!>
!> DTGEVC computes some or all of the right and/or left eigenvectors of
!> a pair of real matrices (S,P), where S is a quasi-triangular matrix
!> and P is upper triangular.  Matrix pairs of this type are produced by
!> the generalized Schur factorization of a matrix pair (A,B):
!>
!>    A = Q*S*Z**T,  B = Q*P*Z**T
!>
!> as computed by DGGHRD + DHGEQZ.
!>
!> The right eigenvector x and the left eigenvector y of (S,P)
!> corresponding to an eigenvalue w are defined by:
!>
!>    S*x = w*P*x,  (y**H)*S = w*(y**H)*P,
!>
!> where y**H denotes the conjugate transpose of y.
!> The eigenvalues are not input to this routine, but are computed
!> directly from the diagonal blocks of S and P.
!>
!> This routine returns the matrices X and/or Y of right and left
!> eigenvectors of (S,P), or the products Z*X and/or Q*Y,
!> where Z and Q are input matrices.
!> If Q and Z are the orthogonal factors from the generalized Schur
!> factorization of a matrix pair (A,B), then Z*X and Q*Y
!> are the matrices of right and left eigenvectors of (A,B).
!>
!> 

SIDE

!>          SIDE is CHARACTER*1
!>          = 'R': compute right eigenvectors only;
!>          = 'L': compute left eigenvectors only;
!>          = 'B': compute both right and left eigenvectors.
!> 

HOWMNY

!>          HOWMNY is CHARACTER*1
!>          = 'A': compute all right and/or left eigenvectors;
!>          = 'B': compute all right and/or left eigenvectors,
!>                 backtransformed by the matrices in VR and/or VL;
!>          = 'S': compute selected right and/or left eigenvectors,
!>                 specified by the logical array SELECT.
!> 

SELECT

!>          SELECT is LOGICAL array, dimension (N)
!>          If HOWMNY='S', SELECT specifies the eigenvectors to be
!>          computed.  If w(j) is a real eigenvalue, the corresponding
!>          real eigenvector is computed if SELECT(j) is .TRUE..
!>          If w(j) and w(j+1) are the real and imaginary parts of a
!>          complex eigenvalue, the corresponding complex eigenvector
!>          is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,
!>          and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is
!>          set to .FALSE..
!>          Not referenced if HOWMNY = 'A' or 'B'.
!> 

N

!>          N is INTEGER
!>          The order of the matrices S and P.  N >= 0.
!> 

S

!>          S is DOUBLE PRECISION array, dimension (LDS,N)
!>          The upper quasi-triangular matrix S from a generalized Schur
!>          factorization, as computed by DHGEQZ.
!> 

LDS

!>          LDS is INTEGER
!>          The leading dimension of array S.  LDS >= max(1,N).
!> 

P

!>          P is DOUBLE PRECISION array, dimension (LDP,N)
!>          The upper triangular matrix P from a generalized Schur
!>          factorization, as computed by DHGEQZ.
!>          2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks
!>          of S must be in positive diagonal form.
!> 

LDP

!>          LDP is INTEGER
!>          The leading dimension of array P.  LDP >= max(1,N).
!> 

VL

!>          VL is DOUBLE PRECISION array, dimension (LDVL,MM)
!>          On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must
!>          contain an N-by-N matrix Q (usually the orthogonal matrix Q
!>          of left Schur vectors returned by DHGEQZ).
!>          On exit, if SIDE = 'L' or 'B', VL contains:
!>          if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);
!>          if HOWMNY = 'B', the matrix Q*Y;
!>          if HOWMNY = 'S', the left eigenvectors of (S,P) specified by
!>                      SELECT, stored consecutively in the columns of
!>                      VL, in the same order as their eigenvalues.
!>
!>          A complex eigenvector corresponding to a complex eigenvalue
!>          is stored in two consecutive columns, the first holding the
!>          real part, and the second the imaginary part.
!>
!>          Not referenced if SIDE = 'R'.
!> 

LDVL

!>          LDVL is INTEGER
!>          The leading dimension of array VL.  LDVL >= 1, and if
!>          SIDE = 'L' or 'B', LDVL >= N.
!> 

