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</span><span class="comment">*</span><span class="comment">          only calculates the optimal size of the WORK array, returns
</span><span class="comment">*</span><span class="comment">          this value as the first entry of the WORK array, and no error
</span><span class="comment">*</span><span class="comment">          message related to LWORK is issued by <a name="XERBLA.136"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  IWORK   (workspace) INTEGER array, dimension (N + 6)
</span><span class="comment">*</span><span class="comment">          If JOB = 'E', IWORK is not referenced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  INFO    (output) INTEGER
</span><span class="comment">*</span><span class="comment">          =0: Successful exit
</span><span class="comment">*</span><span class="comment">          &lt;0: If INFO = -i, the i-th argument had an illegal value
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Further Details
</span><span class="comment">*</span><span class="comment">  ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  The reciprocal of the condition number of a generalized eigenvalue
</span><span class="comment">*</span><span class="comment">  w = (a, b) is defined as
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">       S(w) = (|u'Av|**2 + |u'Bv|**2)**(1/2) / (norm(u)*norm(v))
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where u and v are the left and right eigenvectors of (A, B)
</span><span class="comment">*</span><span class="comment">  corresponding to w; |z| denotes the absolute value of the complex
</span><span class="comment">*</span><span class="comment">  number, and norm(u) denotes the 2-norm of the vector u.
</span><span class="comment">*</span><span class="comment">  The pair (a, b) corresponds to an eigenvalue w = a/b (= u'Av/u'Bv)
</span><span class="comment">*</span><span class="comment">  of the matrix pair (A, B). If both a and b equal zero, then (A B) is
</span><span class="comment">*</span><span class="comment">  singular and S(I) = -1 is returned.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  An approximate error bound on the chordal distance between the i-th
</span><span class="comment">*</span><span class="comment">  computed generalized eigenvalue w and the corresponding exact
</span><span class="comment">*</span><span class="comment">  eigenvalue lambda is
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">       chord(w, lambda) &lt;= EPS * norm(A, B) / S(I)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where EPS is the machine precision.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  The reciprocal of the condition number DIF(i) of right eigenvector u
</span><span class="comment">*</span><span class="comment">  and left eigenvector v corresponding to the generalized eigenvalue w
</span><span class="comment">*</span><span class="comment">  is defined as follows:
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  a) If the i-th eigenvalue w = (a,b) is real
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Suppose U and V are orthogonal transformations such that
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">                U'*(A, B)*V  = (S, T) = ( a   *  ) ( b  *  )  1
</span><span class="comment">*</span><span class="comment">                                        ( 0  S22 ),( 0 T22 )  n-1
</span><span class="comment">*</span><span class="comment">                                          1  n-1     1 n-1
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Then the reciprocal condition number DIF(i) is
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">                Difl((a, b), (S22, T22)) = sigma-min( Zl ),
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     where sigma-min(Zl) denotes the smallest singular value of the
</span><span class="comment">*</span><span class="comment">     2(n-1)-by-2(n-1) matrix
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">         Zl = [ kron(a, In-1)  -kron(1, S22) ]
</span><span class="comment">*</span><span class="comment">              [ kron(b, In-1)  -kron(1, T22) ] .
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Here In-1 is the identity matrix of size n-1. kron(X, Y) is the
</span><span class="comment">*</span><span class="comment">     Kronecker product between the matrices X and Y.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Note that if the default method for computing DIF(i) is wanted
</span><span class="comment">*</span><span class="comment">     (see <a name="SLATDF.195"></a><a href="slatdf.f.html#SLATDF.1">SLATDF</a>), then the parameter DIFDRI (see below) should be
</span><span class="comment">*</span><span class="comment">     changed from 3 to 4 (routine <a name="SLATDF.196"></a><a href="slatdf.f.html#SLATDF.1">SLATDF</a>(IJOB = 2 will be used)).
