SRC/zlaqr4.f(3)

NAME
SRC/zlaqr4.f

SYNOPSIS

  Functions/Subroutines
subroutine zlaqr4 (wantt, wantz, n, ilo, ihi, h, ldh, w, iloz, ihiz, z, ldz, work, lwork, info)
ZLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Function/Subroutine Documentation

  subroutine zlaqr4 (logical wantt, logical wantz, integer n, integer ilo, integer ihi, complex*16, dimension( ldh, * ) h, integer ldh, complex*16, dimension( * ) w, integer iloz, integer ihiz, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, integer info)

ZLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Purpose:

    !>
    !>    ZLAQR4 implements one level of recursion for ZLAQR0.
    !>    It is a complete implementation of the small bulge multi-shift
    !>    QR algorithm.  It may be called by ZLAQR0 and, for large enough
    !>    deflation window size, it may be called by ZLAQR3.  This
    !>    subroutine is identical to ZLAQR0 except that it calls ZLAQR2
    !>    instead of ZLAQR3.
    !>
    !>    ZLAQR4 computes the eigenvalues of a Hessenberg matrix H
    !>    and, optionally, the matrices T and Z from the Schur decomposition
    !>    H = Z T Z**H, where T is an upper triangular matrix (the
    !>    Schur form), and Z is the unitary matrix of Schur vectors.
    !>
    !>    Optionally Z may be postmultiplied into an input unitary
    !>    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 unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.
    !>

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 ZGEBAL, and then passed to ZGEHRD when the
    !>           matrix output by ZGEBAL 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 COMPLEX*16 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 triangular matrix T from the Schur
    !>           decomposition (the Schur form). If INFO = 0 and WANT 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).
    !>

    W

    !>          W is COMPLEX*16 array, dimension (N)
    !>           The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored
    !>           in W(ILO:IHI). 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 W(i) = H(i,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 COMPLEX*16 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 COMPLEX*16 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 ZLAQR4 does a workspace query.
    !>           In this case, ZLAQR4 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
    !>             =  0:  successful exit
    !>             > 0:  if INFO = i, ZLAQR4 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 unitary 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 unitary 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 245 of file zlaqr4.f.

Author

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