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NAME

       LAPACK-3  -  SLAQR4 compute 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

SYNOPSIS

       SUBROUTINE SLAQR4( WANTT,  WANTZ,  N,  ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK,
                          LWORK, INFO )

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

           LOGICAL        WANTT, WANTZ

           REAL           H( LDH, * ), WI( * ), WORK( * ), WR( * ), Z( LDZ, * )

PURPOSE

          SLAQR4 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

        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 SGEBAL, and then passed to SGEHRD when the
              matrix output by SGEBAL 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) REAL 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 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 WANTT 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).

        WR    (output) REAL array, dimension (IHI)
              WI    (output) REAL array, dimension (IHI)
              The real and imaginary parts, respectively, of the computed
              eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
              and WI(ILO:IHI). 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 WANTT is .TRUE., then
              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).

        ILOZ     (input) INTEGER
                 IHIZ     (input) INTEGER
                 Specify the rows of Z to which transformations must be
                 applied if WANTZ is .TRUE..
                 1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.

        Z     (input/output) REAL 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) REAL 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 SLAQR4 does a workspace query.
              In this case, SLAQR4 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, SLAQR4 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 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 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 orthogonal 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                            SLAQR4(3lapack)