Provided by: liblapack-doc_3.12.0-3build1.1_all bug

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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