*> \brief \b SGEJSV
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE SGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
* M, N, A, LDA, SVA, U, LDU, V, LDV,
* WORK, LWORK, IWORK, INFO )
*
* .. Scalar Arguments ..
* IMPLICIT NONE
* INTEGER INFO, LDA, LDU, LDV, LWORK, M, N
* ..
* .. Array Arguments ..
* REAL A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ),
* $ WORK( LWORK )
* INTEGER IWORK( * )
* CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> SGEJSV computes the singular value decomposition (SVD) of a real M-by-N
*> matrix [A], where M >= N. The SVD of [A] is written as
*>
*> [A] = [U] * [SIGMA] * [V]^t,
*>
*> where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
*> diagonal elements, [U] is an M-by-N (or M-by-M) orthonormal matrix, and
*> [V] is an N-by-N orthogonal 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. The matrices [U] and [V]
*> are computed and stored in the arrays U and V, respectively. The diagonal
*> of [SIGMA] is computed and stored in the array SVA.
*> SGEJSV can sometimes compute tiny singular values and their singular vectors much
*> more accurately than other SVD routines, see below under Further Details.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBA
*> \verbatim
*> JOBA is CHARACTER*1
*> Specifies the level of accuracy:
*> = 'C': This option works well (high relative accuracy) if A = B * D,
*> with well-conditioned B and arbitrary diagonal matrix D.
*> The accuracy cannot be spoiled by COLUMN scaling. The
*> accuracy of the computed output depends on the condition of
*> B, and the procedure aims at the best theoretical accuracy.
*> The relative error max_{i=1:N}|d sigma_i| / sigma_i is
*> bounded by f(M,N)*epsilon* cond(B), independent of D.
*> The input matrix is preprocessed with the QRF with column
*> pivoting. This initial preprocessing and preconditioning by
*> a rank revealing QR factorization is common for all values of
*> JOBA. Additional actions are specified as follows:
*> = 'E': Computation as with 'C' with an additional estimate of the
*> condition number of B. It provides a realistic error bound.
*> = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
*> D1, D2, and well-conditioned matrix C, this option gives
*> higher accuracy than the 'C' option. If the structure of the
*> input matrix is not known, and relative accuracy is
*> desirable, then this option is advisable. The input matrix A
*> is preprocessed with QR factorization with FULL (row and
*> column) pivoting.
*> = 'G': Computation as with 'F' with an additional estimate of the
*> condition number of B, where A=D*B. If A has heavily weighted
*> rows, then using this condition number gives too pessimistic
*> error bound.
*> = 'A': Small singular values are the noise and the matrix is treated
*> as numerically rank deficient. The error in the computed
*> singular values is bounded by f(m,n)*epsilon*||A||.
*> The computed SVD A = U * S * V^t restores A up to
*> f(m,n)*epsilon*||A||.
*> This gives the procedure the licence to discard (set to zero)
*> all singular values below N*epsilon*||A||.
*> = 'R': Similar as in 'A'. Rank revealing property of the initial
*> QR factorization is used do reveal (using triangular factor)
*> a gap sigma_{r+1} < epsilon * sigma_r in which case the
*> numerical RANK is declared to be r. The SVD is computed with
*> absolute error bounds, but more accurately than with 'A'.
*> \endverbatim
*>
*> \param[in] JOBU
*> \verbatim
*> JOBU is CHARACTER*1
*> Specifies whether to compute the columns of U:
*> = 'U': N columns of U are returned in the array U.
*> = 'F': full set of M left sing. vectors is returned in the array U.
*> = 'W': U may be used as workspace of length M*N. See the description
*> of U.
*> = 'N': U is not computed.
*> \endverbatim
*>
*> \param[in] JOBV
*> \verbatim
*> JOBV is CHARACTER*1
*> Specifies whether to compute the matrix V:
*> = 'V': N columns of V are returned in the array V; Jacobi rotations
*> are not explicitly accumulated.
*> = 'J': N columns of V are returned in the array V, but they are
*> computed as the product of Jacobi rotations. This option is
*> allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.
*> = 'W': V may be used as workspace of length N*N. See the description
*> of V.
*> = 'N': V is not computed.
*> \endverbatim
*>
*> \param[in] JOBR
*> \verbatim
*> JOBR is CHARACTER*1
*> Specifies the RANGE for the singular values. Issues the licence to
*> set to zero small positive singular values if they are outside
*> specified range. If A .NE. 0 is scaled so that the largest singular
*> value of c*A is around SQRT(BIG), BIG=SLAMCH('O'), then JOBR issues
*> the licence to kill columns of A whose norm in c*A is less than
*> SQRT(SFMIN) (for JOBR = 'R'), or less than SMALL=SFMIN/EPSLN,
*> where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').
*> = 'N': Do not kill small columns of c*A. This option assumes that
*> BLAS and QR factorizations and triangular solvers are
*> implemented to work in that range. If the condition of A
*> is greater than BIG, use SGESVJ.
*> = 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)]
*> (roughly, as described above). This option is recommended.
*> ===========================
*> For computing the singular values in the FULL range [SFMIN,BIG]
*> use SGESVJ.
*> \endverbatim
*>
*> \param[in] JOBT
*> \verbatim
*> JOBT is CHARACTER*1
*> If the matrix is square then the procedure may determine to use
*> transposed A if A^t seems to be better with respect to convergence.
*> If the matrix is not square, JOBT is ignored. This is subject to
*> changes in the future.
*> The decision is based on two values of entropy over the adjoint
*> orbit of A^t * A. See the descriptions of WORK(6) and WORK(7).
*> = 'T': transpose if entropy test indicates possibly faster
*> convergence of Jacobi process if A^t is taken as input. If A is
*> replaced with A^t, then the row pivoting is included automatically.
*> = 'N': do not speculate.
*> This option can be used to compute only the singular values, or the
*> full SVD (U, SIGMA and V). For only one set of singular vectors
*> (U or V), the caller should provide both U and V, as one of the
*> matrices is used as workspace if the matrix A is transposed.
*> The implementer can easily remove this constraint and make the
*> code more complicated. See the descriptions of U and V.
*> \endverbatim
*>
*> \param[in] JOBP
*> \verbatim
*> JOBP is CHARACTER*1
*> Issues the licence to introduce structured perturbations to drown
*> denormalized numbers. This licence should be active if the
*> denormals are poorly implemented, causing slow computation,
*> especially in cases of fast convergence (!). For details see [1,2].
*> For the sake of simplicity, this perturbations are included only
*> when the full SVD or only the singular values are requested. The
*> implementer/user can easily add the perturbation for the cases of
*> computing one set of singular vectors.
*> = 'P': introduce perturbation
*> = 'N': do not perturb
*> \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, the M-by-N matrix A.
*> \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,
*> - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
*> computation SVA contains Euclidean column norms of the
*> iterated matrices in the array A.
*> - For WORK(1) .NE. WORK(2): The singular values of A are
*> (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
*> sigma_max(A) overflows or if small singular values have been
*> saved from underflow by scaling the input matrix A.
*> - If JOBR='R' then some of the singular values may be returned
*> as exact zeros obtained by "set to zero" because they are
*> below the numerical rank threshold or are denormalized numbers.
*> \endverbatim
*>
*> \param[out] U
*> \verbatim
*> U is REAL array, dimension ( LDU, N )
*> If JOBU = 'U', then U contains on exit the M-by-N matrix of
*> the left singular vectors.
*> If JOBU = 'F', then U contains on exit the M-by-M matrix of
*> the left singular vectors, including an ONB
*> of the orthogonal complement of the Range(A).
*> If JOBU = 'W' .AND. (JOBV = 'V' .AND. JOBT = 'T' .AND. M = N),
*> then U is used as workspace if the procedure
*> replaces A with A^t. In that case, [V] is computed
*> in U as left singular vectors of A^t and then
*> copied back to the V array. This 'W' option is just
*> a reminder to the caller that in this case U is
*> reserved as workspace of length N*N.
*> If JOBU = 'N' U is not referenced, unless JOBT='T'.
*> \endverbatim
*>
*> \param[in] LDU
*> \verbatim
*> LDU is INTEGER
*> The leading dimension of the array U, LDU >= 1.
*> IF JOBU = 'U' or 'F' or 'W', then LDU >= M.
*> \endverbatim
*>
*> \param[out] V
*> \verbatim
*> V is REAL array, dimension ( LDV, N )
*> If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
*> the right singular vectors;
*> If JOBV = 'W', AND (JOBU = 'U' AND JOBT = 'T' AND M = N),
*> then V is used as workspace if the pprocedure
*> replaces A with A^t. In that case, [U] is computed
*> in V as right singular vectors of A^t and then
*> copied back to the U array. This 'W' option is just
*> a reminder to the caller that in this case V is
*> reserved as workspace of length N*N.
