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   *> \brief \b DTGSNA
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DTGSNA + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dtgsna.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dtgsna.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dtgsna.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
   *                          LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
   *                          IWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       CHARACTER          HOWMNY, JOB
   *       INTEGER            INFO, LDA, LDB, LDVL, LDVR, LWORK, M, MM, N
   *       ..
   *       .. Array Arguments ..
   *       LOGICAL            SELECT( * )
   *       INTEGER            IWORK( * )
   *       DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), DIF( * ), S( * ),
   *      $                   VL( LDVL, * ), VR( LDVR, * ), WORK( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> DTGSNA estimates reciprocal condition numbers for specified
   *> eigenvalues and/or eigenvectors of a matrix pair (A, B) in
   *> generalized real Schur canonical form (or of any matrix pair
   *> (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where
   *> Z**T denotes the transpose of Z.
   *>
   *> (A, B) must be in generalized real Schur form (as returned by DGGES),
   *> i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
   *> blocks. B is upper triangular.
   *>
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOB
   *> \verbatim
   *>          JOB is CHARACTER*1
   *>          Specifies whether condition numbers are required for
   *>          eigenvalues (S) or eigenvectors (DIF):
   *>          = 'E': for eigenvalues only (S);
   *>          = 'V': for eigenvectors only (DIF);
   *>          = 'B': for both eigenvalues and eigenvectors (S and DIF).
   *> \endverbatim
   *>
   *> \param[in] HOWMNY
   *> \verbatim
   *>          HOWMNY is CHARACTER*1
   *>          = 'A': compute condition numbers for all eigenpairs;
   *>          = 'S': compute condition numbers for selected eigenpairs
   *>                 specified by the array SELECT.
   *> \endverbatim
   *>
   *> \param[in] SELECT
   *> \verbatim
   *>          SELECT is LOGICAL array, dimension (N)
   *>          If HOWMNY = 'S', SELECT specifies the eigenpairs for which
   *>          condition numbers are required. To select condition numbers
   *>          for the eigenpair corresponding to a real eigenvalue w(j),
   *>          SELECT(j) must be set to .TRUE.. To select condition numbers
   *>          corresponding to a complex conjugate pair of eigenvalues w(j)
   *>          and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be
   *>          set to .TRUE..
   *>          If HOWMNY = 'A', SELECT is not referenced.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>          The order of the square matrix pair (A, B). N >= 0.
   *> \endverbatim
   *>
   *> \param[in] A
   *> \verbatim
   *>          A is DOUBLE PRECISION array, dimension (LDA,N)
   *>          The upper quasi-triangular matrix A in the pair (A,B).
   *> \endverbatim
   *>
   *> \param[in] LDA
   *> \verbatim
   *>          LDA is INTEGER
   *>          The leading dimension of the array A. LDA >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in] B
   *> \verbatim
   *>          B is DOUBLE PRECISION array, dimension (LDB,N)
   *>          The upper triangular matrix B in the pair (A,B).
   *> \endverbatim
   *>
   *> \param[in] LDB
   *> \verbatim
   *>          LDB is INTEGER
   *>          The leading dimension of the array B. LDB >= max(1,N).
   *> \endverbatim
   *>
   *> \param[in] VL
   *> \verbatim
   *>          VL is DOUBLE PRECISION array, dimension (LDVL,M)
   *>          If JOB = 'E' or 'B', VL must contain left eigenvectors of
   *>          (A, B), corresponding to the eigenpairs specified by HOWMNY
   *>          and SELECT. The eigenvectors must be stored in consecutive
   *>          columns of VL, as returned by DTGEVC.
   *>          If JOB = 'V', VL is not referenced.
   *> \endverbatim
   *>
   *> \param[in] LDVL
   *> \verbatim
   *>          LDVL is INTEGER
   *>          The leading dimension of the array VL. LDVL >= 1.
   *>          If JOB = 'E' or 'B', LDVL >= N.
   *> \endverbatim
   *>
   *> \param[in] VR
   *> \verbatim
   *>          VR is DOUBLE PRECISION array, dimension (LDVR,M)
   *>          If JOB = 'E' or 'B', VR must contain right eigenvectors of
   *>          (A, B), corresponding to the eigenpairs specified by HOWMNY
   *>          and SELECT. The eigenvectors must be stored in consecutive
   *>          columns ov VR, as returned by DTGEVC.
   *>          If JOB = 'V', VR is not referenced.
   *> \endverbatim
   *>
   *> \param[in] LDVR
   *> \verbatim
   *>          LDVR is INTEGER
   *>          The leading dimension of the array VR. LDVR >= 1.
