ESMF_DLAQR2 Subroutine

subroutine ESMF_DLAQR2(WANTT, WANTZ, N, KTOP, KBOT, NW, H, LDH, ILOZ, IHIZ, Z, LDZ, NS, ND, SR, SI, V, LDV, NH, T, LDT, NV, WV, LDWV, WORK, LWORK)

\brief \b ESMF_DLAQR2 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation). \htmlonly Download ESMF_DLAQR2 + dependencies [TGZ] [ZIP] [TXT] \endhtmlonly \par Purpose:

\verbatim

ESMF_DLAQR2 is identical to ESMF_DLAQR3 except that it avoids recursion by calling ESMF_DLAHQR instead of ESMF_DLAQR4.

Aggressive early deflation:

This subroutine accepts as input an upper Hessenberg matrix H and performs an orthogonal similarity transformation designed to detect and deflate fully converged eigenvalues from a trailing principal submatrix. On output H has been over- written by a new Hessenberg matrix that is a perturbation of an orthogonal similarity transformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries. \endverbatim \param[in] WANTT \verbatim WANTT is LOGICAL If .TRUE., then the Hessenberg matrix H is fully updated so that the quasi-triangular Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then only enough of H is updated to preserve the eigenvalues. \endverbatim

\param[in] WANTZ \verbatim WANTZ is LOGICAL If .TRUE., then the orthogonal matrix Z is updated so so that the orthogonal Schur factor may be computed (in cooperation with the calling subroutine). If .FALSE., then Z is not referenced. \endverbatim

\param[in] N \verbatim N is INTEGER The order of the matrix H and (if WANTZ is .TRUE.) the order of the orthogonal matrix Z. \endverbatim

\param[in] KTOP \verbatim KTOP is INTEGER It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix. \endverbatim

\param[in] KBOT \verbatim KBOT is INTEGER It is assumed without a check that either KBOT = N or H(KBOT+1,KBOT)=0. KBOT and KTOP together determine an isolated block along the diagonal of the Hessenberg matrix. \endverbatim

\param[in] NW \verbatim NW is INTEGER Deflation window size. 1 .LE. NW .LE. (KBOT-KTOP+1). \endverbatim

\param[in,out] H \verbatim H is DOUBLE PRECISION array, dimension (LDH,N) On input the initial N-by-N section of H stores the Hessenberg matrix undergoing aggressive early deflation. On output H has been transformed by an orthogonal similarity transformation, perturbed, and the returned to Hessenberg form that (it is to be hoped) has some zero subdiagonal entries. \endverbatim

\param[in] LDH \verbatim LDH is INTEGER Leading dimension of H just as declared in the calling subroutine. N .LE. LDH \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 .LE. ILOZ .LE. IHIZ .LE. N. \endverbatim

\param[in,out] Z \verbatim Z is DOUBLE PRECISION array, dimension (LDZ,N) IF WANTZ is .TRUE., then on output, the orthogonal similarity transformation mentioned above has been accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right. If WANTZ is .FALSE., then Z is unreferenced. \endverbatim

\param[in] LDZ \verbatim LDZ is INTEGER The leading dimension of Z just as declared in the calling subroutine. 1 .LE. LDZ. \endverbatim

\param[out] NS \verbatim NS is INTEGER The number of unconverged (ie approximate) eigenvalues returned in SR and SI that may be used as shifts by the calling subroutine. \endverbatim

\param[out] ND \verbatim ND is INTEGER The number of converged eigenvalues uncovered by this subroutine. \endverbatim

\param[out] SR \verbatim SR is DOUBLE PRECISION array, dimension (KBOT) \endverbatim

\param[out] SI \verbatim SI is DOUBLE PRECISION array, dimension (KBOT) On output, the real and imaginary parts of approximate eigenvalues that may be used for shifts are stored in SR(KBOT-ND-NS+1) through SR(KBOT-ND) and SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively. The real and imaginary parts of converged eigenvalues are stored in SR(KBOT-ND+1) through SR(KBOT) and SI(KBOT-ND+1) through SI(KBOT), respectively. \endverbatim

\param[out] V \verbatim V is DOUBLE PRECISION array, dimension (LDV,NW) An NW-by-NW work array. \endverbatim

\param[in] LDV \verbatim LDV is INTEGER The leading dimension of V just as declared in the calling subroutine. NW .LE. LDV \endverbatim

\param[in] NH \verbatim NH is INTEGER The number of columns of T. NH.GE.NW. \endverbatim

\param[out] T \verbatim T is DOUBLE PRECISION array, dimension (LDT,NW) \endverbatim

\param[in] LDT \verbatim LDT is INTEGER The leading dimension of T just as declared in the calling subroutine. NW .LE. LDT \endverbatim

\param[in] NV \verbatim NV is INTEGER The number of rows of work array WV available for workspace. NV.GE.NW. \endverbatim

\param[out] WV \verbatim WV is DOUBLE PRECISION array, dimension (LDWV,NW) \endverbatim

\param[in] LDWV \verbatim LDWV is INTEGER The leading dimension of W just as declared in the calling subroutine. NW .LE. LDV \endverbatim

\param[out] WORK \verbatim WORK is DOUBLE PRECISION array, dimension (LWORK) On exit, WORK(1) is set to an estimate of the optimal value of LWORK for the given values of N, NW, KTOP and KBOT. \endverbatim

\param[in] LWORK \verbatim LWORK is INTEGER The dimension of the work array WORK. LWORK = 2*NW suffices, but greater efficiency may result from larger values of LWORK.

     If LWORK = -1, then a workspace query is assumed; ESMF_DLAQR2
     only estimates the optimal workspace size for the given
     values of N, NW, KTOP and KBOT.  The estimate is returned
     in WORK(1).  No error message related to LWORK is issued
     by ESMF_XERBLA.  Neither H nor Z are accessed.

\endverbatim \author Univ. of Tennessee \author Univ. of California Berkeley \author Univ. of Colorado Denver \author NAG Ltd. \date June 2017 \ingroup doubleOTHERauxiliary \par Contributors:

  Karen Braman and Ralph Byers, Department of Mathematics,
  University of Kansas, USA

Arguments