*> If JOBV = 'N' V is not referenced, unless JOBT='T'.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V, LDV >= 1.
*> If JOBV = 'V' or 'J' or 'W', then LDV >= N.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (LWORK)
*> On exit,
*> WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such
*> that SCALE*SVA(1:N) are the computed singular values
*> of A. (See the description of SVA().)
*> WORK(2) = See the description of WORK(1).
*> WORK(3) = SCONDA is an estimate for the condition number of
*> column equilibrated A. (If JOBA = 'E' or 'G')
*> SCONDA is an estimate of SQRT(||(R^t * R)^(-1)||_1).
*> It is computed using SPOCON. It holds
*> N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
*> where R is the triangular factor from the QRF of A.
*> However, if R is truncated and the numerical rank is
*> determined to be strictly smaller than N, SCONDA is
*> returned as -1, thus indicating that the smallest
*> singular values might be lost.
*>
*> If full SVD is needed, the following two condition numbers are
*> useful for the analysis of the algorithm. They are provided for
*> a developer/implementer who is familiar with the details of
*> the method.
*>
*> WORK(4) = an estimate of the scaled condition number of the
*> triangular factor in the first QR factorization.
*> WORK(5) = an estimate of the scaled condition number of the
*> triangular factor in the second QR factorization.
*> The following two parameters are computed if JOBT = 'T'.
*> They are provided for a developer/implementer who is familiar
*> with the details of the method.
*>
*> WORK(6) = the entropy of A^t*A :: this is the Shannon entropy
*> of diag(A^t*A) / Trace(A^t*A) taken as point in the
*> probability simplex.
*> WORK(7) = the entropy of A*A^t.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> Length of WORK to confirm proper allocation of work space.
*> LWORK depends on the job:
*>
*> If only SIGMA is needed ( JOBU = 'N', JOBV = 'N' ) and
*> -> .. no scaled condition estimate required (JOBE = 'N'):
*> LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.
*> ->> For optimal performance (blocked code) the optimal value
*> is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal
*> block size for DGEQP3 and DGEQRF.
*> In general, optimal LWORK is computed as
*> LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7).
*> -> .. an estimate of the scaled condition number of A is
*> required (JOBA='E', 'G'). In this case, LWORK is the maximum
*> of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7).
*> ->> For optimal performance (blocked code) the optimal value
*> is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7).
*> In general, the optimal length LWORK is computed as
*> LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF),
*> N+N*N+LWORK(DPOCON),7).
*>
*> If SIGMA and the right singular vectors are needed (JOBV = 'V'),
*> -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
*> -> For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
*> where NB is the optimal block size for DGEQP3, DGEQRF, DGELQ,
*> DORMLQ. In general, the optimal length LWORK is computed as
*> LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON),
*> N+LWORK(DGELQ), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)).
*>
*> If SIGMA and the left singular vectors are needed
*> -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
*> -> For optimal performance:
*> if JOBU = 'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
*> if JOBU = 'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7),
*> where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR.
*> In general, the optimal length LWORK is computed as
*> LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON),
*> 2*N+LWORK(DGEQRF), N+LWORK(DORMQR)).
*> Here LWORK(DORMQR) equals N*NB (for JOBU = 'U') or
*> M*NB (for JOBU = 'F').
*>
*> If the full SVD is needed: (JOBU = 'U' or JOBU = 'F') and
*> -> if JOBV = 'V'
*> the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N).
*> -> if JOBV = 'J' the minimal requirement is
*> LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6).
*> -> For optimal performance, LWORK should be additionally
*> larger than N+M*NB, where NB is the optimal block size
*> for DORMQR.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*> IWORK is INTEGER array, dimension (M+3*N).
*> On exit,
*> IWORK(1) = the numerical rank determined after the initial
*> QR factorization with pivoting. See the descriptions
*> of JOBA and JOBR.
*> IWORK(2) = the number of the computed nonzero singular values
*> IWORK(3) = if nonzero, a warning message:
*> If IWORK(3) = 1 then some of the column norms of A
*> were denormalized floats. The requested high accuracy
*> is not warranted by the data.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> < 0: if INFO = -i, then the i-th argument had an illegal value.
*> = 0: successful exit;
*> > 0: SGEJSV did not converge in the maximal allowed number
*> of sweeps. The computed values may be inaccurate.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup realGEsing
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> SGEJSV implements a preconditioned Jacobi SVD algorithm. It uses SGEQP3,
*> SGEQRF, and SGELQF as preprocessors and preconditioners. Optionally, an
*> additional row pivoting can be used as a preprocessor, which in some
*> cases results in much higher accuracy. An example is matrix A with the
*> structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
*> diagonal matrices and C is well-conditioned matrix. In that case, complete
*> pivoting in the first QR factorizations provides accuracy dependent on the
*> condition number of C, and independent of D1, D2. Such higher accuracy is
*> not completely understood theoretically, but it works well in practice.
*> Further, if A can be written as A = B*D, with well-conditioned B and some
*> diagonal D, then the high accuracy is guaranteed, both theoretically and
*> in software, independent of D. For more details see [1], [2].
*> The computational range for the singular values can be the full range
*> ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
*> & LAPACK routines called by SGEJSV are implemented to work in that range.
*> If that is not the case, then the restriction for safe computation with
*> the singular values in the range of normalized IEEE numbers is that the
*> spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
*> overflow. This code (SGEJSV) is best used in this restricted range,
*> meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are
*> returned as zeros. See JOBR for details on this.
*> Further, this implementation is somewhat slower than the one described
*> in [1,2] due to replacement of some non-LAPACK components, and because
*> the choice of some tuning parameters in the iterative part (SGESVJ) is
*> left to the implementer on a particular machine.
*> The rank revealing QR factorization (in this code: SGEQP3) should be
*> implemented as in [3]. We have a new version of SGEQP3 under development
*> that is more robust than the current one in LAPACK, with a cleaner cut in
*> rank deficient cases. It will be available in the SIGMA library [4].
*> If M is much larger than N, it is obvious that the initial QRF with
*> column pivoting can be preprocessed by the QRF without pivoting. That
*> well known trick is not used in SGEJSV because in some cases heavy row
*> weighting can be treated with complete pivoting. The overhead in cases
*> M much larger than N is then only due to pivoting, but the benefits in
*> terms of accuracy have prevailed. The implementer/user can incorporate
*> this extra QRF step easily. The implementer can also improve data movement
*> (matrix transpose, matrix copy, matrix transposed copy) - this
*> implementation of SGEJSV uses only the simplest, naive data movement.
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
*
*> \par References:
* ================
*>
*> \verbatim
*>
*> [1] 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.
*> [2] 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.
*> [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
*> factorization software - a case study.
*> ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
*> LAPACK Working note 176.
*> [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
*> QSVD, (H,K)-SVD computations.
*> Department of Mathematics, University of Zagreb, 2008.
*> \endverbatim
*
*> \par Bugs, examples and comments:
* =================================
*>
*> Please report all bugs and send interesting examples and/or comments to
*> drmac@math.hr. Thank you.
*>
* =====================================================================
SUBROUTINE SGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
$ M, N, A, LDA, SVA, U, LDU, V, LDV,
$ WORK, LWORK, IWORK, 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..--
*
* .. Scalar Arguments ..
IMPLICIT NONE
INTEGER INFO, LDA, LDU, LDV, LWORK, M, N
* ..
* .. Array Arguments ..
REAL A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ),
$ WORK( LWORK )
INTEGER IWORK( * )
CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
* ..
*
* ===========================================================================
*
* .. Local Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
* ..
* .. Local Scalars ..
REAL AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1, COND_OK,
$ CONDR1, CONDR2, ENTRA, ENTRAT, EPSLN, MAXPRJ, SCALEM,
$ SCONDA, SFMIN, SMALL, TEMP1, USCAL1, USCAL2, XSC
INTEGER IERR, N1, NR, NUMRANK, p, q, WARNING
LOGICAL ALMORT, DEFR, ERREST, GOSCAL, JRACC, KILL, LSVEC,
$ L2ABER, L2KILL, L2PERT, L2RANK, L2TRAN,
$ NOSCAL, ROWPIV, RSVEC, TRANSP
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, ALOG, MAX, MIN, FLOAT, NINT, SIGN, SQRT
* ..
* .. External Functions ..
