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.

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.

Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd.

November 2011

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

             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.

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.