zlaqr3.f

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*> \brief \b 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).
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       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 )
*
*       .. Scalar Arguments ..
*       INTEGER            IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
*      $                   LDZ, LWORK, N, ND, NH, NS, NV, NW
*       LOGICAL            WANTT, WANTZ
*       ..
*       .. Array Arguments ..
*       COMPLEX*16         H( LDH, * ), SH( * ), T( LDT, * ), V( LDV, * ),
*      $                   WORK( * ), WV( LDWV, * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*>    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.
*>
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] WANTT
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in] WANTZ
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix H and (if WANTZ is .TRUE.) the
*>          order of the unitary matrix Z.
*> \endverbatim
*>
*> \param[in] KTOP
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in] KBOT
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in] NW
*> \verbatim
*>          NW is INTEGER
*>          Deflation window size.  1 <= NW <= (KBOT-KTOP+1).
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*>          LDH is INTEGER
*>          Leading dimension of H just as declared in the calling
*>          subroutine.  N <= LDH
*> \endverbatim
*>
*> \param[in] ILOZ
*> \verbatim
*>          ILOZ is INTEGER
*> \endverbatim
*>
*> \param[in] IHIZ
*> \verbatim
*>          IHIZ is INTEGER
*>          Specify the rows of Z to which transformations must be
*>          applied if WANTZ is .TRUE.. 1 <= ILOZ <= IHIZ <= N.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of Z just as declared in the
*>          calling subroutine.  1 <= LDZ.
*> \endverbatim
*>
*> \param[out] NS
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[out] ND
*> \verbatim
*>          ND is INTEGER
*>          The number of converged eigenvalues uncovered by this
*>          subroutine.
*> \endverbatim
*>
*> \param[out] SH
*> \verbatim
*>          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).
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*>          V is COMPLEX*16 array, dimension (LDV,NW)
*>          An NW-by-NW work array.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of V just as declared in the
*>          calling subroutine.  NW <= LDV
*> \endverbatim
*>
*> \param[in] NH
*> \verbatim
*>          NH is INTEGER
*>          The number of columns of T.  NH >= NW.
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*>          T is COMPLEX*16 array, dimension (LDT,NW)
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*>          LDT is INTEGER
*>          The leading dimension of T just as declared in the
*>          calling subroutine.  NW <= LDT
*> \endverbatim
*>
*> \param[in] NV
*> \verbatim
*>          NV is INTEGER
*>          The number of rows of work array WV available for
*>          workspace.  NV >= NW.
*> \endverbatim
*>
*> \param[out] WV
*> \verbatim
*>          WV is COMPLEX*16 array, dimension (LDWV,NW)
*> \endverbatim
*>
*> \param[in] LDWV
*> \verbatim
*>          LDWV is INTEGER
*>          The leading dimension of W just as declared in the
*>          calling subroutine.  NW <= LDV
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          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.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          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.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16OTHERauxiliary
*
*> \par Contributors:
*  ==================
*>
*>       Karen Braman and Ralph Byers, Department of Mathematics,
*>       University of Kansas, USA
*>
*  =====================================================================
      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 )
*
*  -- LAPACK auxiliary routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      INTEGER            IHIZ, ILOZ, KBOT, KTOP, LDH, LDT, LDV, LDWV,
     $                   LDZ, LWORK, N, ND, NH, NS, NV, NW
      LOGICAL            WANTT, WANTZ
*     ..
*     .. Array Arguments ..
      COMPLEX*16         H( LDH, * ), SH( * ), T( LDT, * ), V( LDV, * ),
     $                   WORK( * ), WV( LDWV, * ), Z( LDZ, * )
*     ..
*
*  ================================================================
*
*     .. Parameters ..
      COMPLEX*16         ZERO, ONE
      PARAMETER          ( ZERO = ( 0.0d0, 0.0d0 ),
     $                   one = ( 1.0d0, 0.0d0 ) )
      DOUBLE PRECISION   RZERO, RONE
      PARAMETER          ( RZERO = 0.0d0, rone = 1.0d0 )
*     ..
*     .. Local Scalars ..
      COMPLEX*16         BETA, CDUM, S, TAU
      DOUBLE PRECISION   FOO, SAFMAX, SAFMIN, SMLNUM, ULP
      INTEGER            I, IFST, ILST, INFO, INFQR, J, JW, KCOL, KLN,
     $                   knt, krow, kwtop, ltop, lwk1, lwk2, lwk3,
     $                   lwkopt, nmin
*     ..
*     .. External Functions ..
      DOUBLE PRECISION   DLAMCH
      INTEGER            ILAENV
      EXTERNAL           dlamch, ilaenv
*     ..
*     .. External Subroutines ..
      EXTERNAL           dlabad, zcopy, zgehrd, zgemm, zlacpy, zlahqr,
     $                   zlaqr4, zlarf, zlarfg, zlaset, ztrexc, zunmhr
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, dcmplx, dconjg, dimag, int, max, min
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   CABS1
*     ..
