SRC/shseqr.f(3)

NAME
SRC/shseqr.f

SYNOPSIS

  Functions/Subroutines
subroutine shseqr (job, compz, n, ilo, ihi, h, ldh, wr, wi, z, ldz, work, lwork, info)
SHSEQR

Function/Subroutine Documentation

  subroutine shseqr (character job, character compz, integer n, integer ilo, integer ihi, real, dimension( ldh, * ) h, integer ldh, real, dimension( * ) wr, real, dimension( * ) wi, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer info)

SHSEQR

Purpose:

    !>
    !>    SHSEQR 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
    JOB

    !>          JOB is CHARACTER*1
    !>           = 'E':  compute eigenvalues only;
    !>           = 'S':  compute eigenvalues and the Schur form T.
    !>

    COMPZ

    !>          COMPZ is CHARACTER*1
    !>           = 'N':  no Schur vectors are computed;
    !>           = 'I':  Z is initialized to the unit matrix and the matrix Z
    !>                   of Schur vectors of H is returned;
    !>           = 'V':  Z must contain an orthogonal matrix Q on entry, and
    !>                   the product Q*Z is returned.
    !>

    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. ILO and IHI are normally
    !>           set by a previous call to SGEBAL, and then passed to ZGEHRD
    !>           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 JOB = 'S', 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 JOB = 'E', the
    !>           contents of H are unspecified on exit.  (The output value of
    !>           H when INFO > 0 is given under the description of INFO
    !>           below.)
    !>
    !>           Unlike earlier versions of SHSEQR, this subroutine may
    !>           explicitly 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 (N)
    !>

    WI

    !>          WI is REAL array, dimension (N)
    !>
    !>           The real and imaginary parts, respectively, of the computed
    !>           eigenvalues. 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 JOB = 'S', 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).
    !>

    Z

    !>          Z is REAL array, dimension (LDZ,N)
    !>           If COMPZ = 'N', Z is not referenced.
    !>           If COMPZ = 'I', on entry Z need not be set and on exit,
    !>           if INFO = 0, Z contains the orthogonal matrix Z of the Schur
    !>           vectors of H.  If COMPZ = 'V', on entry Z must contain an
    !>           N-by-N matrix Q, which is assumed to be equal to the unit
    !>           matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
    !>           if INFO = 0, Z contains Q*Z.
    !>           Normally Q is the orthogonal matrix generated by SORGHR
    !>           after the call to SGEHRD which formed the Hessenberg matrix
    !>           H. (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 COMPZ = 'I' or
    !>           COMPZ = 'V', then LDZ >= MAX(1,N).  Otherwise, LDZ >= 1.
    !>

    WORK

    !>          WORK is REAL array, dimension (LWORK)
    !>           On exit, if INFO = 0, 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 and delivers very good and sometimes
    !>           optimal performance.  However, LWORK as large as 11*N
    !>           may be required for optimal performance.  A workspace
    !>           query is recommended to determine the optimal workspace
    !>           size.
    !>
    !>           If LWORK = -1, then SHSEQR does a workspace query.
    !>           In this case, SHSEQR 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, the i-th argument had an illegal
    !>                    value
    !>             > 0:  if INFO = i, SHSEQR 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 JOB = 'E', 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 JOB   = 'S', then on exit
    !>
    !>           (*)  (initial value of H)*U  = U*(final value of H)
    !>
    !>                where U is an orthogonal matrix.  The final
    !>                value of H is upper Hessenberg and quasi-triangular
    !>                in rows and columns INFO+1 through IHI.
    !>
    !>                If INFO > 0 and COMPZ = 'V', then on exit
    !>
    !>                  (final value of Z)  =  (initial value of Z)*U
    !>
    !>                where U is the orthogonal matrix in (*) (regard-
    !>                less of the value of JOB.)
    !>
    !>                If INFO > 0 and COMPZ = 'I', then on exit
    !>                      (final value of Z)  = U
    !>                where U is the orthogonal matrix in (*) (regard-
    !>                less of the value of JOB.)
    !>
    !>                If INFO > 0 and COMPZ = 'N', 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

Further Details:

    !>
    !>             Default values supplied by
    !>             ILAENV(ISPEC,'SHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
    !>             It is suggested that these defaults be adjusted in order
    !>             to attain best performance in each particular
    !>             computational environment.
    !>
    !>            ISPEC=12: The SLAHQR vs SLAQR0 crossover point.
    !>                      Default: 75. (Must be at least 11.)
    !>
    !>            ISPEC=13: Recommended deflation window size.
    !>                      This depends on ILO, IHI and NS.  NS is the
    !>                      number of simultaneous shifts returned
    !>                      by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
    !>                      The default for (IHI-ILO+1) <= 500 is NS.
    !>                      The default for (IHI-ILO+1) >  500 is 3*NS/2.
    !>
    !>            ISPEC=14: Nibble crossover point. (See IPARMQ for
    !>                      details.)  Default: 14% of deflation window
    !>                      size.
    !>
    !>            ISPEC=15: Number of simultaneous shifts in a multishift
    !>                      QR iteration.
    !>
    !>                      If IHI-ILO+1 is ...
    !>
    !>                      greater than      ...but less    ... the
    !>                      or equal to ...      than        default is
    !>
    !>                           1               30          NS =   2(+)
    !>                          30               60          NS =   4(+)
    !>                          60              150          NS =  10(+)
    !>                         150              590          NS =  **
    !>                         590             3000          NS =  64
    !>                        3000             6000          NS = 128
    !>                        6000             infinity      NS = 256
    !>
    !>                  (+)  By default some or all matrices of this order
    !>                       are passed to the implicit double shift routine
    !>                       SLAHQR and this parameter is ignored.  See
    !>                       ISPEC=12 above and comments in IPARMQ for
    !>                       details.
    !>
    !>                 (**)  The asterisks (**) indicate an ad-hoc
    !>                       function of N increasing from 10 to 64.
    !>
    !>            ISPEC=16: Select structured matrix multiply.
    !>                      If the number of simultaneous shifts (specified
    !>                      by ISPEC=15) is less than 14, then the default
    !>                      for ISPEC=16 is 0.  Otherwise the default for
    !>                      ISPEC=16 is 2.
    !>

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 314 of file shseqr.f.

Author

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