ESMF_DLAHQR Subroutine

subroutine ESMF_DLAHQR(WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, INFO)

\brief \b ESMF_DLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm. \htmlonly Download ESMF_DLAHQR + dependencies [TGZ] [ZIP] [TXT] \endhtmlonly \par Purpose:

\verbatim

ESMF_DLAHQR is an auxiliary routine called by ESMF_DHSEQR to update the eigenvalues and Schur decomposition already computed by ESMF_DHSEQR, by dealing with the Hessenberg submatrix in rows and columns ILO to IHI. \endverbatim \param[in] WANTT \verbatim WANTT is LOGICAL = .TRUE. : the full Schur form T is required; = .FALSE.: only eigenvalues are required. \endverbatim

\param[in] WANTZ \verbatim WANTZ is LOGICAL = .TRUE. : the matrix of Schur vectors Z is required; = .FALSE.: Schur vectors are not required. \endverbatim

\param[in] N \verbatim N is INTEGER The order of the matrix H. N >= 0. \endverbatim

\param[in] ILO \verbatim ILO is INTEGER \endverbatim

\param[in] IHI \verbatim 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). ESMF_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. \endverbatim

\param[in,out] H \verbatim 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. \endverbatim

\param[in] LDH \verbatim LDH is INTEGER The leading dimension of the array H. LDH >= max(1,N). \endverbatim

\param[out] WR \verbatim WR is DOUBLE PRECISION array, dimension (N) \endverbatim

\param[out] WI \verbatim 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). \endverbatim

\param[in] ILOZ \verbatim ILOZ is INTEGER \endverbatim

\param[in] IHIZ \verbatim IHIZ is INTEGER Specify the rows of Z to which transformations must be applied if WANTZ is .TRUE.. 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. \endverbatim

\param[in,out] Z \verbatim 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 ESMF_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. \endverbatim

\param[in] LDZ \verbatim LDZ is INTEGER The leading dimension of the array Z. LDZ >= max(1,N). \endverbatim

\param[out] INFO \verbatim INFO is INTEGER = 0: successful exit .GT. 0: If INFO = i, ESMF_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.)

\endverbatim \author Univ. of Tennessee \author Univ. of California Berkeley \author Univ. of Colorado Denver \author NAG Ltd. \date December 2016 \ingroup doubleOTHERauxiliary \par Further Details:

\verbatim

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 ESMF_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).

\endverbatim

Arguments