   *>          If JOB = 'E' or 'B', LDVR >= N.
   *> \endverbatim
   *>
   *> \param[out] S
   *> \verbatim
   *>          S is DOUBLE PRECISION array, dimension (MM)
   *>          If JOB = 'E' or 'B', the reciprocal condition numbers of the
   *>          selected eigenvalues, stored in consecutive elements of the
   *>          array. For a complex conjugate pair of eigenvalues two
   *>          consecutive elements of S are set to the same value. Thus
   *>          S(j), DIF(j), and the j-th columns of VL and VR all
   *>          correspond to the same eigenpair (but not in general the
   *>          j-th eigenpair, unless all eigenpairs are selected).
   *>          If JOB = 'V', S is not referenced.
   *> \endverbatim
   *>
   *> \param[out] DIF
   *> \verbatim
   *>          DIF is DOUBLE PRECISION array, dimension (MM)
   *>          If JOB = 'V' or 'B', the estimated reciprocal condition
   *>          numbers of the selected eigenvectors, stored in consecutive
   *>          elements of the array. For a complex eigenvector two
   *>          consecutive elements of DIF are set to the same value. If
   *>          the eigenvalues cannot be reordered to compute DIF(j), DIF(j)
   *>          is set to 0; this can only occur when the true value would be
   *>          very small anyway.
   *>          If JOB = 'E', DIF is not referenced.
   *> \endverbatim
   *>
   *> \param[in] MM
   *> \verbatim
   *>          MM is INTEGER
   *>          The number of elements in the arrays S and DIF. MM >= M.
   *> \endverbatim
   *>
   *> \param[out] M
   *> \verbatim
   *>          M is INTEGER
   *>          The number of elements of the arrays S and DIF used to store
   *>          the specified condition numbers; for each selected real
   *>          eigenvalue one element is used, and for each selected complex
   *>          conjugate pair of eigenvalues, two elements are used.
   *>          If HOWMNY = 'A', M is set to N.
   *> \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,N).
   *>          If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+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 JOB = 'E', IWORK 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
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleOTHERcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  The reciprocal of the condition number of a generalized eigenvalue
   *>  w = (a, b) is defined as
   *>
   *>       S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v))
   *>
   *>  where u and v are the left and right eigenvectors of (A, B)
   *>  corresponding to w; |z| denotes the absolute value of the complex
   *>  number, and norm(u) denotes the 2-norm of the vector u.
   *>  The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv)
   *>  of the matrix pair (A, B). If both a and b equal zero, then (A B) is
   *>  singular and S(I) = -1 is returned.
   *>
   *>  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(A, B) / S(I)
   *>
   *>  where EPS is the machine precision.
   *>
   *>  The reciprocal of the condition number DIF(i) of right eigenvector u
   *>  and left eigenvector v corresponding to the generalized eigenvalue w
   *>  is defined as follows:
   *>
   *>  a) If the i-th eigenvalue w = (a,b) is real
   *>
   *>     Suppose U and V are orthogonal transformations such that
   *>
   *>              U**T*(A, B)*V  = (S, T) = ( a   *  ) ( b  *  )  1
   *>                                        ( 0  S22 ),( 0 T22 )  n-1
   *>                                          1  n-1     1 n-1
   *>
   *>     Then the reciprocal condition number DIF(i) is
   *>
   *>                Difl((a, b), (S22, T22)) = sigma-min( Zl ),
   *>
   *>     where sigma-min(Zl) denotes the smallest singular value of the
   *>     2(n-1)-by-2(n-1) matrix
   *>
   *>         Zl = [ kron(a, In-1)  -kron(1, S22) ]
   *>              [ kron(b, In-1)  -kron(1, T22) ] .
   *>
   *>     Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
   *>     Kronecker product between the matrices X and Y.
   *>
   *>     Note that if the default method for computing DIF(i) is wanted
   *>     (see DLATDF), then the parameter DIFDRI (see below) should be
   *>     changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)).
   *>     See DTGSYL for more details.
   *>
   *>  b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,
   *>
   *>     Suppose U and V are orthogonal transformations such that
   *>
   *>              U**T*(A, B)*V = (S, T) = ( S11  *   ) ( T11  *  )  2
   *>                                       ( 0    S22 ),( 0    T22) n-2
   *>                                         2    n-2     2    n-2
   *>
   *>     and (S11, T11) corresponds to the complex conjugate eigenvalue
   *>     pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
   *>     that
   *>
   *>       U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 )
   *>                      (  0  s22 )                    (  0  t22 )
   *>
   *>     where the generalized eigenvalues w = s11/t11 and
   *>     conjg(w) = s22/t22.
