CXML
        	    
LAPACK version 3.0
dhseqr(3)

PURPOSE
  DHSEQR - compute the eigenvalues of a real upper Hessenberg matrix H and,
  optionally, the matrices T and Z from the Schur decomposition H = Z T Z**T,
  where T is an upper quasi-triangular matrix (the Schur form), and Z is the
  orthogonal matrix of Schur vectors

SYNTAX
  SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ, WORK,
		     LWORK, INFO )

      CHARACTER	     COMPZ, JOB

      INTEGER	     IHI, ILO, INFO, LDH, LDZ, LWORK, N

      DOUBLE	     PRECISION H( LDH, * ), WI( * ), WORK( * ), WR( * ), Z(
		     LDZ, * )

DESCRIPTION
  DHSEQR computes the eigenvalues of a real upper Hessenberg matrix H and,
  optionally, the matrices T and Z from the Schur decomposition H = Z T Z**T,
  where T is an upper quasi-triangular matrix (the Schur form), and Z is the
  orthogonal matrix of Schur vectors.  Optionally Z may be postmultiplied
  into an input orthogonal matrix Q, so that this routine can give the Schur
  factorization of a matrix A which has been reduced to the Hessenberg form H
  by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

ARGUMENTS
  JOB	  (input) CHARACTER*1
	  = 'E':  compute eigenvalues only;
	  = 'S':  compute eigenvalues and the Schur form T.

  COMPZ	  (input) CHARACTER*1
	  = 'N':  no Schur vectors are computed;
	  = 'I':  Z is initialized to the unit matrix and the matrix Z of
	  Schur vectors of H is returned; = 'V':  Z must contain an
	  orthogonal matrix Q on entry, and the product Q*Z is returned.

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

  ILO	  (input) INTEGER
	  IHI	  (input) INTEGER It is assumed that H is already upper
	  triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are
	  normally set by a previous call to DGEBAL, and then passed to
	  SGEHRD when the matrix output by DGEBAL is reduced to Hessenberg
	  form. Otherwise ILO and IHI should be set to 1 and N respectively.
	  1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

  H	  (input/output) DOUBLE PRECISION array, dimension (LDH,N)
	  On entry, the upper Hessenberg matrix H.  On exit, if JOB = 'S', H
	  contains the upper quasi-triangular matrix T from the Schur
	  decomposition (the Schur form); 2-by-2 diagonal blocks
	  (corresponding to complex conjugate pairs of eigenvalues) are
	  returned in standard form, with H(i,i) = H(i+1,i+1) and
	  H(i+1,i)*H(i,i+1) < 0. If JOB = 'E', the contents of H are
	  unspecified on exit.

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

  WR	  (output) DOUBLE PRECISION array, dimension (N)
	  WI	  (output) DOUBLE PRECISION array, dimension (N) The real and
	  imaginary parts, respectively, of the computed eigenvalues. If two
	  eigenvalues are computed as a complex conjugate pair, they are
	  stored in consecutive elements of WR and WI, say the i-th and
	  (i+1)th, with WI(i) > 0 and WI(i+1) < 0. If JOB = 'S', the
	  eigenvalues are stored in the same order as on the diagonal of the
	  Schur form returned in H, with WR(i) = H(i,i) and, if
	  H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) =
	  sqrt(H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).

  Z	  (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
	  If COMPZ = 'N': Z is not referenced.
	  If COMPZ = 'I': on entry, Z need not be set, and on exit, Z
	  contains the orthogonal matrix Z of the Schur vectors of H.  If
	  COMPZ = 'V': on entry Z must contain an N-by-N matrix Q, which is
	  assumed to be equal to the unit matrix except for the submatrix
	  Z(ILO:IHI,ILO:IHI); on exit Z contains Q*Z.  Normally Q is the
	  orthogonal matrix generated by DORGHR after the call to DGEHRD
	  which formed the Hessenberg matrix H.

  LDZ	  (input) INTEGER
	  The leading dimension of the array Z.	 LDZ >= max(1,N) if COMPZ =
	  'I' or 'V'; LDZ >= 1 otherwise.

  WORK	  (workspace/output) DOUBLE PRECISION array, dimension (LWORK)
	  On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

  LWORK	  (input) INTEGER
	  The dimension of the array WORK.  LWORK >= max(1,N).

	  If LWORK = -1, then a workspace query is assumed; the routine only
	  calculates the optimal size of the WORK array, returns this value
	  as the first entry of the WORK array, and no error message related
	  to LWORK is issued by XERBLA.

  INFO	  (output) INTEGER
	  = 0:	successful exit
	  < 0:	if INFO = -i, the i-th argument had an illegal value
	  > 0:	if INFO = i, DHSEQR failed to compute all of the eigenvalues
	  in a total of 30*(IHI-ILO+1) iterations; elements 1:ilo-1 and i+1:n
	  of WR and WI contain those eigenvalues which have been successfully
	  computed.