REAL SLAMCH, SNRM2
INTEGER ISAMAX
LOGICAL LSAME
EXTERNAL ISAMAX, LSAME, SLAMCH, SNRM2
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SGELQF, SGEQP3, SGEQRF, SLACPY, SLASCL,
$ SLASET, SLASSQ, SLASWP, SORGQR, SORMLQ,
$ SORMQR, SPOCON, SSCAL, SSWAP, STRSM, XERBLA
*
EXTERNAL SGESVJ
* ..
*
* Test the input arguments
*
LSVEC = LSAME( JOBU, 'U' ) .OR. LSAME( JOBU, 'F' )
JRACC = LSAME( JOBV, 'J' )
RSVEC = LSAME( JOBV, 'V' ) .OR. JRACC
ROWPIV = LSAME( JOBA, 'F' ) .OR. LSAME( JOBA, 'G' )
L2RANK = LSAME( JOBA, 'R' )
L2ABER = LSAME( JOBA, 'A' )
ERREST = LSAME( JOBA, 'E' ) .OR. LSAME( JOBA, 'G' )
L2TRAN = LSAME( JOBT, 'T' )
L2KILL = LSAME( JOBR, 'R' )
DEFR = LSAME( JOBR, 'N' )
L2PERT = LSAME( JOBP, 'P' )
*
IF ( .NOT.(ROWPIV .OR. L2RANK .OR. L2ABER .OR.
$ ERREST .OR. LSAME( JOBA, 'C' ) )) THEN
INFO = - 1
ELSE IF ( .NOT.( LSVEC .OR. LSAME( JOBU, 'N' ) .OR.
$ LSAME( JOBU, 'W' )) ) THEN
INFO = - 2
ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.
$ LSAME( JOBV, 'W' )) .OR. ( JRACC .AND. (.NOT.LSVEC) ) ) THEN
INFO = - 3
ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) ) THEN
INFO = - 4
ELSE IF ( .NOT. ( L2TRAN .OR. LSAME( JOBT, 'N' ) ) ) THEN
INFO = - 5
ELSE IF ( .NOT. ( L2PERT .OR. LSAME( JOBP, 'N' ) ) ) THEN
INFO = - 6
ELSE IF ( M .LT. 0 ) THEN
INFO = - 7
ELSE IF ( ( N .LT. 0 ) .OR. ( N .GT. M ) ) THEN
INFO = - 8
ELSE IF ( LDA .LT. M ) THEN
INFO = - 10
ELSE IF ( LSVEC .AND. ( LDU .LT. M ) ) THEN
INFO = - 13
ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN
INFO = - 15
ELSE IF ( (.NOT.(LSVEC .OR. RSVEC .OR. ERREST).AND.
$ (LWORK .LT. MAX(7,4*N+1,2*M+N))) .OR.
$ (.NOT.(LSVEC .OR. RSVEC) .AND. ERREST .AND.
$ (LWORK .LT. MAX(7,4*N+N*N,2*M+N))) .OR.
$ (LSVEC .AND. (.NOT.RSVEC) .AND. (LWORK .LT. MAX(7,2*M+N,4*N+1)))
$ .OR.
$ (RSVEC .AND. (.NOT.LSVEC) .AND. (LWORK .LT. MAX(7,2*M+N,4*N+1)))
$ .OR.
$ (LSVEC .AND. RSVEC .AND. (.NOT.JRACC) .AND.
$ (LWORK.LT.MAX(2*M+N,6*N+2*N*N)))
$ .OR. (LSVEC .AND. RSVEC .AND. JRACC .AND.
$ LWORK.LT.MAX(2*M+N,4*N+N*N,2*N+N*N+6)))
$ THEN
INFO = - 17
ELSE
* #:)
INFO = 0
END IF
*
IF ( INFO .NE. 0 ) THEN
* #:(
CALL XERBLA( 'SGEJSV', - INFO )
RETURN
END IF
*
* Quick return for void matrix (Y3K safe)
* #:)
IF ( ( M .EQ. 0 ) .OR. ( N .EQ. 0 ) ) THEN
IWORK(1:3) = 0
WORK(1:7) = 0
RETURN
ENDIF
*
* Determine whether the matrix U should be M x N or M x M
*
IF ( LSVEC ) THEN
N1 = N
IF ( LSAME( JOBU, 'F' ) ) N1 = M
END IF
*
* Set numerical parameters
*
*! NOTE: Make sure SLAMCH() does not fail on the target architecture.
*
EPSLN = SLAMCH('Epsilon')
SFMIN = SLAMCH('SafeMinimum')
SMALL = SFMIN / EPSLN
BIG = SLAMCH('O')
* BIG = ONE / SFMIN
*
* Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N
*
*(!) If necessary, scale SVA() to protect the largest norm from
* overflow. It is possible that this scaling pushes the smallest
* column norm left from the underflow threshold (extreme case).
*
SCALEM = ONE / SQRT(FLOAT(M)*FLOAT(N))
NOSCAL = .TRUE.
GOSCAL = .TRUE.
DO 1874 p = 1, N
AAPP = ZERO
AAQQ = ONE
CALL SLASSQ( M, A(1,p), 1, AAPP, AAQQ )
IF ( AAPP .GT. BIG ) THEN
INFO = - 9
CALL XERBLA( 'SGEJSV', -INFO )
RETURN
END IF
AAQQ = SQRT(AAQQ)
IF ( ( AAPP .LT. (BIG / AAQQ) ) .AND. NOSCAL ) THEN
SVA(p) = AAPP * AAQQ
ELSE
NOSCAL = .FALSE.
SVA(p) = AAPP * ( AAQQ * SCALEM )
IF ( GOSCAL ) THEN
GOSCAL = .FALSE.
CALL SSCAL( p-1, SCALEM, SVA, 1 )
END IF
END IF
1874 CONTINUE
*
IF ( NOSCAL ) SCALEM = ONE
*
AAPP = ZERO
AAQQ = BIG
DO 4781 p = 1, N
AAPP = MAX( AAPP, SVA(p) )
IF ( SVA(p) .NE. ZERO ) AAQQ = MIN( AAQQ, SVA(p) )
4781 CONTINUE
*
* Quick return for zero M x N matrix
* #:)
IF ( AAPP .EQ. ZERO ) THEN
IF ( LSVEC ) CALL SLASET( 'G', M, N1, ZERO, ONE, U, LDU )
IF ( RSVEC ) CALL SLASET( 'G', N, N, ZERO, ONE, V, LDV )
WORK(1) = ONE
WORK(2) = ONE
IF ( ERREST ) WORK(3) = ONE
IF ( LSVEC .AND. RSVEC ) THEN
WORK(4) = ONE
WORK(5) = ONE
END IF
IF ( L2TRAN ) THEN
WORK(6) = ZERO
WORK(7) = ZERO
END IF
IWORK(1) = 0
IWORK(2) = 0
IWORK(3) = 0
RETURN
END IF
*
* Issue warning if denormalized column norms detected. Override the
* high relative accuracy request. Issue licence to kill columns
* (set them to zero) whose norm is less than sigma_max / BIG (roughly).
* #:(
WARNING = 0
IF ( AAQQ .LE. SFMIN ) THEN
L2RANK = .TRUE.
L2KILL = .TRUE.
WARNING = 1
END IF
*
* Quick return for one-column matrix
* #:)
IF ( N .EQ. 1 ) THEN
*
IF ( LSVEC ) THEN
CALL SLASCL( 'G',0,0,SVA(1),SCALEM, M,1,A(1,1),LDA,IERR )
CALL SLACPY( 'A', M, 1, A, LDA, U, LDU )
* computing all M left singular vectors of the M x 1 matrix
IF ( N1 .NE. N ) THEN
CALL SGEQRF( M, N, U,LDU, WORK, WORK(N+1),LWORK-N,IERR )
CALL SORGQR( M,N1,1, U,LDU,WORK,WORK(N+1),LWORK-N,IERR )
CALL SCOPY( M, A(1,1), 1, U(1,1), 1 )
END IF
END IF
IF ( RSVEC ) THEN
V(1,1) = ONE
END IF
IF ( SVA(1) .LT. (BIG*SCALEM) ) THEN
SVA(1) = SVA(1) / SCALEM
SCALEM = ONE
END IF
WORK(1) = ONE / SCALEM
WORK(2) = ONE
IF ( SVA(1) .NE. ZERO ) THEN
IWORK(1) = 1
IF ( ( SVA(1) / SCALEM) .GE. SFMIN ) THEN
IWORK(2) = 1
ELSE
IWORK(2) = 0
END IF
ELSE
IWORK(1) = 0
IWORK(2) = 0
END IF
IWORK(3) = 0
IF ( ERREST ) WORK(3) = ONE
IF ( LSVEC .AND. RSVEC ) THEN
WORK(4) = ONE
WORK(5) = ONE
END IF
IF ( L2TRAN ) THEN
WORK(6) = ZERO
WORK(7) = ZERO
END IF
RETURN
*
END IF
*
TRANSP = .FALSE.
