TGEVC - 5.2 English - 68552

TGEVC computes some or all of the right and/or left eigenvectors

of a pair of real matrices.

Purpose:

    TGEVC 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 SGGHRD + SHGEQZ.

    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 tranpose 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).

Parameters:

• SIDE – [in]

  SIDE is CHARACTER*1

  = ‘R’: compute right eigenvectors only;

  = ‘L’: compute left eigenvectors only;

  = ‘B’: compute both right and left eigenvectors.

• HOWMNY – [in]

  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 – [in]

  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 – [in]

  N is INTEGER

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

• S – [in]

  S is REAL array, dimension (LDS,N)

  The upper quasi-triangular matrix S from a generalized Schur factorization, as computed by SHGEQZ.

• LDS – [in]

  LDS is INTEGER

  The leading dimension of array S. LDS >= fla_max(1,N).

• P – [in]

  P is REAL array, dimension (LDP,N)

  The upper triangular matrix P from a generalized Schur factorization, as computed by SHGEQZ. 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S must be in positive diagonal form.

• LDP – [in]

  LDP is INTEGER

  The leading dimension of array P. LDP >= fla_max(1,N).

• VL – [inout]

  VL is REAL 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 SHGEQZ).

  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 – [in]

  LDVL is INTEGER

  The leading dimension of array VL. LDVL >= 1, and if SIDE = ‘L’ or ‘B’, LDVL >= N.

• VR – [inout]

  VR is REAL 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 SHGEQZ).

  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 – [in]

  LDVR is INTEGER

  The leading dimension of the array VR. LDVR >= 1, and if SIDE = ‘R’ or ‘B’, LDVR >= N.

• MM – [in]

  MM is INTEGER

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

• M – [out]

  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 – [out] WORK is REAL array, dimension (6*N)

• INFO – [out]

  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.