zlaqr4 (3) - Linux Manuals

NAME

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 (logicalWANTT, logicalWANTZ, integerN, integerILO, integerIHI, complex*16, dimension( ldh, * )H, integerLDH, complex*16, dimension( * )W, integerILOZ, integerIHIZ, complex*16, dimension( ldz, * )Z, integerLDZ, complex*16, dimension( * )WORK, integerLWORK, integerINFO)

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 .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 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.GT.0, then 1.LE.ILO.LE.IHI.LE.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.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).


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 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. 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.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 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 .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 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
           .GT. 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 .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 unitary 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 unitary 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:

September 2012

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 247 of file zlaqr4.f.

Author

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