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cgesvj.f
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*> \brief <b> CGESVJ </b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CGESVJ + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgesvj.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgesvj.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgesvj.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
* LDV, CWORK, LWORK, RWORK, LRWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, LDV, LWORK, LRWORK, M, MV, N
* CHARACTER*1 JOBA, JOBU, JOBV
* ..
* .. Array Arguments ..
* COMPLEX A( LDA, * ), V( LDV, * ), CWORK( LWORK )
* REAL RWORK( LRWORK ), SVA( N )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CGESVJ computes the singular value decomposition (SVD) of a complex
*> M-by-N matrix A, where M >= N. The SVD of A is written as
*> [++] [xx] [x0] [xx]
*> A = U * SIGMA * V^*, [++] = [xx] * [ox] * [xx]
*> [++] [xx]
*> where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
*> matrix, and V is an N-by-N unitary matrix. The diagonal elements
*> of SIGMA are the singular values of A. The columns of U and V are the
*> left and the right singular vectors of A, respectively.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBA
*> \verbatim
*> JOBA is CHARACTER*1
*> Specifies the structure of A.
*> = 'L': The input matrix A is lower triangular;
*> = 'U': The input matrix A is upper triangular;
*> = 'G': The input matrix A is general M-by-N matrix, M >= N.
*> \endverbatim
*>
*> \param[in] JOBU
*> \verbatim
*> JOBU is CHARACTER*1
*> Specifies whether to compute the left singular vectors
*> (columns of U):
*> = 'U' or 'F': The left singular vectors corresponding to the nonzero
*> singular values are computed and returned in the leading
*> columns of A. See more details in the description of A.
*> The default numerical orthogonality threshold is set to
*> approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=SLAMCH('E').
*> = 'C': Analogous to JOBU='U', except that user can control the
*> level of numerical orthogonality of the computed left
*> singular vectors. TOL can be set to TOL = CTOL*EPS, where
*> CTOL is given on input in the array WORK.
*> No CTOL smaller than ONE is allowed. CTOL greater
*> than 1 / EPS is meaningless. The option 'C'
*> can be used if M*EPS is satisfactory orthogonality
*> of the computed left singular vectors, so CTOL=M could
*> save few sweeps of Jacobi rotations.
*> See the descriptions of A and WORK(1).
*> = 'N': The matrix U is not computed. However, see the
*> description of A.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*> JOBV is CHARACTER*1
*> Specifies whether to compute the right singular vectors, that
*> is, the matrix V:
*> = 'V' or 'J': the matrix V is computed and returned in the array V
*> = 'A': the Jacobi rotations are applied to the MV-by-N
*> array V. In other words, the right singular vector
*> matrix V is not computed explicitly; instead it is
*> applied to an MV-by-N matrix initially stored in the
*> first MV rows of V.
*> = 'N': the matrix V is not computed and the array V is not
*> referenced
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the input matrix A. 1/SLAMCH('E') > 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, the M-by-N matrix A.
*> On exit,
*> If JOBU = 'U' .OR. JOBU = 'C':
*> If INFO = 0 :
*> RANKA orthonormal columns of U are returned in the
*> leading RANKA columns of the array A. Here RANKA <= N
*> is the number of computed singular values of A that are
*> above the underflow threshold SLAMCH('S'). The singular
*> vectors corresponding to underflowed or zero singular
*> values are not computed. The value of RANKA is returned
*> in the array RWORK as RANKA=NINT(RWORK(2)). Also see the
*> descriptions of SVA and RWORK. The computed columns of U
*> are mutually numerically orthogonal up to approximately
*> TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'),
*> see the description of JOBU.
*> If INFO > 0,
*> the procedure CGESVJ did not converge in the given number
*> of iterations (sweeps). In that case, the computed
*> columns of U may not be orthogonal up to TOL. The output
*> U (stored in A), SIGMA (given by the computed singular
*> values in SVA(1:N)) and V is still a decomposition of the
*> input matrix A in the sense that the residual
*> || A - SCALE * U * SIGMA * V^* ||_2 / ||A||_2 is small.
