CXML
        	    
LAPACK version 3.0
chsein(3)

PURPOSE
  CHSEIN - use inverse iteration to find specified right and/or left
  eigenvectors of a complex upper Hessenberg matrix H

SYNTAX
  SUBROUTINE CHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL, LDVL, VR,
		     LDVR, MM, M, WORK, RWORK, IFAILL, IFAILR, INFO )

      CHARACTER	     EIGSRC, INITV, SIDE

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

      LOGICAL	     SELECT( * )

      INTEGER	     IFAILL( * ), IFAILR( * )

      REAL	     RWORK( * )

      COMPLEX	     H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ), W( * ), WORK(
		     * )

DESCRIPTION
  CHSEIN uses inverse iteration to find specified right and/or left
  eigenvectors of a complex upper Hessenberg matrix H.	The right eigenvector
  x and the left eigenvector y of the matrix H corresponding to an eigenvalue
  w are defined by:

	       H * x = w * x,	  y**h * H = w * y**h

  where y**h denotes the conjugate transpose of the vector y.

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

  EIGSRC  (input) CHARACTER*1
	  Specifies the source of eigenvalues supplied in W:
	  = 'Q': the eigenvalues were found using CHSEQR; thus, if H has zero
	  subdiagonal elements, and so is block-triangular, then the j-th
	  eigenvalue can be assumed to be an eigenvalue of the block
	  containing the j-th row/column.  This property allows CHSEIN to
	  perform inverse iteration on just one diagonal block.	 = 'N': no
	  assumptions are made on the correspondence between eigenvalues and
	  diagonal blocks.  In this case, CHSEIN must always perform inverse
	  iteration using the whole matrix H.

  INITV	  (input) CHARACTER*1
	  = 'N': no initial vectors are supplied;
	  = 'U': user-supplied initial vectors are stored in the arrays VL
	  and/or VR.

  SELECT  (input) LOGICAL array, dimension (N)
	  Specifies the eigenvectors to be computed. To select the
	  eigenvector corresponding to the eigenvalue W(j), SELECT(j) must be
	  set to .TRUE..

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

  H	  (input) COMPLEX array, dimension (LDH,N)
	  The upper Hessenberg matrix H.

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

  W	  (input/output) COMPLEX array, dimension (N)
	  On entry, the eigenvalues of H.  On exit, the real parts of W may
	  have been altered since close eigenvalues are perturbed slightly in
	  searching for independent eigenvectors.

  VL	  (input/output) COMPLEX array, dimension (LDVL,MM)
	  On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must contain
	  starting vectors for the inverse iteration for the left
	  eigenvectors; the starting vector for each eigenvector must be in
	  the same column in which the eigenvector will be stored.  On exit,
	  if SIDE = 'L' or 'B', the left eigenvectors specified by SELECT
	  will be 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 INITV = 'U' and SIDE = 'R' or 'B', VR must contain
	  starting vectors for the inverse iteration for the right
	  eigenvectors; the starting vector for each eigenvector must be in
	  the same column in which the eigenvector will be stored.  On exit,
	  if SIDE = 'R' or 'B', the right eigenvectors specified by SELECT
	  will be 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 required to store
	  the eigenvectors (= the number of .TRUE. elements in SELECT).

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

  RWORK	  (workspace) REAL array, dimension (N)

  IFAILL  (output) INTEGER array, dimension (MM)
	  If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left eigenvector in
	  the i-th column of VL (corresponding to the eigenvalue w(j)) failed
	  to converge; IFAILL(i) = 0 if the eigenvector converged
	  satisfactorily.  If SIDE = 'R', IFAILL is not referenced.

  IFAILR  (output) INTEGER array, dimension (MM)
	  If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right eigenvector in
	  the i-th column of VR (corresponding to the eigenvalue w(j)) failed
	  to converge; IFAILR(i) = 0 if the eigenvector converged
	  satisfactorily.  If SIDE = 'L', IFAILR is not referenced.

  INFO	  (output) INTEGER
	  = 0:	successful exit
	  < 0:	if INFO = -i, the i-th argument had an illegal value
	  > 0:	if INFO = i, i is the number of eigenvectors which failed to
	  converge; see IFAILL and IFAILR for further details.

FURTHER DETAILS
  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|.