slaqr4.f - Man Page

subroutine slaqr4 (wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, iloz, ihiz, z, ldz, work, lwork, info)
SLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

SLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.  

Purpose:

    SLAQR4 implements one level of recursion for SLAQR0.
    It is a complete implementation of the small bulge multi-shift
    QR algorithm.  It may be called by SLAQR0 and, for large enough
    deflation window size, it may be called by SLAQR3.  This
    subroutine is identical to SLAQR0 except that it calls SLAQR2
    instead of SLAQR3.

    SLAQR4 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 = Q*H*Q**T = (QZ)*T*(QZ)**T.

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 1:ILO-1 and IHI+1:N and, if ILO > 1,
           H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
           previous call to SGEBAL, and then passed to SGEHRD when the
           matrix output by SGEBAL is reduced to Hessenberg form.
           Otherwise, ILO and IHI should be set to 1 and N,
           respectively.  If N > 0, then 1 <= ILO <= IHI <= N.
           If N = 0, then ILO = 1 and IHI = 0.

H

          H is REAL 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) < 0. If INFO = 0 and WANTT is
           .FALSE., then the contents of H are unspecified on exit.
           (The output value of H when INFO > 0 is given under the
           description of INFO below.)

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

LDH

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

WR

          WR is REAL array, dimension (IHI)

WI

          WI is REAL 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) > 0 and WI(i+1) < 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).

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 REAL 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 > 0 is given under
           the description of INFO below.)

LDZ

          LDZ is INTEGER
           The leading dimension of the array Z.  if WANTZ is .TRUE.
           then LDZ >= MAX(1,IHIZ).  Otherwise, LDZ >= 1.

WORK

          WORK is REAL array, dimension LWORK
           On exit, if LWORK = -1, WORK(1) returns an estimate of
           the optimal value for LWORK.

LWORK

          LWORK is INTEGER
           The dimension of the array WORK.  LWORK >= 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 SLAQR4 does a workspace query.
           In this case, SLAQR4 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 XERBLA.  Neither H nor Z are accessed.

INFO

          INFO is INTEGER
 \verbatim
          INFO is INTEGER
             = 0:  successful exit
             > 0:  if INFO = i, SLAQR4 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 > 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 > 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 > 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 > 0 and WANTZ is .FALSE., then Z is not
                accessed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

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.

Definition at line 263 of file slaqr4.f.