NAME

       dlahqr.f -

SYNOPSIS

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

Function/Subroutine Documentation

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

       Purpose:

               DLAHQR is an auxiliary routine called by DHSEQR to update the
               eigenvalues and Schur decomposition already computed by DHSEQR, 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 quasi-triangular in
                     rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
                     ILO = 1). DLAHQR 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 DOUBLE PRECISION array, dimension (LDH,N)
                     On entry, the upper Hessenberg matrix H.
                     On exit, if INFO is zero and if WANTT is .TRUE., H is upper
                     quasi-triangular in rows and columns ILO:IHI, with any
                     2-by-2 diagonal blocks in standard form. If INFO is zero
                     and WANTT is .FALSE., the contents of H are unspecified on
                     exit.  The output state of H if INFO is nonzero is given
                     below under the description of INFO.

           LDH

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

           WR

                     WR is DOUBLE PRECISION array, dimension (N)

           WI

                     WI is DOUBLE PRECISION array, dimension (N)
                     The real and imaginary parts, respectively, of the computed
                     eigenvalues ILO to IHI are stored in the corresponding
                     elements of WR and WI. 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) > 0 and WI(i+1) < 0. If WANTT is .TRUE., 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

                     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 DOUBLE PRECISION array, dimension (LDZ,N)
                     If WANTZ is .TRUE., on entry Z must contain the current
                     matrix Z of transformations accumulated by DHSEQR, 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, DLAHQR failed to compute all the
                             eigenvalues ILO to IHI in a total of 30 iterations
                             per eigenvalue; elements i+1:ihi of WR and WI
                             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

       Further Details:

                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 DLAHQR 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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