!>
!> CTGSNA 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.
!> 

JOB

!>          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).
!> 

HOWMNY

!>          HOWMNY is CHARACTER*1
!>          = 'A': compute condition numbers for all eigenpairs;
!>          = 'S': compute condition numbers for selected eigenpairs
!>                 specified by the array SELECT.
!> 

SELECT

!>          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.
!> 

N

!>          N is INTEGER
!>          The order of the square matrix pair (A, B). N >= 0.
!> 

A

!>          A is COMPLEX array, dimension (LDA,N)
!>          The upper triangular matrix A in the pair (A,B).
!> 

LDA

!>          LDA is INTEGER
!>          The leading dimension of the array A. LDA >= max(1,N).
!> 

B

!>          B is COMPLEX array, dimension (LDB,N)
!>          The upper triangular matrix B in the pair (A, B).
!> 

LDB

!>          LDB is INTEGER
!>          The leading dimension of the array B. LDB >= max(1,N).
!> 

VL

!>          VL is COMPLEX 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 CTGEVC.
!>          If JOB = 'V', VL is not referenced.
!> 

LDVL

!>          LDVL is INTEGER
!>          The leading dimension of the array VL. LDVL >= 1; and
!>          If JOB = 'E' or 'B', LDVL >= N.
!> 

VR

!>          VR is COMPLEX 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 CTGEVC.
!>          If JOB = 'V', VR is not referenced.
!> 

LDVR

!>          LDVR is INTEGER
!>          The leading dimension of the array VR. LDVR >= 1;
!>          If JOB = 'E' or 'B', LDVR >= N.
!> 

S

!>          S is REAL 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

!>          DIF is REAL 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

!>          MM is INTEGER
!>          The number of elements in the arrays S and DIF. MM >= M.
!> 

M

!>          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.
!> 

WORK

!>          WORK is COMPLEX array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!> 

LWORK

!>          LWORK is INTEGER
!>          The dimension of the array WORK. LWORK >= max(1,N).
!>          If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).
!> 

IWORK

!>          IWORK is INTEGER array, dimension (N+2)
!>          If JOB = 'E', IWORK is not referenced.
!> 

INFO

!>          INFO is INTEGER
!>          = 0: Successful exit
!>          < 0: If INFO = -i, the i-th argument had an illegal value
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

!>
!>  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 CLATDF.
!>
!>  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.
!> 

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

!>
!>  [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.
!> 

!>
!> 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.
!>
!> 

JOB

!>          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).
!> 

HOWMNY

!>          HOWMNY is CHARACTER*1
!>          = 'A': compute condition numbers for all eigenpairs;
!>          = 'S': compute condition numbers for selected eigenpairs
!>                 specified by the array SELECT.
!> 

SELECT

!>          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.
!> 

N

!>          N is INTEGER
!>          The order of the square matrix pair (A, B). N >= 0.
!> 

A

!>          A is DOUBLE PRECISION array, dimension (LDA,N)
!>          The upper quasi-triangular matrix A in the pair (A,B).
!> 

LDA

!>          LDA is INTEGER
!>          The leading dimension of the array A. LDA >= max(1,N).
!> 

B

!>          B is DOUBLE PRECISION array, dimension (LDB,N)
!>          The upper triangular matrix B in the pair (A,B).
!> 

LDB

!>          LDB is INTEGER
!>          The leading dimension of the array B. LDB >= max(1,N).
!> 

VL

!>          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.
!> 

LDVL

!>          LDVL is INTEGER
!>          The leading dimension of the array VL. LDVL >= 1.
!>          If JOB = 'E' or 'B', LDVL >= N.
!> 

VR

!>          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.
!> 

LDVR

!>          LDVR is INTEGER
!>          The leading dimension of the array VR. LDVR >= 1.
!>          If JOB = 'E' or 'B', LDVR >= N.
!> 

S

!>          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.
!> 

DIF

!>          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.
!> 

MM

!>          MM is INTEGER
!>          The number of elements in the arrays S and DIF. MM >= M.
!> 

M

!>          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.
!> 

WORK

!>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!> 

LWORK

!>          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.
!> 

IWORK

!>          IWORK is INTEGER array, dimension (N + 6)
!>          If JOB = 'E', IWORK is not referenced.
!> 

INFO

!>          INFO is INTEGER
!>          =0: Successful exit
!>          <0: If INFO = -i, the i-th argument had an illegal value
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

!>
!>  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.
!> 

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

!>
!>  [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.
!> 

!>
!> STGSNA 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 SGGES),
!> i.e. A is block upper triangular with 1-by-1 and 2-by-2 diagonal
!> blocks. B is upper triangular.
!>
!> 

JOB

!>          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).
!> 

HOWMNY

!>          HOWMNY is CHARACTER*1
!>          = 'A': compute condition numbers for all eigenpairs;
!>          = 'S': compute condition numbers for selected eigenpairs
!>                 specified by the array SELECT.
!> 

SELECT

!>          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.
!> 

N

!>          N is INTEGER
!>          The order of the square matrix pair (A, B). N >= 0.
!> 

A

!>          A is REAL array, dimension (LDA,N)
!>          The upper quasi-triangular matrix A in the pair (A,B).
!> 

LDA

!>          LDA is INTEGER
!>          The leading dimension of the array A. LDA >= max(1,N).
!> 

B

!>          B is REAL array, dimension (LDB,N)
!>          The upper triangular matrix B in the pair (A,B).
!> 

LDB

!>          LDB is INTEGER
!>          The leading dimension of the array B. LDB >= max(1,N).
!> 

VL

!>          VL is REAL 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 STGEVC.
!>          If JOB = 'V', VL is not referenced.
!> 

LDVL

!>          LDVL is INTEGER
!>          The leading dimension of the array VL. LDVL >= 1.
!>          If JOB = 'E' or 'B', LDVL >= N.
!> 

VR

!>          VR is REAL 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 STGEVC.
!>          If JOB = 'V', VR is not referenced.
!> 

LDVR

!>          LDVR is INTEGER
!>          The leading dimension of the array VR. LDVR >= 1.
!>          If JOB = 'E' or 'B', LDVR >= N.
!> 

S

!>          S is REAL 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

!>          DIF is REAL 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

!>          MM is INTEGER
!>          The number of elements in the arrays S and DIF. MM >= M.
!> 

M

!>          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.
!> 

WORK

!>          WORK is REAL array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!> 

LWORK

!>          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.
!> 

IWORK

!>          IWORK is INTEGER array, dimension (N + 6)
!>          If JOB = 'E', IWORK is not referenced.
!> 

INFO

!>          INFO is INTEGER
!>          =0: Successful exit
!>          <0: If INFO = -i, the i-th argument had an illegal value
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

!>
!>  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 SLATDF), then the parameter DIFDRI (see below) should be
!>     changed from 3 to 4 (routine SLATDF(IJOB = 2 will be used)).
!>     See STGSYL 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
!>     SLATDF), then the parameter DIFDRI (see below) should be changed
!>     from 3 to 4 (routine SLATDF(IJOB = 2 will be used)). See STGSYL
!>     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.
!> 

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

!>
!>  [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.
!> 

!>
!> 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.
!> 

JOB

!>          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).
!> 

HOWMNY

!>          HOWMNY is CHARACTER*1
!>          = 'A': compute condition numbers for all eigenpairs;
!>          = 'S': compute condition numbers for selected eigenpairs
!>                 specified by the array SELECT.
!> 

SELECT

!>          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.
!> 

N

!>          N is INTEGER
!>          The order of the square matrix pair (A, B). N >= 0.
!> 

A

!>          A is COMPLEX*16 array, dimension (LDA,N)
!>          The upper triangular matrix A in the pair (A,B).
!> 

LDA

!>          LDA is INTEGER
!>          The leading dimension of the array A. LDA >= max(1,N).
!> 

B

!>          B is COMPLEX*16 array, dimension (LDB,N)
!>          The upper triangular matrix B in the pair (A, B).
!> 

LDB

!>          LDB is INTEGER
!>          The leading dimension of the array B. LDB >= max(1,N).
!> 

VL

!>          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.
!> 

LDVL

!>          LDVL is INTEGER
!>          The leading dimension of the array VL. LDVL >= 1; and
!>          If JOB = 'E' or 'B', LDVL >= N.
!> 

VR

!>          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.
!> 

LDVR

!>          LDVR is INTEGER
!>          The leading dimension of the array VR. LDVR >= 1;
!>          If JOB = 'E' or 'B', LDVR >= N.
!> 

S

!>          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.
!> 

DIF

!>          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.
!> 

MM

!>          MM is INTEGER
!>          The number of elements in the arrays S and DIF. MM >= M.
!> 

M

!>          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.
!> 

WORK

!>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
!>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
!> 

LWORK

!>          LWORK is INTEGER
!>          The dimension of the array WORK. LWORK >= max(1,N).
!>          If JOB = 'V' or 'B', LWORK >= max(1,2*N*N).
!> 

IWORK

!>          IWORK is INTEGER array, dimension (N+2)
!>          If JOB = 'E', IWORK is not referenced.
!> 

INFO

!>          INFO is INTEGER
!>          = 0: Successful exit
!>          < 0: If INFO = -i, the i-th argument had an illegal value
!> 

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

!>
!>  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.
!> 

Bo Kagstrom and Peter Poromaa, Department of Computing Science, Umea University, S-901 87 Umea, Sweden.

!>
!>  [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.
!>