dlaqr4.f(3)

NAME
dlaqr4.f

SYNOPSIS

  Functions/Subroutines
subroutine dlaqr4 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)
DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Function/Subroutine Documentation

  subroutine dlaqr4 (logical WANTT, logical WANTZ, integer N, integer ILO, integer IHI, double precision, dimension( ldh, * ) H, integer LDH, double precision, dimension( * ) WR, double precision, dimension( * ) WI, integer ILOZ, integer IHIZ, double precision, dimension( ldz, * ) Z, integer LDZ, double precision, dimension( * ) WORK, integer LWORK, integer INFO)

DLAQR4 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

Purpose:

        DLAQR4 implements one level of recursion for DLAQR0.
        It is a complete implementation of the small bulge multi-shift
        QR algorithm.  It may be called by DLAQR0 and, for large enough
        deflation window size, it may be called by DLAQR3.  This
        subroutine is identical to DLAQR0 except that it calls DLAQR2
        instead of DLAQR3.
        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 = 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 .GE. 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.GT.1,
               H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
               previous call to DGEBAL, and then passed to DGEHRD when the
               matrix output by 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.

    H

              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.

    LDH

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

    WR

              WR is DOUBLE PRECISION array, dimension (IHI)

    WI

              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).

    ILOZ

              ILOZ is INTEGER

    IHIZ

              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.

    Z

              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.)

    LDZ

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

    WORK

              WORK is DOUBLE PRECISION 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 .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 DLAQR4 does a workspace query.
               In this case, 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 XERBLA.  Neither H nor Z are accessed.

    INFO

              INFO is INTEGER
                 =  0:  successful exit
               .GT. 0:  if INFO = i, 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.

Author:
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Date:
    December 2016

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 265 of file dlaqr4.f.

Author

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