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version 1.7, 2010/12/21 13:53:51 version 1.8, 2011/11/21 20:43:16
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   *> \brief \b ZLAQR4
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download ZLAQR4 + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlaqr4.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlaqr4.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlaqr4.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE ZLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
   *                          IHIZ, Z, LDZ, WORK, LWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
   *       LOGICAL            WANTT, WANTZ
   *       ..
   *       .. Array Arguments ..
   *       COMPLEX*16         H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *>    ZLAQR4 implements one level of recursion for ZLAQR0.
   *>    It is a complete implementation of the small bulge multi-shift
   *>    QR algorithm.  It may be called by ZLAQR0 and, for large enough
   *>    deflation window size, it may be called by ZLAQR3.  This
   *>    subroutine is identical to ZLAQR0 except that it calls ZLAQR2
   *>    instead of ZLAQR3.
   *>
   *>    ZLAQR4 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.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] WANTT
   *> \verbatim
   *>          WANTT is LOGICAL
   *>          = .TRUE. : the full Schur form T is required;
   *>          = .FALSE.: only eigenvalues are required.
   *> \endverbatim
   *>
   *> \param[in] WANTZ
   *> \verbatim
   *>          WANTZ is LOGICAL
   *>          = .TRUE. : the matrix of Schur vectors Z is required;
   *>          = .FALSE.: Schur vectors are not required.
   *> \endverbatim
   *>
   *> \param[in] N
   *> \verbatim
   *>          N is INTEGER
   *>           The order of the matrix H.  N .GE. 0.
   *> \endverbatim
   *>
   *> \param[in] ILO
   *> \verbatim
   *>          ILO is INTEGER
   *> \endverbatim
   *>
   *> \param[in] IHI
   *> \verbatim
   *>          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 ZGEBAL, and then passed to ZGEHRD when the
   *>           matrix output by ZGEBAL 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.
   *> \endverbatim
   *>
   *> \param[in,out] H
   *> \verbatim
   *>          H is COMPLEX*16 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.
   *> \endverbatim
   *>
   *> \param[in] LDH
   *> \verbatim
   *>          LDH is INTEGER
   *>           The leading dimension of the array H. LDH .GE. max(1,N).
   *> \endverbatim
   *>
   *> \param[out] W
   *> \verbatim
   *>          W is COMPLEX*16 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).
   *> \endverbatim
   *>
   *> \param[in] ILOZ
   *> \verbatim
   *>          ILOZ is INTEGER
   *> \endverbatim
   *>
   *> \param[in] IHIZ
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in,out] Z
   *> \verbatim
   *>          Z is COMPLEX*16 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.)
   *> \endverbatim
   *>
   *> \param[in] LDZ
   *> \verbatim
   *>          LDZ is INTEGER
   *>           The leading dimension of the array Z.  if WANTZ is .TRUE.
   *>           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is COMPLEX*16 array, dimension LWORK
   *>           On exit, if LWORK = -1, WORK(1) returns an estimate of
   *>           the optimal value for LWORK.
   *> \endverbatim
   *>
   *> \param[in] LWORK
   *> \verbatim
   *>          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 ZLAQR4 does a workspace query.
   *>           In this case, ZLAQR4 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.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          INFO is INTEGER
   *>             =  0:  successful exit
   *>           .GT. 0:  if INFO = i, ZLAQR4 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.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup complex16OTHERauxiliary
   *
   *> \par Contributors:
   *  ==================
   *>
   *>       Karen Braman and Ralph Byers, Department of Mathematics,
   *>       University of Kansas, USA
   *
   *> \par 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.
   *> \n
   *>       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.
   *>
   *  =====================================================================
       SUBROUTINE ZLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,        SUBROUTINE ZLAQR4( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ,
      $                   IHIZ, Z, LDZ, WORK, LWORK, INFO )       $                   IHIZ, Z, LDZ, WORK, LWORK, INFO )
 *  *
 *  -- LAPACK auxiliary routine (version 3.2) --  *  -- LAPACK auxiliary routine (version 3.4.0) --
 *     Univ. of Tennessee, Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..  *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
 *     November 2006  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
   *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N        INTEGER            IHI, IHIZ, ILO, ILOZ, INFO, LDH, LDZ, LWORK, N
Line 13 Line 260
       COMPLEX*16         H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )        COMPLEX*16         H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
 *     ..  *     ..
 *  *
 *     This subroutine implements one level of recursion for ZLAQR0.  *  ================================================================
 *     It is a complete implementation of the small bulge multi-shift  
 *     QR algorithm.  It may be called by ZLAQR0 and, for large enough  
 *     deflation window size, it may be called by ZLAQR3.  This  
 *     subroutine is identical to ZLAQR0 except that it calls ZLAQR2  
 *     instead of ZLAQR3.  
 *  
 *     Purpose  
 *     =======  
 *  
 *     ZLAQR4 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.  
 *  
 *     Arguments  
 *     =========  
 *  
 *     WANTT   (input) LOGICAL  
 *          = .TRUE. : the full Schur form T is required;  
 *          = .FALSE.: only eigenvalues are required.  
 *  
 *     WANTZ   (input) LOGICAL  
 *          = .TRUE. : the matrix of Schur vectors Z is required;  
 *          = .FALSE.: Schur vectors are not required.  
 *  
 *     N     (input) INTEGER  
 *           The order of the matrix H.  N .GE. 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 and, if ILO.GT.1,  
 *           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a  
 *           previous call to ZGEBAL, and then passed to ZGEHRD when the  
 *           matrix output by ZGEBAL 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     (input/output) COMPLEX*16 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   (input) INTEGER  
 *           The leading dimension of the array H. LDH .GE. max(1,N).  
 *  
 *     W        (output) COMPLEX*16 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).  
 *  
 *     Z     (input/output) COMPLEX*16 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   (input) INTEGER  
 *           The leading dimension of the array Z.  if WANTZ is .TRUE.  
 *           then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.  
 *  
 *     WORK  (workspace/output) COMPLEX*16 array, dimension LWORK  
 *           On exit, if LWORK = -1, WORK(1) returns an estimate of  
 *           the optimal value for LWORK.  
 *  
 *     LWORK (input) 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 ZLAQR4 does a workspace query.  
 *           In this case, ZLAQR4 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  (output) INTEGER  
 *             =  0:  successful exit  
 *           .GT. 0:  if INFO = i, ZLAQR4 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.  
 *  
 *     ================================================================  
 *     Based on contributions by  
 *        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.  
 *  *
 *     ================================================================  
 *     .. Parameters ..  *     .. Parameters ..
 *  *
 *     ==== Matrices of order NTINY or smaller must be processed by  *     ==== Matrices of order NTINY or smaller must be processed by
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  Added in v.1.8

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