cgsvj0.f

Go to the documentation of this file.

*> \brief \b CGSVJ0 pre-processor for the routine cgesvj.
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CGSVJ0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgsvj0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgsvj0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgsvj0.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE CGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS,
*                          SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
*
*       .. Scalar Arguments ..
*       INTEGER            INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP
*       REAL               EPS, SFMIN, TOL
*       CHARACTER*1        JOBV
*       ..
*       .. Array Arguments ..
*       COMPLEX            A( LDA, * ), D( N ), V( LDV, * ), WORK( LWORK )
*       REAL               SVA( N )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CGSVJ0 is called from CGESVJ as a pre-processor and that is its main
*> purpose. It applies Jacobi rotations in the same way as CGESVJ does, but
*> it does not check convergence (stopping criterion). Few tuning
*> parameters (marked by [TP]) are available for the implementer.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBV
*> \verbatim
*>          JOBV is CHARACTER*1
*>          Specifies whether the output from this procedure is used
*>          to compute the matrix V:
*>          = 'V': the product of the Jacobi rotations is accumulated
*>                 by postmultiplying the N-by-N array V.
*>                (See the description of V.)
*>          = 'A': the product of the Jacobi rotations is accumulated
*>                 by postmultiplying the MV-by-N array V.
*>                (See the descriptions of MV and V.)
*>          = 'N': the Jacobi rotations are not accumulated.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*>          M is INTEGER
*>          The number of rows of the input matrix A.  M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of columns of the input matrix A.
*>          M >= N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*>          A is COMPLEX array, dimension (LDA,N)
*>          On entry, M-by-N matrix A, such that A*diag(D) represents
*>          the input matrix.
*>          On exit,
*>          A_onexit * diag(D_onexit) represents the input matrix A*diag(D)
*>          post-multiplied by a sequence of Jacobi rotations, where the
*>          rotation threshold and the total number of sweeps are given in
*>          TOL and NSWEEP, respectively.
*>          (See the descriptions of D, TOL and NSWEEP.)
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,M).
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is COMPLEX array, dimension (N)
*>          The array D accumulates the scaling factors from the complex scaled
*>          Jacobi rotations.
*>          On entry, A*diag(D) represents the input matrix.
*>          On exit, A_onexit*diag(D_onexit) represents the input matrix
*>          post-multiplied by a sequence of Jacobi rotations, where the
*>          rotation threshold and the total number of sweeps are given in
*>          TOL and NSWEEP, respectively.
*>          (See the descriptions of A, TOL and NSWEEP.)
*> \endverbatim
*>
*> \param[in,out] SVA
*> \verbatim
*>          SVA is REAL array, dimension (N)
*>          On entry, SVA contains the Euclidean norms of the columns of
*>          the matrix A*diag(D).
*>          On exit, SVA contains the Euclidean norms of the columns of
*>          the matrix A_onexit*diag(D_onexit).
*> \endverbatim
*>
*> \param[in] MV
*> \verbatim
*>          MV is INTEGER
*>          If JOBV = 'A', then MV rows of V are post-multiplied by a
*>                           sequence of Jacobi rotations.
*>          If JOBV = 'N',   then MV is not referenced.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*>          V is COMPLEX array, dimension (LDV,N)
*>          If JOBV = 'V' then N rows of V are post-multiplied by a
*>                           sequence of Jacobi rotations.
*>          If JOBV = 'A' then MV rows of V are post-multiplied by a
*>                           sequence of Jacobi rotations.
*>          If JOBV = 'N',   then V is not referenced.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*>          LDV is INTEGER
*>          The leading dimension of the array V,  LDV >= 1.
*>          If JOBV = 'V', LDV >= N.
*>          If JOBV = 'A', LDV >= MV.
*> \endverbatim
*>
*> \param[in] EPS
*> \verbatim
*>          EPS is REAL
*>          EPS = SLAMCH('Epsilon')
*> \endverbatim
*>
*> \param[in] SFMIN
*> \verbatim
*>          SFMIN is REAL
*>          SFMIN = SLAMCH('Safe Minimum')
*> \endverbatim
*>
*> \param[in] TOL
*> \verbatim
*>          TOL is REAL
*>          TOL is the threshold for Jacobi rotations. For a pair
*>          A(:,p), A(:,q) of pivot columns, the Jacobi rotation is
*>          applied only if ABS(COS(angle(A(:,p),A(:,q)))) > TOL.
