ztgsen.f

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*> \brief \b ZTGSEN
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE ZTGSEN( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB,
*                          ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF,
*                          WORK, LWORK, IWORK, LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       LOGICAL            WANTQ, WANTZ
*       INTEGER            IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK,
*      $                   M, N
*       DOUBLE PRECISION   PL, PR
*       ..
*       .. Array Arguments ..
*       LOGICAL            SELECT( * )
*       INTEGER            IWORK( * )
*       DOUBLE PRECISION   DIF( * )
*       COMPLEX*16         A( LDA, * ), ALPHA( * ), B( LDB, * ),
*      $                   BETA( * ), Q( LDQ, * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZTGSEN reorders the generalized Schur decomposition of a complex
*> matrix pair (A, B) (in terms of an unitary equivalence trans-
*> formation Q**H * (A, B) * Z), so that a selected cluster of eigenvalues
*> appears in the leading diagonal blocks of the pair (A,B). The leading
*> columns of Q and Z form unitary bases of the corresponding left and
*> right eigenspaces (deflating subspaces). (A, B) must be in
*> generalized Schur canonical form, that is, A and B are both upper
*> triangular.
*>
*> ZTGSEN also computes the generalized eigenvalues
*>
*>          w(j)= ALPHA(j) / BETA(j)
*>
*> of the reordered matrix pair (A, B).
*>
*> Optionally, the routine computes estimates of reciprocal condition
*> numbers for eigenvalues and eigenspaces. These are Difu[(A11,B11),
*> (A22,B22)] and Difl[(A11,B11), (A22,B22)], i.e. the separation(s)
*> between the matrix pairs (A11, B11) and (A22,B22) that correspond to
*> the selected cluster and the eigenvalues outside the cluster, resp.,
*> and norms of "projections" onto left and right eigenspaces w.r.t.
*> the selected cluster in the (1,1)-block.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] IJOB
*> \verbatim
*>          IJOB is INTEGER
*>          Specifies whether condition numbers are required for the
*>          cluster of eigenvalues (PL and PR) or the deflating subspaces
*>          (Difu and Difl):
*>           =0: Only reorder w.r.t. SELECT. No extras.
*>           =1: Reciprocal of norms of "projections" onto left and right
*>               eigenspaces w.r.t. the selected cluster (PL and PR).
*>           =2: Upper bounds on Difu and Difl. F-norm-based estimate
*>               (DIF(1:2)).
*>           =3: Estimate of Difu and Difl. 1-norm-based estimate
*>               (DIF(1:2)).
*>               About 5 times as expensive as IJOB = 2.
*>           =4: Compute PL, PR and DIF (i.e. 0, 1 and 2 above): Economic
*>               version to get it all.
*>           =5: Compute PL, PR and DIF (i.e. 0, 1 and 3 above)
*> \endverbatim
*>
*> \param[in] WANTQ
*> \verbatim
*>          WANTQ is LOGICAL
*>          .TRUE. : update the left transformation matrix Q;
*>          .FALSE.: do not update Q.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*>          WANTZ is LOGICAL
*>          .TRUE. : update the right transformation matrix Z;
*>          .FALSE.: do not update Z.
*> \endverbatim
*>
*> \param[in] SELECT
*> \verbatim
*>          SELECT is LOGICAL array, dimension (N)
*>          SELECT specifies the eigenvalues in the selected cluster. To
*>          select an eigenvalue w(j), SELECT(j) must be set to
*>          .TRUE..
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrices A and B. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is COMPLEX*16 array, dimension(LDA,N)
*>          On entry, the upper triangular matrix A, in generalized
*>          Schur canonical form.
*>          On exit, A is overwritten by the reordered matrix A.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*>          B is COMPLEX*16 array, dimension(LDB,N)
*>          On entry, the upper triangular matrix B, in generalized
*>          Schur canonical form.
*>          On exit, B is overwritten by the reordered matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] ALPHA
*> \verbatim
*>          ALPHA is COMPLEX*16 array, dimension (N)
*> \endverbatim
*>
*> \param[out] BETA
*> \verbatim
*>          BETA is COMPLEX*16 array, dimension (N)
*>
*>          The diagonal elements of A and B, respectively,
*>          when the pair (A,B) has been reduced to generalized Schur
*>          form.  ALPHA(i)/BETA(i) i=1,...,N are the generalized
*>          eigenvalues.
