j) is zero, then the j-th eigenvalue is real; if
*>          positive, then the j-th and (j+1)-st eigenvalues are a
*>          complex conjugate pair, with ALPHAI(j+1) negative.
*>
*>          Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(j)/BETA(j)
*>          may easily over- or underflow, and BETA(j) may even be zero.
*>          Thus, the user should avoid naively computing the ratio.
*>          However, ALPHAR and ALPHAI will be always less than and
*>          usually comparable with norm(A) in magnitude, and BETA always
*>          less than and usually comparable with norm(B).
*> \endverbatim
*>
*> \param[out] VSL
*> \verbatim
*>          VSL is DOUBLE PRECISION array, dimension (LDVSL,N)
*>          If JOBVSL = 'V', VSL will contain the left Schur vectors.
*>          Not referenced if JOBVSL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSL
*> \verbatim
*>          LDVSL is INTEGER
*>          The leading dimension of the matrix VSL. LDVSL >=1, and
*>          if JOBVSL = 'V', LDVSL >= N.
*> \endverbatim
*>
*> \param[out] VSR
*> \verbatim
*>          VSR is DOUBLE PRECISION array, dimension (LDVSR,N)
*>          If JOBVSR = 'V', VSR will contain the right Schur vectors.
*>          Not referenced if JOBVSR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVSR
*> \verbatim
*>          LDVSR is INTEGER
*>          The leading dimension of the matrix VSR. LDVSR >= 1, and
*>          if JOBVSR = 'V', LDVSR >= N.
*> \endverbatim
*>
*> \param[out] RCONDE
*> \verbatim
*>          RCONDE is DOUBLE PRECISION array, dimension ( 2 )
*>          If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the
*>          reciprocal condition numbers for the average of the selected
*>          eigenvalues.
*>          Not referenced if SENSE = 'N' or 'V'.
*> \endverbatim
*>
*> \param[out] RCONDV
*> \verbatim
*>          RCONDV is DOUBLE PRECISION array, dimension ( 2 )
*>          If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the
*>          reciprocal condition numbers for the selected deflating
*>          subspaces.
*>          Not referenced if SENSE = 'N' or 'E'.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',
*>          LWORK >= max( 8*N, 6*N+16, 2*SDIM*(N-SDIM) ), else
*>          LWORK >= max( 8*N, 6*N+16 ).
*>          Note that 2*SDIM*(N-SDIM) <= N*N/2.
*>          Note also that an error is only returned if
*>          LWORK < max( 8*N, 6*N+16), but if SENSE = 'E' or 'V' or 'B'
*>          this may not be large enough.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the bound on the optimal size of the WORK
*>          array and the minimum size of the IWORK array, returns these
*>          values as the first entries of the WORK and IWORK arrays, and
*>          no error message related to LWORK or LIWORK is issued by
*>          XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise
*>          LIWORK >= N+6.
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the bound on the optimal size of the
*>          WORK array and the minimum size of the IWORK array, returns
*>          these values as the first entries of the WORK and IWORK
*>          arrays, and no error message related to LWORK or LIWORK is
*>          issued by XERBLA.
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*>          BWORK is LOGICAL array, dimension (N)
*>          Not referenced if SORT = 'N'.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value.
*>          = 1,...,N:
*>                The QZ iteration failed.  (A,B) are not in Schur
*>                form, but ALPHAR(j), ALPHAI(j), and BETA(j) should
*>                be correct for j=INFO+1,...,N.
*>          > N:  =N+1: other than QZ iteration failed in DHGEQZ
*>                =N+2: after reordering, roundoff changed values of
*>                      some complex eigenvalues so that leading
*>                      eigenvalues in the Generalized Schur form no
*>                      longer satisfy SELCTG=.TRUE.  This could also
*>                      be caused due to scaling.
*>                =N+3: reordering failed in DTGSEN.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleGEeigen
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  An approximate (asymptotic) bound on the average absolute error of
*>  the selected eigenvalues is
*>
*>       EPS * norm((A, B)) / RCONDE( 1 ).
*>
*>  An approximate (asymptotic) bound on the maximum angular error in
*>  the computed deflating subspaces is
*>
*>       EPS * norm((A, B)) / RCONDV( 2 ).
