shseqr.f man page

shseqr.f —



subroutine shseqr (JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z, LDZ, WORK, LWORK, INFO)

Function/Subroutine Documentation

subroutine shseqr (characterJOB, characterCOMPZ, integerN, integerILO, integerIHI, real, dimension( ldh, * )H, integerLDH, real, dimension( * )WR, real, dimension( * )WI, real, dimension( ldz, * )Z, integerLDZ, real, dimension( * )WORK, integerLWORK, integerINFO)



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.



 = 'E':  compute eigenvalues only;
 = 'S':  compute eigenvalues and the Schur form T.


 = '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.


 The order of the matrix H.  N .GE. 0.





 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.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
 If N = 0, then ILO = 1 and IHI = 0.


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).LT.0. 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

 Unlike earlier versions of SHSEQR, 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.


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


WR is REAL array, dimension (N)


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) .GT. 0 and
 WI(i+1) .LT. 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 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.GT.0 is given under
 the description of INFO below.)


 The leading dimension of the array Z.  if COMPZ = 'I' or
 COMPZ = 'V', then LDZ.GE.MAX(1,N).  Otherwize, LDZ.GE.1.


WORK is REAL array, dimension (LWORK)
 On exit, if INFO = 0, WORK(1) returns an estimate of
 the optimal value for LWORK.


 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

 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.


   =  0:  successful exit
 .LT. 0:  if INFO = -i, the i-th argument had an illegal
 .GT. 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 .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 an orthogonal matrix.  The final
      value of H is upper Hessenberg and quasi-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 orthogonal 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 orthogonal matrix in (*) (regard-
      less of the value of JOB.)

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


Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.


November 2011


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

Further Details:

 Default values supplied by
 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).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

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

     (**)  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.


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 316 of file shseqr.f.


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Referenced By

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

LAPACK Version 3.4.2 Sat Nov 16 2013