Annotation of rpl/lapack/lapack/dgejsv.f, revision 1.6

1.1       bertrand    1:       SUBROUTINE DGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
1.6     ! bertrand    2:      $                   M, N, A, LDA, SVA, U, LDU, V, LDV,
        !             3:      $                   WORK, LWORK, IWORK, INFO )
1.1       bertrand    4: *
1.6     ! bertrand    5: *  -- LAPACK routine (version 3.3.1)                                    --
1.1       bertrand    6: *
                      7: *  -- Contributed by Zlatko Drmac of the University of Zagreb and     --
                      8: *  -- Kresimir Veselic of the Fernuniversitaet Hagen                  --
1.6     ! bertrand    9: *  -- April 2011                                                      --
1.1       bertrand   10: *
                     11: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                     12: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
                     13: *
                     14: * This routine is also part of SIGMA (version 1.23, October 23. 2008.)
                     15: * SIGMA is a library of algorithms for highly accurate algorithms for
                     16: * computation of SVD, PSVD, QSVD, (H,K)-SVD, and for solution of the
                     17: * eigenvalue problems Hx = lambda M x, H M x = lambda x with H, M > 0.
                     18: *
                     19: *     .. Scalar Arguments ..
                     20:       IMPLICIT    NONE
                     21:       INTEGER     INFO, LDA, LDU, LDV, LWORK, M, N
                     22: *     ..
                     23: *     .. Array Arguments ..
                     24:       DOUBLE PRECISION A( LDA, * ), SVA( N ), U( LDU, * ), V( LDV, * ),
1.6     ! bertrand   25:      $            WORK( LWORK )
1.1       bertrand   26:       INTEGER     IWORK( * )
                     27:       CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
                     28: *     ..
                     29: *
                     30: *  Purpose
                     31: *  =======
                     32: *
                     33: *  DGEJSV computes the singular value decomposition (SVD) of a real M-by-N
                     34: *  matrix [A], where M >= N. The SVD of [A] is written as
                     35: *
                     36: *               [A] = [U] * [SIGMA] * [V]^t,
                     37: *
                     38: *  where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
                     39: *  diagonal elements, [U] is an M-by-N (or M-by-M) orthonormal matrix, and
                     40: *  [V] is an N-by-N orthogonal matrix. The diagonal elements of [SIGMA] are
                     41: *  the singular values of [A]. The columns of [U] and [V] are the left and
                     42: *  the right singular vectors of [A], respectively. The matrices [U] and [V]
                     43: *  are computed and stored in the arrays U and V, respectively. The diagonal
                     44: *  of [SIGMA] is computed and stored in the array SVA.
                     45: *
                     46: *  Arguments
                     47: *  =========
                     48: *
                     49: *  JOBA    (input) CHARACTER*1
                     50: *        Specifies the level of accuracy:
                     51: *       = 'C': This option works well (high relative accuracy) if A = B * D,
                     52: *             with well-conditioned B and arbitrary diagonal matrix D.
                     53: *             The accuracy cannot be spoiled by COLUMN scaling. The
                     54: *             accuracy of the computed output depends on the condition of
                     55: *             B, and the procedure aims at the best theoretical accuracy.
                     56: *             The relative error max_{i=1:N}|d sigma_i| / sigma_i is
                     57: *             bounded by f(M,N)*epsilon* cond(B), independent of D.
                     58: *             The input matrix is preprocessed with the QRF with column
                     59: *             pivoting. This initial preprocessing and preconditioning by
                     60: *             a rank revealing QR factorization is common for all values of
                     61: *             JOBA. Additional actions are specified as follows:
                     62: *       = 'E': Computation as with 'C' with an additional estimate of the
                     63: *             condition number of B. It provides a realistic error bound.
                     64: *       = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
                     65: *             D1, D2, and well-conditioned matrix C, this option gives
                     66: *             higher accuracy than the 'C' option. If the structure of the
                     67: *             input matrix is not known, and relative accuracy is
                     68: *             desirable, then this option is advisable. The input matrix A
                     69: *             is preprocessed with QR factorization with FULL (row and
                     70: *             column) pivoting.
                     71: *       = 'G'  Computation as with 'F' with an additional estimate of the
                     72: *             condition number of B, where A=D*B. If A has heavily weighted
                     73: *             rows, then using this condition number gives too pessimistic
                     74: *             error bound.
                     75: *       = 'A': Small singular values are the noise and the matrix is treated
                     76: *             as numerically rank defficient. The error in the computed
                     77: *             singular values is bounded by f(m,n)*epsilon*||A||.
                     78: *             The computed SVD A = U * S * V^t restores A up to
                     79: *             f(m,n)*epsilon*||A||.
                     80: *             This gives the procedure the licence to discard (set to zero)
                     81: *             all singular values below N*epsilon*||A||.
                     82: *       = 'R': Similar as in 'A'. Rank revealing property of the initial
                     83: *             QR factorization is used do reveal (using triangular factor)
                     84: *             a gap sigma_{r+1} < epsilon * sigma_r in which case the
                     85: *             numerical RANK is declared to be r. The SVD is computed with
                     86: *             absolute error bounds, but more accurately than with 'A'.
                     87: *
                     88: *  JOBU    (input) CHARACTER*1
                     89: *        Specifies whether to compute the columns of U:
                     90: *       = 'U': N columns of U are returned in the array U.
                     91: *       = 'F': full set of M left sing. vectors is returned in the array U.
                     92: *       = 'W': U may be used as workspace of length M*N. See the description
                     93: *             of U.
                     94: *       = 'N': U is not computed.
