CXML
        	    
LAPACK version 3.0
dhsein(3)

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

SYNTAX
  SUBROUTINE DHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, WR, WI, VL,
		     LDVL, VR, LDVR, MM, M, WORK, IFAILL, IFAILR, INFO )

      CHARACTER	     EIGSRC, INITV, SIDE

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

      LOGICAL	     SELECT( * )

      INTEGER	     IFAILL( * ), IFAILR( * )

      DOUBLE	     PRECISION H( LDH, * ), VL( LDVL, * ), VR( LDVR, * ), WI(
		     * ), WORK( * ), WR( * )

DESCRIPTION
  DHSEIN uses inverse iteration to find specified right and/or left
  eigenvectors of a real 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 (WR,WI):
	  = 'Q': the eigenvalues were found using DHSEQR; 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 DHSEIN 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, DHSEIN 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/output) LOGICAL array, dimension (N)
	  Specifies the eigenvectors to be computed. To select the real
	  eigenvector corresponding to a real eigenvalue WR(j), SELECT(j)
	  must be set to .TRUE.. To select the complex eigenvector
	  corresponding to a complex eigenvalue (WR(j),WI(j)), with complex
	  conjugate (WR(j+1),WI(j+1)), either SELECT(j) or SELECT(j+1) or
	  both must be set to

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

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

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

  WR	  (input/output) DOUBLE PRECISION array, dimension (N)
	  WI	  (input) DOUBLE PRECISION array, dimension (N) On entry, the
	  real and imaginary parts of the eigenvalues of H; a complex
	  conjugate pair of eigenvalues must be stored in consecutive
	  elements of WR and WI.  On exit, WR may have been altered since
	  close eigenvalues are perturbed slightly in searching for
	  independent eigenvectors.

  VL	  (input/output) DOUBLE PRECISION 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(s) 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. 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.  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) DOUBLE PRECISION 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(s) 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. 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.  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; each selected real eigenvector occupies one
	  column and each selected complex eigenvector occupies two columns.

  WORK	  (workspace) DOUBLE PRECISION array, dimension ((N+2)*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 the i-th and (i+1)th columns of VL hold a
	  complex eigenvector, then IFAILL(i) and IFAILL(i+1) are set to the
	  same value.  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 the i-th and (i+1)th columns of VR hold a
	  complex eigenvector, then IFAILR(i) and IFAILR(i+1) are set to the
	  same value.  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|.