NAME

       zlahqr.f -

SYNOPSIS

   Functions/Subroutines
       subroutine zlahqr (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, INFO)
           ZLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix,
           using the double-shift/single-shift QR algorithm.

Function/Subroutine Documentation

   subroutine zlahqr (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI,
       complex*16, dimension( ldh, * ) H, integer LDH, complex*16, dimension( * ) W, integer
       ILOZ, integer IHIZ, complex*16, dimension( ldz, * ) Z, integer LDZ, integer INFO)
       ZLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix,
       using the double-shift/single-shift QR algorithm.

       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.

       Parameters:
           WANTT

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

           WANTZ

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

           N

                     N is INTEGER
                     The order of the matrix H.  N >= 0.

           ILO

                     ILO is INTEGER

           IHI

                     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

                     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

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

           W

                     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

                     ILOZ is INTEGER

           IHIZ

                     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

                     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

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

           INFO

                     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.)

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           November 2015

       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).

Author

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