*     .. Statement Function definitions ..
      cabs1( cdum ) = abs( dble( cdum ) ) + abs( dimag( cdum ) )
*     ..
*     .. Executable Statements ..
*
*     ==== Estimate optimal workspace. ====
*
      jw = min( nw, kbot-ktop+1 )
      IF( jw.LE.2 ) THEN
         lwkopt = 1
      ELSE
*
*        ==== Workspace query call to ZGEHRD ====
*
         CALL zgehrd( jw, 1, jw-1, t, ldt, work, work, -1, info )
         lwk1 = int( work( 1 ) )
*
*        ==== Workspace query call to ZUNMHR ====
*
         CALL zunmhr( 'R', 'N', jw, jw, 1, jw-1, t, ldt, work, v, ldv,
     $                work, -1, info )
         lwk2 = int( work( 1 ) )
*
*        ==== Workspace query call to ZLAQR4 ====
*
         CALL zlaqr4( .true., .true., jw, 1, jw, t, ldt, sh, 1, jw, v,
     $                ldv, work, -1, infqr )
         lwk3 = int( work( 1 ) )
*
*        ==== Optimal workspace ====
*
         lwkopt = max( jw+max( lwk1, lwk2 ), lwk3 )
      END IF
*
*     ==== Quick return in case of workspace query. ====
*
      IF( lwork.EQ.-1 ) THEN
         work( 1 ) = dcmplx( lwkopt, 0 )
         RETURN
      END IF
*
*     ==== Nothing to do ...
*     ... for an empty active block ... ====
      ns = 0
      nd = 0
      work( 1 ) = one
      IF( ktop.GT.kbot )
     $   RETURN
*     ... nor for an empty deflation window. ====
      IF( nw.LT.1 )
     $   RETURN
*
*     ==== Machine constants ====
*
      safmin = dlamch( 'SAFE MINIMUM' )
      safmax = rone / safmin
      CALL dlabad( safmin, safmax )
      ulp = dlamch( 'PRECISION' )
      smlnum = safmin*( dble( n ) / ulp )
*
*     ==== Setup deflation window ====
*
      jw = min( nw, kbot-ktop+1 )
      kwtop = kbot - jw + 1
      IF( kwtop.EQ.ktop ) THEN
         s = zero
      ELSE
         s = h( kwtop, kwtop-1 )
      END IF
*
      IF( kbot.EQ.kwtop ) THEN
*
*        ==== 1-by-1 deflation window: not much to do ====
*
         sh( kwtop ) = h( kwtop, kwtop )
         ns = 1
         nd = 0
         IF( cabs1( s ).LE.max( smlnum, ulp*cabs1( h( kwtop,
     $       kwtop ) ) ) ) THEN
            ns = 0
            nd = 1
            IF( kwtop.GT.ktop )
     $         h( kwtop, kwtop-1 ) = zero
         END IF
         work( 1 ) = one
         RETURN
      END IF
*
*     ==== Convert to spike-triangular form.  (In case of a
*     .    rare QR failure, this routine continues to do
*     .    aggressive early deflation using that part of
*     .    the deflation window that converged using INFQR
*     .    here and there to keep track.) ====
*
      CALL zlacpy( 'U', jw, jw, h( kwtop, kwtop ), ldh, t, ldt )
      CALL zcopy( jw-1, h( kwtop+1, kwtop ), ldh+1, t( 2, 1 ), ldt+1 )
*
      CALL zlaset( 'A', jw, jw, zero, one, v, ldv )
      nmin = ilaenv( 12, 'ZLAQR3', 'SV', jw, 1, jw, lwork )
      IF( jw.GT.nmin ) THEN
         CALL zlaqr4( .true., .true., jw, 1, jw, t, ldt, sh( kwtop ), 1,
     $                jw, v, ldv, work, lwork, infqr )
      ELSE
         CALL zlahqr( .true., .true., jw, 1, jw, t, ldt, sh( kwtop ), 1,
     $                jw, v, ldv, infqr )
      END IF
*
*     ==== Deflation detection loop ====
*
      ns = jw
      ilst = infqr + 1
      DO 10 knt = infqr + 1, jw
*
*        ==== Small spike tip deflation test ====
*
         foo = cabs1( t( ns, ns ) )
         IF( foo.EQ.rzero )
     $      foo = cabs1( s )
         IF( cabs1( s )*cabs1( v( 1, ns ) ).LE.max( smlnum, ulp*foo ) )
     $        THEN
*
*           ==== One more converged eigenvalue ====
*
            ns = ns - 1
         ELSE
*
*           ==== One undeflatable eigenvalue.  Move it up out of the
*           .    way.   (ZTREXC can not fail in this case.) ====
*
            ifst = ns
            CALL ztrexc( 'V', jw, t, ldt, v, ldv, ifst, ilst, info )
            ilst = ilst + 1
         END IF
   10 CONTINUE
*
*        ==== Return to Hessenberg form ====
*
      IF( ns.EQ.0 )
     $   s = zero
*
