DHSEQR(3)      LAPACK routine of NEC Numeric Library Collection      DHSEQR(3)



NAME
       DHSEQR

SYNOPSIS
       SUBROUTINE DHSEQR (JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ,
           WORK, LWORK, INFO)



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)
                     JOB is CHARACTER*1
                      = 'E':  compute eigenvalues only;
                      = 'S':  compute eigenvalues and the Schur form T.

           COMPZ     (input)
                     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         (input)
                     N is INTEGER
                      The order of the matrix H.  N .GE. 0.

           ILO       (input)
                     ILO is INTEGER

           IHI       (input)
                     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         (input/output)
                     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       (input)
                     LDH is INTEGER
                      The leading dimension of the array H. LDH .GE. max(1,N).

           WR        (output)
                     WR is DOUBLE PRECISION array, dimension (N)

           WI        (output)
                     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         (input/output)
                     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       (input)
                     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      (output)
                     WORK is DOUBLE PRECISION array, dimension (LWORK)
                      On exit, if INFO = 0, WORK(1) returns an estimate of
                      the optimal value for LWORK.

           LWORK     (input)
                     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      (output)
                     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.








FURTHER DETAILS
                        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.



LAPACK routine                  31 October 2017                      DHSEQR(3)