laqr3(3)

NAME
laqr3 - laqr3: step in hseqr

SYNOPSIS

  Functions

subroutine claqr3 (wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sh, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork)
CLAQR3 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
subroutine dlaqr3 (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)
DLAQR3 performs the orthogonal similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).
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).
subroutine zlaqr3 (wantt, wantz, n, ktop, kbot, nw, h, ldh, iloz, ihiz, z, ldz, ns, nd, sh, v, ldv, nh, t, ldt, nv, wv, ldwv, work, lwork)
ZLAQR3 performs the unitary similarity transformation of a Hessenberg matrix to detect and deflate fully converged eigenvalues from a trailing principal submatrix (aggressive early deflation).

Detailed Description

Function Documentation

  subroutine claqr3 (logical wantt, logical wantz, integer n, integer ktop, integer kbot, integer nw, complex, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, complex, dimension( ldz, * ) z, integer ldz, integer ns, integer nd, complex, dimension( * ) sh, complex, dimension( ldv, * ) v, integer ldv, integer nh, complex, dimension( ldt, * ) t, integer ldt, integer nv, complex, dimension( ldwv, * ) wv, integer ldwv, complex, dimension( * ) work, integer lwork)

CLAQR3 performs the unitary 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:
    !>
    !>    CLAQR3 accepts as input an upper Hessenberg matrix
    !>    H and performs an unitary 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 unitary 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 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 unitary matrix Z is updated so
    !>          so that the unitary 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 unitary 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 COMPLEX 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 a unitary
    !>          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 COMPLEX array, dimension (LDZ,N)
    !>          IF WANTZ is .TRUE., then on output, the unitary
    !>          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.
    !>

    SH

    !>          SH is COMPLEX array, dimension (KBOT)
    !>          On output, approximate eigenvalues that may
    !>          be used for shifts are stored in SH(KBOT-ND-NS+1)
    !>          through SR(KBOT-ND).  Converged eigenvalues are
    !>          stored in SH(KBOT-ND+1) through SH(KBOT).
    !>

    V

    !>          V is COMPLEX 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 COMPLEX 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 COMPLEX 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 COMPLEX 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; CLAQR3
    !>          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

  subroutine dlaqr3 (logical wantt, logical wantz, integer n, integer ktop, integer kbot, integer nw, double precision, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, double precision, dimension( ldz, * ) z, integer ldz, integer ns, integer nd, double precision, dimension( * ) sr, double precision, dimension( * ) si, double precision, dimension( ldv, * ) v, integer ldv, integer nh, double precision, dimension( ldt, * ) t, integer ldt, integer nv, double precision, dimension( ldwv, * ) wv, integer ldwv, double precision, dimension( * ) work, integer lwork)

DLAQR3 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:
    !>
    !>    DLAQR3 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (KBOT)
    !>

    SI

    !>          SI is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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; DLAQR3
    !>          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

  subroutine slaqr3 (logical wantt, logical wantz, integer n, integer ktop, integer kbot, integer nw, real, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, real, dimension( ldz, * ) z, integer ldz, integer ns, integer nd, real, dimension( * ) sr, real, dimension( * ) si, real, dimension( ldv, * ) v, integer ldv, integer nh, real, dimension( ldt, * ) t, integer ldt, integer nv, real, dimension( ldwv, * ) wv, integer ldwv, real, dimension( * ) work, integer 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).

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

  subroutine zlaqr3 (logical wantt, logical wantz, integer n, integer ktop, integer kbot, integer nw, complex*16, dimension( ldh, * ) h, integer ldh, integer iloz, integer ihiz, complex*16, dimension( ldz, * ) z, integer ldz, integer ns, integer nd, complex*16, dimension( * ) sh, complex*16, dimension( ldv, * ) v, integer ldv, integer nh, complex*16, dimension( ldt, * ) t, integer ldt, integer nv, complex*16, dimension( ldwv, * ) wv, integer ldwv, complex*16, dimension( * ) work, integer lwork)

ZLAQR3 performs the unitary 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:
    !>
    !>    ZLAQR3 accepts as input an upper Hessenberg matrix
    !>    H and performs an unitary 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 unitary 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 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 unitary matrix Z is updated so
    !>          so that the unitary 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 unitary 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 COMPLEX*16 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 a unitary
    !>          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 COMPLEX*16 array, dimension (LDZ,N)
    !>          IF WANTZ is .TRUE., then on output, the unitary
    !>          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.
    !>

    SH

    !>          SH is COMPLEX*16 array, dimension (KBOT)
    !>          On output, approximate eigenvalues that may
    !>          be used for shifts are stored in SH(KBOT-ND-NS+1)
    !>          through SR(KBOT-ND).  Converged eigenvalues are
    !>          stored in SH(KBOT-ND+1) through SH(KBOT).
    !>

    V

    !>          V is COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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; ZLAQR3
    !>          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

Author

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