L2TRAN = L2TRAN .AND. ( M .EQ. N )
*
AATMAX = -ONE
AATMIN = BIG
IF ( ROWPIV .OR. L2TRAN ) THEN
*
* Compute the row norms, needed to determine row pivoting sequence
* (in the case of heavily row weighted A, row pivoting is strongly
* advised) and to collect information needed to compare the
* structures of A * A^t and A^t * A (in the case L2TRAN.EQ..TRUE.).
*
IF ( L2TRAN ) THEN
DO 1950 p = 1, M
XSC = ZERO
TEMP1 = ONE
CALL SLASSQ( N, A(p,1), LDA, XSC, TEMP1 )
* SLASSQ gets both the ell_2 and the ell_infinity norm
* in one pass through the vector
WORK(M+N+p) = XSC * SCALEM
WORK(N+p) = XSC * (SCALEM*SQRT(TEMP1))
AATMAX = MAX( AATMAX, WORK(N+p) )
IF (WORK(N+p) .NE. ZERO) AATMIN = MIN(AATMIN,WORK(N+p))
1950 CONTINUE
ELSE
DO 1904 p = 1, M
WORK(M+N+p) = SCALEM*ABS( A(p,ISAMAX(N,A(p,1),LDA)) )
AATMAX = MAX( AATMAX, WORK(M+N+p) )
AATMIN = MIN( AATMIN, WORK(M+N+p) )
1904 CONTINUE
END IF
*
END IF
*
* For square matrix A try to determine whether A^t would be better
* input for the preconditioned Jacobi SVD, with faster convergence.
* The decision is based on an O(N) function of the vector of column
* and row norms of A, based on the Shannon entropy. This should give
* the right choice in most cases when the difference actually matters.
* It may fail and pick the slower converging side.
*
ENTRA = ZERO
ENTRAT = ZERO
IF ( L2TRAN ) THEN
*
XSC = ZERO
TEMP1 = ONE
CALL SLASSQ( N, SVA, 1, XSC, TEMP1 )
TEMP1 = ONE / TEMP1
*
ENTRA = ZERO
DO 1113 p = 1, N
BIG1 = ( ( SVA(p) / XSC )**2 ) * TEMP1
IF ( BIG1 .NE. ZERO ) ENTRA = ENTRA + BIG1 * ALOG(BIG1)
1113 CONTINUE
ENTRA = - ENTRA / ALOG(FLOAT(N))
*
* Now, SVA().^2/Trace(A^t * A) is a point in the probability simplex.
* It is derived from the diagonal of A^t * A. Do the same with the
* diagonal of A * A^t, compute the entropy of the corresponding
* probability distribution. Note that A * A^t and A^t * A have the
* same trace.
*
ENTRAT = ZERO
DO 1114 p = N+1, N+M
BIG1 = ( ( WORK(p) / XSC )**2 ) * TEMP1
IF ( BIG1 .NE. ZERO ) ENTRAT = ENTRAT + BIG1 * ALOG(BIG1)
1114 CONTINUE
ENTRAT = - ENTRAT / ALOG(FLOAT(M))
*
* Analyze the entropies and decide A or A^t. Smaller entropy
* usually means better input for the algorithm.
*
TRANSP = ( ENTRAT .LT. ENTRA )
*
* If A^t is better than A, transpose A.
*
IF ( TRANSP ) THEN
* In an optimal implementation, this trivial transpose
* should be replaced with faster transpose.
DO 1115 p = 1, N - 1
DO 1116 q = p + 1, N
TEMP1 = A(q,p)
A(q,p) = A(p,q)
A(p,q) = TEMP1
1116 CONTINUE
1115 CONTINUE
DO 1117 p = 1, N
WORK(M+N+p) = SVA(p)
SVA(p) = WORK(N+p)
1117 CONTINUE
TEMP1 = AAPP
AAPP = AATMAX
AATMAX = TEMP1
TEMP1 = AAQQ
AAQQ = AATMIN
AATMIN = TEMP1
KILL = LSVEC
LSVEC = RSVEC
RSVEC = KILL
IF ( LSVEC ) N1 = N
*
ROWPIV = .TRUE.
END IF
*
END IF
* END IF L2TRAN
*
* Scale the matrix so that its maximal singular value remains less
* than SQRT(BIG) -- the matrix is scaled so that its maximal column
* has Euclidean norm equal to SQRT(BIG/N). The only reason to keep
* SQRT(BIG) instead of BIG is the fact that SGEJSV uses LAPACK and
* BLAS routines that, in some implementations, are not capable of
* working in the full interval [SFMIN,BIG] and that they may provoke
* overflows in the intermediate results. If the singular values spread
* from SFMIN to BIG, then SGESVJ will compute them. So, in that case,
* one should use SGESVJ instead of SGEJSV.
*
BIG1 = SQRT( BIG )
TEMP1 = SQRT( BIG / FLOAT(N) )
*
CALL SLASCL( 'G', 0, 0, AAPP, TEMP1, N, 1, SVA, N, IERR )
IF ( AAQQ .GT. (AAPP * SFMIN) ) THEN
AAQQ = ( AAQQ / AAPP ) * TEMP1
ELSE
AAQQ = ( AAQQ * TEMP1 ) / AAPP
END IF
TEMP1 = TEMP1 * SCALEM
CALL SLASCL( 'G', 0, 0, AAPP, TEMP1, M, N, A, LDA, IERR )
*
* To undo scaling at the end of this procedure, multiply the
* computed singular values with USCAL2 / USCAL1.
*
USCAL1 = TEMP1
USCAL2 = AAPP
*
IF ( L2KILL ) THEN
* L2KILL enforces computation of nonzero singular values in
* the restricted range of condition number of the initial A,
* sigma_max(A) / sigma_min(A) approx. SQRT(BIG)/SQRT(SFMIN).
XSC = SQRT( SFMIN )
ELSE
XSC = SMALL
*
* Now, if the condition number of A is too big,
* sigma_max(A) / sigma_min(A) .GT. SQRT(BIG/N) * EPSLN / SFMIN,
* as a precaution measure, the full SVD is computed using SGESVJ
* with accumulated Jacobi rotations. This provides numerically
* more robust computation, at the cost of slightly increased run
* time. Depending on the concrete implementation of BLAS and LAPACK
* (i.e. how they behave in presence of extreme ill-conditioning) the
* implementor may decide to remove this switch.
IF ( ( AAQQ.LT.SQRT(SFMIN) ) .AND. LSVEC .AND. RSVEC ) THEN
JRACC = .TRUE.
END IF
*
END IF
IF ( AAQQ .LT. XSC ) THEN
DO 700 p = 1, N
IF ( SVA(p) .LT. XSC ) THEN
CALL SLASET( 'A', M, 1, ZERO, ZERO, A(1,p), LDA )
SVA(p) = ZERO
END IF
700 CONTINUE
END IF
*
* Preconditioning using QR factorization with pivoting
*
IF ( ROWPIV ) THEN
* Optional row permutation (Bjoerck row pivoting):
* A result by Cox and Higham shows that the Bjoerck's
* row pivoting combined with standard column pivoting
* has similar effect as Powell-Reid complete pivoting.
* The ell-infinity norms of A are made nonincreasing.
DO 1952 p = 1, M - 1
q = ISAMAX( M-p+1, WORK(M+N+p), 1 ) + p - 1
IWORK(2*N+p) = q
IF ( p .NE. q ) THEN
TEMP1 = WORK(M+N+p)
WORK(M+N+p) = WORK(M+N+q)
WORK(M+N+q) = TEMP1
END IF
1952 CONTINUE
CALL SLASWP( N, A, LDA, 1, M-1, IWORK(2*N+1), 1 )
END IF
*
* End of the preparation phase (scaling, optional sorting and
* transposing, optional flushing of small columns).
*
* Preconditioning
*
* If the full SVD is needed, the right singular vectors are computed
* from a matrix equation, and for that we need theoretical analysis
* of the Businger-Golub pivoting. So we use SGEQP3 as the first RR QRF.
* In all other cases the first RR QRF can be chosen by other criteria
* (eg speed by replacing global with restricted window pivoting, such
* as in SGEQPX from TOMS # 782). Good results will be obtained using
* SGEQPX with properly (!) chosen numerical parameters.
