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## clahqr()

subroutine clahqr | ( | logical | wantt,
|  | logical | wantz,
|  | integer | n,
|  | integer | ilo,
|  | integer | ihi,
|  | complex, dimension( ldh, * ) | h,
|  | integer | ldh,
|  | complex, dimension( * ) | w,
|  | integer | iloz,
|  | integer | ihiz,
|  | complex, dimension( ldz, * ) | z,
|  | integer | ldz,
|  | integer | info )

CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Download CLAHQR + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !>    CLAHQR is an auxiliary routine called by CHSEQR to update the
      !>    eigenvalues and Schur decomposition already computed by CHSEQR, by
      !>    dealing with the Hessenberg submatrix in rows and columns ILO to
      !>    IHI.
      !>

Parameters

  [in]

  WANTT

      !>          WANTT is LOGICAL
      !>          = .TRUE. : the full Schur form T is required;
      !>          = .FALSE.: only eigenvalues are required.
      !>

  [in]

  WANTZ

      !>          WANTZ is LOGICAL
      !>          = .TRUE. : the matrix of Schur vectors Z is required;
      !>          = .FALSE.: Schur vectors are not required.
      !>

  [in]

  N

      !>          N is INTEGER
      !>          The order of the matrix H.  N >= 0.
      !>

  [in]

  ILO

      !>          ILO is INTEGER
      !>

  [in]

  IHI

      !>          IHI is INTEGER
      !>          It is assumed that H is already upper triangular in rows and
      !>          columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless ILO = 1).
      !>          CLAHQR works primarily with the Hessenberg submatrix in rows
      !>          and columns ILO to IHI, but applies transformations to all of
      !>          H if WANTT is .TRUE..
      !>          1 <= ILO <= max(1,IHI); IHI <= N.
      !>

  [in,out]

  H

      !>          H is COMPLEX array, dimension (LDH,N)
      !>          On entry, the upper Hessenberg matrix H.
      !>          On exit, if INFO is zero and if WANTT is .TRUE., then H
      !>          is upper triangular in rows and columns ILO:IHI.  If INFO
      !>          is zero and if WANTT is .FALSE., then the contents of H
      !>          are unspecified on exit.  The output state of H in case
      !>          INF is positive is below under the description of INFO.
      !>

  [in]

  LDH

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

  [out]

  W

      !>          W is COMPLEX array, dimension (N)
      !>          The computed eigenvalues ILO to IHI are stored in the
      !>          corresponding elements of W. If WANTT is .TRUE., 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).
      !>

  [in]

  ILOZ

      !>          ILOZ is INTEGER
      !>

  [in]

  IHIZ

      !>          IHIZ is INTEGER
      !>          Specify the rows of Z to which transformations must be
      !>          applied if WANTZ is .TRUE..
      !>          1 <= ILOZ <= ILO; IHI <= IHIZ <= N.
      !>

  [in,out]

  Z

      !>          Z is COMPLEX array, dimension (LDZ,N)
      !>          If WANTZ is .TRUE., on entry Z must contain the current
      !>          matrix Z of transformations accumulated by CHSEQR, and on
      !>          exit Z has been updated; transformations are applied only to
      !>          the submatrix Z(ILOZ:IHIZ,ILO:IHI).
      !>          If WANTZ is .FALSE., Z is not referenced.
      !>

  [in]

  LDZ

      !>          LDZ is INTEGER
      !>          The leading dimension of the array Z. LDZ >= max(1,N).
      !>

  [out]

  INFO

      !>          INFO is INTEGER
      !>           = 0:  successful exit
      !>           > 0:  if INFO = i, CLAHQR failed to compute all the
      !>                  eigenvalues ILO to IHI in a total of 30 iterations
      !>                  per eigenvalue; elements i+1:ihi of W contain
      !>                  those eigenvalues which have been successfully
      !>                  computed.
      !>
      !>                  If INFO > 0 and WANTT is .FALSE., then on exit,
      !>                  the remaining unconverged eigenvalues are the
      !>                  eigenvalues of the upper Hessenberg matrix
      !>                  rows and columns ILO through INFO of the final,
      !>                  output value of H.
      !>
      !>                  If INFO > 0 and WANTT is .TRUE., 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 triangular in
      !>                  rows and columns INFO+1 through IHI.
      !>
      !>                  If INFO > 0 and WANTZ is .TRUE., then on exit
      !>                      (final value of Z)  = (initial value of Z)*U
      !>                  where U is the orthogonal matrix in (*)
      !>                  (regardless of the value of WANTT.)
      !>

Author
  Univ. of Tennessee
  Univ. of California Berkeley
  Univ. of Colorado Denver
  NAG Ltd.

Contributors:

      !>
      !>     02-96 Based on modifications by
      !>     David Day, Sandia National Laboratory, USA
      !>
      !>     12-04 Further modifications by
      !>     Ralph Byers, University of Kansas, USA
      !>     This is a modified version of CLAHQR from LAPACK version 3.0.
      !>     It is (1) more robust against overflow and underflow and
      !>     (2) adopts the more conservative Ahues & Tisseur stopping
      !>     criterion (LAWN 122, 1997).
      !>