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claqr4.f(3)				      LAPACK				      claqr4.f(3)

NAME
       claqr4.f -

SYNOPSIS
   Functions/Subroutines
       subroutine claqr4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK,
	   INFO)
	   CLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices
	   from the Schur decomposition.

Function/Subroutine Documentation
   subroutine claqr4 (logicalWANTT, logicalWANTZ, integerN, integerILO, integerIHI, complex,
       dimension( ldh, * )H, integerLDH, complex, dimension( * )W, integerILOZ, integerIHIZ,
       complex, dimension( ldz, * )Z, integerLDZ, complex, dimension( * )WORK, integerLWORK,
       integerINFO)
       CLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from
       the Schur decomposition.

       Purpose:

	       CLAQR4 implements one level of recursion for CLAQR0.
	       It is a complete implementation of the small bulge multi-shift
	       QR algorithm.  It may be called by CLAQR0 and, for large enough
	       deflation window size, it may be called by CLAQR3.  This
	       subroutine is identical to CLAQR0 except that it calls CLAQR2
	       instead of CLAQR3.

	       CLAQR4 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:
	   WANTT

		     WANTT is LOGICAL
		     = .TRUE. : the full Schur form T is required;
		     = .FALSE.: only eigenvalues are required.

	   WANTZ

		     WANTZ is LOGICAL
		     = .TRUE. : the matrix of Schur vectors Z is required;
		     = .FALSE.: Schur vectors are not required.

	   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 and, if ILO.GT.1,
		      H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
		      previous call to CGEBAL, and then passed to CGEHRD 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 WANTT is .TRUE., then H
		      contains the upper triangular matrix T from the Schur
		      decomposition (the Schur form). If INFO = 0 and WANT is
		      .FALSE., then 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.)

		      This subroutine may explicitly set 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 of H(ILO:IHI,ILO:IHI) are stored
		      in W(ILO:IHI). If WANTT is .TRUE., then 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).

	   ILOZ

		     ILOZ is INTEGER

	   IHIZ

		     IHIZ is INTEGER
		      Specify the rows of Z to which transformations must be
		      applied if WANTZ is .TRUE..
		      1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.

	   Z

		     Z is COMPLEX array, dimension (LDZ,IHI)
		      If WANTZ is .FALSE., then Z is not referenced.
		      If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is
		      replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the
		      orthogonal Schur factor of H(ILO:IHI,ILO:IHI).
		      (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 WANTZ is .TRUE.
		      then LDZ.GE.MAX(1,IHIZ).	Otherwize, LDZ.GE.1.

	   WORK

		     WORK is COMPLEX array, dimension LWORK
		      On exit, if LWORK = -1, 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, but LWORK typically as large as 6*N may
		      be required for optimal performance.  A workspace query
		      to determine the optimal workspace size is recommended.

		      If LWORK = -1, then CLAQR4 does a workspace query.
		      In this case, CLAQR4 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

	    batim
		     INFO is INTEGER
			=  0:  successful exit
		      .GT. 0:  if INFO = i, CLAQR4 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 WANT is .FALSE., 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 WANTT is .TRUE., 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 WANTZ is .TRUE., then on exit

			     (final value of Z(ILO:IHI,ILOZ:IHIZ)
			      =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U

			   where U is the unitary matrix in (*) (regard-
			   less of the value of WANTT.)

			   If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
			   accessed.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   September 2012

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

       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 249 of file claqr4.f.

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

Version 3.4.2				 Tue Sep 25 2012			      claqr4.f(3)
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