zlahqr.f(3)

NAME
zlahqr.f -

SYNOPSIS

  Functions/Subroutines
subroutine zlahqr (WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, INFO)
ZLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.

Function/Subroutine Documentation

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

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

Purpose:

        ZLAHQR 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:
    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 >= 0.

    ILO

              ILO is INTEGER

    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).
              ZLAHQR 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.

    H

              H is COMPLEX*16 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.

    LDH

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

    W

              W is COMPLEX*16 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).

    ILOZ

              ILOZ is INTEGER

    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.

    Z

              Z is COMPLEX*16 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.

    LDZ

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

    INFO

              INFO is INTEGER
               =   0: successful exit
              .GT. 0: if INFO = i, ZLAHQR 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 .GT. 0 and WANTT is .FALSE., then on exit,
                      the remaining unconverged eigenvalues are the
                      eigenvalues of the upper Hessenberg matrix
                      rows and columns ILO thorugh 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 an orthognal 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)  = (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.

Date:
    September 2012

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 ZLAHQR 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).

Definition at line 195 of file zlahqr.f.

Author

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