*>
*>  See LAPACK User's Guide, section 4.11 for more information.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE dggesx( JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA,
     $                   B, LDB, SDIM, ALPHAR, ALPHAI, BETA, VSL, LDVSL,
     $                   VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, IWORK,
     $                   LIWORK, BWORK, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBVSL, JOBVSR, SENSE, SORT
      INTEGER            INFO, LDA, LDB, LDVSL, LDVSR, LIWORK, LWORK, N,
     $                   SDIM
*     ..
*     .. Array Arguments ..
      LOGICAL            BWORK( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
     $                   b( ldb, * ), beta( * ), rconde( 2 ),
     $                   rcondv( 2 ), vsl( ldvsl, * ), vsr( ldvsr, * ),
     $                   work( * )
*     ..
*     .. Function Arguments ..
      LOGICAL            SELCTG
      EXTERNAL           SELCTG
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0d+0, one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            CURSL, ILASCL, ILBSCL, ILVSL, ILVSR, LASTSL,
     $                   LQUERY, LST2SL, WANTSB, WANTSE, WANTSN, WANTST,
     $                   WANTSV
      INTEGER            I, ICOLS, IERR, IHI, IJOB, IJOBVL, IJOBVR,
     $                   ILEFT, ILO, IP, IRIGHT, IROWS, ITAU, IWRK,
     $                   LIWMIN, LWRK, MAXWRK, MINWRK
      DOUBLE PRECISION   ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS, PL,
     $                   PR, SAFMAX, SAFMIN, SMLNUM
*     ..
*     .. Local Arrays ..
      DOUBLE PRECISION   DIF( 2 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           dgeqrf, dggbak, dggbal, dgghrd, dhgeqz, dlabad,
     $                   dlacpy, dlascl, dlaset, dorgqr, dormqr, dtgsen,
     $                   xerbla
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ILAENV
      DOUBLE PRECISION   DLAMCH, DLANGE
      EXTERNAL           lsame, ilaenv, dlamch, dlange
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, sqrt
*     ..
*     .. Executable Statements ..
*
*     Decode the input arguments
*
      IF( lsame( jobvsl, 'N' ) ) THEN
         ijobvl = 1
         ilvsl = .false.
      ELSE IF( lsame( jobvsl, 'V' ) ) THEN
         ijobvl = 2
         ilvsl = .true.
      ELSE
         ijobvl = -1
         ilvsl = .false.
      END IF
*
      IF( lsame( jobvsr, 'N' ) ) THEN
         ijobvr = 1
         ilvsr = .false.
      ELSE IF( lsame( jobvsr, 'V' ) ) THEN
         ijobvr = 2
         ilvsr = .true.
      ELSE
         ijobvr = -1
         ilvsr = .false.
      END IF
*
      wantst = lsame( sort, 'S' )
      wantsn = lsame( sense, 'N' )
      wantse = lsame( sense, 'E' )
      wantsv = lsame( sense, 'V' )
      wantsb = lsame( sense, 'B' )
      lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
      IF( wantsn ) THEN
         ijob = 0
      ELSE IF( wantse ) THEN
         ijob = 1
      ELSE IF( wantsv ) THEN
         ijob = 2
      ELSE IF( wantsb ) THEN
         ijob = 4
      END IF
*
*     Test the input arguments
*
      info = 0
      IF( ijobvl.LE.0 ) THEN
         info = -1
      ELSE IF( ijobvr.LE.0 ) THEN
         info = -2
      ELSE IF( ( .NOT.wantst ) .AND. ( .NOT.lsame( sort, 'N' ) ) ) THEN
         info = -3
      ELSE IF( .NOT.( wantsn .OR. wantse .OR. wantsv .OR. wantsb ) .OR.
     $         ( .NOT.wantst .AND. .NOT.wantsn ) ) THEN
         info = -5
      ELSE IF( n.LT.0 ) THEN
         info = -6
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -8
      ELSE IF( ldb.LT.max( 1, n ) ) THEN
         info = -10
      ELSE IF( ldvsl.LT.1 .OR. ( ilvsl .AND. ldvsl.LT.n ) ) THEN
         info = -16
      ELSE IF( ldvsr.LT.1 .OR. ( ilvsr .AND. ldvsr.LT.n ) ) THEN
         info = -18
      END IF
*
*     Compute workspace
*      (Note: Comments in the code beginning "Workspace:" describe the
*       minimal amount of workspace needed at that point in the code,
*       as well as the preferred amount for good performance.