*> \endverbatim
*>
*> \param[in,out] Q
*> \verbatim
*>          Q is COMPLEX*16 array, dimension (LDQ,N)
*>          On entry, if WANTQ = .TRUE., Q is an N-by-N matrix.
*>          On exit, Q has been postmultiplied by the left unitary
*>          transformation matrix which reorder (A, B); The leading M
*>          columns of Q form orthonormal bases for the specified pair of
*>          left eigenspaces (deflating subspaces).
*>          If WANTQ = .FALSE., Q is not referenced.
*> \endverbatim
*>
*> \param[in] LDQ
*> \verbatim
*>          LDQ is INTEGER
*>          The leading dimension of the array Q. LDQ >= 1.
*>          If WANTQ = .TRUE., LDQ >= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is COMPLEX*16 array, dimension (LDZ,N)
*>          On entry, if WANTZ = .TRUE., Z is an N-by-N matrix.
*>          On exit, Z has been postmultiplied by the left unitary
*>          transformation matrix which reorder (A, B); The leading M
*>          columns of Z form orthonormal bases for the specified pair of
*>          left eigenspaces (deflating subspaces).
*>          If WANTZ = .FALSE., Z is not referenced.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z. LDZ >= 1.
*>          If WANTZ = .TRUE., LDZ >= N.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The dimension of the specified pair of left and right
*>          eigenspaces, (deflating subspaces) 0 <= M <= N.
*> \endverbatim
*>
*> \param[out] PL
*> \verbatim
*>          PL is DOUBLE PRECISION
*> \endverbatim
*>
*> \param[out] PR
*> \verbatim
*>          PR is DOUBLE PRECISION
*>
*>          If IJOB = 1, 4 or 5, PL, PR are lower bounds on the
*>          reciprocal  of the norm of "projections" onto left and right
*>          eigenspace with respect to the selected cluster.
*>          0 < PL, PR <= 1.
*>          If M = 0 or M = N, PL = PR  = 1.
*>          If IJOB = 0, 2 or 3 PL, PR are not referenced.
*> \endverbatim
*>
*> \param[out] DIF
*> \verbatim
*>          DIF is DOUBLE PRECISION array, dimension (2).
*>          If IJOB >= 2, DIF(1:2) store the estimates of Difu and Difl.
*>          If IJOB = 2 or 4, DIF(1:2) are F-norm-based upper bounds on
*>          Difu and Difl. If IJOB = 3 or 5, DIF(1:2) are 1-norm-based
*>          estimates of Difu and Difl, computed using reversed
*>          communication with ZLACN2.
*>          If M = 0 or N, DIF(1:2) = F-norm([A, B]).
*>          If IJOB = 0 or 1, DIF is not referenced.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX*16 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 >=  1
*>          If IJOB = 1, 2 or 4, LWORK >=  2*M*(N-M)
*>          If IJOB = 3 or 5, LWORK >=  4*M*(N-M)
*>
*>          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 (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK. LIWORK >= 1.
*>          If IJOB = 1, 2 or 4, LIWORK >=  N+2;
*>          If IJOB = 3 or 5, LIWORK >= MAX(N+2, 2*M*(N-M));
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the optimal size of the IWORK array,
*>          returns this value as the first entry of the IWORK array, and
*>          no error message related to LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>            =0: Successful exit.
*>            <0: If INFO = -i, the i-th argument had an illegal value.
*>            =1: Reordering of (A, B) failed because the transformed
*>                matrix pair (A, B) would be too far from generalized
*>                Schur form; the problem is very ill-conditioned.
*>                (A, B) may have been partially reordered.
*>                If requested, 0 is returned in DIF(*), PL and PR.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup complex16OTHERcomputational
*
*> \par Further Details:
*  =====================
*>
*> \verbatim
*>
*>  ZTGSEN first collects the selected eigenvalues by computing unitary
*>  U and W that move them to the top left corner of (A, B). In other
*>  words, the selected eigenvalues are the eigenvalues of (A11, B11) in
*>
*>              U**H*(A, B)*W = (A11 A12) (B11 B12) n1
*>                              ( 0  A22),( 0  B22) n2
*>                                n1  n2    n1  n2
*>
*>  where N = n1+n2 and U**H means the conjugate transpose of U. The first
*>  n1 columns of U and W span the specified pair of left and right
*>  eigenspaces (deflating subspaces) of (A, B).