   *>
   *>     Then the reciprocal condition number DIF(i) is bounded by
   *>
   *>         min( d1, max( 1, |real(s11)/real(s22)| )*d2 )
   *>
   *>     where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
   *>     Z1 is the complex 2-by-2 matrix
   *>
   *>              Z1 =  [ s11  -s22 ]
   *>                    [ t11  -t22 ],
   *>
   *>     This is done by computing (using real arithmetic) the
   *>     roots of the characteristical polynomial det(Z1**T * Z1 - lambda I),
   *>     where Z1**T denotes the transpose of Z1 and det(X) denotes
   *>     the determinant of X.
   *>
   *>     and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
   *>     upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)
   *>
   *>              Z2 = [ kron(S11**T, In-2)  -kron(I2, S22) ]
   *>                   [ kron(T11**T, In-2)  -kron(I2, T22) ]
   *>
   *>     Note that if the default method for computing DIF is wanted (see
   *>     DLATDF), then the parameter DIFDRI (see below) should be changed
   *>     from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL
   *>     for more details.
   *>
   *>  For each eigenvalue/vector specified by SELECT, DIF stores a
   *>  Frobenius norm-based estimate of Difl.
   *>
   *>  An approximate error bound for the i-th computed eigenvector VL(i) or
   *>  VR(i) is given by
   *>
   *>             EPS * norm(A, B) / DIF(i).
   *>
   *>  See ref. [2-3] for more details and further references.
   *> \endverbatim
   *
   *> \par Contributors:
   *  ==================
   *>
   *>     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
   *>     Umea University, S-901 87 Umea, Sweden.
   *
   *> \par References:
   *  ================
   *>
   *> \verbatim
   *>
   *>  [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.
   *>
   *>  [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.
   *>
   *>  [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.
   *> \endverbatim
   *>
   *  =====================================================================
       SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,        SUBROUTINE DTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,
      $                   LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,       $                   LDVL, VR, LDVR, S, DIF, MM, M, WORK, LWORK,
      $                   IWORK, INFO )       $                   IWORK, INFO )
 *  *
 *  -- LAPACK routine (version 3.3.1) --  *  -- LAPACK computational routine (version 3.4.0) --
 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
 *  -- April 2011                                                      --  *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       CHARACTER          HOWMNY, JOB        CHARACTER          HOWMNY, JOB
Line 18 Line 397
      $                   VL( LDVL, * ), VR( LDVR, * ), WORK( * )       $                   VL( LDVL, * ), VR( LDVR, * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  DTGSNA estimates reciprocal condition numbers for specified  
 *  eigenvalues and/or eigenvectors of a matrix pair (A, B) in  
 *  generalized real Schur canonical form (or of any matrix pair  
 *  (Q*A*Z**T, Q*B*Z**T) with orthogonal matrices Q and Z, where  
 *  Z**T denotes the transpose of Z.  
 *  
 *  (A, B) must be in generalized real Schur form (as returned by DGGES),  
 *  i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal  
 *  blocks. B is upper triangular.  
 *  
 *  
 *  Arguments  
 *  =========  
 *  
 *  JOB     (input) CHARACTER*1  
 *          Specifies whether condition numbers are required for  
 *          eigenvalues (S) or eigenvectors (DIF):  
 *          = 'E': for eigenvalues only (S);  
 *          = 'V': for eigenvectors only (DIF);  
 *          = 'B': for both eigenvalues and eigenvectors (S and DIF).  
 *  
 *  HOWMNY  (input) CHARACTER*1  
 *          = 'A': compute condition numbers for all eigenpairs;  
 *          = 'S': compute condition numbers for selected eigenpairs  
 *                 specified by the array SELECT.  
 *  
 *  SELECT  (input) LOGICAL array, dimension (N)  
 *          If HOWMNY = 'S', SELECT specifies the eigenpairs for which  
 *          condition numbers are required. To select condition numbers  
 *          for the eigenpair corresponding to a real eigenvalue w(j),  
 *          SELECT(j) must be set to .TRUE.. To select condition numbers  
 *          corresponding to a complex conjugate pair of eigenvalues w(j)  
 *          and w(j+1), either SELECT(j) or SELECT(j+1) or both, must be  
 *          set to .TRUE..  
 *          If HOWMNY = 'A', SELECT is not referenced.  
