slahqr.f - Man Page

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.

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:

December 2016

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 209 of file slahqr.f.