dggevx.f

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*> \brief <b> DGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
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*> [TXT]</a>
*
*  Definition:
*  ===========
*
*       SUBROUTINE DGGEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB,
*                          ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
*                          IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
*                          RCONDV, WORK, LWORK, IWORK, BWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          BALANC, JOBVL, JOBVR, SENSE
*       INTEGER            IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
*       DOUBLE PRECISION   ABNRM, BBNRM
*       ..
*       .. Array Arguments ..
*       LOGICAL            BWORK( * )
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
*      $                   B( LDB, * ), BETA( * ), LSCALE( * ),
*      $                   RCONDE( * ), RCONDV( * ), RSCALE( * ),
*      $                   VL( LDVL, * ), VR( LDVR, * ), WORK( * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> DGGEVX computes for a pair of N-by-N real nonsymmetric matrices (A,B)
*> the generalized eigenvalues, and optionally, the left and/or right
*> generalized eigenvectors.
*>
*> Optionally also, it computes a balancing transformation to improve
*> the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
*> LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for
*> the eigenvalues (RCONDE), and reciprocal condition numbers for the
*> right eigenvectors (RCONDV).
*>
*> A generalized eigenvalue for a pair of matrices (A,B) is a scalar
*> lambda or a ratio alpha/beta = lambda, such that A - lambda*B is
*> singular. It is usually represented as the pair (alpha,beta), as
*> there is a reasonable interpretation for beta=0, and even for both
*> being zero.
*>
*> The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
*> of (A,B) satisfies
*>
*>                  A * v(j) = lambda(j) * B * v(j) .
*>
*> The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
*> of (A,B) satisfies
*>
*>                  u(j)**H * A  = lambda(j) * u(j)**H * B.
*>
*> where u(j)**H is the conjugate-transpose of u(j).
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] BALANC
*> \verbatim
*>          BALANC is CHARACTER*1
*>          Specifies the balance option to be performed.
*>          = 'N':  do not diagonally scale or permute;
*>          = 'P':  permute only;
*>          = 'S':  scale only;
*>          = 'B':  both permute and scale.
*>          Computed reciprocal condition numbers will be for the
*>          matrices after permuting and/or balancing. Permuting does
*>          not change condition numbers (in exact arithmetic), but
*>          balancing does.
*> \endverbatim
*>
*> \param[in] JOBVL
*> \verbatim
*>          JOBVL is CHARACTER*1
*>          = 'N':  do not compute the left generalized eigenvectors;
*>          = 'V':  compute the left generalized eigenvectors.
*> \endverbatim
*>
*> \param[in] JOBVR
*> \verbatim
*>          JOBVR is CHARACTER*1
*>          = 'N':  do not compute the right generalized eigenvectors;
*>          = 'V':  compute the right generalized eigenvectors.
*> \endverbatim
*>
*> \param[in] SENSE
*> \verbatim
*>          SENSE is CHARACTER*1
*>          Determines which reciprocal condition numbers are computed.
*>          = 'N': none are computed;
*>          = 'E': computed for eigenvalues only;
*>          = 'V': computed for eigenvectors only;
*>          = 'B': computed for eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A, B, VL, and VR.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is DOUBLE PRECISION array, dimension (LDA, N)
*>          On entry, the matrix A in the pair (A,B).
*>          On exit, A has been overwritten. If JOBVL='V' or JOBVR='V'
*>          or both, then A contains the first part of the real Schur
*>          form of the "balanced" versions of the input A and B.
*> \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 matrix B in the pair (A,B).
*>          On exit, B has been overwritten. If JOBVL='V' or JOBVR='V'
*>          or both, then B contains the second part of the real Schur
*>          form of the "balanced" versions of the input A and B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of B.  LDB >= max(1,N).