* Any improvement of SGEQP3 improves overall performance of SGEJSV.
*
* A * P1 = Q1 * [ R1^t 0]^t:
DO 1963 p = 1, N
* .. all columns are free columns
IWORK(p) = 0
1963 CONTINUE
CALL SGEQP3( M,N,A,LDA, IWORK,WORK, WORK(N+1),LWORK-N, IERR )
*
* The upper triangular matrix R1 from the first QRF is inspected for
* rank deficiency and possibilities for deflation, or possible
* ill-conditioning. Depending on the user specified flag L2RANK,
* the procedure explores possibilities to reduce the numerical
* rank by inspecting the computed upper triangular factor. If
* L2RANK or L2ABER are up, then SGEJSV will compute the SVD of
* A + dA, where ||dA|| <= f(M,N)*EPSLN.
*
NR = 1
IF ( L2ABER ) THEN
* Standard absolute error bound suffices. All sigma_i with
* sigma_i < N*EPSLN*||A|| are flushed to zero. This is an
* aggressive enforcement of lower numerical rank by introducing a
* backward error of the order of N*EPSLN*||A||.
TEMP1 = SQRT(FLOAT(N))*EPSLN
DO 3001 p = 2, N
IF ( ABS(A(p,p)) .GE. (TEMP1*ABS(A(1,1))) ) THEN
NR = NR + 1
ELSE
GO TO 3002
END IF
3001 CONTINUE
3002 CONTINUE
ELSE IF ( L2RANK ) THEN
* .. similarly as above, only slightly more gentle (less aggressive).
* Sudden drop on the diagonal of R1 is used as the criterion for
* close-to-rank-deficient.
TEMP1 = SQRT(SFMIN)
DO 3401 p = 2, N
IF ( ( ABS(A(p,p)) .LT. (EPSLN*ABS(A(p-1,p-1))) ) .OR.
$ ( ABS(A(p,p)) .LT. SMALL ) .OR.
$ ( L2KILL .AND. (ABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3402
NR = NR + 1
3401 CONTINUE
3402 CONTINUE
*
ELSE
* The goal is high relative accuracy. However, if the matrix
* has high scaled condition number the relative accuracy is in
* general not feasible. Later on, a condition number estimator
* will be deployed to estimate the scaled condition number.
* Here we just remove the underflowed part of the triangular
* factor. This prevents the situation in which the code is
* working hard to get the accuracy not warranted by the data.
TEMP1 = SQRT(SFMIN)
DO 3301 p = 2, N
IF ( ( ABS(A(p,p)) .LT. SMALL ) .OR.
$ ( L2KILL .AND. (ABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3302
NR = NR + 1
3301 CONTINUE
3302 CONTINUE
*
END IF
*
ALMORT = .FALSE.
IF ( NR .EQ. N ) THEN
MAXPRJ = ONE
DO 3051 p = 2, N
TEMP1 = ABS(A(p,p)) / SVA(IWORK(p))
MAXPRJ = MIN( MAXPRJ, TEMP1 )
3051 CONTINUE
IF ( MAXPRJ**2 .GE. ONE - FLOAT(N)*EPSLN ) ALMORT = .TRUE.
END IF
*
*
SCONDA = - ONE
CONDR1 = - ONE
CONDR2 = - ONE
*
IF ( ERREST ) THEN
IF ( N .EQ. NR ) THEN
IF ( RSVEC ) THEN
* .. V is available as workspace
CALL SLACPY( 'U', N, N, A, LDA, V, LDV )
DO 3053 p = 1, N
TEMP1 = SVA(IWORK(p))
CALL SSCAL( p, ONE/TEMP1, V(1,p), 1 )
3053 CONTINUE
CALL SPOCON( 'U', N, V, LDV, ONE, TEMP1,
$ WORK(N+1), IWORK(2*N+M+1), IERR )
ELSE IF ( LSVEC ) THEN
* .. U is available as workspace
CALL SLACPY( 'U', N, N, A, LDA, U, LDU )
DO 3054 p = 1, N
TEMP1 = SVA(IWORK(p))
CALL SSCAL( p, ONE/TEMP1, U(1,p), 1 )
3054 CONTINUE
CALL SPOCON( 'U', N, U, LDU, ONE, TEMP1,
$ WORK(N+1), IWORK(2*N+M+1), IERR )
ELSE
CALL SLACPY( 'U', N, N, A, LDA, WORK(N+1), N )
DO 3052 p = 1, N
TEMP1 = SVA(IWORK(p))
CALL SSCAL( p, ONE/TEMP1, WORK(N+(p-1)*N+1), 1 )
3052 CONTINUE
* .. the columns of R are scaled to have unit Euclidean lengths.
CALL SPOCON( 'U', N, WORK(N+1), N, ONE, TEMP1,
$ WORK(N+N*N+1), IWORK(2*N+M+1), IERR )
END IF
SCONDA = ONE / SQRT(TEMP1)
* SCONDA is an estimate of SQRT(||(R^t * R)^(-1)||_1).
* N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
ELSE
SCONDA = - ONE
END IF
END IF
*
L2PERT = L2PERT .AND. ( ABS( A(1,1)/A(NR,NR) ) .GT. SQRT(BIG1) )
* If there is no violent scaling, artificial perturbation is not needed.
*
* Phase 3:
*
IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN
*
* Singular Values only
*
* .. transpose A(1:NR,1:N)
DO 1946 p = 1, MIN( N-1, NR )
CALL SCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 )
1946 CONTINUE
*
* The following two DO-loops introduce small relative perturbation
* into the strict upper triangle of the lower triangular matrix.
* Small entries below the main diagonal are also changed.
* This modification is useful if the computing environment does not
* provide/allow FLUSH TO ZERO underflow, for it prevents many
* annoying denormalized numbers in case of strongly scaled matrices.
* The perturbation is structured so that it does not introduce any
* new perturbation of the singular values, and it does not destroy
* the job done by the preconditioner.
* The licence for this perturbation is in the variable L2PERT, which
* should be .FALSE. if FLUSH TO ZERO underflow is active.
*
IF ( .NOT. ALMORT ) THEN
*
IF ( L2PERT ) THEN
* XSC = SQRT(SMALL)
XSC = EPSLN / FLOAT(N)
DO 4947 q = 1, NR
TEMP1 = XSC*ABS(A(q,q))
DO 4949 p = 1, N
IF ( ( (p.GT.q) .AND. (ABS(A(p,q)).LE.TEMP1) )
$ .OR. ( p .LT. q ) )
$ A(p,q) = SIGN( TEMP1, A(p,q) )
4949 CONTINUE
4947 CONTINUE
ELSE
CALL SLASET( 'U', NR-1,NR-1, ZERO,ZERO, A(1,2),LDA )
END IF
*
* .. second preconditioning using the QR factorization
*
CALL SGEQRF( N,NR, A,LDA, WORK, WORK(N+1),LWORK-N, IERR )
*
* .. and transpose upper to lower triangular
DO 1948 p = 1, NR - 1
CALL SCOPY( NR-p, A(p,p+1), LDA, A(p+1,p), 1 )
1948 CONTINUE
*
END IF
*
* Row-cyclic Jacobi SVD algorithm with column pivoting
*
* .. again some perturbation (a "background noise") is added
* to drown denormals
IF ( L2PERT ) THEN
* XSC = SQRT(SMALL)
XSC = EPSLN / FLOAT(N)
DO 1947 q = 1, NR
TEMP1 = XSC*ABS(A(q,q))
DO 1949 p = 1, NR
IF ( ( (p.GT.q) .AND. (ABS(A(p,q)).LE.TEMP1) )
$ .OR. ( p .LT. q ) )
$ A(p,q) = SIGN( TEMP1, A(p,q) )
1949 CONTINUE
1947 CONTINUE
ELSE
CALL SLASET( 'U', NR-1, NR-1, ZERO, ZERO, A(1,2), LDA )
END IF
*
* .. and one-sided Jacobi rotations are started on a lower
* triangular matrix (plus perturbation which is ignored in
* the part which destroys triangular form (confusing?!))