*> \endverbatim
*>
*> \param[in] NSWEEP
*> \verbatim
*>          NSWEEP is INTEGER
*>          NSWEEP is the number of sweeps of Jacobi rotations to be
*>          performed.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          LWORK is the dimension of WORK. LWORK >= M.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit.
*>          < 0:  if INFO = -i, then the i-th argument had an illegal value
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup gsvj0
*
*> \par Further Details:
*  =====================
*>
*> CGSVJ0 is used just to enable CGESVJ to call a simplified version of
*> itself to work on a submatrix of the original matrix.
*>
*> \par Contributor:
*  ==================
*>
*> Zlatko Drmac (Zagreb, Croatia)
*>
*> \par Bugs, Examples and Comments:
*  =================================
*>
*> Please report all bugs and send interesting test examples and comments to
*> drmac@math.hr. Thank you.
*
*  =====================================================================
      SUBROUTINE cgsvj0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS,
     $                   SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
*
*  -- LAPACK computational routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
      IMPLICIT NONE
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP
      REAL               EPS, SFMIN, TOL
      CHARACTER*1        JOBV
*     ..
*     .. Array Arguments ..
      COMPLEX            A( LDA, * ), D( N ), V( LDV, * ), WORK( LWORK )
      REAL               SVA( N )
*     ..
*
*  =====================================================================
*
*     .. Local Parameters ..
      REAL               ZERO, HALF, ONE
      parameter( zero = 0.0e0, half = 0.5e0, one = 1.0e0)
      COMPLEX      CZERO,                  CONE
      parameter( czero = (0.0e0, 0.0e0), cone = (1.0e0, 0.0e0) )
*     ..
*     .. Local Scalars ..
      COMPLEX            AAPQ, OMPQ
      REAL               AAPP, AAPP0, AAPQ1, AAQQ, APOAQ, AQOAP, BIG,
     $                   bigtheta, cs, mxaapq, mxsinj, rootbig, rooteps,
     $                   rootsfmin, roottol, small, sn, t, temp1, theta,
     $                   thsign
      INTEGER            BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1,
     $                   iswrot, jbc, jgl, kbl, lkahead, mvl, nbl,
     $                   notrot, p, pskipped, q, rowskip, swband
      LOGICAL            APPLV, ROTOK, RSVEC
*     ..
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC abs, max, conjg, real, min, sign, sqrt
*     ..
*     .. External Functions ..
      REAL               SCNRM2
      COMPLEX            CDOTC
      INTEGER            ISAMAX
      LOGICAL            LSAME
      EXTERNAL           isamax, lsame, cdotc, scnrm2
*     ..
*     ..
*     .. External Subroutines ..
*     ..
*     from BLAS
      EXTERNAL           ccopy, crot, cswap, caxpy
*     from LAPACK
      EXTERNAL           clascl, classq, xerbla
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      applv = lsame( jobv, 'A' )
      rsvec = lsame( jobv, 'V' )
      IF( .NOT.( rsvec .OR. applv .OR. lsame( jobv, 'N' ) ) ) THEN
         info = -1
      ELSE IF( m.LT.0 ) THEN
         info = -2
      ELSE IF( ( n.LT.0 ) .OR. ( n.GT.m ) ) THEN
         info = -3
      ELSE IF( lda.LT.m ) THEN
         info = -5
      ELSE IF( ( rsvec.OR.applv ) .AND. ( mv.LT.0 ) ) THEN
         info = -8
      ELSE IF( ( rsvec.AND.( ldv.LT.n ) ).OR.