</span><span class="comment">*</span><span class="comment">     See <a name="STGSYL.197"></a><a href="stgsyl.f.html#STGSYL.1">STGSYL</a> for more details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  b) If the i-th and (i+1)-th eigenvalues are complex conjugate pair,
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Suppose U and V are orthogonal transformations such that
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">                U'*(A, B)*V = (S, T) = ( S11  *   ) ( T11  *  )  2
</span><span class="comment">*</span><span class="comment">                                       ( 0    S22 ),( 0    T22) n-2
</span><span class="comment">*</span><span class="comment">                                         2    n-2     2    n-2
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     and (S11, T11) corresponds to the complex conjugate eigenvalue
</span><span class="comment">*</span><span class="comment">     pair (w, conjg(w)). There exist unitary matrices U1 and V1 such
</span><span class="comment">*</span><span class="comment">     that
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">         U1'*S11*V1 = ( s11 s12 )   and U1'*T11*V1 = ( t11 t12 )
</span><span class="comment">*</span><span class="comment">                      (  0  s22 )                    (  0  t22 )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     where the generalized eigenvalues w = s11/t11 and
</span><span class="comment">*</span><span class="comment">     conjg(w) = s22/t22.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Then the reciprocal condition number DIF(i) is bounded by
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">         min( d1, max( 1, |real(s11)/real(s22)| )*d2 )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     where, d1 = Difl((s11, t11), (s22, t22)) = sigma-min(Z1), where
</span><span class="comment">*</span><span class="comment">     Z1 is the complex 2-by-2 matrix
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">              Z1 =  [ s11  -s22 ]
</span><span class="comment">*</span><span class="comment">                    [ t11  -t22 ],
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     This is done by computing (using real arithmetic) the
</span><span class="comment">*</span><span class="comment">     roots of the characteristical polynomial det(Z1' * Z1 - lambda I),
</span><span class="comment">*</span><span class="comment">     where Z1' denotes the conjugate transpose of Z1 and det(X) denotes
</span><span class="comment">*</span><span class="comment">     the determinant of X.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     and d2 is an upper bound on Difl((S11, T11), (S22, T22)), i.e. an
</span><span class="comment">*</span><span class="comment">     upper bound on sigma-min(Z2), where Z2 is (2n-2)-by-(2n-2)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">              Z2 = [ kron(S11', In-2)  -kron(I2, S22) ]
</span><span class="comment">*</span><span class="comment">                   [ kron(T11', In-2)  -kron(I2, T22) ]
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Note that if the default method for computing DIF is wanted (see
</span><span class="comment">*</span><span class="comment">     <a name="SLATDF.239"></a><a href="slatdf.f.html#SLATDF.1">SLATDF</a>), then the parameter DIFDRI (see below) should be changed
</span><span class="comment">*</span><span class="comment">     from 3 to 4 (routine <a name="SLATDF.240"></a><a href="slatdf.f.html#SLATDF.1">SLATDF</a>(IJOB = 2 will be used)). See <a name="STGSYL.240"></a><a href="stgsyl.f.html#STGSYL.1">STGSYL</a>
</span><span class="comment">*</span><span class="comment">     for more details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  For each eigenvalue/vector specified by SELECT, DIF stores a
</span><span class="comment">*</span><span class="comment">  Frobenius norm-based estimate of Difl.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  An approximate error bound for the i-th computed eigenvector VL(i) or
</span><span class="comment">*</span><span class="comment">  VR(i) is given by
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">             EPS * norm(A, B) / DIF(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  See ref. [2-3] for more details and further references.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Based on contributions by
</span><span class="comment">*</span><span class="comment">     Bo Kagstrom and Peter Poromaa, Department of Computing Science,
</span><span class="comment">*</span><span class="comment">     Umea University, S-901 87 Umea, Sweden.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  References
</span><span class="comment">*</span><span class="comment">  ==========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  [1] B. Kagstrom; A Direct Method for Reordering Eigenvalues in the
</span><span class="comment">*</span><span class="comment">      Generalized Real Schur Form of a Regular Matrix Pair (A, B), in
</span><span class="comment">*</span><span class="comment">      M.S. Moonen et al (eds), Linear Algebra for Large Scale and
</span><span class="comment">*</span><span class="comment">      Real-Time Applications, Kluwer Academic Publ. 1993, pp 195-218.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  [2] B. Kagstrom and P. Poromaa; Computing Eigenspaces with Specified
</span><span class="comment">*</span><span class="comment">      Eigenvalues of a Regular Matrix Pair (A, B) and Condition
</span><span class="comment">*</span><span class="comment">      Estimation: Theory, Algorithms and Software,
</span><span class="comment">*</span><span class="comment">      Report UMINF - 94.04, Department of Computing Science, Umea
</span><span class="comment">*</span><span class="comment">      University, S-901 87 Umea, Sweden, 1994. Also as LAPACK Working
</span><span class="comment">*</span><span class="comment">      Note 87. To appear in Numerical Algorithms, 1996.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  [3] B. Kagstrom and P. Poromaa, LAPACK-Style Algorithms and Software
</span><span class="comment">*</span><span class="comment">      for Solving the Generalized Sylvester Equation and Estimating the
</span><span class="comment">*</span><span class="comment">      Separation between Regular Matrix Pairs, Report UMINF - 93.23,
</span><span class="comment">*</span><span class="comment">      Department of Computing Science, Umea University, S-901 87 Umea,
</span><span class="comment">*</span><span class="comment">      Sweden, December 1993, Revised April 1994, Also as LAPACK Working
</span><span class="comment">*</span><span class="comment">      Note 75.  To appear in ACM Trans. on Math. Software, Vol 22,
</span><span class="comment">*</span><span class="comment">      No 1, 1996.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Parameters ..
</span>      INTEGER            DIFDRI
      PARAMETER          ( DIFDRI = 3 )
      REAL               ZERO, ONE, TWO, FOUR