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chseqr.f(3)				      LAPACK				      chseqr.f(3)

NAME
       chseqr.f -

SYNOPSIS
   Functions/Subroutines
       subroutine chseqr (JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO)
	   CHSEQR

Function/Subroutine Documentation
   subroutine chseqr (characterJOB, characterCOMPZ, integerN, integerILO, integerIHI, complex,
       dimension( ldh, * )H, integerLDH, complex, dimension( * )W, complex, dimension( ldz, * )Z,
       integerLDZ, complex, dimension( * )WORK, integerLWORK, integerINFO)
       CHSEQR

       Purpose:

	       CHSEQR computes the eigenvalues of a Hessenberg matrix H
	       and, optionally, the matrices T and Z from the Schur decomposition
	       H = Z T Z**H, where T is an upper triangular matrix (the
	       Schur form), and Z is the unitary matrix of Schur vectors.

	       Optionally Z may be postmultiplied into an input unitary
	       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 unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.

       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 unitary 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 CGEBAL, and then passed to ZGEHRD
		      when the matrix output by CGEBAL 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 COMPLEX array, dimension (LDH,N)
		      On entry, the upper Hessenberg matrix H.
		      On exit, if INFO = 0 and JOB = 'S', H contains the upper
		      triangular matrix T from the Schur decomposition (the
		      Schur form). 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 CHSEQR, 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).

	   W

		     W is COMPLEX array, dimension (N)
		      The computed eigenvalues. If JOB = 'S', the eigenvalues are
		      stored in the same order as on the diagonal of the Schur
		      form returned in H, with W(i) = H(i,i).

	   Z

		     Z is COMPLEX 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 unitary 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 unitary matrix generated by CUNGHR
		      after the call to CGEHRD 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 COMPLEX 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 CHSEQR does a workspace query.
		      In this case, CHSEQR 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, CHSEQR 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 a unitary matrix.  The final
			   value of  H is upper Hessenberg and 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 unitary 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 unitary 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,'CHSEQR',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 CLAHQR vs CLAQR0 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
				  CLAHQR 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 299 of file chseqr.f.

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

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