CLAQR4(3)      LAPACK routine of NEC Numeric Library Collection      CLAQR4(3)



NAME
       CLAQR4

SYNOPSIS
       SUBROUTINE CLAQR4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z,
           LDZ, WORK, LWORK, INFO)



PURPOSE
               CLAQR4 implements one level of recursion for CLAQR0.
               It is a complete implementation of the small bulge multi-shift
               QR algorithm.  It may be called by CLAQR0 and, for large enough
               deflation window size, it may be called by CLAQR3.  This
               subroutine is identical to CLAQR0 except that it calls CLAQR2
               instead of CLAQR3.

               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)
                     WANTT is LOGICAL
                     = .TRUE. : the full Schur form T is required;
                     = .FALSE.: only eigenvalues are required.

           WANTZ     (input)
                     WANTZ is LOGICAL
                     = .TRUE. : the matrix of Schur vectors Z is required;
                     = .FALSE.: Schur vectors are not required.

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

           W         (output)
                     W is 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).

           ILOZ      (input)
                     ILOZ is INTEGER

           IHIZ      (input)
                     IHIZ is 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)
                     Z is 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)
                     LDZ is INTEGER
                      The leading dimension of the array Z.  if WANTZ is .TRUE.
                      then LDZ.GE.MAX(1,IHIZ).  Otherwize, LDZ.GE.1.

           WORK      (output)
                     WORK is COMPLEX array, dimension LWORK
                      On exit, if LWORK = -1, 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, 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)
                     INFO is 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 routine                  31 October 2017                      CLAQR4(3)