                     95: *
                     96: *  JOBV    (input) CHARACTER*1
                     97: *        Specifies whether to compute the matrix V:
                     98: *       = 'V': N columns of V are returned in the array V; Jacobi rotations
                     99: *             are not explicitly accumulated.
                    100: *       = 'J': N columns of V are returned in the array V, but they are
                    101: *             computed as the product of Jacobi rotations. This option is
                    102: *             allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.
                    103: *       = 'W': V may be used as workspace of length N*N. See the description
                    104: *             of V.
                    105: *       = 'N': V is not computed.
                    106: *
                    107: *  JOBR    (input) CHARACTER*1
                    108: *        Specifies the RANGE for the singular values. Issues the licence to
                    109: *        set to zero small positive singular values if they are outside
                    110: *        specified range. If A .NE. 0 is scaled so that the largest singular
                    111: *        value of c*A is around DSQRT(BIG), BIG=SLAMCH('O'), then JOBR issues
                    112: *        the licence to kill columns of A whose norm in c*A is less than
                    113: *        DSQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN,
                    114: *        where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E').
                    115: *       = 'N': Do not kill small columns of c*A. This option assumes that
                    116: *             BLAS and QR factorizations and triangular solvers are
                    117: *             implemented to work in that range. If the condition of A
                    118: *             is greater than BIG, use DGESVJ.
                    119: *       = 'R': RESTRICTED range for sigma(c*A) is [DSQRT(SFMIN), DSQRT(BIG)]
                    120: *             (roughly, as described above). This option is recommended.
                    121: *                                            ~~~~~~~~~~~~~~~~~~~~~~~~~~~
                    122: *        For computing the singular values in the FULL range [SFMIN,BIG]
                    123: *        use DGESVJ.
                    124: *
                    125: *  JOBT    (input) CHARACTER*1
                    126: *        If the matrix is square then the procedure may determine to use
                    127: *        transposed A if A^t seems to be better with respect to convergence.
                    128: *        If the matrix is not square, JOBT is ignored. This is subject to
                    129: *        changes in the future.
                    130: *        The decision is based on two values of entropy over the adjoint
                    131: *        orbit of A^t * A. See the descriptions of WORK(6) and WORK(7).
                    132: *       = 'T': transpose if entropy test indicates possibly faster
                    133: *        convergence of Jacobi process if A^t is taken as input. If A is
                    134: *        replaced with A^t, then the row pivoting is included automatically.
                    135: *       = 'N': do not speculate.
                    136: *        This option can be used to compute only the singular values, or the
                    137: *        full SVD (U, SIGMA and V). For only one set of singular vectors
                    138: *        (U or V), the caller should provide both U and V, as one of the
                    139: *        matrices is used as workspace if the matrix A is transposed.
                    140: *        The implementer can easily remove this constraint and make the
                    141: *        code more complicated. See the descriptions of U and V.
                    142: *
                    143: *  JOBP    (input) CHARACTER*1
                    144: *        Issues the licence to introduce structured perturbations to drown
                    145: *        denormalized numbers. This licence should be active if the
                    146: *        denormals are poorly implemented, causing slow computation,
                    147: *        especially in cases of fast convergence (!). For details see [1,2].
                    148: *        For the sake of simplicity, this perturbations are included only
                    149: *        when the full SVD or only the singular values are requested. The
                    150: *        implementer/user can easily add the perturbation for the cases of
                    151: *        computing one set of singular vectors.
                    152: *       = 'P': introduce perturbation
                    153: *       = 'N': do not perturb
                    154: *
                    155: *  M       (input) INTEGER
                    156: *         The number of rows of the input matrix A.  M >= 0.
                    157: *
                    158: *  N       (input) INTEGER
                    159: *         The number of columns of the input matrix A. M >= N >= 0.
                    160: *
                    161: *  A       (input/workspace) DOUBLE PRECISION array, dimension (LDA,N)
                    162: *          On entry, the M-by-N matrix A.
                    163: *
                    164: *  LDA     (input) INTEGER
                    165: *          The leading dimension of the array A.  LDA >= max(1,M).
                    166: *
                    167: *  SVA     (workspace/output) DOUBLE PRECISION array, dimension (N)
                    168: *          On exit,
                    169: *          - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
                    170: *            computation SVA contains Euclidean column norms of the
                    171: *            iterated matrices in the array A.
                    172: *          - For WORK(1) .NE. WORK(2): The singular values of A are
                    173: *            (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
                    174: *            sigma_max(A) overflows or if small singular values have been
                    175: *            saved from underflow by scaling the input matrix A.
                    176: *          - If JOBR='R' then some of the singular values may be returned
                    177: *            as exact zeros obtained by "set to zero" because they are
                    178: *            below the numerical rank threshold or are denormalized numbers.
                    179: *
                    180: *  U       (workspace/output) DOUBLE PRECISION array, dimension ( LDU, N )
                    181: *          If JOBU = 'U', then U contains on exit the M-by-N matrix of
                    182: *                         the left singular vectors.
                    183: *          If JOBU = 'F', then U contains on exit the M-by-M matrix of
                    184: *                         the left singular vectors, including an ONB
                    185: *                         of the orthogonal complement of the Range(A).
                    186: *          If JOBU = 'W'  .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N),
                    187: *                         then U is used as workspace if the procedure
                    188: *                         replaces A with A^t. In that case, [V] is computed
                    189: *                         in U as left singular vectors of A^t and then
                    190: *                         copied back to the V array. This 'W' option is just
                    191: *                         a reminder to the caller that in this case U is
                    192: *                         reserved as workspace of length N*N.
                    193: *          If JOBU = 'N'  U is not referenced.