*> If JOBU = 'N':
*> If INFO = 0 :
*> Note that the left singular vectors are 'for free' in the
*> one-sided Jacobi SVD algorithm. However, if only the
*> singular values are needed, the level of numerical
*> orthogonality of U is not an issue and iterations are
*> stopped when the columns of the iterated matrix are
*> numerically orthogonal up to approximately M*EPS. Thus,
*> on exit, A contains the columns of U scaled with the
*> corresponding singular values.
*> If INFO > 0 :
*> the procedure CGESVJ did not converge in the given number
*> of iterations (sweeps).
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] SVA
*> \verbatim
*> SVA is REAL array, dimension (N)
*> On exit,
*> If INFO = 0 :
*> depending on the value SCALE = RWORK(1), we have:
*> If SCALE = ONE:
*> SVA(1:N) contains the computed singular values of A.
*> During the computation SVA contains the Euclidean column
*> norms of the iterated matrices in the array A.
*> If SCALE .NE. ONE:
*> The singular values of A are SCALE*SVA(1:N), and this
*> factored representation is due to the fact that some of the
*> singular values of A might underflow or overflow.
*>
*> If INFO > 0 :
*> the procedure CGESVJ did not converge in the given number of
*> iterations (sweeps) and SCALE*SVA(1:N) may not be accurate.
*> \endverbatim
*>
*> \param[in] MV
*> \verbatim
*> MV is INTEGER
*> If JOBV = 'A', then the product of Jacobi rotations in CGESVJ
*> is applied to the first MV rows of V. See the description of JOBV.
*> \endverbatim
*>
*> \param[in,out] V
*> \verbatim
*> V is COMPLEX array, dimension (LDV,N)
*> If JOBV = 'V', then V contains on exit the N-by-N matrix of
*> the right singular vectors;
*> If JOBV = 'A', then V contains the product of the computed right
*> singular vector matrix and the initial matrix in
*> the array V.
*> 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', then LDV >= max(1,N).
*> If JOBV = 'A', then LDV >= max(1,MV) .
*> \endverbatim
*>
*> \param[in,out] CWORK
*> \verbatim
*> CWORK is COMPLEX array, dimension (max(1,LWORK))
*> Used as workspace.
*> If on entry LWORK = -1, then a workspace query is assumed and
*> no computation is done; CWORK(1) is set to the minial (and optimal)
*> length of CWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER.
*> Length of CWORK, LWORK >= M+N.
*> \endverbatim
*>
*> \param[in,out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (max(6,LRWORK))
*> On entry,
*> If JOBU = 'C' :
*> RWORK(1) = CTOL, where CTOL defines the threshold for convergence.
*> The process stops if all columns of A are mutually
*> orthogonal up to CTOL*EPS, EPS=SLAMCH('E').
*> It is required that CTOL >= ONE, i.e. it is not
*> allowed to force the routine to obtain orthogonality
*> below EPSILON.
*> On exit,
*> RWORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
*> are the computed singular values of A.
*> (See description of SVA().)
*> RWORK(2) = NINT(RWORK(2)) is the number of the computed nonzero
*> singular values.
*> RWORK(3) = NINT(RWORK(3)) is the number of the computed singular
*> values that are larger than the underflow threshold.
*> RWORK(4) = NINT(RWORK(4)) is the number of sweeps of Jacobi
*> rotations needed for numerical convergence.
*> RWORK(5) = max_{i.NE.j} |COS(A(:,i),A(:,j))| in the last sweep.
*> This is useful information in cases when CGESVJ did
*> not converge, as it can be used to estimate whether
*> the output is still useful and for post festum analysis.
*> RWORK(6) = the largest absolute value over all sines of the
*> Jacobi rotation angles in the last sweep. It can be
*> useful for a post festum analysis.
*> If on entry LRWORK = -1, then a workspace query is assumed and
*> no computation is done; RWORK(1) is set to the minial (and optimal)
*> length of RWORK.
*> \endverbatim
*>
*> \param[in] LRWORK
*> \verbatim
*> LRWORK is INTEGER
*> Length of RWORK, LRWORK >= MAX(6,N).
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, then the i-th argument had an illegal value
*> > 0: CGESVJ did not converge in the maximal allowed number
*> (NSWEEP=30) of sweeps. The output may still be useful.
*> See the description of RWORK.
