slaqr3.f - Man Page

subroutine slaqr3 (wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sr, si, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork)
SLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

SLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).  

Purpose:

    Aggressive early deflation:

    SLAQR3 accepts as input an upper Hessenberg matrix
    H and performs an orthogonal similarity transformation
    designed to detect and deflate fully converged eigenvalues from
    a trailing principal submatrix.  On output H has been over-
    written by a new Hessenberg matrix that is a perturbation of
    an orthogonal similarity transformation of H.  It is to be
    hoped that the final version of H has many zero subdiagonal
    entries.

Parameters

WANTT

          WANTT is LOGICAL
          If .TRUE., then the Hessenberg matrix H is fully updated
          so that the quasi-triangular Schur factor may be
          computed (in cooperation with the calling subroutine).
          If .FALSE., then only enough of H is updated to preserve
          the eigenvalues.

WANTZ

          WANTZ is LOGICAL
          If .TRUE., then the orthogonal matrix Z is updated so
          so that the orthogonal Schur factor may be computed
          (in cooperation with the calling subroutine).
          If .FALSE., then Z is not referenced.

N

          N is INTEGER
          The order of the matrix H and (if WANTZ is .TRUE.) the
          order of the orthogonal matrix Z.

KTOP

          KTOP is INTEGER
          It is assumed that either KTOP = 1 or H(KTOP,KTOP-1)=0.
          KBOT and KTOP together determine an isolated block
          along the diagonal of the Hessenberg matrix.

KBOT

          KBOT is INTEGER
          It is assumed without a check that either
          KBOT = N or H(KBOT+1,KBOT)=0.  KBOT and KTOP together
          determine an isolated block along the diagonal of the
          Hessenberg matrix.

NW

          NW is INTEGER
          Deflation window size.  1 <= NW <= (KBOT-KTOP+1).

H

          H is REAL array, dimension (LDH,N)
          On input the initial N-by-N section of H stores the
          Hessenberg matrix undergoing aggressive early deflation.
          On output H has been transformed by an orthogonal
          similarity transformation, perturbed, and the returned
          to Hessenberg form that (it is to be hoped) has some
          zero subdiagonal entries.

LDH

          LDH is INTEGER
          Leading dimension of H just as declared in the calling
          subroutine.  N <= LDH

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 <= IHIZ <= N.

Z

          Z is REAL array, dimension (LDZ,N)
          IF WANTZ is .TRUE., then on output, the orthogonal
          similarity transformation mentioned above has been
          accumulated into Z(ILOZ:IHIZ,ILOZ:IHIZ) from the right.
          If WANTZ is .FALSE., then Z is unreferenced.

LDZ

          LDZ is INTEGER
          The leading dimension of Z just as declared in the
          calling subroutine.  1 <= LDZ.

NS

          NS is INTEGER
          The number of unconverged (ie approximate) eigenvalues
          returned in SR and SI that may be used as shifts by the
          calling subroutine.

ND

          ND is INTEGER
          The number of converged eigenvalues uncovered by this
          subroutine.

SR

          SR is REAL array, dimension (KBOT)

SI

          SI is REAL array, dimension (KBOT)
          On output, the real and imaginary parts of approximate
          eigenvalues that may be used for shifts are stored in
          SR(KBOT-ND-NS+1) through SR(KBOT-ND) and
          SI(KBOT-ND-NS+1) through SI(KBOT-ND), respectively.
          The real and imaginary parts of converged eigenvalues
          are stored in SR(KBOT-ND+1) through SR(KBOT) and
          SI(KBOT-ND+1) through SI(KBOT), respectively.

V

          V is REAL array, dimension (LDV,NW)
          An NW-by-NW work array.

LDV

          LDV is INTEGER
          The leading dimension of V just as declared in the
          calling subroutine.  NW <= LDV

NH

          NH is INTEGER
          The number of columns of T.  NH >= NW.

T

          T is REAL array, dimension (LDT,NW)

LDT

          LDT is INTEGER
          The leading dimension of T just as declared in the
          calling subroutine.  NW <= LDT

NV

          NV is INTEGER
          The number of rows of work array WV available for
          workspace.  NV >= NW.

WV

          WV is REAL array, dimension (LDWV,NW)

LDWV

          LDWV is INTEGER
          The leading dimension of W just as declared in the
          calling subroutine.  NW <= LDV

WORK

          WORK is REAL array, dimension (LWORK)
          On exit, WORK(1) is set to an estimate of the optimal value
          of LWORK for the given values of N, NW, KTOP and KBOT.

LWORK

          LWORK is INTEGER
          The dimension of the work array WORK.  LWORK = 2*NW
          suffices, but greater efficiency may result from larger
          values of LWORK.

          If LWORK = -1, then a workspace query is assumed; SLAQR3
          only estimates the optimal workspace size for the given
          values of N, NW, KTOP and KBOT.  The estimate is returned
          in WORK(1).  No error message related to LWORK is issued
          by XERBLA.  Neither H nor Z are accessed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

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

Definition at line 272 of file slaqr3.f.