                    194: *
                    195: * LDU      (input) INTEGER
                    196: *          The leading dimension of the array U,  LDU >= 1.
                    197: *          IF  JOBU = 'U' or 'F' or 'W',  then LDU >= M.
                    198: *
                    199: *  V       (workspace/output) DOUBLE PRECISION array, dimension ( LDV, N )
                    200: *          If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
                    201: *                         the right singular vectors;
                    202: *          If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),
                    203: *                         then V is used as workspace if the pprocedure
                    204: *                         replaces A with A^t. In that case, [U] is computed
                    205: *                         in V as right singular vectors of A^t and then
                    206: *                         copied back to the U array. This 'W' option is just
                    207: *                         a reminder to the caller that in this case V is
                    208: *                         reserved as workspace of length N*N.
                    209: *          If JOBV = 'N'  V is not referenced.
                    210: *
                    211: *  LDV     (input) INTEGER
                    212: *          The leading dimension of the array V,  LDV >= 1.
                    213: *          If JOBV = 'V' or 'J' or 'W', then LDV >= N.
                    214: *
                    215: *  WORK    (workspace/output) DOUBLE PRECISION array, dimension at least LWORK.
1.6     ! bertrand  216: *          On exit, if N.GT.0 .AND. M.GT.0 (else not referenced), 
1.1       bertrand  217: *          WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such
                    218: *                    that SCALE*SVA(1:N) are the computed singular values
                    219: *                    of A. (See the description of SVA().)
                    220: *          WORK(2) = See the description of WORK(1).
                    221: *          WORK(3) = SCONDA is an estimate for the condition number of
                    222: *                    column equilibrated A. (If JOBA .EQ. 'E' or 'G')
                    223: *                    SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).
                    224: *                    It is computed using DPOCON. It holds
                    225: *                    N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
                    226: *                    where R is the triangular factor from the QRF of A.
                    227: *                    However, if R is truncated and the numerical rank is
                    228: *                    determined to be strictly smaller than N, SCONDA is
                    229: *                    returned as -1, thus indicating that the smallest
                    230: *                    singular values might be lost.
                    231: *
                    232: *          If full SVD is needed, the following two condition numbers are
                    233: *          useful for the analysis of the algorithm. They are provied for
                    234: *          a developer/implementer who is familiar with the details of
                    235: *          the method.
                    236: *
                    237: *          WORK(4) = an estimate of the scaled condition number of the
                    238: *                    triangular factor in the first QR factorization.
                    239: *          WORK(5) = an estimate of the scaled condition number of the
                    240: *                    triangular factor in the second QR factorization.
                    241: *          The following two parameters are computed if JOBT .EQ. 'T'.
                    242: *          They are provided for a developer/implementer who is familiar
                    243: *          with the details of the method.
                    244: *
                    245: *          WORK(6) = the entropy of A^t*A :: this is the Shannon entropy
                    246: *                    of diag(A^t*A) / Trace(A^t*A) taken as point in the
                    247: *                    probability simplex.
                    248: *          WORK(7) = the entropy of A*A^t.
                    249: *
                    250: *  LWORK   (input) INTEGER
                    251: *          Length of WORK to confirm proper allocation of work space.
                    252: *          LWORK depends on the job:
                    253: *
                    254: *          If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
1.6     ! bertrand  255: *            -> .. no scaled condition estimate required (JOBE.EQ.'N'):
1.1       bertrand  256: *               LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement.
1.6     ! bertrand  257: *               ->> For optimal performance (blocked code) the optimal value
1.1       bertrand  258: *               is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal
1.6     ! bertrand  259: *               block size for DGEQP3 and DGEQRF.
        !           260: *               In general, optimal LWORK is computed as 
        !           261: *               LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 7).        
1.1       bertrand  262: *            -> .. an estimate of the scaled condition number of A is
                    263: *               required (JOBA='E', 'G'). In this case, LWORK is the maximum
1.6     ! bertrand  264: *               of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4*N,7).
        !           265: *               ->> For optimal performance (blocked code) the optimal value 
        !           266: *               is LWORK >= max(2*M+N,3*N+(N+1)*NB, N*N+4*N, 7).
        !           267: *               In general, the optimal length LWORK is computed as
        !           268: *               LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DGEQRF), 
        !           269: *                                                     N+N*N+LWORK(DPOCON),7).
1.1       bertrand  270: *
                    271: *          If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),
1.6     ! bertrand  272: *            -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
        !           273: *            -> For optimal performance, LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
        !           274: *               where NB is the optimal block size for DGEQP3, DGEQRF, DGELQ,
        !           275: *               DORMLQ. In general, the optimal length LWORK is computed as
        !           276: *               LWORK >= max(2*M+N,N+LWORK(DGEQP3), N+LWORK(DPOCON), 
        !           277: *                       N+LWORK(DGELQ), 2*N+LWORK(DGEQRF), N+LWORK(DORMLQ)).
1.1       bertrand  278: *
                    279: *          If SIGMA and the left singular vectors are needed
1.6     ! bertrand  280: *            -> the minimal requirement is LWORK >= max(2*M+N,4*N+1,7).
        !           281: *            -> For optimal performance:
        !           282: *               if JOBU.EQ.'U' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,7),
        !           283: *               if JOBU.EQ.'F' :: LWORK >= max(2*M+N,3*N+(N+1)*NB,N+M*NB,7),
        !           284: *               where NB is the optimal block size for DGEQP3, DGEQRF, DORMQR.
        !           285: *               In general, the optimal length LWORK is computed as
        !           286: *               LWORK >= max(2*M+N,N+LWORK(DGEQP3),N+LWORK(DPOCON),
        !           287: *                        2*N+LWORK(DGEQRF), N+LWORK(DORMQR)). 
