slahqr.f(3)

NAME
slahqr.f -

SYNOPSIS

  Functions/Subroutines

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

Function/Subroutine Documentation

  subroutine slahqr (logicalWANTT, logicalWANTZ, integerN, integerILO, integerIHI, real, dimension( ldh, * )H, integerLDH, real, dimension( * )WR, real, dimension( * )WI, integerILOZ, integerIHIZ, real, dimension( ldz, * )Z, integerLDZ, integerINFO)

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

Purpose:

        SLAHQR is an auxiliary routine called by SHSEQR to update the
        eigenvalues and Schur decomposition already computed by SHSEQR, 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 quasi-triangular in
              rows and columns IHI+1:N, and that H(ILO,ILO-1) = 0 (unless
              ILO = 1). SLAHQR 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 REAL array, dimension (LDH,N)
              On entry, the upper Hessenberg matrix H.
              On exit, if INFO is zero and if WANTT is .TRUE., H is upper
              quasi-triangular in rows and columns ILO:IHI, with any
              2-by-2 diagonal blocks in standard form. If INFO is zero
              and WANTT is .FALSE., the contents of H are unspecified on
              exit.  The output state of H if INFO is nonzero is given
              below under the description of INFO.

    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 ILO to IHI are stored in the corresponding
              elements of WR and WI. 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 WANTT is .TRUE., 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).

    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 REAL array, dimension (LDZ,N)
              If WANTZ is .TRUE., on entry Z must contain the current
              matrix Z of transformations accumulated by SHSEQR, 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, SLAHQR failed to compute all the
                      eigenvalues ILO to IHI in a total of 30 iterations
                      per eigenvalue; elements i+1:ihi of WR and WI
                      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

Further Details:

         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 SLAHQR 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 207 of file slahqr.f.

Author

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