*> \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.  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
*>          ALPHA/BETA. 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] VL
*> \verbatim
*>          VL is DOUBLE PRECISION array, dimension (LDVL,N)
*>          If JOBVL = 'V', the left eigenvectors u(j) are stored one
*>          after another in the columns of VL, in the same order as
*>          their eigenvalues. If the j-th eigenvalue is real, then
*>          u(j) = VL(:,j), the j-th column of VL. If the j-th and
*>          (j+1)-th eigenvalues form a complex conjugate pair, then
*>          u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
*>          Each eigenvector will be scaled so the largest component have
*>          abs(real part) + abs(imag. part) = 1.
*>          Not referenced if JOBVL = 'N'.
*> \endverbatim
*>
*> \param[in] LDVL
*> \verbatim
*>          LDVL is INTEGER
*>          The leading dimension of the matrix VL. LDVL >= 1, and
*>          if JOBVL = 'V', LDVL >= N.
*> \endverbatim
*>
*> \param[out] VR
*> \verbatim
*>          VR is DOUBLE PRECISION array, dimension (LDVR,N)
*>          If JOBVR = 'V', the right eigenvectors v(j) are stored one
*>          after another in the columns of VR, in the same order as
*>          their eigenvalues. If the j-th eigenvalue is real, then
*>          v(j) = VR(:,j), the j-th column of VR. If the j-th and
*>          (j+1)-th eigenvalues form a complex conjugate pair, then
*>          v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
*>          Each eigenvector will be scaled so the largest component have
*>          abs(real part) + abs(imag. part) = 1.
*>          Not referenced if JOBVR = 'N'.
*> \endverbatim
*>
*> \param[in] LDVR
*> \verbatim
*>          LDVR is INTEGER
*>          The leading dimension of the matrix VR. LDVR >= 1, and
*>          if JOBVR = 'V', LDVR >= N.
*> \endverbatim
*>
*> \param[out] ILO
*> \verbatim
*>          ILO is INTEGER
*> \endverbatim
*>
*> \param[out] IHI
*> \verbatim
*>          IHI is INTEGER
*>          ILO and IHI are integer values such that on exit
*>          A(i,j) = 0 and B(i,j) = 0 if i > j and
*>          j = 1,...,ILO-1 or i = IHI+1,...,N.
*>          If BALANC = 'N' or 'S', ILO = 1 and IHI = N.
*> \endverbatim
*>
*> \param[out] LSCALE
*> \verbatim
*>          LSCALE is DOUBLE PRECISION array, dimension (N)
*>          Details of the permutations and scaling factors applied
*>          to the left side of A and B.  If PL(j) is the index of the
*>          row interchanged with row j, and DL(j) is the scaling
*>          factor applied to row j, then
*>            LSCALE(j) = PL(j)  for j = 1,...,ILO-1
*>                      = DL(j)  for j = ILO,...,IHI
*>                      = PL(j)  for j = IHI+1,...,N.
*>          The order in which the interchanges are made is N to IHI+1,
*>          then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] RSCALE
*> \verbatim
*>          RSCALE is DOUBLE PRECISION array, dimension (N)
*>          Details of the permutations and scaling factors applied
*>          to the right side of A and B.  If PR(j) is the index of the
*>          column interchanged with column j, and DR(j) is the scaling
*>          factor applied to column j, then
*>            RSCALE(j) = PR(j)  for j = 1,...,ILO-1
*>                      = DR(j)  for j = ILO,...,IHI
*>                      = PR(j)  for j = IHI+1,...,N
*>          The order in which the interchanges are made is N to IHI+1,
*>          then 1 to ILO-1.
*> \endverbatim
*>
*> \param[out] ABNRM
*> \verbatim
*>          ABNRM is DOUBLE PRECISION
*>          The one-norm of the balanced matrix A.
*> \endverbatim
*>
*> \param[out] BBNRM
*> \verbatim
*>          BBNRM is DOUBLE PRECISION
*>          The one-norm of the balanced matrix B.