        !           288: *               Here LWORK(DORMQR) equals N*NB (for JOBU.EQ.'U') or 
        !           289: *               M*NB (for JOBU.EQ.'F').
        !           290: *               
        !           291: *          If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and 
        !           292: *            -> if JOBV.EQ.'V'  
        !           293: *               the minimal requirement is LWORK >= max(2*M+N,6*N+2*N*N). 
        !           294: *            -> if JOBV.EQ.'J' the minimal requirement is 
        !           295: *               LWORK >= max(2*M+N, 4*N+N*N,2*N+N*N+6).
        !           296: *            -> For optimal performance, LWORK should be additionally
        !           297: *               larger than N+M*NB, where NB is the optimal block size
        !           298: *               for DORMQR.
1.1       bertrand  299: *
                    300: *  IWORK   (workspace/output) INTEGER array, dimension M+3*N.
                    301: *          On exit,
                    302: *          IWORK(1) = the numerical rank determined after the initial
                    303: *                     QR factorization with pivoting. See the descriptions
                    304: *                     of JOBA and JOBR.
                    305: *          IWORK(2) = the number of the computed nonzero singular values
                    306: *          IWORK(3) = if nonzero, a warning message:
                    307: *                     If IWORK(3).EQ.1 then some of the column norms of A
                    308: *                     were denormalized floats. The requested high accuracy
                    309: *                     is not warranted by the data.
                    310: *
                    311: *  INFO    (output) INTEGER
                    312: *           < 0  : if INFO = -i, then the i-th argument had an illegal value.
                    313: *           = 0 :  successfull exit;
                    314: *           > 0 :  DGEJSV  did not converge in the maximal allowed number
                    315: *                  of sweeps. The computed values may be inaccurate.
                    316: *
                    317: *  Further Details
                    318: *  ===============
                    319: *
1.6     ! bertrand  320: *  DGEJSV implements a preconditioned Jacobi SVD algorithm. It uses DGEQP3,
        !           321: *  DGEQRF, and DGELQF as preprocessors and preconditioners. Optionally, an
1.1       bertrand  322: *  additional row pivoting can be used as a preprocessor, which in some
                    323: *  cases results in much higher accuracy. An example is matrix A with the
                    324: *  structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
                    325: *  diagonal matrices and C is well-conditioned matrix. In that case, complete
                    326: *  pivoting in the first QR factorizations provides accuracy dependent on the
                    327: *  condition number of C, and independent of D1, D2. Such higher accuracy is
                    328: *  not completely understood theoretically, but it works well in practice.
                    329: *  Further, if A can be written as A = B*D, with well-conditioned B and some
                    330: *  diagonal D, then the high accuracy is guaranteed, both theoretically and
                    331: *  in software, independent of D. For more details see [1], [2].
                    332: *     The computational range for the singular values can be the full range
                    333: *  ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
                    334: *  & LAPACK routines called by DGEJSV are implemented to work in that range.
                    335: *  If that is not the case, then the restriction for safe computation with
                    336: *  the singular values in the range of normalized IEEE numbers is that the
                    337: *  spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
                    338: *  overflow. This code (DGEJSV) is best used in this restricted range,
1.6     ! bertrand  339: *  meaning that singular values of magnitude below ||A||_2 / DLAMCH('O') are
1.1       bertrand  340: *  returned as zeros. See JOBR for details on this.
                    341: *     Further, this implementation is somewhat slower than the one described
                    342: *  in [1,2] due to replacement of some non-LAPACK components, and because
                    343: *  the choice of some tuning parameters in the iterative part (DGESVJ) is
                    344: *  left to the implementer on a particular machine.
1.6     ! bertrand  345: *     The rank revealing QR factorization (in this code: DGEQP3) should be
        !           346: *  implemented as in [3]. We have a new version of DGEQP3 under development
1.1       bertrand  347: *  that is more robust than the current one in LAPACK, with a cleaner cut in
                    348: *  rank defficient cases. It will be available in the SIGMA library [4].
                    349: *  If M is much larger than N, it is obvious that the inital QRF with
                    350: *  column pivoting can be preprocessed by the QRF without pivoting. That
                    351: *  well known trick is not used in DGEJSV because in some cases heavy row
                    352: *  weighting can be treated with complete pivoting. The overhead in cases
                    353: *  M much larger than N is then only due to pivoting, but the benefits in
                    354: *  terms of accuracy have prevailed. The implementer/user can incorporate
                    355: *  this extra QRF step easily. The implementer can also improve data movement
                    356: *  (matrix transpose, matrix copy, matrix transposed copy) - this
                    357: *  implementation of DGEJSV uses only the simplest, naive data movement.
                    358: *
                    359: *  Contributors
                    360: *
                    361: *  Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
                    362: *
                    363: *  References
                    364: *
                    365: * [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
                    366: *     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
                    367: *     LAPACK Working note 169.
                    368: * [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
                    369: *     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
                    370: *     LAPACK Working note 170.
                    371: * [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
                    372: *     factorization software - a case study.
                    373: *     ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
                    374: *     LAPACK Working note 176.
                    375: * [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
                    376: *     QSVD, (H,K)-SVD computations.
                    377: *     Department of Mathematics, University of Zagreb, 2008.
                    378: *
                    379: *  Bugs, examples and comments
1.6     ! bertrand  380: *
1.1       bertrand  381: *  Please report all bugs and send interesting examples and/or comments to
                    382: *  drmac@math.hr. Thank you.