VR

!>          VR is DOUBLE PRECISION array, dimension (LDVR,MM)
!>          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must
!>          contain an N-by-N matrix Z (usually the orthogonal matrix Z
!>          of right Schur vectors returned by DHGEQZ).
!>
!>          On exit, if SIDE = 'R' or 'B', VR contains:
!>          if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);
!>          if HOWMNY = 'B' or 'b', the matrix Z*X;
!>          if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)
!>                      specified by SELECT, stored consecutively in the
!>                      columns of VR, in the same order as their
!>                      eigenvalues.
!>
!>          A complex eigenvector corresponding to a complex eigenvalue
!>          is stored in two consecutive columns, the first holding the
!>          real part and the second the imaginary part.
!>
!>          Not referenced if SIDE = 'L'.
!> 

LDVR

!>          LDVR is INTEGER
!>          The leading dimension of the array VR.  LDVR >= 1, and if
!>          SIDE = 'R' or 'B', LDVR >= N.
!> 

MM

!>          MM is INTEGER
!>          The number of columns in the arrays VL and/or VR. MM >= M.
!> 

M

!>          M is INTEGER
!>          The number of columns in the arrays VL and/or VR actually
!>          used to store the eigenvectors.  If HOWMNY = 'A' or 'B', M
!>          is set to N.  Each selected real eigenvector occupies one
!>          column and each selected complex eigenvector occupies two
!>          columns.
!> 

WORK

!>          WORK is DOUBLE PRECISION array, dimension (6*N)
!> 

INFO

!>          INFO is INTEGER
!>          = 0:  successful exit.
!>          < 0:  if INFO = -i, the i-th argument had an illegal value.
!>          > 0:  the 2-by-2 block (INFO:INFO+1) does not have a complex
!>                eigenvalue.
!> 

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!>
!>  Allocation of workspace:
!>  ---------- -- ---------
!>
!>     WORK( j ) = 1-norm of j-th column of A, above the diagonal
!>     WORK( N+j ) = 1-norm of j-th column of B, above the diagonal
!>     WORK( 2*N+1:3*N ) = real part of eigenvector
!>     WORK( 3*N+1:4*N ) = imaginary part of eigenvector
!>     WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector
!>     WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector
!>
!>  Rowwise vs. columnwise solution methods:
!>  ------- --  ---------- -------- -------
!>
!>  Finding a generalized eigenvector consists basically of solving the
!>  singular triangular system
!>
!>   (A - w B) x = 0     (for right) or:   (A - w B)**H y = 0  (for left)
!>
!>  Consider finding the i-th right eigenvector (assume all eigenvalues
!>  are real). The equation to be solved is:
!>       n                   i
!>  0 = sum  C(j,k) v(k)  = sum  C(j,k) v(k)     for j = i,. . .,1
!>      k=j                 k=j
!>
!>  where  C = (A - w B)  (The components v(i+1:n) are 0.)
!>
!>  The  method is:
!>
!>  (1)  v(i) := 1
!>  for j = i-1,. . .,1:
!>                          i
!>      (2) compute  s = - sum C(j,k) v(k)   and
!>                        k=j+1
!>
!>      (3) v(j) := s / C(j,j)
!>
!>  Step 2 is sometimes called the  step, since it is an
!>  inner product between the j-th row and the portion of the eigenvector
!>  that has been computed so far.
!>
!>  The  method consists basically in doing the sums
!>  for all the rows in parallel.  As each v(j) is computed, the
!>  contribution of v(j) times the j-th column of C is added to the
!>  partial sums.  Since FORTRAN arrays are stored columnwise, this has
!>  the advantage that at each step, the elements of C that are accessed
!>  are adjacent to one another, whereas with the rowwise method, the
!>  elements accessed at a step are spaced LDS (and LDP) words apart.
!>
!>  When finding left eigenvectors, the matrix in question is the
!>  transpose of the one in storage, so the rowwise method then
!>  actually accesses columns of A and B at each step, and so is the
!>  preferred method.
!>