ESMF_DLAQR4 Subroutine

subroutine ESMF_DLAQR4(WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)

\brief \b ESMF_DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition. \htmlonly Download ESMF_DLAQR4 + dependencies [TGZ] [ZIP] [TXT] \endhtmlonly \par Purpose:

\verbatim

ESMF_DLAQR4 implements one level of recursion for ESMF_DLAQR0. It is a complete implementation of the small bulge multi-shift QR algorithm. It may be called by ESMF_DLAQR0 and, for large enough deflation window size, it may be called by ESMF_DLAQR3. This subroutine is identical to ESMF_DLAQR0 except that it calls ESMF_DLAQR2 instead of ESMF_DLAQR3.

ESMF_DLAQR4 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the Schur decomposition H = Z T Z**T, where T is an upper quasi-triangular matrix (the Schur form), and Z is the orthogonal matrix of Schur vectors.

Optionally Z may be postmultiplied into an input orthogonal matrix Q so that this routine can give the Schur factorization of a matrix A which has been reduced to the Hessenberg form H by the orthogonal matrix Q: A = QHQT = (QZ)T(QZ)T. \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 .GE. 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 triangular in rows and columns 1:ILO-1 and IHI+1:N and, if ILO.GT.1, H(ILO,ILO-1) is zero. ILO and IHI are normally set by a previous call to ESMF_DGEBAL, and then passed to ESMF_DGEHRD when the matrix output by ESMF_DGEBAL is reduced to Hessenberg form. Otherwise, ILO and IHI should be set to 1 and N, respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0. \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 = 0 and WANTT is .TRUE., then H contains the upper quasi-triangular matrix T from the Schur decomposition (the Schur form); 2-by-2 diagonal blocks (corresponding to complex conjugate pairs of eigenvalues) are returned in standard form, with H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is .FALSE., then the contents of H are unspecified on exit. (The output value of H when INFO.GT.0 is given under the description of INFO below.)

      This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
      j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

\endverbatim

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

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

\param[out] WI \verbatim WI is DOUBLE PRECISION array, dimension (IHI) The real and imaginary parts, respectively, of the computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI) and WI(ILO:IHI). 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) .GT. 0 and WI(i+1) .LT. 0. If WANTT is .TRUE., then 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 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N. \endverbatim

\param[in,out] Z \verbatim Z is DOUBLE PRECISION array, dimension (LDZ,IHI) If WANTZ is .FALSE., then Z is not referenced. If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the orthogonal Schur factor of H(ILO:IHI,ILO:IHI). (The output value of Z when INFO.GT.0 is given under the description of INFO below.) \endverbatim

\param[in] LDZ \verbatim LDZ is INTEGER The leading dimension of the array Z. if WANTZ is .TRUE. then LDZ.GE.MAX(1,IHIZ). Otherwize, LDZ.GE.1. \endverbatim

\param[out] WORK \verbatim WORK is DOUBLE PRECISION array, dimension LWORK On exit, if LWORK = -1, WORK(1) returns an estimate of the optimal value for LWORK. \endverbatim

\param[in] LWORK \verbatim LWORK is INTEGER The dimension of the array WORK. LWORK .GE. max(1,N) is sufficient, but LWORK typically as large as 6*N may be required for optimal performance. A workspace query to determine the optimal workspace size is recommended.

      If LWORK = -1, then ESMF_DLAQR4 does a workspace query.
      In this case, ESMF_DLAQR4 checks the input parameters and
      estimates the optimal workspace size for the given
      values of N, ILO and IHI.  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

\param[out] INFO \verbatim INFO is INTEGER = 0: successful exit .GT. 0: if INFO = i, ESMF_DLAQR4 failed to compute all of the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed. (Failures are rare.)

           If INFO .GT. 0 and WANT is .FALSE., then on exit,
           the remaining unconverged eigenvalues are the eigen-
           values of the upper Hessenberg matrix rows and
           columns ILO through 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 a orthogonal 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(ILO:IHI,ILOZ:IHIZ)
              =  (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U

           where U is the orthogonal matrix in (*) (regard-
           less of the value of WANTT.)

           If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
           accessed.

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

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

\par References:

  K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
  Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
  Performance, SIAM Journal of Matrix Analysis, volume 23, pages
  929--947, 2002.

\n K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948–973, 2002.

Arguments