                    383: *
1.6     ! bertrand  384: *  ===========================================================================
1.1       bertrand  385: *
                    386: *     .. Local Parameters ..
                    387:       DOUBLE PRECISION   ZERO,  ONE
                    388:       PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
                    389: *     ..
                    390: *     .. Local Scalars ..
                    391:       DOUBLE PRECISION AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1, COND_OK,
1.6     ! bertrand  392:      $        CONDR1, CONDR2, ENTRA,  ENTRAT, EPSLN,  MAXPRJ, SCALEM,
        !           393:      $        SCONDA, SFMIN,  SMALL,  TEMP1,  USCAL1, USCAL2, XSC
1.1       bertrand  394:       INTEGER IERR,   N1,     NR,     NUMRANK,        p, q,   WARNING
                    395:       LOGICAL ALMORT, DEFR,   ERREST, GOSCAL, JRACC,  KILL,   LSVEC,
1.6     ! bertrand  396:      $        L2ABER, L2KILL, L2PERT, L2RANK, L2TRAN,
        !           397:      $        NOSCAL, ROWPIV, RSVEC,  TRANSP
1.1       bertrand  398: *     ..
                    399: *     .. Intrinsic Functions ..
                    400:       INTRINSIC DABS,  DLOG, DMAX1, DMIN1, DBLE,
1.6     ! bertrand  401:      $          MAX0, MIN0, IDNINT,  DSIGN,  DSQRT
1.1       bertrand  402: *     ..
                    403: *     .. External Functions ..
                    404:       DOUBLE PRECISION  DLAMCH, DNRM2
                    405:       INTEGER   IDAMAX
                    406:       LOGICAL   LSAME
                    407:       EXTERNAL  IDAMAX, LSAME, DLAMCH, DNRM2
                    408: *     ..
                    409: *     .. External Subroutines ..
                    410:       EXTERNAL  DCOPY,  DGELQF, DGEQP3, DGEQRF, DLACPY, DLASCL,
1.6     ! bertrand  411:      $          DLASET, DLASSQ, DLASWP, DORGQR, DORMLQ,
        !           412:      $          DORMQR, DPOCON, DSCAL,  DSWAP,  DTRSM,  XERBLA
1.1       bertrand  413: *
                    414:       EXTERNAL  DGESVJ
                    415: *     ..
                    416: *
                    417: *     Test the input arguments
                    418: *
                    419:       LSVEC  = LSAME( JOBU, 'U' ) .OR. LSAME( JOBU, 'F' )
                    420:       JRACC  = LSAME( JOBV, 'J' )
                    421:       RSVEC  = LSAME( JOBV, 'V' ) .OR. JRACC
                    422:       ROWPIV = LSAME( JOBA, 'F' ) .OR. LSAME( JOBA, 'G' )
                    423:       L2RANK = LSAME( JOBA, 'R' )
                    424:       L2ABER = LSAME( JOBA, 'A' )
                    425:       ERREST = LSAME( JOBA, 'E' ) .OR. LSAME( JOBA, 'G' )
                    426:       L2TRAN = LSAME( JOBT, 'T' )
                    427:       L2KILL = LSAME( JOBR, 'R' )
                    428:       DEFR   = LSAME( JOBR, 'N' )
                    429:       L2PERT = LSAME( JOBP, 'P' )
                    430: *
                    431:       IF ( .NOT.(ROWPIV .OR. L2RANK .OR. L2ABER .OR.
1.6     ! bertrand  432:      $     ERREST .OR. LSAME( JOBA, 'C' ) )) THEN
1.1       bertrand  433:          INFO = - 1
                    434:       ELSE IF ( .NOT.( LSVEC  .OR. LSAME( JOBU, 'N' ) .OR.
1.6     ! bertrand  435:      $                             LSAME( JOBU, 'W' )) ) THEN
1.1       bertrand  436:          INFO = - 2
                    437:       ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.
1.6     ! bertrand  438:      $   LSAME( JOBV, 'W' )) .OR. ( JRACC .AND. (.NOT.LSVEC) ) ) THEN
1.1       bertrand  439:          INFO = - 3
                    440:       ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) )    THEN
                    441:          INFO = - 4
                    442:       ELSE IF ( .NOT. ( L2TRAN .OR. LSAME( JOBT, 'N' ) ) ) THEN
                    443:          INFO = - 5
                    444:       ELSE IF ( .NOT. ( L2PERT .OR. LSAME( JOBP, 'N' ) ) ) THEN
                    445:          INFO = - 6
                    446:       ELSE IF ( M .LT. 0 ) THEN
                    447:          INFO = - 7
                    448:       ELSE IF ( ( N .LT. 0 ) .OR. ( N .GT. M ) ) THEN
                    449:          INFO = - 8
                    450:       ELSE IF ( LDA .LT. M ) THEN
                    451:          INFO = - 10
                    452:       ELSE IF ( LSVEC .AND. ( LDU .LT. M ) ) THEN
                    453:          INFO = - 13
                    454:       ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN
                    455:          INFO = - 14
                    456:       ELSE IF ( (.NOT.(LSVEC .OR. RSVEC .OR. ERREST).AND.
                    457:      &                           (LWORK .LT. MAX0(7,4*N+1,2*M+N))) .OR.
1.6     ! bertrand  458:      & (.NOT.(LSVEC .OR. RSVEC) .AND. ERREST .AND.
1.1       bertrand  459:      &                         (LWORK .LT. MAX0(7,4*N+N*N,2*M+N))) .OR.
1.6     ! bertrand  460:      & (LSVEC .AND. (.NOT.RSVEC) .AND. (LWORK .LT. MAX0(7,2*M+N,4*N+1)))
        !           461:      & .OR.
