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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