     $         ( applv.AND.( ldv.LT.mv ) ) ) THEN
         info = -10
      ELSE IF( tol.LE.eps ) THEN
         info = -13
      ELSE IF( nsweep.LT.0 ) THEN
         info = -14
      ELSE IF( lwork.LT.m ) THEN
         info = -16
      ELSE
         info = 0
      END IF
*
*     #:(
      IF( info.NE.0 ) THEN
         CALL xerbla( 'CGSVJ0', -info )
         RETURN
      END IF
*
      IF( rsvec ) THEN
         mvl = n
      ELSE IF( applv ) THEN
         mvl = mv
      END IF
      rsvec = rsvec .OR. applv
 
      rooteps = sqrt( eps )
      rootsfmin = sqrt( sfmin )
      small = sfmin / eps
      big = one / sfmin
      rootbig = one / rootsfmin
      bigtheta = one / rooteps
      roottol = sqrt( tol )
*
*     .. Row-cyclic Jacobi SVD algorithm with column pivoting ..
*
      emptsw = ( n*( n-1 ) ) / 2
      notrot = 0
*
*     .. Row-cyclic pivot strategy with de Rijk's pivoting ..
*
 
      swband = 0
*[TP] SWBAND is a tuning parameter [TP]. It is meaningful and effective
*     if CGESVJ is used as a computational routine in the preconditioned
*     Jacobi SVD algorithm CGEJSV. For sweeps i=1:SWBAND the procedure
*     works on pivots inside a band-like region around the diagonal.
*     The boundaries are determined dynamically, based on the number of
*     pivots above a threshold.
*
      kbl = min( 8, n )
*[TP] KBL is a tuning parameter that defines the tile size in the
*     tiling of the p-q loops of pivot pairs. In general, an optimal
*     value of KBL depends on the matrix dimensions and on the
*     parameters of the computer's memory.
*
      nbl = n / kbl
      IF( ( nbl*kbl ).NE.n )nbl = nbl + 1
*
      blskip = kbl**2
*[TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL.
*
      rowskip = min( 5, kbl )
*[TP] ROWSKIP is a tuning parameter.
*
      lkahead = 1
*[TP] LKAHEAD is a tuning parameter.
*
*     Quasi block transformations, using the lower (upper) triangular
*     structure of the input matrix. The quasi-block-cycling usually
*     invokes cubic convergence. Big part of this cycle is done inside
*     canonical subspaces of dimensions less than M.
*
*
*     .. Row-cyclic pivot strategy with de Rijk's pivoting ..
*
      DO 1993 i = 1, nsweep
*
*     .. go go go ...
*
         mxaapq = zero
         mxsinj = zero
         iswrot = 0
*
         notrot = 0
         pskipped = 0
*
*     Each sweep is unrolled using KBL-by-KBL tiles over the pivot pairs
*     1 <= p < q <= N. This is the first step toward a blocked implementation
*     of the rotations. New implementation, based on block transformations,
*     is under development.
*
         DO 2000 ibr = 1, nbl
*
            igl = ( ibr-1 )*kbl + 1
*
            DO 1002 ir1 = 0, min( lkahead, nbl-ibr )
*
               igl = igl + ir1*kbl
*
               DO 2001 p = igl, min( igl+kbl-1, n-1 )
*
*     .. de Rijk's pivoting
*
                  q = isamax( n-p+1, sva( p ), 1 ) + p - 1
                  IF( p.NE.q ) THEN
                     CALL cswap( m, a( 1, p ), 1, a( 1, q ), 1 )
                     IF( rsvec )CALL cswap( mvl, v( 1, p ), 1,
     $                                           v( 1, q ), 1 )
                     temp1 = sva( p )
                     sva( p ) = sva( q )
                     sva( q ) = temp1
                     aapq = d(p)
                     d(p) = d(q)
                     d(q) = aapq
                  END IF
*
                  IF( ir1.EQ.0 ) THEN
*
*        Column norms are periodically updated by explicit
*        norm computation.
*        Caveat:
*        Unfortunately, some BLAS implementations compute SNCRM2(M,A(1,p),1)
*        as SQRT(S=CDOTC(M,A(1,p),1,A(1,p),1)), which may cause the result to
*        overflow for ||A(:,p)||_2 > SQRT(overflow_threshold), and to
*        underflow for ||A(:,p)||_2 < SQRT(underflow_threshold).