        !           462:      & (RSVEC .AND. (.NOT.LSVEC) .AND. (LWORK .LT. MAX0(7,2*M+N,4*N+1)))
        !           463:      & .OR.
        !           464:      & (LSVEC .AND. RSVEC .AND. (.NOT.JRACC) .AND. 
        !           465:      &                          (LWORK.LT.MAX0(2*M+N,6*N+2*N*N)))
        !           466:      & .OR. (LSVEC .AND. RSVEC .AND. JRACC .AND.
        !           467:      &                          LWORK.LT.MAX0(2*M+N,4*N+N*N,2*N+N*N+6)))
1.1       bertrand  468:      &   THEN
                    469:          INFO = - 17
                    470:       ELSE
                    471: *        #:)
                    472:          INFO = 0
                    473:       END IF
                    474: *
                    475:       IF ( INFO .NE. 0 ) THEN
                    476: *       #:(
                    477:          CALL XERBLA( 'DGEJSV', - INFO )
1.6     ! bertrand  478:          RETURN
1.1       bertrand  479:       END IF
                    480: *
                    481: *     Quick return for void matrix (Y3K safe)
                    482: * #:)
                    483:       IF ( ( M .EQ. 0 ) .OR. ( N .EQ. 0 ) ) RETURN
                    484: *
                    485: *     Determine whether the matrix U should be M x N or M x M
                    486: *
                    487:       IF ( LSVEC ) THEN
                    488:          N1 = N
                    489:          IF ( LSAME( JOBU, 'F' ) ) N1 = M
                    490:       END IF
                    491: *
                    492: *     Set numerical parameters
                    493: *
                    494: *!    NOTE: Make sure DLAMCH() does not fail on the target architecture.
                    495: *
                    496:       EPSLN = DLAMCH('Epsilon')
                    497:       SFMIN = DLAMCH('SafeMinimum')
                    498:       SMALL = SFMIN / EPSLN
                    499:       BIG   = DLAMCH('O')
                    500: *     BIG   = ONE / SFMIN
                    501: *
                    502: *     Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N
                    503: *
                    504: *(!)  If necessary, scale SVA() to protect the largest norm from
                    505: *     overflow. It is possible that this scaling pushes the smallest
                    506: *     column norm left from the underflow threshold (extreme case).
                    507: *
                    508:       SCALEM  = ONE / DSQRT(DBLE(M)*DBLE(N))
                    509:       NOSCAL  = .TRUE.
                    510:       GOSCAL  = .TRUE.
                    511:       DO 1874 p = 1, N
                    512:          AAPP = ZERO
1.4       bertrand  513:          AAQQ = ONE
1.1       bertrand  514:          CALL DLASSQ( M, A(1,p), 1, AAPP, AAQQ )
                    515:          IF ( AAPP .GT. BIG ) THEN
                    516:             INFO = - 9
                    517:             CALL XERBLA( 'DGEJSV', -INFO )
                    518:             RETURN
                    519:          END IF
                    520:          AAQQ = DSQRT(AAQQ)
                    521:          IF ( ( AAPP .LT. (BIG / AAQQ) ) .AND. NOSCAL  ) THEN
                    522:             SVA(p)  = AAPP * AAQQ
                    523:          ELSE
                    524:             NOSCAL  = .FALSE.
                    525:             SVA(p)  = AAPP * ( AAQQ * SCALEM )
                    526:             IF ( GOSCAL ) THEN
                    527:                GOSCAL = .FALSE.
                    528:                CALL DSCAL( p-1, SCALEM, SVA, 1 )
                    529:             END IF
                    530:          END IF
                    531:  1874 CONTINUE
                    532: *
                    533:       IF ( NOSCAL ) SCALEM = ONE
                    534: *
                    535:       AAPP = ZERO
                    536:       AAQQ = BIG
                    537:       DO 4781 p = 1, N
                    538:          AAPP = DMAX1( AAPP, SVA(p) )
                    539:          IF ( SVA(p) .NE. ZERO ) AAQQ = DMIN1( AAQQ, SVA(p) )
                    540:  4781 CONTINUE
                    541: *
                    542: *     Quick return for zero M x N matrix
                    543: * #:)
                    544:       IF ( AAPP .EQ. ZERO ) THEN
                    545:          IF ( LSVEC ) CALL DLASET( 'G', M, N1, ZERO, ONE, U, LDU )
                    546:          IF ( RSVEC ) CALL DLASET( 'G', N, N,  ZERO, ONE, V, LDV )
                    547:          WORK(1) = ONE
                    548:          WORK(2) = ONE
                    549:          IF ( ERREST ) WORK(3) = ONE
                    550:          IF ( LSVEC .AND. RSVEC ) THEN
                    551:             WORK(4) = ONE
                    552:             WORK(5) = ONE
                    553:          END IF
                    554:          IF ( L2TRAN ) THEN
                    555:             WORK(6) = ZERO
                    556:             WORK(7) = ZERO
                    557:          END IF
                    558:          IWORK(1) = 0
                    559:          IWORK(2) = 0
1.6     ! bertrand  560:          IWORK(3) = 0
1.1       bertrand  561:          RETURN
                    562:       END IF
                    563: *
                    564: *     Issue warning if denormalized column norms detected. Override the
                    565: *     high relative accuracy request. Issue licence to kill columns
                    566: *     (set them to zero) whose norm is less than sigma_max / BIG (roughly).
                    567: * #:(
                    568:       WARNING = 0
                    569:       IF ( AAQQ .LE. SFMIN ) THEN
                    570:          L2RANK = .TRUE.
                    571:          L2KILL = .TRUE.
