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version 1.8, 2010/12/21 13:53:27 version 1.9, 2011/11/21 20:42:53
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   *> \brief \b DHSEQR
   *
   *  =========== DOCUMENTATION ===========
   *
   * Online html documentation available at 
   *            http://www.netlib.org/lapack/explore-html/ 
   *
   *> \htmlonly
   *> Download DHSEQR + dependencies 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dhseqr.f"> 
   *> [TGZ]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dhseqr.f"> 
   *> [ZIP]</a> 
   *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dhseqr.f"> 
   *> [TXT]</a>
   *> \endhtmlonly 
   *
   *  Definition:
   *  ===========
   *
   *       SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
   *                          LDZ, WORK, LWORK, INFO )
   * 
   *       .. Scalar Arguments ..
   *       INTEGER            IHI, ILO, INFO, LDH, LDZ, LWORK, N
   *       CHARACTER          COMPZ, JOB
   *       ..
   *       .. Array Arguments ..
   *       DOUBLE PRECISION   H( LDH, * ), WI( * ), WORK( * ), WR( * ),
   *      $                   Z( LDZ, * )
   *       ..
   *  
   *
   *> \par Purpose:
   *  =============
   *>
   *> \verbatim
   *>
   *>    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.
   *> \endverbatim
   *
   *  Arguments:
   *  ==========
   *
   *> \param[in] JOB
   *> \verbatim
   *>          JOB is CHARACTER*1
   *>           = 'E':  compute eigenvalues only;
   *>           = 'S':  compute eigenvalues and the Schur form T.
   *> \endverbatim
   *>
   *> \param[in] COMPZ
   *> \verbatim
   *>          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.
   *> \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. 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.
   *> \endverbatim
   *>
   *> \param[in,out] H
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[in] LDH
   *> \verbatim
   *>          LDH is INTEGER
   *>           The leading dimension of the array H. LDH .GE. max(1,N).
   *> \endverbatim
   *>
   *> \param[out] WR
   *> \verbatim
   *>          WR is DOUBLE PRECISION array, dimension (N)
   *> \endverbatim
   *>
   *> \param[out] WI
   *> \verbatim
   *>          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).
   *> \endverbatim
   *>
   *> \param[in,out] Z
   *> \verbatim
   *>          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.)
   *> \endverbatim
   *>
   *> \param[in] LDZ
   *> \verbatim
   *>          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.
   *> \endverbatim
   *>
   *> \param[out] WORK
   *> \verbatim
   *>          WORK is DOUBLE PRECISION array, dimension (LWORK)
   *>           On exit, if INFO = 0, 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 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.
   *> \endverbatim
   *>
   *> \param[out] INFO
   *> \verbatim
   *>          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.
   *> \endverbatim
   *
   *  Authors:
   *  ========
   *
   *> \author Univ. of Tennessee 
   *> \author Univ. of California Berkeley 
   *> \author Univ. of Colorado Denver 
   *> \author NAG Ltd. 
   *
   *> \date November 2011
   *
   *> \ingroup doubleOTHERcomputational
   *
   *> \par Contributors:
   *  ==================
   *>
   *>       Karen Braman and Ralph Byers, Department of Mathematics,
   *>       University of Kansas, USA
   *
   *> \par Further Details:
   *  =====================
   *>
   *> \verbatim
   *>
   *>             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.
   *> \endverbatim
   *
   *> \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 DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,        SUBROUTINE DHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
      $                   LDZ, WORK, LWORK, INFO )       $                   LDZ, WORK, LWORK, INFO )
 *  *
 *  -- LAPACK computational routine (version 3.2.2) --  *  -- LAPACK computational 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,    --
 *     June 2010  *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
   *     November 2011
 *  *
 *     .. Scalar Arguments ..  *     .. Scalar Arguments ..
       INTEGER            IHI, ILO, INFO, LDH, LDZ, LWORK, N        INTEGER            IHI, ILO, INFO, LDH, LDZ, LWORK, N
Line 13 Line 329
       DOUBLE PRECISION   H( LDH, * ), WI( * ), WORK( * ), WR( * ),        DOUBLE PRECISION   H( LDH, * ), WI( * ), WORK( * ), WR( * ),
      $                   Z( LDZ, * )       $                   Z( LDZ, * )
 *     ..  *     ..
 *     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.  
 *  
 *     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 .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. ILO and IHI are normally  
 *           set by a previous call to DGEBAL, and then passed to DGEHRD  
 *           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     (input/output) 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   (input) INTEGER  
 *           The leading dimension of the array H. LDH .GE. 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) .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     (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,  
 *           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   (input) 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  (workspace/output) DOUBLE PRECISION array, dimension (LWORK)  
 *           On exit, if INFO = 0, 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 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  (output) 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.  
 *  
 *     ================================================================  
 *             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.  
 *  
 *     ================================================================  
 *     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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