*> \endverbatim
*>
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup complexGEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane
*> rotations. In the case of underflow of the tangent of the Jacobi angle, a
*> modified Jacobi transformation of Drmac [3] is used. Pivot strategy uses
*> column interchanges of de Rijk [1]. The relative accuracy of the computed
*> singular values and the accuracy of the computed singular vectors (in
*> angle metric) is as guaranteed by the theory of Demmel and Veselic [2].
*> The condition number that determines the accuracy in the full rank case
*> is essentially min_{D=diag} kappa(A*D), where kappa(.) is the
*> spectral condition number. The best performance of this Jacobi SVD
*> procedure is achieved if used in an accelerated version of Drmac and
*> Veselic [4,5], and it is the kernel routine in the SIGMA library [6].
*> Some tunning parameters (marked with [TP]) are available for the
*> implementer.
*> The computational range for the nonzero singular values is the machine
*> number interval ( UNDERFLOW , OVERFLOW ). In extreme cases, even
*> denormalized singular values can be computed with the corresponding
*> gradual loss of accurate digits.
*> \endverbatim
*
*> \par Contributor:
* ==================
*>
*> \verbatim
*>
*> ============
*>
*> Zlatko Drmac (Zagreb, Croatia)
*>
*> \endverbatim
*
*> \par References:
* ================
*>
*> \verbatim
*>
*> [1] P. P. M. De Rijk: A one-sided Jacobi algorithm for computing the
*> singular value decomposition on a vector computer.
*> SIAM J. Sci. Stat. Comp., Vol. 10 (1998), pp. 359-371.
*> [2] J. Demmel and K. Veselic: Jacobi method is more accurate than QR.
*> [3] Z. Drmac: Implementation of Jacobi rotations for accurate singular
*> value computation in floating point arithmetic.
*> SIAM J. Sci. Comp., Vol. 18 (1997), pp. 1200-1222.
*> [4] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
*> LAPACK Working note 169.
*> [5] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
*> SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
*> LAPACK Working note 170.
*> [6] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
*> QSVD, (H,K)-SVD computations.
*> Department of Mathematics, University of Zagreb, 2008, 2015.
*> \endverbatim
*
*> \par Bugs, examples and comments:
* =================================
*>
*> \verbatim
*> ===========================
*> Please report all bugs and send interesting test examples and comments to
*> [email protected]. Thank you.
*> \endverbatim
*>
* =====================================================================
SUBROUTINE CGESVJ( JOBA, JOBU, JOBV, M, N, A, LDA, SVA, MV, V,
$ LDV, CWORK, LWORK, RWORK, LRWORK, 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..--
* June 2016
*
IMPLICIT NONE
* .. Scalar Arguments ..
INTEGER INFO, LDA, LDV, LWORK, LRWORK, M, MV, N
CHARACTER*1 JOBA, JOBU, JOBV
* ..
* .. Array Arguments ..
COMPLEX A( LDA, * ), V( LDV, * ), CWORK( LWORK )
REAL RWORK( LRWORK ), 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) )
INTEGER NSWEEP
PARAMETER ( NSWEEP = 30 )
* ..
* .. Local Scalars ..
COMPLEX AAPQ, OMPQ
REAL AAPP, AAPP0, AAPQ1, AAQQ, APOAQ, AQOAP, BIG,
$ BIGTHETA, CS, CTOL, EPSLN, MXAAPQ,
$ MXSINJ, ROOTBIG, ROOTEPS, ROOTSFMIN, ROOTTOL,
$ SKL, SFMIN, SMALL, SN, T, TEMP1, THETA, THSIGN, TOL
INTEGER BLSKIP, EMPTSW, i, ibr, IERR, igl, IJBLSK, ir1,
$ ISWROT, jbc, jgl, KBL, LKAHEAD, MVL, N2, N34,
$ N4, NBL, NOTROT, p, PSKIPPED, q, ROWSKIP, SWBAND
LOGICAL APPLV, GOSCALE, LOWER, LQUERY, LSVEC, NOSCALE, ROTOK,
$ RSVEC, UCTOL, UPPER
* ..
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX, MIN, CONJG, REAL, SIGN, SQRT
* ..
* .. External Functions ..
* ..