*       NB refers to the optimal block size for the immediately
*       following subroutine, as returned by ILAENV.)
*
      IF( info.EQ.0 ) THEN
         IF( n.GT.0) THEN
            minwrk = max( 8*n, 6*n + 16 )
            maxwrk = minwrk - n +
     $               n*ilaenv( 1, 'DGEQRF', ' ', N, 1, N, 0 )
            MAXWRK = MAX( MAXWRK, MINWRK - N +
     $               N*ILAENV( 1, 'dormqr', ' ', N, 1, N, -1 ) )
            IF( ILVSL ) THEN
               MAXWRK = MAX( MAXWRK, MINWRK - N +
     $                  N*ILAENV( 1, 'dorgqr', ' ', N, 1, N, -1 ) )
            END IF
            LWRK = MAXWRK
.GE.            IF( IJOB1 )
     $         LWRK = MAX( LWRK, N*N/2 )
         ELSE
            MINWRK = 1
            MAXWRK = 1
            LWRK   = 1
         END IF
         WORK( 1 ) = LWRK
.OR..EQ.         IF( WANTSN  N0 ) THEN
            LIWMIN = 1
         ELSE
            LIWMIN = N + 6
         END IF
         IWORK( 1 ) = LIWMIN
*
.LT..AND..NOT.         IF( LWORKMINWRK  LQUERY ) THEN
            INFO = -22
.LT..AND..NOT.         ELSE IF( LIWORKLIWMIN   LQUERY ) THEN
            INFO = -24
         END IF
      END IF
*
.NE.      IF( INFO0 ) THEN
         CALL XERBLA( 'dggesx', -INFO )
         RETURN
      ELSE IF (LQUERY) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
.EQ.      IF( N0 ) THEN
         SDIM = 0
         RETURN
      END IF
*
*     Get machine constants
*
      EPS = DLAMCH( 'p' )
      SAFMIN = DLAMCH( 's' )
      SAFMAX = ONE / SAFMIN
      CALL DLABAD( SAFMIN, SAFMAX )
      SMLNUM = SQRT( SAFMIN ) / EPS
      BIGNUM = ONE / SMLNUM
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      ANRM = DLANGE( 'm', N, N, A, LDA, WORK )
      ILASCL = .FALSE.
.GT..AND..LT.      IF( ANRMZERO  ANRMSMLNUM ) THEN
         ANRMTO = SMLNUM
         ILASCL = .TRUE.
.GT.      ELSE IF( ANRMBIGNUM ) THEN
         ANRMTO = BIGNUM
         ILASCL = .TRUE.
      END IF
      IF( ILASCL )
     $   CALL DLASCL( 'g', 0, 0, ANRM, ANRMTO, N, N, A, LDA, IERR )
*
*     Scale B if max element outside range [SMLNUM,BIGNUM]
*
      BNRM = DLANGE( 'm', N, N, B, LDB, WORK )
      ILBSCL = .FALSE.
.GT..AND..LT.      IF( BNRMZERO  BNRMSMLNUM ) THEN
         BNRMTO = SMLNUM
         ILBSCL = .TRUE.
.GT.      ELSE IF( BNRMBIGNUM ) THEN
         BNRMTO = BIGNUM
         ILBSCL = .TRUE.