*> \endverbatim
*>
*> \param[out] RCONDE
*> \verbatim
*>          RCONDE is DOUBLE PRECISION array, dimension (N)
*>          If SENSE = 'E' or 'B', the reciprocal condition numbers of
*>          the eigenvalues, stored in consecutive elements of the array.
*>          For a complex conjugate pair of eigenvalues two consecutive
*>          elements of RCONDE are set to the same value. Thus RCONDE(j),
*>          RCONDV(j), and the j-th columns of VL and VR all correspond
*>          to the j-th eigenpair.
*>          If SENSE = 'N or 'V', RCONDE is not referenced.
*> \endverbatim
*>
*> \param[out] RCONDV
*> \verbatim
*>          RCONDV is DOUBLE PRECISION array, dimension (N)
*>          If SENSE = 'V' or 'B', the estimated reciprocal condition
*>          numbers of the eigenvectors, stored in consecutive elements
*>          of the array. For a complex eigenvector two consecutive
*>          elements of RCONDV are set to the same value. If the
*>          eigenvalues cannot be reordered to compute RCONDV(j),
*>          RCONDV(j) is set to 0; this can only occur when the true
*>          value would be very small anyway.
*>          If SENSE = 'N' or 'E', RCONDV is not referenced.
*> \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. LWORK >= max(1,2*N).
*>          If BALANC = 'S' or 'B', or JOBVL = 'V', or JOBVR = 'V',
*>          LWORK >= max(1,6*N).
*>          If SENSE = 'E' or 'B', LWORK >= max(1,10*N).
*>          If SENSE = 'V' or 'B', LWORK >= 2*N*N+8*N+16.
*>
*>          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.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (N+6)
*>          If SENSE = 'E', IWORK is not referenced.
*> \endverbatim
*>
*> \param[out] BWORK
*> \verbatim
*>          BWORK is LOGICAL array, dimension (N)
*>          If SENSE = 'N', BWORK is not referenced.
*> \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.  No eigenvectors have been
*>                calculated, 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: error return from DTGEVC.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup ggevx
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  Balancing a matrix pair (A,B) includes, first, permuting rows and
*>  columns to isolate eigenvalues, second, applying diagonal similarity
*>  transformation to the rows and columns to make the rows and columns
*>  as close in norm as possible. The computed reciprocal condition
*>  numbers correspond to the balanced matrix. Permuting rows and columns
*>  will not change the condition numbers (in exact arithmetic) but
*>  diagonal scaling will.  For further explanation of balancing, see
*>  section 4.11.1.2 of LAPACK Users' Guide.
*>
*>  An approximate error bound on the chordal distance between the i-th
*>  computed generalized eigenvalue w and the corresponding exact
*>  eigenvalue lambda is
*>
*>       chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I)
*>
*>  An approximate error bound for the angle between the i-th computed
*>  eigenvector VL(i) or VR(i) is given by
*>
*>       EPS * norm(ABNRM, BBNRM) / DIF(i).
*>
*>  For further explanation of the reciprocal condition numbers RCONDE
*>  and RCONDV, see section 4.11 of LAPACK User's Guide.
*> \endverbatim
*>
*  =====================================================================
      SUBROUTINE dggevx( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B,
     $                   LDB,
     $                   ALPHAR, ALPHAI, BETA, VL, LDVL, VR, LDVR, ILO,
     $                   IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE,
     $                   RCONDV, WORK, LWORK, IWORK, 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          BALANC, JOBVL, JOBVR, SENSE
      INTEGER            IHI, ILO, INFO, LDA, LDB, LDVL, LDVR, LWORK, N
      DOUBLE PRECISION   ABNRM, BBNRM
*     ..
*     .. Array Arguments ..
