zhseqr.f man page

zhseqr.f —

Synopsis

Functions/Subroutines

subroutine zhseqr (JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO)
ZHSEQR

Function/Subroutine Documentation

subroutine zhseqr (characterJOB, characterCOMPZ, integerN, integerILO, integerIHI, complex*16, dimension( ldh, * )H, integerLDH, complex*16, dimension( * )W, complex*16, dimension( ldz, * )Z, integerLDZ, complex*16, dimension( * )WORK, integerLWORK, integerINFO)

ZHSEQR

Purpose:

ZHSEQR 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)*T*(QZ)**H.

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 unitary matrix Q on entry, and
         the product Q*Z is returned.

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. 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 JOB = 'S', H contains the upper
 triangular matrix T from the Schur decomposition (the
 Schur form). If INFO = 0 and JOB = 'E', 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.)

 Unlike earlier versions of ZHSEQR, this subroutine may
 explicitly 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. If JOB = 'S', 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).

Z

Z is COMPLEX*16 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 unitary 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 unitary matrix generated by ZUNGHR
 after the call to ZGEHRD which formed the Hessenberg matrix
 H. (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 COMPZ = 'I' or
 COMPZ = 'V', then LDZ.GE.MAX(1,N).  Otherwize, LDZ.GE.1.

WORK

WORK is COMPLEX*16 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 .GE. 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 ZHSEQR does a workspace query.
 In this case, ZHSEQR 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
 .LT. 0:  if INFO = -i, the i-th argument had an illegal
          value
 .GT. 0:  if INFO = i, ZHSEQR 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 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 .GT. 0 and JOB   = 'S', 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 COMPZ = 'V', then on exit

        (final value of Z)  =  (initial value of Z)*U

      where U is the unitary matrix in (*) (regard-
      less of the value of JOB.)

      If INFO .GT. 0 and COMPZ = 'I', then on exit
            (final value of Z)  = U
      where U is the unitary matrix in (*) (regard-
      less of the value of JOB.)

      If INFO .GT. 0 and COMPZ = 'N', then Z is not
      accessed.

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2013

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

Further Details:

 Default values supplied by
 ILAENV(ISPEC,'ZHSEQR',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 ZLAHQR vs ZLAQR0 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).LE.500 is NS.
          The default for (IHI-ILO+1).GT.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
           ZLAHQR 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 299 of file zhseqr.f.

Author

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Referenced By

zhseqr(3) is an alias of zhseqr.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK