*> \brief \b SGSVJ0 pre-processor for the routine sgesvj.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download SGSVJ0 + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgsvj0.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgsvj0.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgsvj0.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE SGSVJ0( 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 ..
* REAL A( LDA, * ), SVA( N ), D( N ), V( LDV, * ),
* $ WORK( LWORK )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SGSVJ0 is called from SGESVJ as a pre-processor and that is its main
*> purpose. It applies Jacobi rotations in the same way as SGESVJ 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 postmulyiplying the N-by-N array V.
*> (See the description of V.)
*> = 'A': the product of the Jacobi rotations is accumulated
*> by postmulyiplying 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 REAL array, dimension (LDA,N)
*> On entry, M-by-N matrix A, such that A*diag(D) represents
*> the input matrix.
*> On exit,
*> A_onexit * 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 REAL array, dimension (N)
*> The array D accumulates the scaling factors from the fast 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 onexit*diag(D_onexit).
*> \endverbatim
*>
*> \param[in] MV
*> \verbatim
*> MV is INTEGER
*> If JOBV .EQ. 'A', then MV rows of V are post-multipled by a
*> sequence of Jacobi rotations.
*> If JOBV = 'N', then MV is not referenced.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*> V is REAL array, dimension (LDV,N)
*> If JOBV .EQ. 'V' then N rows of V are post-multipled by a
*> sequence of Jacobi rotations.
*> If JOBV .EQ. 'A' then MV rows of V are post-multipled 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 .GE. N.
*> If JOBV = 'A', LDV .GE. 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)))) .GT. 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 REAL array, dimension (LWORK)
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> LWORK is the dimension of WORK. LWORK .GE. 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.
*
*> \date November 2017
*
*> \ingroup realOTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> SGSVJ0 is used just to enable SGESVJ to call a simplified version of
*> itself to work on a submatrix of the original matrix.
*>
*> \par Contributors:
* ==================
*>
*> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
*>
*> \par Bugs, Examples and Comments:
* =================================
*>
*> Please report all bugs and send interesting test examples and comments to
*> drmac@math.hr. Thank you.
*
* =====================================================================
SUBROUTINE SGSVJ0( JOBV, M, N, A, LDA, D, SVA, MV, V, LDV, EPS,
$ SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.8.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2017
*
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDV, LWORK, M, MV, N, NSWEEP
REAL EPS, SFMIN, TOL
CHARACTER*1 JOBV
* ..
* .. Array Arguments ..
REAL A( LDA, * ), SVA( N ), D( N ), V( LDV, * ),
$ WORK( LWORK )
* ..
*
* =====================================================================
*
* .. Local Parameters ..
REAL ZERO, HALF, ONE
PARAMETER ( ZERO = 0.0E0, HALF = 0.5E0, ONE = 1.0E0)
* ..
* .. Local Scalars ..
REAL AAPP, AAPP0, AAPQ, 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
* ..
* .. Local Arrays ..
REAL FASTR( 5 )
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, FLOAT, MIN, SIGN, SQRT
* ..
* .. External Functions ..
REAL SDOT, SNRM2
INTEGER ISAMAX
LOGICAL LSAME
EXTERNAL ISAMAX, LSAME, SDOT, SNRM2
* ..
* .. External Subroutines ..
EXTERNAL SAXPY, SCOPY, SLASCL, SLASSQ, SROTM, SSWAP,
$ 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( 'SGSVJ0', -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
FASTR( 1 ) = ZERO
*
* .. Row-cyclic pivot strategy with de Rijk's pivoting ..
*
SWBAND = 0
*[TP] SWBAND is a tuning parameter. It is meaningful and effective
* if SGESVJ is used as a computational routine in the preconditioned
* Jacobi SVD algorithm SGESVJ. For sweeps i=1:SWBAND the procedure
* ......
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 ) + 1
*[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.
SWBAND = 0
PSKIPPED = 0
*
DO 1993 i = 1, NSWEEP
* .. go go go ...
*
MXAAPQ = ZERO
MXSINJ = ZERO
ISWROT = 0
*
NOTROT = 0
PSKIPPED = 0
*
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 SSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
IF( RSVEC )CALL SSWAP( MVL, V( 1, p ), 1,
$ V( 1, q ), 1 )
TEMP1 = SVA( p )
SVA( p ) = SVA( q )
SVA( q ) = TEMP1
TEMP1 = D( p )
D( p ) = D( q )
D( q ) = TEMP1
END IF
*
IF( ir1.EQ.0 ) THEN
*
* Column norms are periodically updated by explicit
* norm computation.
* Caveat:
* Some BLAS implementations compute SNRM2(M,A(1,p),1)
* as SQRT(SDOT(M,A(1,p),1,A(1,p),1)), which may result in
* overflow for ||A(:,p)||_2 > SQRT(overflow_threshold), and
* undeflow for ||A(:,p)||_2 < SQRT(underflow_threshold).
* Hence, SNRM2 cannot be trusted, not even in the case when
* the true norm is far from the under(over)flow boundaries.
* If properly implemented SNRM2 is available, the IF-THEN-ELSE
* below should read "AAPP = SNRM2( M, A(1,p), 1 ) * D(p)".
*
IF( ( SVA( p ).LT.ROOTBIG ) .AND.
$ ( SVA( p ).GT.ROOTSFMIN ) ) THEN
SVA( p ) = SNRM2( M, A( 1, p ), 1 )*D( p )
ELSE
TEMP1 = ZERO
AAPP = ONE
CALL SLASSQ( M, A( 1, p ), 1, TEMP1, AAPP )
SVA( p ) = TEMP1*SQRT( AAPP )*D( p )
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 = ( SDOT( M, A( 1, p ), 1, A( 1,
$ q ), 1 )*D( p )*D( q ) / AAQQ )
$ / AAPP
ELSE
CALL SCOPY( M, A( 1, p ), 1, WORK, 1 )
CALL SLASCL( 'G', 0, 0, AAPP, D( p ),
$ M, 1, WORK, LDA, IERR )
AAPQ = SDOT( M, WORK, 1, A( 1, q ),
$ 1 )*D( q ) / AAQQ
END IF
ELSE
ROTOK = AAPP.LE.( AAQQ / SMALL )
IF( AAPP.GT.( SMALL / AAQQ ) ) THEN
AAPQ = ( SDOT( M, A( 1, p ), 1, A( 1,
$ q ), 1 )*D( p )*D( q ) / AAQQ )
$ / AAPP
ELSE
CALL SCOPY( M, A( 1, q ), 1, WORK, 1 )
CALL SLASCL( 'G', 0, 0, AAQQ, D( q ),
$ M, 1, WORK, LDA, IERR )
AAPQ = SDOT( M, WORK, 1, A( 1, p ),
$ 1 )*D( p ) / AAPP
END IF
END IF
*
MXAAPQ = MAX( MXAAPQ, ABS( AAPQ ) )
*
* TO rotate or NOT to rotate, THAT is the question ...
*
IF( ABS( AAPQ ).GT.TOL ) THEN
*
* .. rotate
* 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 ) / AAPQ
*
IF( ABS( THETA ).GT.BIGTHETA ) THEN
*
T = HALF / THETA
FASTR( 3 ) = T*D( p ) / D( q )
FASTR( 4 ) = -T*D( q ) / D( p )
CALL SROTM( M, A( 1, p ), 1,
$ A( 1, q ), 1, FASTR )
IF( RSVEC )CALL SROTM( MVL,
$ V( 1, p ), 1,
$ V( 1, q ), 1,
$ FASTR )
SVA( q ) = AAQQ*SQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
AAPP = AAPP*SQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
MXSINJ = MAX( MXSINJ, ABS( T ) )
*
ELSE
*
* .. choose correct signum for THETA and rotate
*
THSIGN = -SIGN( ONE, AAPQ )
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*AAPQ ) )
AAPP = AAPP*SQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
*
APOAQ = D( p ) / D( q )
AQOAP = D( q ) / D( p )
IF( D( p ).GE.ONE ) THEN
IF( D( q ).GE.ONE ) THEN
FASTR( 3 ) = T*APOAQ
FASTR( 4 ) = -T*AQOAP
D( p ) = D( p )*CS
D( q ) = D( q )*CS
CALL SROTM( M, A( 1, p ), 1,
$ A( 1, q ), 1,
$ FASTR )
IF( RSVEC )CALL SROTM( MVL,
$ V( 1, p ), 1, V( 1, q ),
$ 1, FASTR )
ELSE
CALL SAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL SAXPY( M, CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
D( p ) = D( p )*CS
D( q ) = D( q ) / CS
IF( RSVEC ) THEN
CALL SAXPY( MVL, -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL SAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
END IF
ELSE
IF( D( q ).GE.ONE ) THEN
CALL SAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL SAXPY( M, -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
D( p ) = D( p ) / CS
D( q ) = D( q )*CS
IF( RSVEC ) THEN
CALL SAXPY( MVL, T*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
CALL SAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
ELSE
IF( D( p ).GE.D( q ) ) THEN
CALL SAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL SAXPY( M, CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
D( p ) = D( p )*CS
D( q ) = D( q ) / CS
IF( RSVEC ) THEN
CALL SAXPY( MVL,
$ -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL SAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
ELSE
CALL SAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL SAXPY( M,
$ -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
D( p ) = D( p ) / CS
D( q ) = D( q )*CS
IF( RSVEC ) THEN
CALL SAXPY( MVL,
$ T*APOAQ, V( 1, p ),
$ 1, V( 1, q ), 1 )
CALL SAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
END IF
END IF
END IF
END IF
*
ELSE
* .. have to use modified Gram-Schmidt like transformation
CALL SCOPY( M, A( 1, p ), 1, WORK, 1 )
CALL SLASCL( 'G', 0, 0, AAPP, ONE, M,
$ 1, WORK, LDA, IERR )
CALL SLASCL( 'G', 0, 0, AAQQ, ONE, M,
$ 1, A( 1, q ), LDA, IERR )
TEMP1 = -AAPQ*D( p ) / D( q )
CALL SAXPY( M, TEMP1, WORK, 1,
$ A( 1, q ), 1 )
CALL SLASCL( 'G', 0, 0, ONE, AAQQ, M,
$ 1, A( 1, q ), LDA, IERR )
SVA( q ) = AAQQ*SQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
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 ) = SNRM2( M, A( 1, q ), 1 )*
$ D( q )
ELSE
T = ZERO
AAQQ = ONE
CALL SLASSQ( M, A( 1, q ), 1, T,
$ AAQQ )
SVA( q ) = T*SQRT( AAQQ )*D( q )
END IF
END IF
IF( ( AAPP / AAPP0 ).LE.ROOTEPS ) THEN
IF( ( AAPP.LT.ROOTBIG ) .AND.
$ ( AAPP.GT.ROOTSFMIN ) ) THEN
AAPP = SNRM2( M, A( 1, p ), 1 )*
$ D( p )
ELSE
T = ZERO
AAPP = ONE
CALL SLASSQ( M, A( 1, p ), 1, T,
$ AAPP )
AAPP = T*SQRT( AAPP )*D( p )
END IF
SVA( p ) = AAPP
END IF
*
ELSE
* A(:,p) and A(:,q) already numerically orthogonal
IF( ir1.EQ.0 )NOTROT = NOTROT + 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 = ( SDOT( M, A( 1, p ), 1, A( 1,
$ q ), 1 )*D( p )*D( q ) / AAQQ )
$ / AAPP
ELSE
CALL SCOPY( M, A( 1, p ), 1, WORK, 1 )
CALL SLASCL( 'G', 0, 0, AAPP, D( p ),
$ M, 1, WORK, LDA, IERR )
AAPQ = SDOT( M, WORK, 1, A( 1, q ),
$ 1 )*D( q ) / 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 = ( SDOT( M, A( 1, p ), 1, A( 1,
$ q ), 1 )*D( p )*D( q ) / AAQQ )
$ / AAPP
ELSE
CALL SCOPY( M, A( 1, q ), 1, WORK, 1 )
CALL SLASCL( 'G', 0, 0, AAQQ, D( q ),
$ M, 1, WORK, LDA, IERR )
AAPQ = SDOT( M, WORK, 1, A( 1, p ),
$ 1 )*D( p ) / AAPP
END IF
END IF
*
MXAAPQ = MAX( MXAAPQ, ABS( AAPQ ) )
*
* TO rotate or NOT to rotate, THAT is the question ...
*
IF( ABS( AAPQ ).GT.TOL ) THEN
NOTROT = 0
* ROTATED = ROTATED + 1
PSKIPPED = 0
ISWROT = ISWROT + 1
*
IF( ROTOK ) THEN
*
AQOAP = AAQQ / AAPP
APOAQ = AAPP / AAQQ
THETA = -HALF*ABS( AQOAP-APOAQ ) / AAPQ
IF( AAQQ.GT.AAPP0 )THETA = -THETA
*
IF( ABS( THETA ).GT.BIGTHETA ) THEN
T = HALF / THETA
FASTR( 3 ) = T*D( p ) / D( q )
FASTR( 4 ) = -T*D( q ) / D( p )
CALL SROTM( M, A( 1, p ), 1,
$ A( 1, q ), 1, FASTR )
IF( RSVEC )CALL SROTM( MVL,
$ V( 1, p ), 1,
$ V( 1, q ), 1,
$ FASTR )
SVA( q ) = AAQQ*SQRT( MAX( ZERO,
$ ONE+T*APOAQ*AAPQ ) )
AAPP = AAPP*SQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
MXSINJ = MAX( MXSINJ, ABS( T ) )
ELSE
*
* .. choose correct signum for THETA and rotate
*
THSIGN = -SIGN( ONE, AAPQ )
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*AAPQ ) )
AAPP = AAPP*SQRT( MAX( ZERO,
$ ONE-T*AQOAP*AAPQ ) )
*
APOAQ = D( p ) / D( q )
AQOAP = D( q ) / D( p )
IF( D( p ).GE.ONE ) THEN
*
IF( D( q ).GE.ONE ) THEN
FASTR( 3 ) = T*APOAQ
FASTR( 4 ) = -T*AQOAP
D( p ) = D( p )*CS
D( q ) = D( q )*CS
CALL SROTM( M, A( 1, p ), 1,
$ A( 1, q ), 1,
$ FASTR )
IF( RSVEC )CALL SROTM( MVL,
$ V( 1, p ), 1, V( 1, q ),
$ 1, FASTR )
ELSE
CALL SAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL SAXPY( M, CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
IF( RSVEC ) THEN
CALL SAXPY( MVL, -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL SAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
D( p ) = D( p )*CS
D( q ) = D( q ) / CS
END IF
ELSE
IF( D( q ).GE.ONE ) THEN
CALL SAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL SAXPY( M, -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
IF( RSVEC ) THEN
CALL SAXPY( MVL, T*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
CALL SAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
D( p ) = D( p ) / CS
D( q ) = D( q )*CS
ELSE
IF( D( p ).GE.D( q ) ) THEN
CALL SAXPY( M, -T*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
CALL SAXPY( M, CS*SN*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
D( p ) = D( p )*CS
D( q ) = D( q ) / CS
IF( RSVEC ) THEN
CALL SAXPY( MVL,
$ -T*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
CALL SAXPY( MVL,
$ CS*SN*APOAQ,
$ V( 1, p ), 1,
$ V( 1, q ), 1 )
END IF
ELSE
CALL SAXPY( M, T*APOAQ,
$ A( 1, p ), 1,
$ A( 1, q ), 1 )
CALL SAXPY( M,
$ -CS*SN*AQOAP,
$ A( 1, q ), 1,
$ A( 1, p ), 1 )
D( p ) = D( p ) / CS
D( q ) = D( q )*CS
IF( RSVEC ) THEN
CALL SAXPY( MVL,
$ T*APOAQ, V( 1, p ),
$ 1, V( 1, q ), 1 )
CALL SAXPY( MVL,
$ -CS*SN*AQOAP,
$ V( 1, q ), 1,