*        Hence, SCNRM2 cannot be trusted, not even in the case when
*        the true norm is far from the under(over)flow boundaries.
*        If properly implemented SCNRM2 is available, the IF-THEN-ELSE-END IF
*        below should be replaced with "AAPP = SCNRM2( M, A(1,p), 1 )".
*
                     IF( ( sva( p ).LT.rootbig ) .AND.
     $                    ( sva( p ).GT.rootsfmin ) ) THEN
                        sva( p ) = scnrm2( m, a( 1, p ), 1 )
                     ELSE
                        temp1 = zero
                        aapp = one
                        CALL classq( m, a( 1, p ), 1, temp1, aapp )
                        sva( p ) = temp1*sqrt( aapp )
                     END IF
                     aapp = sva( p )
                  ELSE
                     aapp = sva( p )
                  END IF
*
                  IF( aapp.GT.zero ) THEN
*
                     pskipped = 0
*
                     DO 2002 q = p + 1, min( igl+kbl-1, n )
*
                        aaqq = sva( q )
*
                        IF( aaqq.GT.zero ) THEN
*
                           aapp0 = aapp
                           IF( aaqq.GE.one ) THEN
                              rotok = ( small*aapp ).LE.aaqq
                              IF( aapp.LT.( big / aaqq ) ) THEN
                                 aapq = ( cdotc( m, a( 1, p ), 1,
     $                                   a( 1, q ), 1 ) / aaqq ) / aapp
                              ELSE
                                 CALL ccopy( m, a( 1, p ), 1,
     $                                        work, 1 )
                                 CALL clascl( 'G', 0, 0, aapp, one,
     $                                m, 1, work, lda, ierr )
                                 aapq = cdotc( m, work, 1,
     $                                   a( 1, q ), 1 ) / aaqq
                              END IF
                           ELSE
                              rotok = aapp.LE.( aaqq / small )
                              IF( aapp.GT.( small / aaqq ) ) THEN
                                 aapq = ( cdotc( m, a( 1, p ), 1,
     $                                    a( 1, q ), 1 ) / aapp ) / aaqq
                              ELSE
                                 CALL ccopy( m, a( 1, q ), 1,
     $                                        work, 1 )
                                 CALL clascl( 'G', 0, 0, aaqq,
     $                                         one, m, 1,
     $                                         work, lda, ierr )
                                 aapq = cdotc( m, a( 1, p ), 1,
     $                                   work, 1 ) / aapp
                              END IF
                           END IF
*
*                           AAPQ = AAPQ * CONJG( CWORK(p) ) * CWORK(q)
                           aapq1  = -abs(aapq)
                           mxaapq = max( mxaapq, -aapq1 )
*
*        TO rotate or NOT to rotate, THAT is the question ...
*
                           IF( abs( aapq1 ).GT.tol ) THEN
                              ompq = aapq / abs(aapq)
*
*           .. rotate
*[RTD]      ROTATED = ROTATED + ONE
*
                              IF( ir1.EQ.0 ) THEN
                                 notrot = 0
                                 pskipped = 0
                                 iswrot = iswrot + 1
                              END IF
*
                              IF( rotok ) THEN
*
                                 aqoap = aaqq / aapp
                                 apoaq = aapp / aaqq
                                 theta = -half*abs( aqoap-apoaq )/aapq1
*
                                 IF( abs( theta ).GT.bigtheta ) THEN
*
                                    t  = half / theta
                                    cs = one
 
                                    CALL crot( m, a(1,p), 1, a(1,q), 1,
     $                                          cs, conjg(ompq)*t )
                                    IF ( rsvec ) THEN
                                        CALL crot( mvl, v(1,p), 1,
     $                                  v(1,q), 1, cs, conjg(ompq)*t )
                                    END IF
 
                                    sva( q ) = aaqq*sqrt( max( zero,
     $                                          one+t*apoaq*aapq1 ) )
                                    aapp = aapp*sqrt( max( zero,
     $                                          one-t*aqoap*aapq1 ) )