      LOGICAL            BWORK( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), ALPHAI( * ), ALPHAR( * ),
     $                   B( LDB, * ), BETA( * ), LSCALE( * ),
     $                   rconde( * ), rcondv( * ), rscale( * ),
     $                   vl( ldvl, * ), vr( ldvr, * ), work( * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      PARAMETER          ( ZERO = 0.0d+0, one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            ILASCL, ILBSCL, ILV, ILVL, ILVR, LQUERY, NOSCL,
     $                   pair, wantsb, wantse, wantsn, wantsv
      CHARACTER          CHTEMP
      INTEGER            I, ICOLS, IERR, IJOBVL, IJOBVR, IN, IROWS,
     $                   itau, iwrk, iwrk1, j, jc, jr, m, maxwrk,
     $                   minwrk, mm
      DOUBLE PRECISION   ANRM, ANRMTO, BIGNUM, BNRM, BNRMTO, EPS,
     $                   smlnum, temp
*     ..
*     .. Local Arrays ..
      LOGICAL            LDUMMA( 1 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           DGEQRF, DGGBAK, DGGBAL, DGGHRD, DHGEQZ,
     $                   dlacpy,
     $                   dlascl, dlaset, dorgqr, dormqr, dtgevc, dtgsna,
     $                   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( jobvl, 'N' ) ) THEN
         ijobvl = 1
         ilvl = .false.
      ELSE IF( lsame( jobvl, 'V' ) ) THEN
         ijobvl = 2
         ilvl = .true.
      ELSE
         ijobvl = -1
         ilvl = .false.
      END IF
*
      IF( lsame( jobvr, 'N' ) ) THEN
         ijobvr = 1
         ilvr = .false.
      ELSE IF( lsame( jobvr, 'V' ) ) THEN
         ijobvr = 2
         ilvr = .true.
      ELSE
         ijobvr = -1
         ilvr = .false.
      END IF
      ilv = ilvl .OR. ilvr
*
      noscl  = lsame( balanc, 'N' ) .OR. lsame( balanc, 'P' )
      wantsn = lsame( sense, 'N' )
      wantse = lsame( sense, 'E' )
      wantsv = lsame( sense, 'V' )
      wantsb = lsame( sense, 'B' )
*
*     Test the input arguments
*
      info = 0
      lquery = ( lwork.EQ.-1 )
      IF( .NOT.( lsame( balanc, 'N' ) .OR. lsame( balanc,
     $    'S' ) .OR.
     $                      lsame( balanc, 'P' ) .OR.
     $                      lsame( balanc, 'B' ) ) )
     $     THEN
         info = -1
      ELSE IF( ijobvl.LE.0 ) THEN
         info = -2
      ELSE IF( ijobvr.LE.0 ) THEN
         info = -3
      ELSE IF( .NOT.( wantsn .OR. wantse .OR. wantsb .OR. wantsv ) )
     $          THEN
         info = -4
      ELSE IF( n.LT.0 ) THEN
         info = -5
      ELSE IF( lda.LT.max( 1, n ) ) THEN
         info = -7
      ELSE IF( ldb.LT.max( 1, n ) ) THEN
         info = -9
      ELSE IF( ldvl.LT.1 .OR. ( ilvl .AND. ldvl.LT.n ) ) THEN
         info = -14
      ELSE IF( ldvr.LT.1 .OR. ( ilvr .AND. ldvr.LT.n ) ) THEN
         info = -16
      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. The workspace is
*       computed assuming ILO = 1 and IHI = N, the worst case.)