*>
*>  If (A, B) has been obtained from the generalized real Schur
*>  decomposition of a matrix pair (C, D) = Q*(A, B)*Z**H, then the
*>  reordered generalized Schur form of (C, D) is given by
*>
*>           (C, D) = (Q*U)*(U**H *(A, B)*W)*(Z*W)**H,
*>
*>  and the first n1 columns of Q*U and Z*W span the corresponding
*>  deflating subspaces of (C, D) (Q and Z store Q*U and Z*W, resp.).
*>
*>  Note that if the selected eigenvalue is sufficiently ill-conditioned,
*>  then its value may differ significantly from its value before
*>  reordering.
*>
*>  The reciprocal condition numbers of the left and right eigenspaces
*>  spanned by the first n1 columns of U and W (or Q*U and Z*W) may
*>  be returned in DIF(1:2), corresponding to Difu and Difl, resp.
*>
*>  The Difu and Difl are defined as:
*>
*>       Difu[(A11, B11), (A22, B22)] = sigma-min( Zu )
*>  and
*>       Difl[(A11, B11), (A22, B22)] = Difu[(A22, B22), (A11, B11)],
*>
*>  where sigma-min(Zu) is the smallest singular value of the
*>  (2*n1*n2)-by-(2*n1*n2) matrix
*>
*>       Zu = [ kron(In2, A11)  -kron(A22**H, In1) ]
*>            [ kron(In2, B11)  -kron(B22**H, In1) ].
*>
*>  Here, Inx is the identity matrix of size nx and A22**H is the
*>  conjugate transpose of A22. kron(X, Y) is the Kronecker product between
*>  the matrices X and Y.
*>
*>  When DIF(2) is small, small changes in (A, B) can cause large changes
*>  in the deflating subspace. An approximate (asymptotic) bound on the
*>  maximum angular error in the computed deflating subspaces is
*>
*>       EPS * norm((A, B)) / DIF(2),
*>
*>  where EPS is the machine precision.
*>
*>  The reciprocal norm of the projectors on the left and right
*>  eigenspaces associated with (A11, B11) may be returned in PL and PR.
*>  They are computed as follows. First we compute L and R so that
*>  P*(A, B)*Q is block diagonal, where
*>
*>       P = ( I -L ) n1           Q = ( I R ) n1
*>           ( 0  I ) n2    and        ( 0 I ) n2
*>             n1 n2                    n1 n2
*>
*>  and (L, R) is the solution to the generalized Sylvester equation
*>
*>       A11*R - L*A22 = -A12
*>       B11*R - L*B22 = -B12
*>
*>  Then PL = (F-norm(L)**2+1)**(-1/2) and PR = (F-norm(R)**2+1)**(-1/2).
*>  An approximate (asymptotic) bound on the average absolute error of
*>  the selected eigenvalues is
*>
*>       EPS * norm((A, B)) / PL.
*>
*>  There are also global error bounds which valid for perturbations up
*>  to a certain restriction:  A lower bound (x) on the smallest
*>  F-norm(E,F) for which an eigenvalue of (A11, B11) may move and
*>  coalesce with an eigenvalue of (A22, B22) under perturbation (E,F),
*>  (i.e. (A + E, B + F), is
*>
*>   x = min(Difu,Difl)/((1/(PL*PL)+1/(PR*PR))**(1/2)+2*max(1/PL,1/PR)).
*>
*>  An approximate bound on x can be computed from DIF(1:2), PL and PR.
*>
*>  If y = ( F-norm(E,F) / x) <= 1, the angles between the perturbed
*>  (L', R') and unperturbed (L, R) left and right deflating subspaces
*>  associated with the selected cluster in the (1,1)-blocks can be
*>  bounded as
*>
*>   max-angle(L, L') <= arctan( y * PL / (1 - y * (1 - PL * PL)**(1/2))
*>   max-angle(R, R') <= arctan( y * PR / (1 - y * (1 - PR * PR)**(1/2))
*>
*>  See LAPACK User's Guide section 4.11 or the following references
*>  for more information.
