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



NAME
       ZLAHQR

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



PURPOSE
               ZLAHQR is an auxiliary routine called by CHSEQR to update the
               eigenvalues and Schur decomposition already computed by CHSEQR, by
               dealing with the Hessenberg submatrix in rows and columns ILO to
               IHI.




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 >= 0.

           ILO       (input)
                     ILO is INTEGER

           IHI       (input)
                     IHI is INTEGER
                     It is assumed that H is already upper triangular in rows and
                     columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1).
                     ZLAHQR works primarily with the Hessenberg submatrix in rows
                     and columns ILO to IHI, but applies transformations to all of
                     H if WANTT is .TRUE..
                     1 <= ILO <= max(1,IHI); IHI <= N.

           H         (input/output)
                     H is COMPLEX*16 array, dimension (LDH,N)
                     On entry, the upper Hessenberg matrix H.
                     On exit, if INFO is zero and if WANTT is .TRUE., then H
                     is upper triangular in rows and columns ILO:IHI.  If INFO
                     is zero and if WANTT is .FALSE., then the contents of H
                     are unspecified on exit.  The output state of H in case
                     INF is positive is below under the description of INFO.

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

           W         (output)
                     W is COMPLEX*16 array, dimension (N)
                     The computed eigenvalues ILO to IHI are stored in the
                     corresponding elements of W. If WANTT is .TRUE., 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 <= ILOZ <= ILO; IHI <= IHIZ <= N.

           Z         (input/output)
                     Z is COMPLEX*16 array, dimension (LDZ,N)
                     If WANTZ is .TRUE., on entry Z must contain the current
                     matrix Z of transformations accumulated by CHSEQR, and on
                     exit Z has been updated; transformations are applied only to
                     the submatrix Z(ILOZ:IHIZ,ILO:IHI).
                     If WANTZ is .FALSE., Z is not referenced.

           LDZ       (input)
                     LDZ is INTEGER
                     The leading dimension of the array Z. LDZ >= max(1,N).

           INFO      (output)
                     INFO is INTEGER
                      =   0: successful exit
                     .GT. 0: if INFO = i, ZLAHQR failed to compute all the
                             eigenvalues ILO to IHI in a total of 30 iterations
                             per eigenvalue; elements i+1:ihi of W contain
                             those eigenvalues which have been successfully
                             computed.

                             If INFO .GT. 0 and WANTT is .FALSE., then on exit,
                             the remaining unconverged eigenvalues are the
                             eigenvalues of the upper Hessenberg matrix
                             rows and columns ILO thorugh 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 orthognal 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)  = (initial value of Z)*U
                             where U is the orthogonal matrix in (*)
                             (regardless of the value of WANTT.)





       Contributors:


                02-96 Based on modifications by
                David Day, Sandia National Laboratory, USA

                12-04 Further modifications by
                Ralph Byers, University of Kansas, USA
                This is a modified version of ZLAHQR from LAPACK version 3.0.
                It is (1) more robust against overflow and underflow and
                (2) adopts the more conservative Ahues & Tisseur stopping
                criterion (LAWN 122, 1997).



LAPACK routine                  31 October 2017                      ZLAHQR(3)