*
      IF( info.EQ.0 ) THEN
         IF( n.EQ.0 ) THEN
            minwrk = 1
            maxwrk = 1
         ELSE
            IF( noscl .AND. .NOT.ilv ) THEN
               minwrk = 2*n
            ELSE
               minwrk = 6*n
            END IF
            IF( wantse .OR. wantsb ) THEN
               minwrk = 10*n
            END IF
            IF( wantsv .OR. wantsb ) THEN
               minwrk = max( minwrk, 2*n*( n + 4 ) + 16 )
            END IF
            maxwrk = minwrk
            maxwrk = max( maxwrk,
     $                    n + n*ilaenv( 1, 'DGEQRF', ' ', n, 1, n,
     $                                  0 ) )
            maxwrk = max( maxwrk,
     $                    n + n*ilaenv( 1, 'DORMQR', ' ', n, 1, n,
     $                                  0 ) )
            IF( ilvl ) THEN
               maxwrk = max( maxwrk, n +
     $                       n*ilaenv( 1, 'DORGQR', ' ', n, 1, n,
     $                                 0 ) )
            END IF
         END IF
         work( 1 ) = maxwrk
*
         IF( lwork.LT.minwrk .AND. .NOT.lquery ) THEN
            info = -26
         END IF
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'DGGEVX', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   RETURN
*
*
*     Get machine constants
*
      eps = dlamch( 'P' )
      smlnum = dlamch( 'S' )
      bignum = one / smlnum
      smlnum = sqrt( smlnum ) / eps
      bignum = one / smlnum
*
*     Scale A if max element outside range [SMLNUM,BIGNUM]
*
      anrm = dlange( 'M', n, n, a, lda, work )
      ilascl = .false.
      IF( anrm.GT.zero .AND. anrm.LT.smlnum ) THEN
         anrmto = smlnum
         ilascl = .true.
      ELSE IF( anrm.GT.bignum ) 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.
      IF( bnrm.GT.zero .AND. bnrm.LT.smlnum ) THEN
         bnrmto = smlnum
         ilbscl = .true.
      ELSE IF( bnrm.GT.bignum ) THEN
         bnrmto = bignum
         ilbscl = .true.
      END IF
      IF( ilbscl )
     $   CALL dlascl( 'G', 0, 0, bnrm, bnrmto, n, n, b, ldb, ierr )
*
*     Permute and/or balance the matrix pair (A,B)
*     (Workspace: need 6*N if BALANC = 'S' or 'B', 1 otherwise)
*
      CALL dggbal( balanc, n, a, lda, b, ldb, ilo, ihi, lscale,
     $             rscale,
     $             work, ierr )
*
*     Compute ABNRM and BBNRM
*
      abnrm = dlange( '1', n, n, a, lda, work( 1 ) )
      IF( ilascl ) THEN
         work( 1 ) = abnrm
         CALL dlascl( 'G', 0, 0, anrmto, anrm, 1, 1, work( 1 ), 1,
     $                ierr )
         abnrm = work( 1 )
      END IF
*
      bbnrm = dlange( '1', n, n, b, ldb, work( 1 ) )
      IF( ilbscl ) THEN
         work( 1 ) = bbnrm
         CALL dlascl( 'G', 0, 0, bnrmto, bnrm, 1, 1, work( 1 ), 1,
     $                ierr )
         bbnrm = work( 1 )
      END IF
*
*     Reduce B to triangular form (QR decomposition of B)
*     (Workspace: need N, prefer N*NB )
*
      irows = ihi + 1 - ilo
      IF( ilv .OR. .NOT.wantsn ) THEN
         icols = n + 1 - ilo
      ELSE
         icols = irows
      END IF
      itau = 1
      iwrk = itau + irows
      CALL dgeqrf( irows, icols, b( ilo, ilo ), ldb, work( itau ),
     $             work( iwrk ), lwork+1-iwrk, ierr )
*
*     Apply the orthogonal transformation to 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 VL and/or VR
*     (Workspace: need N, prefer N*NB)
*
      IF( ilvl ) THEN
         CALL dlaset( 'Full', n, n, zero, one, vl, ldvl )
         IF( irows.GT.1 ) THEN
            CALL dlacpy( 'L', irows-1, irows-1, b( ilo+1, ilo ), ldb,
     $                   vl( ilo+1, ilo ), ldvl )
         END IF
         CALL dorgqr( irows, irows, irows, vl( ilo, ilo ), ldvl,
     $                work( itau ), work( iwrk ), lwork+1-iwrk, ierr )
      END IF
*
      IF( ilvr )
     $   CALL dlaset( 'Full', n, n, zero, one, vr, ldvr )
*
*     Reduce to generalized Hessenberg form
*     (Workspace: none needed)
*
      IF( ilv .OR. .NOT.wantsn ) THEN
*
*        Eigenvectors requested -- work on whole matrix.