* from BLAS
REAL SCNRM2
COMPLEX CDOTC
EXTERNAL CDOTC, SCNRM2
INTEGER ISAMAX
EXTERNAL ISAMAX
* from LAPACK
REAL SLAMCH
EXTERNAL SLAMCH
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
* ..
* from BLAS
EXTERNAL CCOPY, CROT, CSSCAL, CSWAP, CAXPY
* from LAPACK
EXTERNAL CLASCL, CLASET, CLASSQ, SLASCL, XERBLA
EXTERNAL CGSVJ0, CGSVJ1
* ..
* .. Executable Statements ..
*
* Test the input arguments
*
LSVEC = LSAME( JOBU, 'U' ) .OR. LSAME( JOBU, 'F' )
UCTOL = LSAME( JOBU, 'C' )
RSVEC = LSAME( JOBV, 'V' ) .OR. LSAME( JOBV, 'J' )
APPLV = LSAME( JOBV, 'A' )
UPPER = LSAME( JOBA, 'U' )
LOWER = LSAME( JOBA, 'L' )
*
LQUERY = ( LWORK .EQ. -1 ) .OR. ( LRWORK .EQ. -1 )
IF( .NOT.( UPPER .OR. LOWER .OR. LSAME( JOBA, 'G' ) ) ) THEN
INFO = -1
ELSE IF( .NOT.( LSVEC .OR. UCTOL .OR. LSAME( JOBU, 'N' ) ) ) THEN
INFO = -2
ELSE IF( .NOT.( RSVEC .OR. APPLV .OR. LSAME( JOBV, 'N' ) ) ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( ( N.LT.0 ) .OR. ( N.GT.M ) ) THEN
INFO = -5
ELSE IF( LDA.LT.M ) THEN
INFO = -7
ELSE IF( MV.LT.0 ) THEN
INFO = -9
ELSE IF( ( RSVEC .AND. ( LDV.LT.N ) ) .OR.
$ ( APPLV .AND. ( LDV.LT.MV ) ) ) THEN
INFO = -11
ELSE IF( UCTOL .AND. ( RWORK( 1 ).LE.ONE ) ) THEN
INFO = -12
ELSE IF( LWORK.LT.( M+N ) .AND. ( .NOT.LQUERY ) ) THEN
INFO = -13
ELSE IF( LRWORK.LT.MAX( N, 6 ) .AND. ( .NOT.LQUERY ) ) THEN
INFO = -15
ELSE
INFO = 0
END IF
*
* #:(
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CGESVJ', -INFO )
RETURN
ELSE IF ( LQUERY ) THEN
CWORK(1) = M + N
RWORK(1) = MAX( N, 6 )
RETURN
END IF
*
* #:) Quick return for void matrix
*
IF( ( M.EQ.0 ) .OR. ( N.EQ.0 ) )RETURN
*
* Set numerical parameters
* The stopping criterion for Jacobi rotations is
*
* max_{i<>j}|A(:,i)^* * A(:,j)| / (||A(:,i)||*||A(:,j)||) < CTOL*EPS
*
* where EPS is the round-off and CTOL is defined as follows:
*
IF( UCTOL ) THEN
* ... user controlled
CTOL = RWORK( 1 )
ELSE
* ... default
IF( LSVEC .OR. RSVEC .OR. APPLV ) THEN
CTOL = SQRT( REAL( M ) )
ELSE
CTOL = REAL( M )
END IF
END IF
* ... and the machine dependent parameters are
*[!] (Make sure that SLAMCH() works properly on the target machine.)
*
EPSLN = SLAMCH( 'Epsilon' )
ROOTEPS = SQRT( EPSLN )
SFMIN = SLAMCH( 'SafeMinimum' )
ROOTSFMIN = SQRT( SFMIN )
SMALL = SFMIN / EPSLN
* BIG = SLAMCH( 'Overflow' )
BIG = ONE / SFMIN
ROOTBIG = ONE / ROOTSFMIN
* LARGE = BIG / SQRT( REAL( M*N ) )
BIGTHETA = ONE / ROOTEPS
*
TOL = CTOL*EPSLN
ROOTTOL = SQRT( TOL )
*
IF( REAL( M )*EPSLN.GE.ONE ) THEN
INFO = -4
CALL XERBLA( 'CGESVJ', -INFO )
RETURN
END IF
*
* Initialize the right singular vector matrix.