                    572:          WARNING = 1
                    573:       END IF
                    574: *
                    575: *     Quick return for one-column matrix
                    576: * #:)
                    577:       IF ( N .EQ. 1 ) THEN
                    578: *
                    579:          IF ( LSVEC ) THEN
                    580:             CALL DLASCL( 'G',0,0,SVA(1),SCALEM, M,1,A(1,1),LDA,IERR )
                    581:             CALL DLACPY( 'A', M, 1, A, LDA, U, LDU )
                    582: *           computing all M left singular vectors of the M x 1 matrix
                    583:             IF ( N1 .NE. N  ) THEN
                    584:                CALL DGEQRF( M, N, U,LDU, WORK, WORK(N+1),LWORK-N,IERR )
                    585:                CALL DORGQR( M,N1,1, U,LDU,WORK,WORK(N+1),LWORK-N,IERR )
                    586:                CALL DCOPY( M, A(1,1), 1, U(1,1), 1 )
                    587:             END IF
                    588:          END IF
                    589:          IF ( RSVEC ) THEN
                    590:              V(1,1) = ONE
                    591:          END IF
                    592:          IF ( SVA(1) .LT. (BIG*SCALEM) ) THEN
                    593:             SVA(1)  = SVA(1) / SCALEM
                    594:             SCALEM  = ONE
                    595:          END IF
                    596:          WORK(1) = ONE / SCALEM
                    597:          WORK(2) = ONE
                    598:          IF ( SVA(1) .NE. ZERO ) THEN
                    599:             IWORK(1) = 1
                    600:             IF ( ( SVA(1) / SCALEM) .GE. SFMIN ) THEN
                    601:                IWORK(2) = 1
                    602:             ELSE
                    603:                IWORK(2) = 0
                    604:             END IF
                    605:          ELSE
                    606:             IWORK(1) = 0
                    607:             IWORK(2) = 0
                    608:          END IF
                    609:          IF ( ERREST ) WORK(3) = ONE
                    610:          IF ( LSVEC .AND. RSVEC ) THEN
                    611:             WORK(4) = ONE
                    612:             WORK(5) = ONE
                    613:          END IF
                    614:          IF ( L2TRAN ) THEN
                    615:             WORK(6) = ZERO
                    616:             WORK(7) = ZERO
                    617:          END IF
                    618:          RETURN
                    619: *
                    620:       END IF
                    621: *
                    622:       TRANSP = .FALSE.
                    623:       L2TRAN = L2TRAN .AND. ( M .EQ. N )
                    624: *
                    625:       AATMAX = -ONE
                    626:       AATMIN =  BIG
                    627:       IF ( ROWPIV .OR. L2TRAN ) THEN
                    628: *
                    629: *     Compute the row norms, needed to determine row pivoting sequence
                    630: *     (in the case of heavily row weighted A, row pivoting is strongly
                    631: *     advised) and to collect information needed to compare the
                    632: *     structures of A * A^t and A^t * A (in the case L2TRAN.EQ..TRUE.).
                    633: *
                    634:          IF ( L2TRAN ) THEN
                    635:             DO 1950 p = 1, M
                    636:                XSC   = ZERO
1.4       bertrand  637:                TEMP1 = ONE
1.1       bertrand  638:                CALL DLASSQ( N, A(p,1), LDA, XSC, TEMP1 )
                    639: *              DLASSQ gets both the ell_2 and the ell_infinity norm
                    640: *              in one pass through the vector
                    641:                WORK(M+N+p)  = XSC * SCALEM
                    642:                WORK(N+p)    = XSC * (SCALEM*DSQRT(TEMP1))
                    643:                AATMAX = DMAX1( AATMAX, WORK(N+p) )
                    644:                IF (WORK(N+p) .NE. ZERO) AATMIN = DMIN1(AATMIN,WORK(N+p))
                    645:  1950       CONTINUE
                    646:          ELSE
                    647:             DO 1904 p = 1, M
                    648:                WORK(M+N+p) = SCALEM*DABS( A(p,IDAMAX(N,A(p,1),LDA)) )
                    649:                AATMAX = DMAX1( AATMAX, WORK(M+N+p) )
                    650:                AATMIN = DMIN1( AATMIN, WORK(M+N+p) )
                    651:  1904       CONTINUE
                    652:          END IF
                    653: *
                    654:       END IF
                    655: *
                    656: *     For square matrix A try to determine whether A^t  would be  better
                    657: *     input for the preconditioned Jacobi SVD, with faster convergence.
                    658: *     The decision is based on an O(N) function of the vector of column
                    659: *     and row norms of A, based on the Shannon entropy. This should give
                    660: *     the right choice in most cases when the difference actually matters.
                    661: *     It may fail and pick the slower converging side.
                    662: *
                    663:       ENTRA  = ZERO
                    664:       ENTRAT = ZERO
                    665:       IF ( L2TRAN ) THEN
                    666: *
                    667:          XSC   = ZERO
1.4       bertrand  668:          TEMP1 = ONE
1.1       bertrand  669:          CALL DLASSQ( N, SVA, 1, XSC, TEMP1 )
                    670:          TEMP1 = ONE / TEMP1
                    671: *
                    672:          ENTRA = ZERO
                    673:          DO 1113 p = 1, N
                    674:             BIG1  = ( ( SVA(p) / XSC )**2 ) * TEMP1
                    675:             IF ( BIG1 .NE. ZERO ) ENTRA = ENTRA + BIG1 * DLOG(BIG1)
                    676:  1113    CONTINUE
                    677:          ENTRA = - ENTRA / DLOG(DBLE(N))
                    678: *
                    679: *        Now, SVA().^2/Trace(A^t * A) is a point in the probability simplex.