                                    mxsinj = max( mxsinj, abs( t ) )
*
                                 ELSE
*
*                 .. choose correct signum for THETA and rotate
*
                                    thsign = -sign( one, aapq1 )
                                    t = one / ( theta+thsign*
     $                                   sqrt( one+theta*theta ) )
                                    cs = sqrt( one / ( one+t*t ) )
                                    sn = t*cs
*
                                    mxsinj = max( mxsinj, abs( sn ) )
                                    sva( q ) = aaqq*sqrt( max( zero,
     $                                          one+t*apoaq*aapq1 ) )
                                    aapp = aapp*sqrt( max( zero,
     $                                      one-t*aqoap*aapq1 ) )
*
                                    CALL crot( m, a(1,p), 1, a(1,q), 1,
     $                                          cs, conjg(ompq)*sn )
                                    IF ( rsvec ) THEN
                                        CALL crot( mvl, v(1,p), 1,
     $                                  v(1,q), 1, cs, conjg(ompq)*sn )
                                    END IF
                                 END IF
                                 d(p) = -d(q) * ompq
*
                                 ELSE
*              .. have to use modified Gram-Schmidt like transformation
                                 CALL ccopy( m, a( 1, p ), 1,
     $                                       work, 1 )
                                 CALL clascl( 'G', 0, 0, aapp, one, m,
     $                                        1, work, lda,
     $                                        ierr )
                                 CALL clascl( 'G', 0, 0, aaqq, one, m,
     $                                        1, a( 1, q ), lda, ierr )
                                 CALL caxpy( m, -aapq, work, 1,
     $                                       a( 1, q ), 1 )
                                 CALL clascl( 'G', 0, 0, one, aaqq, m,
     $                                        1, a( 1, q ), lda, ierr )
                                 sva( q ) = aaqq*sqrt( max( zero,
     $                                      one-aapq1*aapq1 ) )
                                 mxsinj = max( mxsinj, sfmin )
                              END IF
*           END IF ROTOK THEN ... ELSE
*
*           In the case of cancellation in updating SVA(q), SVA(p)
*           recompute SVA(q), SVA(p).
*
                              IF( ( sva( q ) / aaqq )**2.LE.rooteps )
     $                            THEN
                                 IF( ( aaqq.LT.rootbig ) .AND.
     $                               ( aaqq.GT.rootsfmin ) ) THEN
                                    sva( q ) = scnrm2( m, a( 1, q ), 1 )
                                 ELSE
                                    t = zero
                                    aaqq = one
                                    CALL classq( m, a( 1, q ), 1, t,
     $                                           aaqq )
                                    sva( q ) = t*sqrt( aaqq )
                                 END IF
                              END IF
                              IF( ( aapp / aapp0 ).LE.rooteps ) THEN
                                 IF( ( aapp.LT.rootbig ) .AND.
     $                               ( aapp.GT.rootsfmin ) ) THEN
                                    aapp = scnrm2( m, a( 1, p ), 1 )
                                 ELSE
                                    t = zero
                                    aapp = one
                                    CALL classq( m, a( 1, p ), 1, t,
     $                                           aapp )
                                    aapp = t*sqrt( aapp )
                                 END IF
                                 sva( p ) = aapp
                              END IF
*
                           ELSE
*        A(:,p) and A(:,q) already numerically orthogonal
                              IF( ir1.EQ.0 )notrot = notrot + 1
*[RTD]      SKIPPED  = SKIPPED  + 1
                              pskipped = pskipped + 1
                           END IF
                        ELSE
*        A(:,q) is zero column
                           IF( ir1.EQ.0 )notrot = notrot + 1
                           pskipped = pskipped + 1
                        END IF
*
                        IF( ( i.LE.swband ) .AND.