      END IF
      IF( ILBSCL )
     $   CALL DLASCL( 'g', 0, 0, BNRM, BNRMTO, N, N, B, LDB, IERR )
*
*     Permute the matrix to make it more nearly triangular
*     (Workspace: need 6*N + 2*N for permutation parameters)
*
      ILEFT = 1
      IRIGHT = N + 1
      IWRK = IRIGHT + N
      CALL DGGBAL( 'p', N, A, LDA, B, LDB, ILO, IHI, WORK( ILEFT ),
     $             WORK( IRIGHT ), WORK( IWRK ), IERR )
*
*     Reduce B to triangular form (QR decomposition of B)
*     (Workspace: need N, prefer N*NB)
*
      IROWS = IHI + 1 - ILO
      ICOLS = N + 1 - ILO
      ITAU = IWRK
      IWRK = ITAU + IROWS
      CALL DGEQRF( IROWS, ICOLS, B( ILO, ILO ), LDB, WORK( ITAU ),
     $             WORK( IWRK ), LWORK+1-IWRK, IERR )
*
*     Apply the orthogonal transformation to matrix A
*     (Workspace: need N, prefer N*NB)
*
      CALL DORMQR( 'l', 't', IROWS, ICOLS, IROWS, B( ILO, ILO ), LDB,
     $             WORK( ITAU ), A( ILO, ILO ), LDA, WORK( IWRK ),
     $             LWORK+1-IWRK, IERR )
*
*     Initialize VSL
*     (Workspace: need N, prefer N*NB)
*
      IF( ILVSL ) THEN
         CALL DLASET( 'full', N, N, ZERO, ONE, VSL, LDVSL )
.GT.         IF( IROWS1 ) THEN
            CALL DLACPY( 'l', IROWS-1, IROWS-1, B( ILO+1, ILO ), LDB,
     $                   VSL( ILO+1, ILO ), LDVSL )
         END IF
         CALL DORGQR( IROWS, IROWS, IROWS, VSL( ILO, ILO ), LDVSL,
     $                WORK( ITAU ), WORK( IWRK ), LWORK+1-IWRK, IERR )
      END IF
*
*     Initialize VSR
*
      IF( ILVSR )
     $   CALL DLASET( 'full', N, N, ZERO, ONE, VSR, LDVSR )
*
*     Reduce to generalized Hessenberg form
*     (Workspace: none needed)
*
      CALL DGGHRD( JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB, VSL,
     $             LDVSL, VSR, LDVSR, IERR )
*
      SDIM = 0
*
*     Perform QZ algorithm, computing Schur vectors if desired
*     (Workspace: need N)
*
      IWRK = ITAU
      CALL DHGEQZ( 's', JOBVSL, JOBVSR, N, ILO, IHI, A, LDA, B, LDB,
     $             ALPHAR, ALPHAI, BETA, VSL, LDVSL, VSR, LDVSR,
     $             WORK( IWRK ), LWORK+1-IWRK, IERR )
.NE.      IF( IERR0 ) THEN
.GT..AND..LE.         IF( IERR0  IERRN ) THEN
            INFO = IERR
.GT..AND..LE.         ELSE IF( IERRN  IERR2*N ) THEN
            INFO = IERR - N
         ELSE
            INFO = N + 1
         END IF
         GO TO 60
      END IF
*
*     Sort eigenvalues ALPHA/BETA and compute the reciprocal of
*     condition number(s)
*     (Workspace: If IJOB >= 1, need MAX( 8*(N+1), 2*SDIM*(N-SDIM) )
*                 otherwise, need 8*(N+1) )
*
      IF( WANTST ) THEN
*
*        Undo scaling on eigenvalues before SELCTGing
*
         IF( ILASCL ) THEN
            CALL DLASCL( 'g', 0, 0, ANRMTO, ANRM, N, 1, ALPHAR, N,
     $                   IERR )
            CALL DLASCL( 'g', 0, 0, ANRMTO, ANRM, N, 1, ALPHAI, N,
     $                   IERR )
         END IF
         IF( ILBSCL )
     $      CALL DLASCL( 'g', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
*
*        Select eigenvalues
*
         DO 10 i = 1, n
            bwork( i ) = selctg( alphar( i ), alphai( i ), beta( i ) )
   10    CONTINUE
*
*        Reorder eigenvalues, transform Generalized Schur vectors, and
*        compute reciprocal condition numbers
*
         CALL dtgsen( ijob, ilvsl, ilvsr, bwork, n, a, lda, b, ldb,
     $                alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr,
     $                sdim, pl, pr, dif, work( iwrk ), lwork-iwrk+1,
     $                iwork, liwork, ierr )
*
         IF( ijob.GE.1 )
     $      maxwrk = max( maxwrk, 2*sdim*( n-sdim ) )
         IF( ierr.EQ.-22 ) THEN
*
*            not enough real workspace
*
            info = -22
         ELSE
            IF( ijob.EQ.1 .OR. ijob.EQ.4 ) THEN
               rconde( 1 ) = pl
               rconde( 2 ) = pr
            END IF
            IF( ijob.EQ.2 .OR. ijob.EQ.4 ) THEN
               rcondv( 1 ) = dif( 1 )
               rcondv( 2 ) = dif( 2 )
            END IF
            IF( ierr.EQ.1 )
     $         info = n + 3
         END IF
*
      END IF
*
*     Apply permutation to VSL and VSR
*     (Workspace: none needed)
*
      IF( ilvsl )
     $   CALL dggbak( 'P', 'L', n, ilo, ihi, work( ileft ),
     $                work( iright ), n, vsl, ldvsl, ierr )
*
      IF( ilvsr )
     $   CALL dggbak( 'P', 'R', n, ilo, ihi, work( ileft ),
     $                work( iright ), n, vsr, ldvsr, ierr )
*
*     Check if unscaling would cause over/underflow, if so, rescale
*     (ALPHAR(I),ALPHAI(I),BETA(I)) so BETA(I) is on the order of
*     B(I,I) and ALPHAR(I) and ALPHAI(I) are on the order of A(I,I)
*
      IF( ilascl ) THEN
         DO 20 i = 1, n
            IF( alphai( i ).NE.zero ) THEN
               IF( ( alphar( i ) / safmax ).GT.( anrmto / anrm ) .OR.