*
IF( RSVEC ) THEN
MVL = N
CALL CLASET( 'A', MVL, N, CZERO, CONE, V, LDV )
ELSE IF( APPLV ) THEN
MVL = MV
END IF
RSVEC = RSVEC .OR. APPLV
*
* Initialize SVA( 1:N ) = ( ||A e_i||_2, i = 1:N )
*(!) If necessary, scale A to protect the largest singular value
* from overflow. It is possible that saving the largest singular
* value destroys the information about the small ones.
* This initial scaling is almost minimal in the sense that the
* goal is to make sure that no column norm overflows, and that
* SQRT(N)*max_i SVA(i) does not overflow. If INFinite entries
* in A are detected, the procedure returns with INFO=-6.
*
SKL = ONE / SQRT( REAL( M )*REAL( N ) )
NOSCALE = .TRUE.
GOSCALE = .TRUE.
*
IF( LOWER ) THEN
* the input matrix is M-by-N lower triangular (trapezoidal)
DO 1874 p = 1, N
AAPP = ZERO
AAQQ = ONE
CALL CLASSQ( M-p+1, A( p, p ), 1, AAPP, AAQQ )
IF( AAPP.GT.BIG ) THEN
INFO = -6
CALL XERBLA( 'CGESVJ', -INFO )
RETURN
END IF
AAQQ = SQRT( AAQQ )
IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN
SVA( p ) = AAPP*AAQQ
ELSE
NOSCALE = .FALSE.
SVA( p ) = AAPP*( AAQQ*SKL )
IF( GOSCALE ) THEN
GOSCALE = .FALSE.
DO 1873 q = 1, p - 1
SVA( q ) = SVA( q )*SKL
1873 CONTINUE
END IF
END IF
1874 CONTINUE
ELSE IF( UPPER ) THEN
* the input matrix is M-by-N upper triangular (trapezoidal)
DO 2874 p = 1, N
AAPP = ZERO
AAQQ = ONE
CALL CLASSQ( p, A( 1, p ), 1, AAPP, AAQQ )
IF( AAPP.GT.BIG ) THEN
INFO = -6
CALL XERBLA( 'CGESVJ', -INFO )
RETURN
END IF
AAQQ = SQRT( AAQQ )
IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN
SVA( p ) = AAPP*AAQQ
ELSE
NOSCALE = .FALSE.
SVA( p ) = AAPP*( AAQQ*SKL )
IF( GOSCALE ) THEN
GOSCALE = .FALSE.
DO 2873 q = 1, p - 1
SVA( q ) = SVA( q )*SKL
2873 CONTINUE
END IF
END IF
2874 CONTINUE
ELSE
* the input matrix is M-by-N general dense
DO 3874 p = 1, N
AAPP = ZERO
AAQQ = ONE
CALL CLASSQ( M, A( 1, p ), 1, AAPP, AAQQ )
IF( AAPP.GT.BIG ) THEN
INFO = -6
CALL XERBLA( 'CGESVJ', -INFO )
RETURN
END IF
AAQQ = SQRT( AAQQ )
IF( ( AAPP.LT.( BIG / AAQQ ) ) .AND. NOSCALE ) THEN
SVA( p ) = AAPP*AAQQ
ELSE
NOSCALE = .FALSE.
SVA( p ) = AAPP*( AAQQ*SKL )
IF( GOSCALE ) THEN
GOSCALE = .FALSE.
DO 3873 q = 1, p - 1
SVA( q ) = SVA( q )*SKL
3873 CONTINUE
END IF
END IF
3874 CONTINUE
END IF
*
IF( NOSCALE )SKL = ONE
*
* Move the smaller part of the spectrum from the underflow threshold
*(!) Start by determining the position of the nonzero entries of the
* array SVA() relative to ( SFMIN, BIG ).