     $                      ( pskipped.GT.rowskip ) ) THEN
                           IF( ir1.EQ.0 )aapp = -aapp
                           notrot = 0
                           GO TO 2103
                        END IF
*
 2002                CONTINUE
*     END q-LOOP
*
 2103                CONTINUE
*     bailed out of q-loop
*
                     sva( p ) = aapp
*
                  ELSE
                     sva( p ) = aapp
                     IF( ( ir1.EQ.0 ) .AND. ( aapp.EQ.zero ) )
     $                   notrot = notrot + min( igl+kbl-1, n ) - p
                  END IF
*
 2001          CONTINUE
*     end of the p-loop
*     end of doing the block ( ibr, ibr )
 1002       CONTINUE
*     end of ir1-loop
*
* ... go to the off diagonal blocks
*
            igl = ( ibr-1 )*kbl + 1
*
            DO 2010 jbc = ibr + 1, nbl
*
               jgl = ( jbc-1 )*kbl + 1
*
*        doing the block at ( ibr, jbc )
*
               ijblsk = 0
               DO 2100 p = igl, min( igl+kbl-1, n )
*
                  aapp = sva( p )
                  IF( aapp.GT.zero ) THEN
*
                     pskipped = 0
*
                     DO 2200 q = jgl, min( jgl+kbl-1, n )
*
                        aaqq = sva( q )
                        IF( aaqq.GT.zero ) THEN
                           aapp0 = aapp
*
*     .. M x 2 Jacobi SVD ..
*
*        Safe Gram matrix computation
*
                           IF( aaqq.GE.one ) THEN
                              IF( aapp.GE.aaqq ) THEN
                                 rotok = ( small*aapp ).LE.aaqq
                              ELSE
                                 rotok = ( small*aaqq ).LE.aapp
                              END IF
                              IF( aapp.LT.( big / aaqq ) ) THEN
                                 aapq = ( cdotc( m, a( 1, p ), 1,
     $                                  a( 1, q ), 1 ) / aaqq ) / aapp
                              ELSE
                                 CALL ccopy( m, a( 1, p ), 1,
     $                                       work, 1 )
                                 CALL clascl( 'G', 0, 0, aapp,
     $                                        one, m, 1,
     $                                        work, lda, ierr )
                                 aapq = cdotc( m, work, 1,
     $                                  a( 1, q ), 1 ) / aaqq
                              END IF
                           ELSE
                              IF( aapp.GE.aaqq ) THEN
                                 rotok = aapp.LE.( aaqq / small )
                              ELSE
                                 rotok = aaqq.LE.( aapp / small )
                              END IF
                              IF( aapp.GT.( small / aaqq ) ) THEN
                                 aapq = ( cdotc( m, a( 1, p ), 1,
     $                                 a( 1, q ), 1 ) / max(aaqq,aapp) )
     $                                               / min(aaqq,aapp)
                              ELSE
                                 CALL ccopy( m, a( 1, q ), 1,
     $                                       work, 1 )
                                 CALL clascl( 'G', 0, 0, aaqq,
     $                                        one, m, 1,
     $                                        work, lda, ierr )
                                 aapq = cdotc( m, a( 1, p ), 1,
     $                                  work, 1 ) / aapp
                              END IF
                           END IF
*
*                           AAPQ = AAPQ * CONJG(CWORK(p))*CWORK(q)
                           aapq1  = -abs(aapq)
                           mxaapq = max( mxaapq, -aapq1 )
*
*        TO rotate or NOT to rotate, THAT is the question ...