      IF( ns.LT.jw ) THEN
*
*        ==== sorting the diagonal of T improves accuracy for
*        .    graded matrices.  ====
*
         DO 30 i = infqr + 1, ns
            ifst = i
            DO 20 j = i + 1, ns
               IF( cabs1( t( j, j ) ).GT.cabs1( t( ifst, ifst ) ) )
     $            ifst = j
   20       CONTINUE
            ilst = i
            IF( ifst.NE.ilst )
     $         CALL ztrexc( 'V', jw, t, ldt, v, ldv, ifst, ilst, info )
   30    CONTINUE
      END IF
*
*     ==== Restore shift/eigenvalue array from T ====
*
      DO 40 i = infqr + 1, jw
         sh( kwtop+i-1 ) = t( i, i )
   40 CONTINUE
*
*
      IF( ns.LT.jw .OR. s.EQ.zero ) THEN
         IF( ns.GT.1 .AND. s.NE.zero ) THEN
*
*           ==== Reflect spike back into lower triangle ====
*
            CALL zcopy( ns, v, ldv, work, 1 )
            DO 50 i = 1, ns
               work( i ) = dconjg( work( i ) )
   50       CONTINUE
            beta = work( 1 )
            CALL zlarfg( ns, beta, work( 2 ), 1, tau )
            work( 1 ) = one
*
            CALL zlaset( 'L', jw-2, jw-2, zero, zero, t( 3, 1 ), ldt )
*
            CALL zlarf( 'L', ns, jw, work, 1, dconjg( tau ), t, ldt,
     $                  work( jw+1 ) )
            CALL zlarf( 'R', ns, ns, work, 1, tau, t, ldt,
     $                  work( jw+1 ) )
            CALL zlarf( 'R', jw, ns, work, 1, tau, v, ldv,
     $                  work( jw+1 ) )
*
            CALL zgehrd( jw, 1, ns, t, ldt, work, work( jw+1 ),
     $                   lwork-jw, info )
         END IF
*
*        ==== Copy updated reduced window into place ====
*
         IF( kwtop.GT.1 )
     $      h( kwtop, kwtop-1 ) = s*dconjg( v( 1, 1 ) )
         CALL zlacpy( 'U', jw, jw, t, ldt, h( kwtop, kwtop ), ldh )
         CALL zcopy( jw-1, t( 2, 1 ), ldt+1, h( kwtop+1, kwtop ),
     $               ldh+1 )
*
*        ==== Accumulate orthogonal matrix in order update
*        .    H and Z, if requested.  ====
*
         IF( ns.GT.1 .AND. s.NE.zero )
     $      CALL zunmhr( 'R', 'N', jw, ns, 1, ns, t, ldt, work, v, ldv,
     $                   work( jw+1 ), lwork-jw, info )
*
*        ==== Update vertical slab in H ====
*
         IF( wantt ) THEN
            ltop = 1
         ELSE
            ltop = ktop
         END IF
         DO 60 krow = ltop, kwtop - 1, nv
            kln = min( nv, kwtop-krow )
            CALL zgemm( 'N', 'N', kln, jw, jw, one, h( krow, kwtop ),
     $                  ldh, v, ldv, zero, wv, ldwv )
            CALL zlacpy( 'A', kln, jw, wv, ldwv, h( krow, kwtop ), ldh )
   60    CONTINUE
*
*        ==== Update horizontal slab in H ====
*
         IF( wantt ) THEN
            DO 70 kcol = kbot + 1, n, nh
               kln = min( nh, n-kcol+1 )
               CALL zgemm( 'C', 'N', jw, kln, jw, one, v, ldv,
     $                     h( kwtop, kcol ), ldh, zero, t, ldt )
               CALL zlacpy( 'A', jw, kln, t, ldt, h( kwtop, kcol ),
     $                      ldh )
   70       CONTINUE
         END IF
*
*        ==== Update vertical slab in Z ====
*
         IF( wantz ) THEN
            DO 80 krow = iloz, ihiz, nv
               kln = min( nv, ihiz-krow+1 )
               CALL zgemm( 'N', 'N', kln, jw, jw, one, z( krow, kwtop ),
     $                     ldz, v, ldv, zero, wv, ldwv )
               CALL zlacpy( 'A', kln, jw, wv, ldwv, z( krow, kwtop ),
     $                      ldz )
   80       CONTINUE
         END IF
      END IF
*
*     ==== Return the number of deflations ... ====
*
      nd = jw - ns
*
*     ==== ... and the number of shifts. (Subtracting
*     .    INFQR from the spike length takes care
*     .    of the case of a rare QR failure while
*     .    calculating eigenvalues of the deflation
*     .    window.)  ====
*
      ns = ns - infqr
*
*      ==== Return optimal workspace. ====
*
      work( 1 ) = dcmplx( lwkopt, 0 )
*
*     ==== End of ZLAQR3 ====
*
      END