$ V( 1, p ), 1 )
END IF
END IF
END IF
END IF
END IF
*
ELSE
IF( AAPP.GT.AAQQ ) THEN
CALL SCOPY( M, A( 1, p ), 1, WORK,
$ 1 )
CALL SLASCL( 'G', 0, 0, AAPP, ONE,
$ M, 1, WORK, LDA, IERR )
CALL SLASCL( 'G', 0, 0, AAQQ, ONE,
$ M, 1, A( 1, q ), LDA,
$ IERR )
TEMP1 = -AAPQ*D( p ) / D( q )
CALL SAXPY( M, TEMP1, WORK, 1,
$ A( 1, q ), 1 )
CALL SLASCL( 'G', 0, 0, ONE, AAQQ,
$ M, 1, A( 1, q ), LDA,
$ IERR )
SVA( q ) = AAQQ*SQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
MXSINJ = MAX( MXSINJ, SFMIN )
ELSE
CALL SCOPY( M, A( 1, q ), 1, WORK,
$ 1 )
CALL SLASCL( 'G', 0, 0, AAQQ, ONE,
$ M, 1, WORK, LDA, IERR )
CALL SLASCL( 'G', 0, 0, AAPP, ONE,
$ M, 1, A( 1, p ), LDA,
$ IERR )
TEMP1 = -AAPQ*D( q ) / D( p )
CALL SAXPY( M, TEMP1, WORK, 1,
$ A( 1, p ), 1 )
CALL SLASCL( 'G', 0, 0, ONE, AAPP,
$ M, 1, A( 1, p ), LDA,
$ IERR )
SVA( p ) = AAPP*SQRT( MAX( ZERO,
$ ONE-AAPQ*AAPQ ) )
MXSINJ = MAX( MXSINJ, SFMIN )
END IF
END IF
* END IF ROTOK THEN ... ELSE
*
* In the case of cancellation in updating SVA(q)
* .. recompute SVA(q)
IF( ( SVA( q ) / AAQQ )**2.LE.ROOTEPS )
$ THEN
IF( ( AAQQ.LT.ROOTBIG ) .AND.
$ ( AAQQ.GT.ROOTSFMIN ) ) THEN
SVA( q ) = SNRM2( M, A( 1, q ), 1 )*
$ D( q )
ELSE
T = ZERO
AAQQ = ONE
CALL SLASSQ( M, A( 1, q ), 1, T,
$ AAQQ )
SVA( q ) = T*SQRT( AAQQ )*D( q )
END IF
END IF
IF( ( AAPP / AAPP0 )**2.LE.ROOTEPS ) THEN
IF( ( AAPP.LT.ROOTBIG ) .AND.
$ ( AAPP.GT.ROOTSFMIN ) ) THEN
AAPP = SNRM2( M, A( 1, p ), 1 )*
$ D( p )
ELSE
T = ZERO
AAPP = ONE
CALL SLASSQ( M, A( 1, p ), 1, T,
$ AAPP )
AAPP = T*SQRT( AAPP )*D( p )
END IF
SVA( p ) = AAPP
END IF
* end of OK rotation
ELSE
NOTROT = NOTROT + 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 ) = SNRM2( M, A( 1, N ), 1 )*D( N )
ELSE
T = ZERO
AAPP = ONE
CALL SLASSQ( M, A( 1, N ), 1, T, AAPP )
SVA( N ) = T*SQRT( AAPP )*D( N )
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.FLOAT( N )*TOL ) .AND.
$ ( FLOAT( 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 comleted the given
* number of iterations.
INFO = NSWEEP - 1
GO TO 1995
1994 CONTINUE
* #:) Reaching this point means that during the i-th sweep all pivots were
* below the given tolerance, causing early exit.
*
INFO = 0
* #:) INFO = 0 confirms successful iterations.
1995 CONTINUE
*
* Sort the vector D.
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
TEMP1 = D( p )
D( p ) = D( q )
D( q ) = TEMP1
CALL SSWAP( M, A( 1, p ), 1, A( 1, q ), 1 )
IF( RSVEC )CALL SSWAP( MVL, V( 1, p ), 1, V( 1, q ), 1 )
END IF
5991 CONTINUE
*
RETURN
* ..
* .. END OF SGSVJ0
* ..
END