*
                           IF( abs( aapq1 ).GT.tol ) THEN
                              ompq = aapq / abs(aapq)
                              notrot = 0
*[RTD]      ROTATED  = ROTATED + 1
                              pskipped = 0
                              iswrot = iswrot + 1
*
                              IF( rotok ) THEN
*
                                 aqoap = aaqq / aapp
                                 apoaq = aapp / aaqq
                                 theta = -half*abs( aqoap-apoaq )/ aapq1
                                 IF( aaqq.GT.aapp0 )theta = -theta
*
                                 IF( abs( theta ).GT.bigtheta ) THEN
                                    t  = half / theta
                                    cs = one
                                    CALL crot( m, a(1,p), 1, a(1,q), 1,
     $                                          cs, conjg(ompq)*t )
                                    IF( rsvec ) THEN
                                        CALL crot( mvl, v(1,p), 1,
     $                                  v(1,q), 1, cs, conjg(ompq)*t )
                                    END IF
                                    sva( q ) = aaqq*sqrt( max( zero,
     $                                         one+t*apoaq*aapq1 ) )
                                    aapp = aapp*sqrt( max( zero,
     $                                     one-t*aqoap*aapq1 ) )
                                    mxsinj = max( mxsinj, abs( t ) )
                                 ELSE
*
*                 .. choose correct signum for THETA and rotate
*
                                    thsign = -sign( one, aapq1 )
                                    IF( aaqq.GT.aapp0 )thsign = -thsign
                                    t = one / ( theta+thsign*
     $                                  sqrt( one+theta*theta ) )
                                    cs = sqrt( one / ( one+t*t ) )
                                    sn = t*cs
                                    mxsinj = max( mxsinj, abs( sn ) )
                                    sva( q ) = aaqq*sqrt( max( zero,
     $                                         one+t*apoaq*aapq1 ) )
                                    aapp = aapp*sqrt( max( zero,
     $                                         one-t*aqoap*aapq1 ) )
*
                                    CALL crot( m, a(1,p), 1, a(1,q), 1,
     $                                          cs, conjg(ompq)*sn )
                                    IF( rsvec ) THEN
                                        CALL crot( mvl, v(1,p), 1,
     $                                  v(1,q), 1, cs, conjg(ompq)*sn )
                                    END IF
                                 END IF
                                 d(p) = -d(q) * ompq
*
                              ELSE
*              .. have to use modified Gram-Schmidt like transformation
                               IF( aapp.GT.aaqq ) THEN
                                    CALL ccopy( m, a( 1, p ), 1,
     $                                          work, 1 )
                                    CALL clascl( 'G', 0, 0, aapp, one,
     $                                           m, 1, work,lda,
     $                                           ierr )
                                    CALL clascl( 'G', 0, 0, aaqq, one,
     $                                           m, 1, a( 1, q ), lda,
     $                                           ierr )
                                    CALL caxpy( m, -aapq, work,
     $                                          1, a( 1, q ), 1 )
                                    CALL clascl( 'G', 0, 0, one, aaqq,
     $                                           m, 1, a( 1, q ), lda,
     $                                           ierr )
                                    sva( q ) = aaqq*sqrt( max( zero,
     $                                         one-aapq1*aapq1 ) )
                                    mxsinj = max( mxsinj, sfmin )
                               ELSE
                                   CALL ccopy( m, a( 1, q ), 1,
     $                                          work, 1 )
                                    CALL clascl( 'G', 0, 0, aaqq, one,
     $                                           m, 1, work,lda,
     $                                           ierr )
                                    CALL clascl( 'G', 0, 0, aapp, one,
     $                                           m, 1, a( 1, p ), lda,
     $                                           ierr )
                                    CALL caxpy( m, -conjg(aapq),
     $                                   work, 1, a( 1, p ), 1 )
                                    CALL clascl( 'G', 0, 0, one, aapp,
     $                                           m, 1, a( 1, p ), lda,
     $                                           ierr )
                                    sva( p ) = aapp*sqrt( max( zero,
     $                                         one-aapq1*aapq1 ) )
                                    mxsinj = max( mxsinj, sfmin )
                               END IF
                              END IF
*           END IF ROTOK THEN ... ELSE
*
*           In the case of cancellation in updating SVA(q), SVA(p)
*           .. recompute SVA(q), SVA(p)
                              IF( ( sva( q ) / aaqq )**2.LE.rooteps )
     $                            THEN
                                 IF( ( aaqq.LT.rootbig ) .AND.
     $                               ( aaqq.GT.rootsfmin ) ) THEN
                                    sva( q ) = scnrm2( m, a( 1, q ), 1)
                                  ELSE
                                    t = zero
                                    aaqq = one
                                    CALL classq( m, a( 1, q ), 1, t,
     $                                           aaqq )
                                    sva( q ) = t*sqrt( aaqq )
                                 END IF
                              END IF
                              IF( ( aapp / aapp0 )**2.LE.rooteps ) THEN
                                 IF( ( aapp.LT.rootbig ) .AND.