*>
*>  Note that if the default method for computing the Frobenius-norm-
*>  based estimate DIF is not wanted (see ZLATDF), then the parameter
*>  IDIFJB (see below) should be changed from 3 to 4 (routine ZLATDF
*>  (IJOB = 2 will be used)). See ZTGSYL for more details.
*> \endverbatim
*
*> \par Contributors:
*  ==================
*>
*>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
*>     Umea University, S-901 87 Umea, Sweden.
*
*> \par References:
*  ================
*>
*>  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
*>      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
*>      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
*>      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
*> \n
*>  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
*>      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
*>      Estimation: Theory, Algorithms and Software, Report
*>      UMINF - 94.04, Department of Computing Science, Umea University,
*>      S-901 87 Umea, Sweden, 1994. Also as LAPACK Working Note 87.
*>      To appear in Numerical Algorithms, 1996.
*> \n
*>  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
*>      for Solving the Generalized Sylvester Equation and Estimating the
*>      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
*>      Department of Computing Science, Umea University, S-901 87 Umea,
*>      Sweden, December 1993, Revised April 1994, Also as LAPACK working
*>      Note 75. To appear in ACM Trans. on Math. Software, Vol 22, No 1,
*>      1996.
*>
*  =====================================================================
      SUBROUTINE ztgsen( IJOB, WANTQ, WANTZ, SELECT, N, A, LDA, B, LDB,
     $                   ALPHA, BETA, Q, LDQ, Z, LDZ, M, PL, PR, DIF,
     $                   WORK, LWORK, IWORK, LIWORK, INFO )
*
*  -- LAPACK computational routine (version 3.7.1) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     June 2016
*
*     .. Scalar Arguments ..
      LOGICAL            WANTQ, WANTZ
      INTEGER            IJOB, INFO, LDA, LDB, LDQ, LDZ, LIWORK, LWORK,
     $                   m, n
      DOUBLE PRECISION   PL, PR
*     ..
*     .. Array Arguments ..
      LOGICAL            SELECT( * )
      INTEGER            IWORK( * )
      DOUBLE PRECISION   DIF( * )
      COMPLEX*16         A( LDA, * ), ALPHA( * ), B( LDB, * ),
     $                   beta( * ), q( ldq, * ), work( * ), z( ldz, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      INTEGER            IDIFJB
      PARAMETER          ( IDIFJB = 3 )
      DOUBLE PRECISION   ZERO, ONE
      parameter( zero = 0.0d+0, one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, SWAP, WANTD, WANTD1, WANTD2, WANTP
      INTEGER            I, IERR, IJB, K, KASE, KS, LIWMIN, LWMIN, MN2,
     $                   n1, n2
      DOUBLE PRECISION   DSCALE, DSUM, RDSCAL, SAFMIN
      COMPLEX*16         TEMP1, TEMP2
*     ..
*     .. Local Arrays ..
      INTEGER            ISAVE( 3 )
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, zlacn2, zlacpy, zlassq, zscal, ztgexc,
     $                   ztgsyl
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dcmplx, dconjg, max, sqrt
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           DLAMCH
*     ..
*     .. Executable Statements ..
*
*     Decode and test the input parameters
*
      info = 0
      lquery = ( lwork.EQ.-1 .OR. liwork.EQ.-1 )
*
      IF( ijob.LT.0 .OR. ijob.GT.5 ) THEN
         info = -1
      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( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
         info = -13
      ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
         info = -15
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZTGSEN', -info )
         RETURN
      END IF
*
      ierr = 0
*
      wantp = ijob.EQ.1 .OR. ijob.GE.4
      wantd1 = ijob.EQ.2 .OR. ijob.EQ.4
      wantd2 = ijob.EQ.3 .OR. ijob.EQ.5
      wantd = wantd1 .OR. wantd2
*
*     Set M to the dimension of the specified pair of deflating
*     subspaces.