*
         CALL dgghrd( jobvl, jobvr, n, ilo, ihi, a, lda, b, ldb, vl,
     $                ldvl, vr, ldvr, ierr )
      ELSE
         CALL dgghrd( 'N', 'N', irows, 1, irows, a( ilo, ilo ), lda,
     $                b( ilo, ilo ), ldb, vl, ldvl, vr, ldvr, ierr )
      END IF
*
*     Perform QZ algorithm (Compute eigenvalues, and optionally, the
*     Schur forms and Schur vectors)
*     (Workspace: need N)
*
      IF( ilv .OR. .NOT.wantsn ) THEN
         chtemp = 'S'
      ELSE
         chtemp = 'E'
      END IF
*
      CALL dhgeqz( chtemp, jobvl, jobvr, n, ilo, ihi, a, lda, b, ldb,
     $             alphar, alphai, beta, vl, ldvl, vr, ldvr, work,
     $             lwork, ierr )
      IF( ierr.NE.0 ) THEN
         IF( ierr.GT.0 .AND. ierr.LE.n ) THEN
            info = ierr
         ELSE IF( ierr.GT.n .AND. ierr.LE.2*n ) THEN
            info = ierr - n
         ELSE
            info = n + 1
         END IF
         GO TO 130
      END IF
*
*     Compute Eigenvectors and estimate condition numbers if desired
*     (Workspace: DTGEVC: need 6*N
*                 DTGSNA: need 2*N*(N+2)+16 if SENSE = 'V' or 'B',
*                         need N otherwise )
*
      IF( ilv .OR. .NOT.wantsn ) THEN
         IF( ilv ) THEN
            IF( ilvl ) THEN
               IF( ilvr ) THEN
                  chtemp = 'B'
               ELSE
                  chtemp = 'L'
               END IF
            ELSE
               chtemp = 'R'
            END IF
*
            CALL dtgevc( chtemp, 'B', ldumma, n, a, lda, b, ldb, vl,
     $                   ldvl, vr, ldvr, n, in, work, ierr )
            IF( ierr.NE.0 ) THEN
               info = n + 2
               GO TO 130
            END IF
         END IF
*
         IF( .NOT.wantsn ) THEN
*
*           compute eigenvectors (DTGEVC) and estimate condition
*           numbers (DTGSNA). Note that the definition of the condition
*           number is not invariant under transformation (u,v) to
*           (Q*u, Z*v), where (u,v) are eigenvectors of the generalized
*           Schur form (S,T), Q and Z are orthogonal matrices. In order
*           to avoid using extra 2*N*N workspace, we have to recalculate
*           eigenvectors and estimate one condition numbers at a time.
*
            pair = .false.
            DO 20 i = 1, n
*
               IF( pair ) THEN
                  pair = .false.
                  GO TO 20
               END IF
               mm = 1
               IF( i.LT.n ) THEN
                  IF( a( i+1, i ).NE.zero ) THEN
                     pair = .true.
                     mm = 2
                  END IF
               END IF
*
               DO 10 j = 1, n
                  bwork( j ) = .false.
   10          CONTINUE
               IF( mm.EQ.1 ) THEN
                  bwork( i ) = .true.
               ELSE IF( mm.EQ.2 ) THEN
                  bwork( i ) = .true.
                  bwork( i+1 ) = .true.