     $                               ( aapp.GT.rootsfmin ) ) THEN
                                    aapp = scnrm2( m, a( 1, p ), 1 )
                                 ELSE
                                    t = zero
                                    aapp = one
                                    CALL classq( m, a( 1, p ), 1, t,
     $                                           aapp )
                                    aapp = t*sqrt( aapp )
                                 END IF
                                 sva( p ) = aapp
                              END IF
*              end of OK rotation
                           ELSE
                              notrot = notrot + 1
*[RTD]      SKIPPED  = SKIPPED  + 1
                              pskipped = pskipped + 1
                              ijblsk = ijblsk + 1
                           END IF
                        ELSE
                           notrot = notrot + 1
                           pskipped = pskipped + 1
                           ijblsk = ijblsk + 1
                        END IF
*
                        IF( ( i.LE.swband ) .AND. ( ijblsk.GE.blskip ) )
     $                      THEN
                           sva( p ) = aapp
                           notrot = 0
                           GO TO 2011
                        END IF
                        IF( ( i.LE.swband ) .AND.
     $                      ( pskipped.GT.rowskip ) ) THEN
                           aapp = -aapp
                           notrot = 0
                           GO TO 2203
                        END IF
*
 2200                CONTINUE
*        end of the q-loop
 2203                CONTINUE
*
                     sva( p ) = aapp
*
                  ELSE
*
                     IF( aapp.EQ.zero )notrot = notrot +
     $                   min( jgl+kbl-1, n ) - jgl + 1
                     IF( aapp.LT.zero )notrot = 0
*
                  END IF
*
 2100          CONTINUE
*     end of the p-loop
 2010       CONTINUE
*     end of the jbc-loop
 2011       CONTINUE
*2011 bailed out of the jbc-loop
            DO 2012 p = igl, min( igl+kbl-1, n )
               sva( p ) = abs( sva( p ) )
 2012       CONTINUE
***
 2000    CONTINUE
*2000 :: end of the ibr-loop
*
*     .. update SVA(N)
         IF( ( sva( n ).LT.rootbig ) .AND. ( sva( n ).GT.rootsfmin ) )
     $       THEN
            sva( n ) = scnrm2( m, a( 1, n ), 1 )
         ELSE
            t = zero
            aapp = one
            CALL classq( m, a( 1, n ), 1, t, aapp )
            sva( n ) = t*sqrt( aapp )
         END IF
*
*     Additional steering devices
*
         IF( ( i.LT.swband ) .AND. ( ( mxaapq.LE.roottol ) .OR.
     $       ( iswrot.LE.n ) ) )swband = i
*
         IF( ( i.GT.swband+1 ) .AND. ( mxaapq.LT.sqrt( real( n ) )*
     $       tol ) .AND. ( real( n )*mxaapq*mxsinj.LT.tol ) ) THEN
            GO TO 1994
         END IF
*
         IF( notrot.GE.emptsw )GO TO 1994
*
 1993 CONTINUE
*     end i=1:NSWEEP loop
*
* #:( Reaching this point means that the procedure has not converged.
      info = nsweep - 1
      GO TO 1995
*
 1994 CONTINUE
* #:) Reaching this point means numerical convergence after the i-th
*     sweep.
*
      info = 0
* #:) INFO = 0 confirms successful iterations.
 1995 CONTINUE
*
*     Sort the vector SVA() of column norms.
      DO 5991 p = 1, n - 1
         q = isamax( n-p+1, sva( p ), 1 ) + p - 1
         IF( p.NE.q ) THEN
            temp1 = sva( p )
            sva( p ) = sva( q )
            sva( q ) = temp1
            aapq = d( p )
            d( p ) = d( q )
            d( q ) = aapq
            CALL cswap( m, a( 1, p ), 1, a( 1, q ), 1 )
            IF( rsvec )CALL cswap( mvl, v( 1, p ), 1, v( 1, q ), 1 )
         END IF
 5991 CONTINUE
*
      RETURN
*     ..
*     .. END OF CGSVJ0
*     ..
      END