*
      m = 0
      IF( .NOT.lquery .OR. ijob.NE.0 ) THEN
      DO 10 k = 1, n
         alpha( k ) = a( k, k )
         beta( k ) = b( k, k )
         IF( k.LT.n ) THEN
            IF( SELECT( k ) )
     $         m = m + 1
         ELSE
            IF( SELECT( n ) )
     $         m = m + 1
         END IF
   10 CONTINUE
      END IF
*
      IF( ijob.EQ.1 .OR. ijob.EQ.2 .OR. ijob.EQ.4 ) THEN
         lwmin = max( 1, 2*m*( n-m ) )
         liwmin = max( 1, n+2 )
      ELSE IF( ijob.EQ.3 .OR. ijob.EQ.5 ) THEN
         lwmin = max( 1, 4*m*( n-m ) )
         liwmin = max( 1, 2*m*( n-m ), n+2 )
      ELSE
         lwmin = 1
         liwmin = 1
      END IF
*
      work( 1 ) = lwmin
      iwork( 1 ) = liwmin
*
      IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
         info = -21
      ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
         info = -23
      END IF
*
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZTGSEN', -info )
         RETURN
      ELSE IF( lquery ) THEN
         RETURN
      END IF
*
*     Quick return if possible.
*
      IF( m.EQ.n .OR. m.EQ.0 ) THEN
         IF( wantp ) THEN
            pl = one
            pr = one
         END IF
         IF( wantd ) THEN
            dscale = zero
            dsum = one
            DO 20 i = 1, n
               CALL zlassq( n, a( 1, i ), 1, dscale, dsum )
               CALL zlassq( n, b( 1, i ), 1, dscale, dsum )
   20       CONTINUE
            dif( 1 ) = dscale*sqrt( dsum )
            dif( 2 ) = dif( 1 )
         END IF
         GO TO 70
      END IF
*
*     Get machine constant
*
      safmin = dlamch( 'S' )
*
*     Collect the selected blocks at the top-left corner of (A, B).
*
      ks = 0
      DO 30 k = 1, n
         swap = SELECT( k )
         IF( swap ) THEN
            ks = ks + 1
*
*           Swap the K-th block to position KS. Compute unitary Q
*           and Z that will swap adjacent diagonal blocks in (A, B).
*
            IF( k.NE.ks )
     $         CALL ztgexc( wantq, wantz, n, a, lda, b, ldb, q, ldq, z,
     $                      ldz, k, ks, ierr )
*
            IF( ierr.GT.0 ) THEN
*
*              Swap is rejected: exit.
*
               info = 1
               IF( wantp ) THEN
                  pl = zero
                  pr = zero
               END IF
               IF( wantd ) THEN
                  dif( 1 ) = zero
                  dif( 2 ) = zero
               END IF
               GO TO 70
            END IF
         END IF
   30 CONTINUE
      IF( wantp ) THEN
*
*        Solve generalized Sylvester equation for R and L:
*                   A11 * R - L * A22 = A12
*                   B11 * R - L * B22 = B12
*
         n1 = m
         n2 = n - m
         i = n1 + 1
         CALL zlacpy( 'Full', n1, n2, a( 1, i ), lda, work, n1 )
         CALL zlacpy( 'Full', n1, n2, b( 1, i ), ldb, work( n1*n2+1 ),
     $                n1 )
         ijb = 0
         CALL ztgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda, work,
     $                n1, b, ldb, b( i, i ), ldb, work( n1*n2+1 ), n1,
     $                dscale, dif( 1 ), work( n1*n2*2+1 ),
     $                lwork-2*n1*n2, iwork, ierr )
*
*        Estimate the reciprocal of norms of "projections" onto
*        left and right eigenspaces
*
         rdscal = zero
         dsum = one
         CALL zlassq( n1*n2, work, 1, rdscal, dsum )
         pl = rdscal*sqrt( dsum )
         IF( pl.EQ.zero ) THEN
            pl = one
         ELSE
            pl = dscale / ( sqrt( dscale*dscale / pl+pl )*sqrt( pl ) )
         END IF
         rdscal = zero
         dsum = one
         CALL zlassq( n1*n2, work( n1*n2+1 ), 1, rdscal, dsum )
         pr = rdscal*sqrt( dsum )
         IF( pr.EQ.zero ) THEN
            pr = one
         ELSE
            pr = dscale / ( sqrt( dscale*dscale / pr+pr )*sqrt( pr ) )
         END IF
      END IF
      IF( wantd ) THEN
*
*        Compute estimates Difu and Difl.
*
         IF( wantd1 ) THEN
            n1 = m
            n2 = n - m
            i = n1 + 1
            ijb = idifjb
*
*           Frobenius norm-based Difu estimate.