     $             ( safmin / alphar( i ) ).GT.( anrm / anrmto ) ) THEN
                  work( 1 ) = abs( a( i, i ) / alphar( i ) )
                  beta( i ) = beta( i )*work( 1 )
                  alphar( i ) = alphar( i )*work( 1 )
                  alphai( i ) = alphai( i )*work( 1 )
               ELSE IF( ( alphai( i ) / safmax ).GT.
     $                  ( anrmto / anrm ) .OR.
     $                  ( safmin / alphai( i ) ).GT.( anrm / anrmto ) )
     $                   THEN
                  work( 1 ) = abs( a( i, i+1 ) / alphai( i ) )
                  beta( i ) = beta( i )*work( 1 )
                  alphar( i ) = alphar( i )*work( 1 )
                  alphai( i ) = alphai( i )*work( 1 )
               END IF
            END IF
   20    CONTINUE
      END IF
*
      IF( ilbscl ) THEN
         DO 30 i = 1, n
            IF( alphai( i ).NE.zero ) THEN
               IF( ( beta( i ) / safmax ).GT.( bnrmto / bnrm ) .OR.
     $             ( safmin / beta( i ) ).GT.( bnrm / bnrmto ) ) THEN
                  work( 1 ) = abs( b( i, i ) / beta( i ) )
                  beta( i ) = beta( i )*work( 1 )
                  alphar( i ) = alphar( i )*work( 1 )
                  alphai( i ) = alphai( i )*work( 1 )
               END IF
            END IF
   30    CONTINUE
      END IF
*
*     Undo scaling
*
      IF( ilascl ) THEN
         CALL dlascl( 'H', 0, 0, anrmto, anrm, n, n, a, lda, ierr )
         CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphar, n, ierr )
         CALL dlascl( 'G', 0, 0, anrmto, anrm, n, 1, alphai, n, ierr )
      END IF
*
      IF( ilbscl ) THEN
         CALL dlascl( 'U', 0, 0, bnrmto, bnrm, n, n, b, ldb, ierr )
         CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
      END IF
*
      IF( wantst ) THEN
*
*        Check if reordering is correct
*
         lastsl = .true.
         lst2sl = .true.
         sdim = 0
         ip = 0
         DO 50 i = 1, n
            cursl = selctg( alphar( i ), alphai( i ), beta( i ) )
            IF( alphai( i ).EQ.zero ) THEN
               IF( cursl )
     $            sdim = sdim + 1
               ip = 0
               IF( cursl .AND. .NOT.lastsl )
     $            info = n + 2
            ELSE
               IF( ip.EQ.1 ) THEN
*
*                 Last eigenvalue of conjugate pair
*
                  cursl = cursl .OR. lastsl
                  lastsl = cursl
                  IF( cursl )
     $               sdim = sdim + 2
                  ip = -1
                  IF( cursl .AND. .NOT.lst2sl )
     $               info = n + 2
               ELSE
*
*                 First eigenvalue of conjugate pair
*
                  ip = 1
               END IF
            END IF
            lst2sl = lastsl
            lastsl = cursl
   50    CONTINUE
*
      END IF
*
   60 CONTINUE
*
      work( 1 ) = maxwrk
      iwork( 1 ) = liwmin
*
      RETURN
*
*     End of DGGESX
*
      END