 *  
 *  N       (input) INTEGER  
 *          The order of the square matrix pair (A, B). N >= 0.  
 *  
 *  A       (input) DOUBLE PRECISION array, dimension (LDA,N)  
 *          The upper quasi-triangular matrix A in the pair (A,B).  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A. LDA >= max(1,N).  
 *  
 *  B       (input) DOUBLE PRECISION array, dimension (LDB,N)  
 *          The upper triangular matrix B in the pair (A,B).  
 *  
 *  LDB     (input) INTEGER  
 *          The leading dimension of the array B. LDB >= max(1,N).  
 *  
 *  VL      (input) DOUBLE PRECISION array, dimension (LDVL,M)  
 *          If JOB = 'E' or 'B', VL must contain left eigenvectors of  
 *          (A, B), corresponding to the eigenpairs specified by HOWMNY  
 *          and SELECT. The eigenvectors must be stored in consecutive  
 *          columns of VL, as returned by DTGEVC.  
 *          If JOB = 'V', VL is not referenced.  
 *  
 *  LDVL    (input) INTEGER  
 *          The leading dimension of the array VL. LDVL >= 1.  
 *          If JOB = 'E' or 'B', LDVL >= N.  
 *  
 *  VR      (input) DOUBLE PRECISION array, dimension (LDVR,M)  
 *          If JOB = 'E' or 'B', VR must contain right eigenvectors of  
 *          (A, B), corresponding to the eigenpairs specified by HOWMNY  
 *          and SELECT. The eigenvectors must be stored in consecutive  
 *          columns ov VR, as returned by DTGEVC.  
 *          If JOB = 'V', VR is not referenced.  
 *  
 *  LDVR    (input) INTEGER  
 *          The leading dimension of the array VR. LDVR >= 1.  
 *          If JOB = 'E' or 'B', LDVR >= N.  
 *  
 *  S       (output) DOUBLE PRECISION array, dimension (MM)  
 *          If JOB = 'E' or 'B', the reciprocal condition numbers of the  
 *          selected eigenvalues, stored in consecutive elements of the  
 *          array. For a complex conjugate pair of eigenvalues two  
 *          consecutive elements of S are set to the same value. Thus  
 *          S(j), DIF(j), and the j-th columns of VL and VR all  
 *          correspond to the same eigenpair (but not in general the  
 *          j-th eigenpair, unless all eigenpairs are selected).  
 *          If JOB = 'V', S is not referenced.  
 *  
 *  DIF     (output) DOUBLE PRECISION array, dimension (MM)  
 *          If JOB = 'V' or 'B', the estimated reciprocal condition  
 *          numbers of the selected eigenvectors, stored in consecutive  
 *          elements of the array. For a complex eigenvector two  
 *          consecutive elements of DIF are set to the same value. If  
 *          the eigenvalues cannot be reordered to compute DIF(j), DIF(j)  
 *          is set to 0; this can only occur when the true value would be  
 *          very small anyway.  
 *          If JOB = 'E', DIF is not referenced.  
 *  
 *  MM      (input) INTEGER  
 *          The number of elements in the arrays S and DIF. MM >= M.  
 *  
 *  M       (output) INTEGER  
 *          The number of elements of the arrays S and DIF used to store  
 *          the specified condition numbers; for each selected real  
 *          eigenvalue one element is used, and for each selected complex  
 *          conjugate pair of eigenvalues, two elements are used.  
 *          If HOWMNY = 'A', M is set to N.  
 *  
 *  WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))  
 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.  
 *  
 *  LWORK   (input) INTEGER  
 *          The dimension of the array WORK. LWORK >= max(1,N).  
 *          If JOB = 'V' or 'B' LWORK >= 2*N*(N+2)+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.  
 *  
 *  IWORK   (workspace) INTEGER array, dimension (N + 6)  
 *          If JOB = 'E', IWORK is not referenced.  
 *  
 *  INFO    (output) INTEGER  
 *          =0: Successful exit  
 *          <0: If INFO = -i, the i-th argument had an illegal value  
 *  
 *  
 *  Further Details  
 *  ===============  
 *  
 *  The reciprocal of the condition number of a generalized eigenvalue  
 *  w = (a, b) is defined as  
 *  
 *       S(w) = (|u**TAv|**2 + |u**TBv|**2)**(1/2) / (norm(u)*norm(v))  
 *  
 *  where u and v are the left and right eigenvectors of (A, B)  
 *  corresponding to w; |z| denotes the absolute value of the complex  
 *  number, and norm(u) denotes the 2-norm of the vector u.  
