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   *> \brief \b ZTGSNA
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZTGSNA + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ztgsna.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ztgsna.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ztgsna.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZTGSNA( 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   DIF( * ), S( * )
   *       COMPLEX*16         A( LDA, * ), B( LDB, * ), VL( LDVL, * ),
   *      $                   VR( LDVR, * ), WORK( * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *> ZTGSNA estimates reciprocal condition numbers for specified
   *> eigenvalues and/or eigenvectors of a matrix pair (A, B).
   *>
   *> (A, B) must be in generalized Schur canonical form, that is, A and
   *> B are both 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 corresponding j-th eigenvalue and/or eigenvector,
   *>          SELECT(j) 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 COMPLEX*16 array, dimension (LDA,N)
   *>          The upper 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 COMPLEX*16 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 COMPLEX*16 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 ZTGEVC.
   *>          If JOB = 'V', VL is not referenced.
   *> \endverbatim
   *>
   *> \param[in] LDVL
   *> \verbatim
   *>          LDVL is INTEGER
   *>          The leading dimension of the array VL. LDVL >= 1; and
   *>          If JOB = 'E' or 'B', LDVL >= N.
   *> \endverbatim
   *>
   *> \param[in] VR
   *> \verbatim
   *>          VR is COMPLEX*16 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 of VR, as returned by ZTGEVC.
   *>          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.
   *>          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.
   *>          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.
   *>          For each eigenvalue/vector specified by SELECT, DIF stores
   *>          a Frobenius norm-based estimate of Difl.
   *>          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 eigenvalue
   *>          one element is used. If HOWMNY = 'A', M is set to N.
   *> \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 >= max(1,N).
   *>          If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).
   *> \endverbatim
   *>
   *> \param[out] IWORK
   *> \verbatim
   *>          IWORK is INTEGER array, dimension (N+2)
   *>          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 complex16OTHERcomputational
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>  The reciprocal of the condition number of the i-th generalized
   *>  eigenvalue w = (a, b) is defined as
   *>
   *>          S(I) = (|v**HAu|**2 + |v**HBu|**2)**(1/2) / (norm(u)*norm(v))
   *>
   *>  where u and v are the right and left 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 (= v**HAu/v**HBu) 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 of the right eigenvector u
   *>  and left eigenvector v corresponding to the generalized eigenvalue w
   *>  is defined as follows. Suppose
   *>
   *>                   (A, B) = ( a   *  ) ( b  *  )  1
   *>                            ( 0  A22 ),( 0 B22 )  n-1
   *>                              1  n-1     1 n-1
   *>
   *>  Then the reciprocal condition number DIF(I) is
   *>
   *>          Difl[(a, b), (A22, B22)]  = sigma-min( Zl )
   *>
   *>  where sigma-min(Zl) denotes the smallest singular value of
   *>
   *>         Zl = [ kron(a, In-1) -kron(1, A22) ]
   *>              [ kron(b, In-1) -kron(1, B22) ].
   *>
   *>  Here In-1 is the identity matrix of size n-1 and X**H is the conjugate
   *>  transpose of X. kron(X, Y) is the Kronecker product between the
   *>  matrices X and Y.
   *>
   *>  We approximate the smallest singular value of Zl with an upper
   *>  bound. This is done by ZLATDF.
   *>
   *>  An approximate error bound for a 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 ZTGSNA( JOB, HOWMNY, SELECT, N, A, LDA, B, LDB, VL,        SUBROUTINE ZTGSNA( 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 19 Line 328
      $                   VR( LDVR, * ), WORK( * )       $                   VR( LDVR, * ), WORK( * )
 *     ..  *     ..
 *  *
 *  Purpose  
 *  =======  
 *  
 *  ZTGSNA estimates reciprocal condition numbers for specified  
 *  eigenvalues and/or eigenvectors of a matrix pair (A, B).  
 *  
 *  (A, B) must be in generalized Schur canonical form, that is, A and  
 *  B are both 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 corresponding j-th eigenvalue and/or eigenvector,  
 *          SELECT(j) 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) COMPLEX*16 array, dimension (LDA,N)  
 *          The upper triangular matrix A in the pair (A,B).  
 *  
 *  LDA     (input) INTEGER  
 *          The leading dimension of the array A. LDA >= max(1,N).  
 *  
 *  B       (input) COMPLEX*16 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) COMPLEX*16 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 ZTGEVC.  
 *          If JOB = 'V', VL is not referenced.  
 *  
 *  LDVL    (input) INTEGER  
 *          The leading dimension of the array VL. LDVL >= 1; and  
 *          If JOB = 'E' or 'B', LDVL >= N.  
 *  
 *  VR      (input) COMPLEX*16 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 of VR, as returned by ZTGEVC.  
 *          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.  
 *          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.  
 *          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.  
 *          For each eigenvalue/vector specified by SELECT, DIF stores  
 *          a Frobenius norm-based estimate of Difl.  
 *          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 eigenvalue  
 *          one element is used. If HOWMNY = 'A', M is set to N.  
 *  
 *  WORK    (workspace/output) COMPLEX*16 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 >= max(1,2*N*N).  
 *  
 *  IWORK   (workspace) INTEGER array, dimension (N+2)  
 *          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 the i-th generalized  
 *  eigenvalue w = (a, b) is defined as  
 *  
 *          S(I) = (|v**HAu|**2 + |v**HBu|**2)**(1/2) / (norm(u)*norm(v))  
 *  
 *  where u and v are the right and left 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 (= v**HAu/v**HBu) 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 of the right eigenvector u  
 *  and left eigenvector v corresponding to the generalized eigenvalue w  
 *  is defined as follows. Suppose  
 *  
 *                   (A, B) = ( a   *  ) ( b  *  )  1  
 *                            ( 0  A22 ),( 0 B22 )  n-1  
 *                              1  n-1     1 n-1  
 *  
 *  Then the reciprocal condition number DIF(I) is  
 *  
 *          Difl[(a, b), (A22, B22)]  = sigma-min( Zl )  
 *  
 *  where sigma-min(Zl) denotes the smallest singular value of  
 *  
 *         Zl = [ kron(a, In-1) -kron(1, A22) ]  
 *              [ kron(b, In-1) -kron(1, B22) ].  
 *  
 *  Here In-1 is the identity matrix of size n-1 and X**H is the conjugate  
 *  transpose of X. kron(X, Y) is the Kronecker product between the  
 *  matrices X and Y.  
 *  
 *  We approximate the smallest singular value of Zl with an upper  
 *  bound. This is done by ZLATDF.  
 *  
 *  An approximate error bound for a 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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