*
CALL SGESVJ( 'L', 'NoU', 'NoV', NR, NR, A, LDA, SVA,
$ N, V, LDV, WORK, LWORK, INFO )
*
SCALEM = WORK(1)
NUMRANK = NINT(WORK(2))
*
*
ELSE IF ( RSVEC .AND. ( .NOT. LSVEC ) ) THEN
*
* -> Singular Values and Right Singular Vectors <-
*
IF ( ALMORT ) THEN
*
* .. in this case NR equals N
DO 1998 p = 1, NR
CALL SCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
1998 CONTINUE
CALL SLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
*
CALL SGESVJ( 'L','U','N', N, NR, V,LDV, SVA, NR, A,LDA,
$ WORK, LWORK, INFO )
SCALEM = WORK(1)
NUMRANK = NINT(WORK(2))
ELSE
*
* .. two more QR factorizations ( one QRF is not enough, two require
* accumulated product of Jacobi rotations, three are perfect )
*
CALL SLASET( 'Lower', NR-1, NR-1, ZERO, ZERO, A(2,1), LDA )
CALL SGELQF( NR, N, A, LDA, WORK, WORK(N+1), LWORK-N, IERR)
CALL SLACPY( 'Lower', NR, NR, A, LDA, V, LDV )
CALL SLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
CALL SGEQRF( NR, NR, V, LDV, WORK(N+1), WORK(2*N+1),
$ LWORK-2*N, IERR )
DO 8998 p = 1, NR
CALL SCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )
8998 CONTINUE
CALL SLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
*
CALL SGESVJ( 'Lower', 'U','N', NR, NR, V,LDV, SVA, NR, U,
$ LDU, WORK(N+1), LWORK-N, INFO )
SCALEM = WORK(N+1)
NUMRANK = NINT(WORK(N+2))
IF ( NR .LT. N ) THEN
CALL SLASET( 'A',N-NR, NR, ZERO,ZERO, V(NR+1,1), LDV )
CALL SLASET( 'A',NR, N-NR, ZERO,ZERO, V(1,NR+1), LDV )
CALL SLASET( 'A',N-NR,N-NR,ZERO,ONE, V(NR+1,NR+1), LDV )
END IF
*
CALL SORMLQ( 'Left', 'Transpose', N, N, NR, A, LDA, WORK,
$ V, LDV, WORK(N+1), LWORK-N, IERR )
*
END IF
*
DO 8991 p = 1, N
CALL SCOPY( N, V(p,1), LDV, A(IWORK(p),1), LDA )
8991 CONTINUE
CALL SLACPY( 'All', N, N, A, LDA, V, LDV )
*
IF ( TRANSP ) THEN
CALL SLACPY( 'All', N, N, V, LDV, U, LDU )
END IF
*
ELSE IF ( LSVEC .AND. ( .NOT. RSVEC ) ) THEN
*
* .. Singular Values and Left Singular Vectors ..
*
* .. second preconditioning step to avoid need to accumulate
* Jacobi rotations in the Jacobi iterations.
DO 1965 p = 1, NR
CALL SCOPY( N-p+1, A(p,p), LDA, U(p,p), 1 )
1965 CONTINUE
CALL SLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )
*
CALL SGEQRF( N, NR, U, LDU, WORK(N+1), WORK(2*N+1),
$ LWORK-2*N, IERR )
*
DO 1967 p = 1, NR - 1
CALL SCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )
1967 CONTINUE
CALL SLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )
*
CALL SGESVJ( 'Lower', 'U', 'N', NR,NR, U, LDU, SVA, NR, A,
$ LDA, WORK(N+1), LWORK-N, INFO )
SCALEM = WORK(N+1)
NUMRANK = NINT(WORK(N+2))
*
IF ( NR .LT. M ) THEN
CALL SLASET( 'A', M-NR, NR,ZERO, ZERO, U(NR+1,1), LDU )
IF ( NR .LT. N1 ) THEN
CALL SLASET( 'A',NR, N1-NR, ZERO, ZERO, U(1,NR+1), LDU )
CALL SLASET( 'A',M-NR,N1-NR,ZERO,ONE,U(NR+1,NR+1), LDU )
END IF
END IF
*
CALL SORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,
$ LDU, WORK(N+1), LWORK-N, IERR )
*
IF ( ROWPIV )
$ CALL SLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
*
DO 1974 p = 1, N1
XSC = ONE / SNRM2( M, U(1,p), 1 )
CALL SSCAL( M, XSC, U(1,p), 1 )
1974 CONTINUE
*
IF ( TRANSP ) THEN
CALL SLACPY( 'All', N, N, U, LDU, V, LDV )
END IF
*
ELSE
*
* .. Full SVD ..
*
IF ( .NOT. JRACC ) THEN
*
IF ( .NOT. ALMORT ) THEN
*
* Second Preconditioning Step (QRF [with pivoting])
* Note that the composition of TRANSPOSE, QRF and TRANSPOSE is
* equivalent to an LQF CALL. Since in many libraries the QRF
* seems to be better optimized than the LQF, we do explicit
* transpose and use the QRF. This is subject to changes in an
* optimized implementation of SGEJSV.
*
DO 1968 p = 1, NR
CALL SCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
1968 CONTINUE
*
* .. the following two loops perturb small entries to avoid
* denormals in the second QR factorization, where they are
* as good as zeros. This is done to avoid painfully slow
* computation with denormals. The relative size of the perturbation
* is a parameter that can be changed by the implementer.
* This perturbation device will be obsolete on machines with
* properly implemented arithmetic.
* To switch it off, set L2PERT=.FALSE. To remove it from the
* code, remove the action under L2PERT=.TRUE., leave the ELSE part.
* The following two loops should be blocked and fused with the
* transposed copy above.
*
IF ( L2PERT ) THEN
XSC = SQRT(SMALL)
DO 2969 q = 1, NR
TEMP1 = XSC*ABS( V(q,q) )
DO 2968 p = 1, N
IF ( ( p .GT. q ) .AND. ( ABS(V(p,q)) .LE. TEMP1 )
$ .OR. ( p .LT. q ) )
$ V(p,q) = SIGN( TEMP1, V(p,q) )
IF ( p .LT. q ) V(p,q) = - V(p,q)
2968 CONTINUE
2969 CONTINUE
ELSE
CALL SLASET( 'U', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
END IF
*
* Estimate the row scaled condition number of R1
* (If R1 is rectangular, N > NR, then the condition number
* of the leading NR x NR submatrix is estimated.)
*
CALL SLACPY( 'L', NR, NR, V, LDV, WORK(2*N+1), NR )
DO 3950 p = 1, NR
TEMP1 = SNRM2(NR-p+1,WORK(2*N+(p-1)*NR+p),1)
CALL SSCAL(NR-p+1,ONE/TEMP1,WORK(2*N+(p-1)*NR+p),1)
3950 CONTINUE
CALL SPOCON('Lower',NR,WORK(2*N+1),NR,ONE,TEMP1,
$ WORK(2*N+NR*NR+1),IWORK(M+2*N+1),IERR)
CONDR1 = ONE / SQRT(TEMP1)
* .. here need a second opinion on the condition number
* .. then assume worst case scenario
* R1 is OK for inverse <=> CONDR1 .LT. FLOAT(N)
* more conservative <=> CONDR1 .LT. SQRT(FLOAT(N))
*
COND_OK = SQRT(FLOAT(NR))
*[TP] COND_OK is a tuning parameter.
IF ( CONDR1 .LT. COND_OK ) THEN
* .. the second QRF without pivoting. Note: in an optimized
* implementation, this QRF should be implemented as the QRF
* of a lower triangular matrix.
* R1^t = Q2 * R2
CALL SGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
$ LWORK-2*N, IERR )
*
IF ( L2PERT ) THEN
XSC = SQRT(SMALL)/EPSLN
DO 3959 p = 2, NR
DO 3958 q = 1, p - 1
TEMP1 = XSC * MIN(ABS(V(p,p)),ABS(V(q,q)))
IF ( ABS(V(q,p)) .LE. TEMP1 )
$ V(q,p) = SIGN( TEMP1, V(q,p) )
3958 CONTINUE
3959 CONTINUE
END IF
*
IF ( NR .NE. N )
$ CALL SLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N )
* .. save ...
*
* .. this transposed copy should be better than naive
DO 1969 p = 1, NR - 1
CALL SCOPY( NR-p, V(p,p+1), LDV, V(p+1,p), 1 )
1969 CONTINUE
*
CONDR2 = CONDR1
*
ELSE
*
* .. ill-conditioned case: second QRF with pivoting
* Note that windowed pivoting would be equally good
* numerically, and more run-time efficient. So, in
* an optimal implementation, the next call to SGEQP3
* should be replaced with eg. CALL SGEQPX (ACM TOMS #782)
* with properly (carefully) chosen parameters.
