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dhseqr.f(3)				      LAPACK				      dhseqr.f(3)

NAME
       dhseqr.f -

SYNOPSIS
   Functions/Subroutines
       subroutine dhseqr (JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ, WORK, LWORK, INFO)
	   DHSEQR

Function/Subroutine Documentation
   subroutine dhseqr (characterJOB, characterCOMPZ, integerN, integerILO, integerIHI, double
       precision, dimension( ldh, * )H, integerLDH, double precision, dimension( * )WR, double
       precision, dimension( * )WI, double precision, dimension( ldz, * )Z, integerLDZ, double
       precision, dimension( * )WORK, integerLWORK, integerINFO)
       DHSEQR

       Purpose:

	       DHSEQR computes the eigenvalues of a 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.

       Parameters:
	   JOB

		     JOB is CHARACTER*1
		      = 'E':  compute eigenvalues only;
		      = 'S':  compute eigenvalues and the Schur form T.

	   COMPZ

		     COMPZ is 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

		     N is INTEGER
		      The order of the matrix H.  N .GE. 0.

	   ILO

		     ILO is INTEGER

	   IHI

		     IHI is 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 ZGEHRD
		      when the matrix output by DGEBAL is reduced to Hessenberg
		      form. Otherwise ILO and IHI should be set to 1 and N
		      respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
		      If N = 0, then ILO = 1 and IHI = 0.

	   H

		     H is DOUBLE PRECISION array, dimension (LDH,N)
		      On entry, the upper Hessenberg matrix H.
		      On exit, if INFO = 0 and JOB = 'S', then 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).LT.0. If INFO = 0 and JOB = 'E', the
		      contents of H are unspecified on exit.  (The output value of
		      H when INFO.GT.0 is given under the description of INFO
		      below.)

		      Unlike earlier versions of DHSEQR, this subroutine may
		      explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1
		      or j = IHI+1, IHI+2, ... N.

	   LDH

		     LDH is INTEGER
		      The leading dimension of the array H. LDH .GE. max(1,N).

	   WR

		     WR is DOUBLE PRECISION array, dimension (N)

	   WI

		     WI is 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) .GT. 0 and
		      WI(i+1) .LT. 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

		     Z is 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,
		      if INFO = 0, 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,
		      if INFO = 0, Z contains Q*Z.
		      Normally Q is the orthogonal matrix generated by DORGHR
		      after the call to DGEHRD which formed the Hessenberg matrix
		      H. (The output value of Z when INFO.GT.0 is given under
		      the description of INFO below.)

	   LDZ

		     LDZ is INTEGER
		      The leading dimension of the array Z.  if COMPZ = 'I' or
		      COMPZ = 'V', then LDZ.GE.MAX(1,N).  Otherwize, LDZ.GE.1.

	   WORK

		     WORK is DOUBLE PRECISION array, dimension (LWORK)
		      On exit, if INFO = 0, WORK(1) returns an estimate of
		      the optimal value for LWORK.

	   LWORK

		     LWORK is INTEGER
		      The dimension of the array WORK.	LWORK .GE. max(1,N)
		      is sufficient and delivers very good and sometimes
		      optimal performance.  However, LWORK as large as 11*N
		      may be required for optimal performance.	A workspace
		      query is recommended to determine the optimal workspace
		      size.

		      If LWORK = -1, then DHSEQR does a workspace query.
		      In this case, DHSEQR checks the input parameters and
		      estimates the optimal workspace size for the given
		      values of N, ILO and IHI.  The estimate is returned
		      in WORK(1).  No error message related to LWORK is
		      issued by XERBLA.  Neither H nor Z are accessed.

	   INFO

		     INFO is INTEGER
			=  0:  successful exit
		      .LT. 0:  if INFO = -i, the i-th argument had an illegal
			       value
		      .GT. 0:  if INFO = i, DHSEQR failed to compute all of
			   the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
			   and WI contain those eigenvalues which have been
			   successfully computed.  (Failures are rare.)

			   If INFO .GT. 0 and JOB = 'E', then on exit, the
			   remaining unconverged eigenvalues are the eigen-
			   values of the upper Hessenberg matrix rows and
			   columns ILO through INFO of the final, output
			   value of H.

			   If INFO .GT. 0 and JOB   = 'S', then on exit

		      (*)  (initial value of H)*U  = U*(final value of H)

			   where U is an orthogonal matrix.  The final
			   value of H is upper Hessenberg and quasi-triangular
			   in rows and columns INFO+1 through IHI.

			   If INFO .GT. 0 and COMPZ = 'V', then on exit

			     (final value of Z)  =  (initial value of Z)*U

			   where U is the orthogonal matrix in (*) (regard-
			   less of the value of JOB.)

			   If INFO .GT. 0 and COMPZ = 'I', then on exit
				 (final value of Z)  = U
			   where U is the orthogonal matrix in (*) (regard-
			   less of the value of JOB.)

			   If INFO .GT. 0 and COMPZ = 'N', then Z is not
			   accessed.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   November 2011

       Contributors:
	   Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

       Further Details:

			Default values supplied by
			ILAENV(ISPEC,'DHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
			It is suggested that these defaults be adjusted in order
			to attain best performance in each particular
			computational environment.

		       ISPEC=12: The DLAHQR vs DLAQR0 crossover point.
				 Default: 75. (Must be at least 11.)

		       ISPEC=13: Recommended deflation window size.
				 This depends on ILO, IHI and NS.  NS is the
				 number of simultaneous shifts returned
				 by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
				 The default for (IHI-ILO+1).LE.500 is NS.
				 The default for (IHI-ILO+1).GT.500 is 3*NS/2.

		       ISPEC=14: Nibble crossover point. (See IPARMQ for
				 details.)  Default: 14% of deflation window
				 size.

		       ISPEC=15: Number of simultaneous shifts in a multishift
				 QR iteration.

				 If IHI-ILO+1 is ...

				 greater than	   ...but less	  ... the
				 or equal to ...      than	  default is

				      1 	      30	  NS =	 2(+)
				     30 	      60	  NS =	 4(+)
				     60 	     150	  NS =	10(+)
				    150 	     590	  NS =	**
				    590 	    3000	  NS =	64
				   3000 	    6000	  NS = 128
				   6000 	    infinity	  NS = 256

			     (+)  By default some or all matrices of this order
				  are passed to the implicit double shift routine
				  DLAHQR and this parameter is ignored.  See
				  ISPEC=12 above and comments in IPARMQ for
				  details.

			    (**)  The asterisks (**) indicate an ad-hoc
				  function of N increasing from 10 to 64.

		       ISPEC=16: Select structured matrix multiply.
				 If the number of simultaneous shifts (specified
				 by ISPEC=15) is less than 14, then the default
				 for ISPEC=16 is 0.  Otherwise the default for
				 ISPEC=16 is 2.

       References:
	   K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part I: Maintaining
	   Well Focused Shifts, and Level 3 Performance, SIAM Journal of Matrix Analysis, volume
	   23, pages 929--947, 2002.
	    K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive
	   Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948--973, 2002.

       Definition at line 316 of file dhseqr.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      dhseqr.f(3)
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