*
AAPP = ZERO
AAQQ = BIG
DO 4781 p = 1, N
IF( SVA( p ).NE.ZERO )AAQQ = MIN( AAQQ, SVA( p ) )
AAPP = MAX( AAPP, SVA( p ) )
4781 CONTINUE
*
* #:) Quick return for zero matrix
*
IF( AAPP.EQ.ZERO ) THEN
IF( LSVEC )CALL CLASET( 'G', M, N, CZERO, CONE, A, LDA )
RWORK( 1 ) = ONE
RWORK( 2 ) = ZERO
RWORK( 3 ) = ZERO
RWORK( 4 ) = ZERO
RWORK( 5 ) = ZERO
RWORK( 6 ) = ZERO
RETURN
END IF
*
* #:) Quick return for one-column matrix
*
IF( N.EQ.1 ) THEN
IF( LSVEC )CALL CLASCL( 'G', 0, 0, SVA( 1 ), SKL, M, 1,
$ A( 1, 1 ), LDA, IERR )
RWORK( 1 ) = ONE / SKL
IF( SVA( 1 ).GE.SFMIN ) THEN
RWORK( 2 ) = ONE
ELSE
RWORK( 2 ) = ZERO
END IF
RWORK( 3 ) = ZERO
RWORK( 4 ) = ZERO
RWORK( 5 ) = ZERO
RWORK( 6 ) = ZERO
RETURN
END IF
*
* Protect small singular values from underflow, and try to
* avoid underflows/overflows in computing Jacobi rotations.
*
SN = SQRT( SFMIN / EPSLN )
TEMP1 = SQRT( BIG / REAL( N ) )
IF( ( AAPP.LE.SN ) .OR. ( AAQQ.GE.TEMP1 ) .OR.
$ ( ( SN.LE.AAQQ ) .AND. ( AAPP.LE.TEMP1 ) ) ) THEN
TEMP1 = MIN( BIG, TEMP1 / AAPP )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.LE.TEMP1 ) ) THEN
TEMP1 = MIN( SN / AAQQ, BIG / ( AAPP*SQRT( REAL( N ) ) ) )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE IF( ( AAQQ.GE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN
TEMP1 = MAX( SN / AAQQ, TEMP1 / AAPP )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE IF( ( AAQQ.LE.SN ) .AND. ( AAPP.GE.TEMP1 ) ) THEN
TEMP1 = MIN( SN / AAQQ, BIG / ( SQRT( REAL( N ) )*AAPP ) )
* AAQQ = AAQQ*TEMP1
* AAPP = AAPP*TEMP1
ELSE
TEMP1 = ONE
END IF
*
* Scale, if necessary
*
IF( TEMP1.NE.ONE ) THEN
CALL SLASCL( 'G', 0, 0, ONE, TEMP1, N, 1, SVA, N, IERR )
END IF
SKL = TEMP1*SKL
IF( SKL.NE.ONE ) THEN
CALL CLASCL( JOBA, 0, 0, ONE, SKL, M, N, A, LDA, IERR )
SKL = ONE / SKL
END IF
*
* Row-cyclic Jacobi SVD algorithm with column pivoting
*
EMPTSW = ( N*( N-1 ) ) / 2
NOTROT = 0
DO 1868 q = 1, N
CWORK( q ) = CONE
1868 CONTINUE
*
*
*
SWBAND = 3
*[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.
*
IF( ( LOWER .OR. UPPER ) .AND. ( N.GT.MAX( 64, 4*KBL ) ) ) THEN
*[TP] The number of partition levels and the actual partition are
* tuning parameters.
N4 = N / 4
N2 = N / 2
N34 = 3*N4
IF( APPLV ) THEN
q = 0
ELSE
q = 1
END IF
*
IF( LOWER ) THEN
*
* This works very well on lower triangular matrices, in particular
* in the framework of the preconditioned Jacobi SVD (xGEJSV).
