SRC/slaqr4.f(3)

NAME
SRC/slaqr4.f

SYNOPSIS

  Functions/Subroutines
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.

Function/Subroutine Documentation

  subroutine slaqr4 (logical wantt, logical wantz, integer n, integer ilo, integer ihi, real, dimension( ldh, * ) h, integer ldh, real, dimension( * ) wr, real, dimension( * ) wi, integer iloz, integer ihiz, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer info)

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.

     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.

Definition at line 263 of file slaqr4.f.

Author

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