*
            CALL ztgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda, work,
     $                   n1, b, ldb, b( i, i ), ldb, work( n1*n2+1 ),
     $                   n1, dscale, dif( 1 ), work( n1*n2*2+1 ),
     $                   lwork-2*n1*n2, iwork, ierr )
*
*           Frobenius norm-based Difl estimate.
*
            CALL ztgsyl( 'N', ijb, n2, n1, a( i, i ), lda, a, lda, work,
     $                   n2, b( i, i ), ldb, b, ldb, work( n1*n2+1 ),
     $                   n2, dscale, dif( 2 ), work( n1*n2*2+1 ),
     $                   lwork-2*n1*n2, iwork, ierr )
         ELSE
*
*           Compute 1-norm-based estimates of Difu and Difl using
*           reversed communication with ZLACN2. In each step a
*           generalized Sylvester equation or a transposed variant
*           is solved.
*
            kase = 0
            n1 = m
            n2 = n - m
            i = n1 + 1
            ijb = 0
            mn2 = 2*n1*n2
*
*           1-norm-based estimate of Difu.
*
   40       CONTINUE
            CALL zlacn2( mn2, work( mn2+1 ), work, dif( 1 ), kase,
     $                   isave )
            IF( kase.NE.0 ) THEN
               IF( kase.EQ.1 ) THEN
*
*                 Solve generalized Sylvester equation
*
                  CALL ztgsyl( 'N', ijb, n1, n2, a, lda, a( i, i ), lda,
     $                         work, n1, b, ldb, b( i, i ), ldb,
     $                         work( n1*n2+1 ), n1, dscale, dif( 1 ),
     $                         work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
     $                         ierr )
               ELSE
*
*                 Solve the transposed variant.
*
                  CALL ztgsyl( 'C', ijb, n1, n2, a, lda, a( i, i ), lda,
     $                         work, n1, b, ldb, b( i, i ), ldb,
     $                         work( n1*n2+1 ), n1, dscale, dif( 1 ),
     $                         work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
     $                         ierr )
               END IF
               GO TO 40
            END IF
            dif( 1 ) = dscale / dif( 1 )
*
*           1-norm-based estimate of Difl.
*
   50       CONTINUE
            CALL zlacn2( mn2, work( mn2+1 ), work, dif( 2 ), kase,
     $                   isave )
            IF( kase.NE.0 ) THEN
               IF( kase.EQ.1 ) THEN
*
*                 Solve generalized Sylvester equation
*
                  CALL ztgsyl( 'N', ijb, n2, n1, a( i, i ), lda, a, lda,
     $                         work, n2, b( i, i ), ldb, b, ldb,
     $                         work( n1*n2+1 ), n2, dscale, dif( 2 ),
     $                         work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
     $                         ierr )
               ELSE
*
*                 Solve the transposed variant.
*
                  CALL ztgsyl( 'C', ijb, n2, n1, a( i, i ), lda, a, lda,
     $                         work, n2, b, ldb, b( i, i ), ldb,
     $                         work( n1*n2+1 ), n2, dscale, dif( 2 ),
     $                         work( n1*n2*2+1 ), lwork-2*n1*n2, iwork,
     $                         ierr )
               END IF
               GO TO 50
            END IF
            dif( 2 ) = dscale / dif( 2 )
         END IF
      END IF
*
*     If B(K,K) is complex, make it real and positive (normalization
*     of the generalized Schur form) and Store the generalized
*     eigenvalues of reordered pair (A, B)
*
      DO 60 k = 1, n
         dscale = abs( b( k, k ) )
         IF( dscale.GT.safmin ) THEN
            temp1 = dconjg( b( k, k ) / dscale )
            temp2 = b( k, k ) / dscale
            b( k, k ) = dscale
            CALL zscal( n-k, temp1, b( k, k+1 ), ldb )
            CALL zscal( n-k+1, temp1, a( k, k ), lda )
            IF( wantq )
     $         CALL zscal( n, temp2, q( 1, k ), 1 )
         ELSE
            b( k, k ) = dcmplx( zero, zero )
         END IF
*
         alpha( k ) = a( k, k )
         beta( k ) = b( k, k )
*
   60 CONTINUE
*
   70 CONTINUE
*
      work( 1 ) = lwmin
      iwork( 1 ) = liwmin
*
      RETURN
*
*     End of ZTGSEN
*
      END