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NAME

       LAPACK-3  -  CLAQR4 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 CLAQR4( WANTT,  WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK,
                          INFO )

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

           LOGICAL        WANTT, WANTZ

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

PURPOSE

          CLAQR4 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 CGEBAL, and then passed to CGEHRD when the
              matrix output by CGEBAL 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 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 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 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 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 CLAQR4 does a workspace query.
              In this case, CLAQR4 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, CLAQR4 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.

 LAPACK auxiliary routine (version 3.2)     April 2011                            CLAQR4(3lapack)