               END IF
*
               iwrk = mm*n + 1
               iwrk1 = iwrk + mm*n
*
*              Compute a pair of left and right eigenvectors.
*              (compute workspace: need up to 4*N + 6*N)
*
               IF( wantse .OR. wantsb ) THEN
                  CALL dtgevc( 'B', 'S', bwork, n, a, lda, b, ldb,
     $                         work( 1 ), n, work( iwrk ), n, mm, m,
     $                         work( iwrk1 ), ierr )
                  IF( ierr.NE.0 ) THEN
                     info = n + 2
                     GO TO 130
                  END IF
               END IF
*
               CALL dtgsna( sense, 'S', bwork, n, a, lda, b, ldb,
     $                      work( 1 ), n, work( iwrk ), n, rconde( i ),
     $                      rcondv( i ), mm, m, work( iwrk1 ),
     $                      lwork-iwrk1+1, iwork, ierr )
*
   20       CONTINUE
         END IF
      END IF
*
*     Undo balancing on VL and VR and normalization
*     (Workspace: none needed)
*
      IF( ilvl ) THEN
         CALL dggbak( balanc, 'L', n, ilo, ihi, lscale, rscale, n,
     $                vl,
     $                ldvl, ierr )
*
         DO 70 jc = 1, n
            IF( alphai( jc ).LT.zero )
     $         GO TO 70
            temp = zero
            IF( alphai( jc ).EQ.zero ) THEN
               DO 30 jr = 1, n
                  temp = max( temp, abs( vl( jr, jc ) ) )
   30          CONTINUE
            ELSE
               DO 40 jr = 1, n
                  temp = max( temp, abs( vl( jr, jc ) )+
     $                   abs( vl( jr, jc+1 ) ) )
   40          CONTINUE
            END IF
            IF( temp.LT.smlnum )
     $         GO TO 70
            temp = one / temp
            IF( alphai( jc ).EQ.zero ) THEN
               DO 50 jr = 1, n
                  vl( jr, jc ) = vl( jr, jc )*temp
   50          CONTINUE
            ELSE
               DO 60 jr = 1, n
                  vl( jr, jc ) = vl( jr, jc )*temp
                  vl( jr, jc+1 ) = vl( jr, jc+1 )*temp
   60          CONTINUE
            END IF
   70    CONTINUE
      END IF
      IF( ilvr ) THEN
         CALL dggbak( balanc, 'R', n, ilo, ihi, lscale, rscale, n,
     $                vr,
     $                ldvr, ierr )
         DO 120 jc = 1, n
            IF( alphai( jc ).LT.zero )
     $         GO TO 120
            temp = zero
            IF( alphai( jc ).EQ.zero ) THEN
               DO 80 jr = 1, n
                  temp = max( temp, abs( vr( jr, jc ) ) )
   80          CONTINUE
            ELSE
               DO 90 jr = 1, n
                  temp = max( temp, abs( vr( jr, jc ) )+
     $                   abs( vr( jr, jc+1 ) ) )
   90          CONTINUE
            END IF
            IF( temp.LT.smlnum )
     $         GO TO 120
            temp = one / temp
            IF( alphai( jc ).EQ.zero ) THEN
               DO 100 jr = 1, n
                  vr( jr, jc ) = vr( jr, jc )*temp
  100          CONTINUE
            ELSE
               DO 110 jr = 1, n
                  vr( jr, jc ) = vr( jr, jc )*temp
                  vr( jr, jc+1 ) = vr( jr, jc+1 )*temp
  110          CONTINUE
            END IF
  120    CONTINUE
      END IF
*
*     Undo scaling if necessary
*
  130 CONTINUE
*
      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 ) THEN
         CALL dlascl( 'G', 0, 0, bnrmto, bnrm, n, 1, beta, n, ierr )
      END IF
*
      work( 1 ) = maxwrk
      RETURN
*
*     End of DGGEVX
*
      SUBROUTINE dggevx( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, …
      END