* The idea is simple:
* [+ 0 0 0] Note that Jacobi transformations of [0 0]
* [+ + 0 0] [0 0]
* [+ + x 0] actually work on [x 0] [x 0]
* [+ + x x] [x x]. [x x]
*
CALL CGSVJ0( JOBV, M-N34, N-N34, A( N34+1, N34+1 ), LDA,
$ CWORK( N34+1 ), SVA( N34+1 ), MVL,
$ V( N34*q+1, N34+1 ), LDV, EPSLN, SFMIN, TOL,
$ 2, CWORK( N+1 ), LWORK-N, IERR )
CALL CGSVJ0( JOBV, M-N2, N34-N2, A( N2+1, N2+1 ), LDA,
$ CWORK( N2+1 ), SVA( N2+1 ), MVL,
$ V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 2,
$ CWORK( N+1 ), LWORK-N, IERR )
CALL CGSVJ1( JOBV, M-N2, N-N2, N4, A( N2+1, N2+1 ), LDA,
$ CWORK( N2+1 ), SVA( N2+1 ), MVL,
$ V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 1,
$ CWORK( N+1 ), LWORK-N, IERR )
*
CALL CGSVJ0( JOBV, M-N4, N2-N4, A( N4+1, N4+1 ), LDA,
$ CWORK( N4+1 ), SVA( N4+1 ), MVL,
$ V( N4*q+1, N4+1 ), LDV, EPSLN, SFMIN, TOL, 1,
$ CWORK( N+1 ), LWORK-N, IERR )
*
CALL CGSVJ0( JOBV, M, N4, A, LDA, CWORK, SVA, MVL, V, LDV,
$ EPSLN, SFMIN, TOL, 1, CWORK( N+1 ), LWORK-N,
$ IERR )
*
CALL CGSVJ1( JOBV, M, N2, N4, A, LDA, CWORK, SVA, MVL, V,
$ LDV, EPSLN, SFMIN, TOL, 1, CWORK( N+1 ),
$ LWORK-N, IERR )
*
*
ELSE IF( UPPER ) THEN
*
*
CALL CGSVJ0( JOBV, N4, N4, A, LDA, CWORK, SVA, MVL, V, LDV,
$ EPSLN, SFMIN, TOL, 2, CWORK( N+1 ), LWORK-N,
$ IERR )
*
CALL CGSVJ0( JOBV, N2, N4, A( 1, N4+1 ), LDA, CWORK( N4+1 ),
$ SVA( N4+1 ), MVL, V( N4*q+1, N4+1 ), LDV,
$ EPSLN, SFMIN, TOL, 1, CWORK( N+1 ), LWORK-N,
$ IERR )
*
CALL CGSVJ1( JOBV, N2, N2, N4, A, LDA, CWORK, SVA, MVL, V,
$ LDV, EPSLN, SFMIN, TOL, 1, CWORK( N+1 ),
$ LWORK-N, IERR )
*
CALL CGSVJ0( JOBV, N2+N4, N4, A( 1, N2+1 ), LDA,
$ CWORK( N2+1 ), SVA( N2+1 ), MVL,
$ V( N2*q+1, N2+1 ), LDV, EPSLN, SFMIN, TOL, 1,
$ CWORK( N+1 ), LWORK-N, IERR )
END IF
*
END IF
*
* .. 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 = CWORK(p)
CWORK(p) = CWORK(q)
CWORK(q) = AAPQ
END IF
*
IF( ir1.EQ.0 ) THEN
*
* Column norms are periodically updated by explicit
* norm computation.
*[!] Caveat:
* Unfortunately, some BLAS implementations compute SCNRM2(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,
$ CWORK(N+1), 1 )
CALL CLASCL( 'G', 0, 0, AAPP, ONE,
$ M, 1, CWORK(N+1), LDA, IERR )
AAPQ = CDOTC( M, CWORK(N+1), 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,
$ CWORK(N+1), 1 )
CALL CLASCL( 'G', 0, 0, AAQQ,
$ ONE, M, 1,
$ CWORK(N+1), LDA, IERR )
AAPQ = CDOTC( M, A(1, p ), 1,
$ CWORK(N+1), 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
CWORK(p) = -CWORK(q) * OMPQ
*
ELSE
* .. have to use modified Gram-Schmidt like transformation
CALL CCOPY( M, A( 1, p ), 1,
$ CWORK(N+1), 1 )
CALL CLASCL( 'G', 0, 0, AAPP, ONE, M,
$ 1, CWORK(N+1), LDA,
$ IERR )
CALL CLASCL( 'G', 0, 0, AAQQ, ONE, M,
$ 1, A( 1, q ), LDA, IERR )
CALL CAXPY( M, -AAPQ, CWORK(N+1), 1,
$ A( 1, q ), 1 )
CALL CLASCL( 'G', 0, 0, ONE, AAQQ, M,
$ 1, A( 1, q ), LDA, IERR )