 *  The pair (a, b) corresponds to an eigenvalue w = a/b (= u**TAv/u**TBv)  
 *  of the matrix pair (A, B). If both a and b equal zero, then (A B) is  
 *  singular and S(I) = -1 is returned.  
 *  
 *  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(A, B) / S(I)  
 *  
 *  where EPS is the machine precision.  
 *  
 *  The reciprocal of the condition number DIF(i) of right eigenvector u  
 *  and left eigenvector v corresponding to the generalized eigenvalue w  
 *  is defined as follows:  
 *  
 *  a) If the i-th eigenvalue w = (a,b) is real  
 *  
 *     Suppose U and V are orthogonal transformations such that  
 *  
 *              U**T*(A, B)*V  = (S, T) = ( a   *  ) ( b  *  )  1  
 *                                        ( 0  S22 ),( 0 T22 )  n-1  
 *                                          1  n-1     1 n-1  
 *  
 *     Then the reciprocal condition number DIF(i) is  
 *  
 *                Difl((a, b), (S22, T22)) = sigma-min( Zl ),  
 *  
 *     where sigma-min(Zl) denotes the smallest singular value of the  
 *     2(n-1)-by-2(n-1) matrix  
 *  
 *         Zl = [ kron(a, In-1)  -kron(1, S22) ]  
 *              [ kron(b, In-1)  -kron(1, T22) ] .  
 *  
 *     Here In-1 is the identity matrix of size n-1. kron(X, Y) is the  
 *     Kronecker product between the matrices X and Y.  
 *  
 *     Note that if the default method for computing DIF(i) is wanted  
 *     (see DLATDF), then the parameter DIFDRI (see below) should be  
 *     changed from 3 to 4 (routine DLATDF(IJOB = 2 will be used)).  
 *     See DTGSYL for more details.  
 *  
 *  b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,  
 *  
 *     Suppose U and V are orthogonal transformations such that  
 *  
 *              U**T*(A, B)*V = (S, T) = ( S11  *   ) ( T11  *  )  2  
 *                                       ( 0    S22 ),( 0    T22) n-2  
 *                                         2    n-2     2    n-2  
 *  
 *     and (S11, T11) corresponds to the complex conjugate eigenvalue  
 *     pair (w, conjg(w)). There exist unitary matrices U1 and V1 such  
 *     that  
 *  
 *       U1**T*S11*V1 = ( s11 s12 ) and U1**T*T11*V1 = ( t11 t12 )  
 *                      (  0  s22 )                    (  0  t22 )  
 *  
 *     where the generalized eigenvalues w = s11/t11 and  
 *     conjg(w) = s22/t22.  
 *  
 *     Then the reciprocal condition number DIF(i) is bounded by  
 *  
 *         min( d1, max( 1, |real(s11)/real(s22)| )*d2 )  
 *  
 *     where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where  
 *     Z1 is the complex 2-by-2 matrix  
 *  
 *              Z1 =  [ s11  -s22 ]  
 *                    [ t11  -t22 ],  
 *  
 *     This is done by computing (using real arithmetic) the  
 *     roots of the characteristical polynomial det(Z1**T * Z1 - lambda I),  
 *     where Z1**T denotes the transpose of Z1 and det(X) denotes  
 *     the determinant of X.  
 *  
 *     and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an  
 *     upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)  
 *  
 *              Z2 = [ kron(S11**T, In-2)  -kron(I2, S22) ]  
 *                   [ kron(T11**T, In-2)  -kron(I2, T22) ]  
 *  
 *     Note that if the default method for computing DIF is wanted (see  
 *     DLATDF), then the parameter DIFDRI (see below) should be changed  
 *     from 3 to 4 (routine DLATDF(IJOB = 2 will be used)). See DTGSYL  
 *     for more details.  
 *  
 *  For each eigenvalue/vector specified by SELECT, DIF stores a  
 *  Frobenius norm-based estimate of Difl.  
 *  
 *  An approximate error bound for the i-th computed eigenvector VL(i) or  
 *  VR(i) is given by  
 *  
 *             EPS * norm(A, B) / DIF(i).  
 *  
 *  See ref. [2-3] for more details and further references.  
 *  
 *  Based on contributions by  
 *     Bo Kagstrom and Peter Poromaa, Department of Computing Science,  
 *     Umea University, S-901 87 Umea, Sweden.  
 *  
 *  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.  
 *  
 *  [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.  
 *  
 *  [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.  
 *  
 *  =====================================================================  *  =====================================================================
 *  *
 *     .. Parameters ..  *     .. Parameters ..
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