CTREVC (3lapack)

compute some or all of the right and/or left eigenvectors of a complex upper triangular matrix T

SYNOPSIS

  SUBROUTINE CTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM,
                     M, WORK, RWORK, INFO )

      CHARACTER      HOWMNY, SIDE

      INTEGER        INFO, LDT, LDVL, LDVR, M, MM, N

      LOGICAL        SELECT( * )

      REAL           RWORK( * )

      COMPLEX        T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * )

PURPOSE

  CTREVC computes some or all of the right and/or left eigenvectors of a
  complex upper triangular matrix T.

  The right eigenvector x and the left eigenvector y of T corresponding to an
  eigenvalue w are defined by:

               T*x = w*x,     y'*T = w*y'

  where y' denotes the conjugate transpose of the vector y.

  If all eigenvectors are requested, the routine may either return the
  matrices X and/or Y of right or left eigenvectors of T, or the products Q*X
  and/or Q*Y, where Q is an input unitary
  matrix. If T was obtained from the Schur factorization of an original
  matrix A = Q*T*Q', then Q*X and Q*Y are the matrices of right or left
  eigenvectors of A.

ARGUMENTS

  SIDE    (input) CHARACTER*1
          = 'R':  compute right eigenvectors only;
          = 'L':  compute left eigenvectors only;
          = 'B':  compute both right and left eigenvectors.

  HOWMNY  (input) CHARACTER*1
          = 'A':  compute all right and/or left eigenvectors;
          = 'B':  compute all right and/or left eigenvectors, and
          backtransform them using the input matrices supplied in VR and/or
          VL; = 'S':  compute selected right and/or left eigenvectors,
          specified by the logical array SELECT.

  SELECT  (input) LOGICAL array, dimension (N)
          If HOWMNY = 'S', SELECT specifies the eigenvectors to be computed.
          If HOWMNY = 'A' or 'B', SELECT is not referenced.  To select the
          eigenvector corresponding to the j-th eigenvalue, SELECT(j) must be
          set to .TRUE..

  N       (input) INTEGER
          The order of the matrix T. N >= 0.

  T       (input/output) COMPLEX array, dimension (LDT,N)
          The upper triangular matrix T.  T is modified, but restored on
          exit.

  LDT     (input) INTEGER
          The leading dimension of the array T. LDT >= max(1,N).

  VL      (input/output) COMPLEX 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 unitary matrix Q of Schur vectors
          returned by CHSEQR).  On exit, if SIDE = 'L' or 'B', VL contains:
          if HOWMNY = 'A', the matrix Y of left eigenvectors of T; if HOWMNY
          = 'B', the matrix Q*Y; if HOWMNY = 'S', the left eigenvectors of T
          specified by SELECT, stored consecutively in the columns of VL, in
          the same order as their eigenvalues.  If SIDE = 'R', VL is not
          referenced.

  LDVL    (input) INTEGER
          The leading dimension of the array VL.  LDVL >= max(1,N) if SIDE =
          'L' or 'B'; LDVL >= 1 otherwise.

  VR      (input/output) COMPLEX array, dimension (LDVR,MM)
          On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must contain an
          N-by-N matrix Q (usually the unitary matrix Q of Schur vectors
          returned by CHSEQR).  On exit, if SIDE = 'R' or 'B', VR contains:
          if HOWMNY = 'A', the matrix X of right eigenvectors of T; if HOWMNY
          = 'B', the matrix Q*X; if HOWMNY = 'S', the right eigenvectors of T
          specified by SELECT, stored consecutively in the columns of VR, in
          the same order as their eigenvalues.  If SIDE = 'L', VR is not
          referenced.

  LDVR    (input) INTEGER
          The leading dimension of the array VR.  LDVR >= max(1,N) if SIDE =
          'R' or 'B'; LDVR >= 1 otherwise.

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

  M       (output) 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 eigenvector occupies one column.

  WORK    (workspace) COMPLEX array, dimension (2*N)

  RWORK   (workspace) REAL array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value

FURTHER DETAILS

  The algorithm used in this program is basically backward (forward)
  substitution, with scaling to make the the code robust against possible
  overflow.

  Each eigenvector is normalized so that the element of largest magnitude has
  magnitude 1; here the magnitude of a complex number (x,y) is taken to be
  |x| + |y|.