*
* R1^t * P2 = Q2 * R2
DO 3003 p = 1, NR
IWORK(N+p) = 0
3003 CONTINUE
CALL SGEQP3( N, NR, V, LDV, IWORK(N+1), WORK(N+1),
$ WORK(2*N+1), LWORK-2*N, IERR )
** CALL SGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
** $ LWORK-2*N, IERR )
IF ( L2PERT ) THEN
XSC = SQRT(SMALL)
DO 3969 p = 2, NR
DO 3968 q = 1, p - 1
TEMP1 = XSC * MIN(ABS(V(p,p)),ABS(V(q,q)))
IF ( ABS(V(q,p)) .LE. TEMP1 )
$ V(q,p) = SIGN( TEMP1, V(q,p) )
3968 CONTINUE
3969 CONTINUE
END IF
*
CALL SLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N )
*
IF ( L2PERT ) THEN
XSC = SQRT(SMALL)
DO 8970 p = 2, NR
DO 8971 q = 1, p - 1
TEMP1 = XSC * MIN(ABS(V(p,p)),ABS(V(q,q)))
V(p,q) = - SIGN( TEMP1, V(q,p) )
8971 CONTINUE
8970 CONTINUE
ELSE
CALL SLASET( 'L',NR-1,NR-1,ZERO,ZERO,V(2,1),LDV )
END IF
* Now, compute R2 = L3 * Q3, the LQ factorization.
CALL SGELQF( NR, NR, V, LDV, WORK(2*N+N*NR+1),
$ WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, IERR )
* .. and estimate the condition number
CALL SLACPY( 'L',NR,NR,V,LDV,WORK(2*N+N*NR+NR+1),NR )
DO 4950 p = 1, NR
TEMP1 = SNRM2( p, WORK(2*N+N*NR+NR+p), NR )
CALL SSCAL( p, ONE/TEMP1, WORK(2*N+N*NR+NR+p), NR )
4950 CONTINUE
CALL SPOCON( 'L',NR,WORK(2*N+N*NR+NR+1),NR,ONE,TEMP1,
$ WORK(2*N+N*NR+NR+NR*NR+1),IWORK(M+2*N+1),IERR )
CONDR2 = ONE / SQRT(TEMP1)
*
IF ( CONDR2 .GE. COND_OK ) THEN
* .. save the Householder vectors used for Q3
* (this overwrites the copy of R2, as it will not be
* needed in this branch, but it does not overwritte the
* Huseholder vectors of Q2.).
CALL SLACPY( 'U', NR, NR, V, LDV, WORK(2*N+1), N )
* .. and the rest of the information on Q3 is in
* WORK(2*N+N*NR+1:2*N+N*NR+N)
END IF
*
END IF
*
IF ( L2PERT ) THEN
XSC = SQRT(SMALL)
DO 4968 q = 2, NR
TEMP1 = XSC * V(q,q)
DO 4969 p = 1, q - 1
* V(p,q) = - SIGN( TEMP1, V(q,p) )
V(p,q) = - SIGN( TEMP1, V(p,q) )
4969 CONTINUE
4968 CONTINUE
ELSE
CALL SLASET( 'U', NR-1,NR-1, ZERO,ZERO, V(1,2), LDV )
END IF
*
* Second preconditioning finished; continue with Jacobi SVD
* The input matrix is lower trinagular.
*
* Recover the right singular vectors as solution of a well
* conditioned triangular matrix equation.
*
IF ( CONDR1 .LT. COND_OK ) THEN
*
CALL SGESVJ( 'L','U','N',NR,NR,V,LDV,SVA,NR,U,
$ LDU,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,INFO )
SCALEM = WORK(2*N+N*NR+NR+1)
NUMRANK = NINT(WORK(2*N+N*NR+NR+2))
DO 3970 p = 1, NR
CALL SCOPY( NR, V(1,p), 1, U(1,p), 1 )
CALL SSCAL( NR, SVA(p), V(1,p), 1 )
3970 CONTINUE
* .. pick the right matrix equation and solve it
*
IF ( NR .EQ. N ) THEN
* :)) .. best case, R1 is inverted. The solution of this matrix
* equation is Q2*V2 = the product of the Jacobi rotations
* used in SGESVJ, premultiplied with the orthogonal matrix
* from the second QR factorization.
CALL STRSM( 'L','U','N','N', NR,NR,ONE, A,LDA, V,LDV )
ELSE
* .. R1 is well conditioned, but non-square. Transpose(R2)
* is inverted to get the product of the Jacobi rotations
* used in SGESVJ. The Q-factor from the second QR
* factorization is then built in explicitly.
CALL STRSM('L','U','T','N',NR,NR,ONE,WORK(2*N+1),
$ N,V,LDV)
IF ( NR .LT. N ) THEN
CALL SLASET('A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV)
CALL SLASET('A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV)
CALL SLASET('A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV)
END IF
CALL SORMQR('L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
$ V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR)
END IF
*
ELSE IF ( CONDR2 .LT. COND_OK ) THEN
*
* :) .. the input matrix A is very likely a relative of
* the Kahan matrix :)
* The matrix R2 is inverted. The solution of the matrix equation
* is Q3^T*V3 = the product of the Jacobi rotations (appplied to
* the lower triangular L3 from the LQ factorization of
* R2=L3*Q3), pre-multiplied with the transposed Q3.
CALL SGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR, U,
$ LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )
SCALEM = WORK(2*N+N*NR+NR+1)
NUMRANK = NINT(WORK(2*N+N*NR+NR+2))
DO 3870 p = 1, NR
CALL SCOPY( NR, V(1,p), 1, U(1,p), 1 )
CALL SSCAL( NR, SVA(p), U(1,p), 1 )
3870 CONTINUE
CALL STRSM('L','U','N','N',NR,NR,ONE,WORK(2*N+1),N,U,LDU)
* .. apply the permutation from the second QR factorization
DO 873 q = 1, NR
DO 872 p = 1, NR
WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
872 CONTINUE
DO 874 p = 1, NR
U(p,q) = WORK(2*N+N*NR+NR+p)
874 CONTINUE
873 CONTINUE
IF ( NR .LT. N ) THEN
CALL SLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )
CALL SLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )
CALL SLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )
END IF
CALL SORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
$ V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
ELSE
* Last line of defense.
* #:( This is a rather pathological case: no scaled condition
* improvement after two pivoted QR factorizations. Other
* possibility is that the rank revealing QR factorization
* or the condition estimator has failed, or the COND_OK
* is set very close to ONE (which is unnecessary). Normally,
* this branch should never be executed, but in rare cases of
* failure of the RRQR or condition estimator, the last line of
* defense ensures that SGEJSV completes the task.
* Compute the full SVD of L3 using SGESVJ with explicit
* accumulation of Jacobi rotations.
CALL SGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR, U,
$ LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )
SCALEM = WORK(2*N+N*NR+NR+1)
NUMRANK = NINT(WORK(2*N+N*NR+NR+2))
IF ( NR .LT. N ) THEN
CALL SLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )
CALL SLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )
CALL SLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )
END IF
CALL SORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
$ V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
*
CALL SORMLQ( 'L', 'T', NR, NR, NR, WORK(2*N+1), N,
$ WORK(2*N+N*NR+1), U, LDU, WORK(2*N+N*NR+NR+1),
$ LWORK-2*N-N*NR-NR, IERR )
DO 773 q = 1, NR
DO 772 p = 1, NR
WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
772 CONTINUE
DO 774 p = 1, NR
U(p,q) = WORK(2*N+N*NR+NR+p)
774 CONTINUE
773 CONTINUE
*
END IF
*
* Permute the rows of V using the (column) permutation from the
* first QRF. Also, scale the columns to make them unit in
* Euclidean norm. This applies to all cases.
*
TEMP1 = SQRT(FLOAT(N)) * EPSLN
DO 1972 q = 1, N
DO 972 p = 1, N
WORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
972 CONTINUE
DO 973 p = 1, N
V(p,q) = WORK(2*N+N*NR+NR+p)
973 CONTINUE
XSC = ONE / SNRM2( N, V(1,q), 1 )
IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
$ CALL SSCAL( N, XSC, V(1,q), 1 )
1972 CONTINUE
* At this moment, V contains the right singular vectors of A.
* Next, assemble the left singular vector matrix U (M x N).
IF ( NR .LT. M ) THEN
CALL SLASET( 'A', M-NR, NR, ZERO, ZERO, U(NR+1,1), LDU )
IF ( NR .LT. N1 ) THEN
CALL SLASET('A',NR,N1-NR,ZERO,ZERO,U(1,NR+1),LDU)
CALL SLASET('A',M-NR,N1-NR,ZERO,ONE,U(NR+1,NR+1),LDU)
END IF
END IF
*
* The Q matrix from the first QRF is built into the left singular
* matrix U. This applies to all cases.
