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NAME

       LAPACK-3  -  ZHSEQR compute 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

SYNOPSIS

       SUBROUTINE ZHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO )

           INTEGER        IHI, ILO, INFO, LDH, LDZ, LWORK, N

           CHARACTER      COMPZ, JOB

           COMPLEX*16     H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )

PURPOSE

          ZHSEQR 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

        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 unitary 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 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 JOB = 'S', H contains the upper
              triangular matrix T from the Schur decomposition (the
              Schur form). 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 ZHSEQR, 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).

        W        (output) COMPLEX*16 array, dimension (N)
                 The computed eigenvalues. If JOB = 'S', 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,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 unitary 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 unitary matrix generated by ZUNGHR
              after the call to ZGEHRD 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) COMPLEX*16 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 ZHSEQR does a workspace query.
              In this case, ZHSEQR 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, ZHSEQR 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 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 COMPZ = 'V', then on exit
             (final value of Z)  =  (initial value of Z)*U
             where U is the unitary 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 unitary matrix in (*) (regard-
             less of the value of JOB.)
             If INFO .GT. 0 and COMPZ = 'N', then Z is not
             accessed.

 LAPACK computational routine (version 3.2.2April 2011                            ZHSEQR(3lapack)