*
CALL SORMQR( 'Left', 'No_Tr', M, N1, N, A, LDA, WORK, U,
$ LDU, WORK(N+1), LWORK-N, IERR )
* The columns of U are normalized. The cost is O(M*N) flops.
TEMP1 = SQRT(FLOAT(M)) * EPSLN
DO 1973 p = 1, NR
XSC = ONE / SNRM2( M, U(1,p), 1 )
IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
$ CALL SSCAL( M, XSC, U(1,p), 1 )
1973 CONTINUE
*
* If the initial QRF is computed with row pivoting, the left
* singular vectors must be adjusted.
*
IF ( ROWPIV )
$ CALL SLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
*
ELSE
*
* .. the initial matrix A has almost orthogonal columns and
* the second QRF is not needed
*
CALL SLACPY( 'Upper', N, N, A, LDA, WORK(N+1), N )
IF ( L2PERT ) THEN
XSC = SQRT(SMALL)
DO 5970 p = 2, N
TEMP1 = XSC * WORK( N + (p-1)*N + p )
DO 5971 q = 1, p - 1
WORK(N+(q-1)*N+p)=-SIGN(TEMP1,WORK(N+(p-1)*N+q))
5971 CONTINUE
5970 CONTINUE
ELSE
CALL SLASET( 'Lower',N-1,N-1,ZERO,ZERO,WORK(N+2),N )
END IF
*
CALL SGESVJ( 'Upper', 'U', 'N', N, N, WORK(N+1), N, SVA,
$ N, U, LDU, WORK(N+N*N+1), LWORK-N-N*N, INFO )
*
SCALEM = WORK(N+N*N+1)
NUMRANK = NINT(WORK(N+N*N+2))
DO 6970 p = 1, N
CALL SCOPY( N, WORK(N+(p-1)*N+1), 1, U(1,p), 1 )
CALL SSCAL( N, SVA(p), WORK(N+(p-1)*N+1), 1 )
6970 CONTINUE
*
CALL STRSM( 'Left', 'Upper', 'NoTrans', 'No UD', N, N,
$ ONE, A, LDA, WORK(N+1), N )
DO 6972 p = 1, N
CALL SCOPY( N, WORK(N+p), N, V(IWORK(p),1), LDV )
6972 CONTINUE
TEMP1 = SQRT(FLOAT(N))*EPSLN
DO 6971 p = 1, N
XSC = ONE / SNRM2( N, V(1,p), 1 )
IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
$ CALL SSCAL( N, XSC, V(1,p), 1 )
6971 CONTINUE
*
* Assemble the left singular vector matrix U (M x N).
*
IF ( N .LT. M ) THEN
CALL SLASET( 'A', M-N, N, ZERO, ZERO, U(N+1,1), LDU )
IF ( N .LT. N1 ) THEN
CALL SLASET( 'A',N, N1-N, ZERO, ZERO, U(1,N+1),LDU )
CALL SLASET( 'A',M-N,N1-N, ZERO, ONE,U(N+1,N+1),LDU )
END IF
END IF
CALL SORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,
$ LDU, WORK(N+1), LWORK-N, IERR )
TEMP1 = SQRT(FLOAT(M))*EPSLN
DO 6973 p = 1, N1
XSC = ONE / SNRM2( M, U(1,p), 1 )
IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
$ CALL SSCAL( M, XSC, U(1,p), 1 )
6973 CONTINUE
*
IF ( ROWPIV )
$ CALL SLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
*
END IF
*
* end of the >> almost orthogonal case << in the full SVD
*
ELSE
*
* This branch deploys a preconditioned Jacobi SVD with explicitly
* accumulated rotations. It is included as optional, mainly for
* experimental purposes. It does perform well, and can also be used.
* In this implementation, this branch will be automatically activated
* if the condition number sigma_max(A) / sigma_min(A) is predicted
* to be greater than the overflow threshold. This is because the
* a posteriori computation of the singular vectors assumes robust
* implementation of BLAS and some LAPACK procedures, capable of working
* in presence of extreme values. Since that is not always the case, ...
*
DO 7968 p = 1, NR
CALL SCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
7968 CONTINUE
*
IF ( L2PERT ) THEN
XSC = SQRT(SMALL/EPSLN)
DO 5969 q = 1, NR
TEMP1 = XSC*ABS( V(q,q) )
DO 5968 p = 1, N
IF ( ( p .GT. q ) .AND. ( ABS(V(p,q)) .LE. TEMP1 )
$ .OR. ( p .LT. q ) )
$ V(p,q) = SIGN( TEMP1, V(p,q) )
IF ( p .LT. q ) V(p,q) = - V(p,q)
5968 CONTINUE
5969 CONTINUE
ELSE
CALL SLASET( 'U', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
END IF
CALL SGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
$ LWORK-2*N, IERR )
CALL SLACPY( 'L', N, NR, V, LDV, WORK(2*N+1), N )
*
DO 7969 p = 1, NR
CALL SCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 )
7969 CONTINUE
IF ( L2PERT ) THEN
XSC = SQRT(SMALL/EPSLN)
DO 9970 q = 2, NR
DO 9971 p = 1, q - 1
TEMP1 = XSC * MIN(ABS(U(p,p)),ABS(U(q,q)))
U(p,q) = - SIGN( TEMP1, U(q,p) )
9971 CONTINUE
9970 CONTINUE
ELSE
CALL SLASET('U', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )
END IF
CALL SGESVJ( 'L', 'U', 'V', NR, NR, U, LDU, SVA,
$ N, V, LDV, WORK(2*N+N*NR+1), LWORK-2*N-N*NR, INFO )
SCALEM = WORK(2*N+N*NR+1)
NUMRANK = NINT(WORK(2*N+N*NR+2))
IF ( NR .LT. N ) THEN
CALL SLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )
CALL SLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )
CALL SLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )
END IF
CALL SORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
$ V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
*
* Permute the rows of V using the (column) permutation from the
* first QRF. Also, scale the columns to make them unit in
* Euclidean norm. This applies to all cases.
*
TEMP1 = SQRT(FLOAT(N)) * EPSLN
DO 7972 q = 1, N
DO 8972 p = 1, N
WORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
8972 CONTINUE
DO 8973 p = 1, N
V(p,q) = WORK(2*N+N*NR+NR+p)
8973 CONTINUE
XSC = ONE / SNRM2( N, V(1,q), 1 )
IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
$ CALL SSCAL( N, XSC, V(1,q), 1 )
7972 CONTINUE
*
* At this moment, V contains the right singular vectors of A.
* Next, assemble the left singular vector matrix U (M x N).
*
IF ( NR .LT. M ) THEN
CALL SLASET( 'A', M-NR, NR, ZERO, ZERO, U(NR+1,1), LDU )
IF ( NR .LT. N1 ) THEN
CALL SLASET( 'A',NR, N1-NR, ZERO, ZERO, U(1,NR+1),LDU )
CALL SLASET( 'A',M-NR,N1-NR, ZERO, ONE,U(NR+1,NR+1),LDU )
END IF
END IF
*
CALL SORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,
$ LDU, WORK(N+1), LWORK-N, IERR )
*
IF ( ROWPIV )
$ CALL SLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
*
*
END IF
IF ( TRANSP ) THEN
* .. swap U and V because the procedure worked on A^t
DO 6974 p = 1, N
CALL SSWAP( N, U(1,p), 1, V(1,p), 1 )
6974 CONTINUE
END IF
*
END IF
* end of the full SVD
*
* Undo scaling, if necessary (and possible)
*
IF ( USCAL2 .LE. (BIG/SVA(1))*USCAL1 ) THEN
CALL SLASCL( 'G', 0, 0, USCAL1, USCAL2, NR, 1, SVA, N, IERR )
USCAL1 = ONE
USCAL2 = ONE
END IF
*
IF ( NR .LT. N ) THEN
DO 3004 p = NR+1, N
SVA(p) = ZERO
3004 CONTINUE
END IF
*
WORK(1) = USCAL2 * SCALEM
WORK(2) = USCAL1
IF ( ERREST ) WORK(3) = SCONDA
IF ( LSVEC .AND. RSVEC ) THEN
WORK(4) = CONDR1
WORK(5) = CONDR2
END IF
IF ( L2TRAN ) THEN
WORK(6) = ENTRA
WORK(7) = ENTRAT
END IF
*
IWORK(1) = NR
IWORK(2) = NUMRANK
IWORK(3) = WARNING
*
RETURN
* ..
* .. END OF SGEJSV
* ..
END
*