                    680: *        It is derived from the diagonal of  A^t * A.  Do the same with the
                    681: *        diagonal of A * A^t, compute the entropy of the corresponding
                    682: *        probability distribution. Note that A * A^t and A^t * A have the
                    683: *        same trace.
                    684: *
                    685:          ENTRAT = ZERO
                    686:          DO 1114 p = N+1, N+M
                    687:             BIG1 = ( ( WORK(p) / XSC )**2 ) * TEMP1
                    688:             IF ( BIG1 .NE. ZERO ) ENTRAT = ENTRAT + BIG1 * DLOG(BIG1)
                    689:  1114    CONTINUE
                    690:          ENTRAT = - ENTRAT / DLOG(DBLE(M))
                    691: *
                    692: *        Analyze the entropies and decide A or A^t. Smaller entropy
                    693: *        usually means better input for the algorithm.
                    694: *
                    695:          TRANSP = ( ENTRAT .LT. ENTRA )
                    696: *
                    697: *        If A^t is better than A, transpose A.
                    698: *
                    699:          IF ( TRANSP ) THEN
                    700: *           In an optimal implementation, this trivial transpose
                    701: *           should be replaced with faster transpose.
                    702:             DO 1115 p = 1, N - 1
                    703:                DO 1116 q = p + 1, N
                    704:                    TEMP1 = A(q,p)
                    705:                   A(q,p) = A(p,q)
                    706:                   A(p,q) = TEMP1
                    707:  1116          CONTINUE
                    708:  1115       CONTINUE
                    709:             DO 1117 p = 1, N
                    710:                WORK(M+N+p) = SVA(p)
                    711:                SVA(p)      = WORK(N+p)
                    712:  1117       CONTINUE
                    713:             TEMP1  = AAPP
                    714:             AAPP   = AATMAX
                    715:             AATMAX = TEMP1
                    716:             TEMP1  = AAQQ
                    717:             AAQQ   = AATMIN
                    718:             AATMIN = TEMP1
                    719:             KILL   = LSVEC
                    720:             LSVEC  = RSVEC
                    721:             RSVEC  = KILL
1.4       bertrand  722:             IF ( LSVEC ) N1 = N
1.1       bertrand  723: *
                    724:             ROWPIV = .TRUE.
                    725:          END IF
                    726: *
                    727:       END IF
                    728: *     END IF L2TRAN
                    729: *
                    730: *     Scale the matrix so that its maximal singular value remains less
                    731: *     than DSQRT(BIG) -- the matrix is scaled so that its maximal column
                    732: *     has Euclidean norm equal to DSQRT(BIG/N). The only reason to keep
                    733: *     DSQRT(BIG) instead of BIG is the fact that DGEJSV uses LAPACK and
                    734: *     BLAS routines that, in some implementations, are not capable of
                    735: *     working in the full interval [SFMIN,BIG] and that they may provoke
                    736: *     overflows in the intermediate results. If the singular values spread
                    737: *     from SFMIN to BIG, then DGESVJ will compute them. So, in that case,
                    738: *     one should use DGESVJ instead of DGEJSV.
                    739: *
                    740:       BIG1   = DSQRT( BIG )
                    741:       TEMP1  = DSQRT( BIG / DBLE(N) )
                    742: *
                    743:       CALL DLASCL( 'G', 0, 0, AAPP, TEMP1, N, 1, SVA, N, IERR )
                    744:       IF ( AAQQ .GT. (AAPP * SFMIN) ) THEN
                    745:           AAQQ = ( AAQQ / AAPP ) * TEMP1
                    746:       ELSE
                    747:           AAQQ = ( AAQQ * TEMP1 ) / AAPP
                    748:       END IF
                    749:       TEMP1 = TEMP1 * SCALEM
                    750:       CALL DLASCL( 'G', 0, 0, AAPP, TEMP1, M, N, A, LDA, IERR )
                    751: *
                    752: *     To undo scaling at the end of this procedure, multiply the
                    753: *     computed singular values with USCAL2 / USCAL1.
                    754: *
                    755:       USCAL1 = TEMP1
                    756:       USCAL2 = AAPP
                    757: *
                    758:       IF ( L2KILL ) THEN
                    759: *        L2KILL enforces computation of nonzero singular values in
                    760: *        the restricted range of condition number of the initial A,
                    761: *        sigma_max(A) / sigma_min(A) approx. DSQRT(BIG)/DSQRT(SFMIN).
                    762:          XSC = DSQRT( SFMIN )
                    763:       ELSE
                    764:          XSC = SMALL
                    765: *
                    766: *        Now, if the condition number of A is too big,
                    767: *        sigma_max(A) / sigma_min(A) .GT. DSQRT(BIG/N) * EPSLN / SFMIN,
                    768: *        as a precaution measure, the full SVD is computed using DGESVJ
                    769: *        with accumulated Jacobi rotations. This provides numerically
                    770: *        more robust computation, at the cost of slightly increased run
                    771: *        time. Depending on the concrete implementation of BLAS and LAPACK
                    772: *        (i.e. how they behave in presence of extreme ill-conditioning) the
                    773: *        implementor may decide to remove this switch.
                    774:          IF ( ( AAQQ.LT.DSQRT(SFMIN) ) .AND. LSVEC .AND. RSVEC ) THEN
                    775:             JRACC = .TRUE.
                    776:          END IF
                    777: *
                    778:       END IF
                    779:       IF ( AAQQ .LT. XSC ) THEN
                    780:          DO 700 p = 1, N
                    781:             IF ( SVA(p) .LT. XSC ) THEN
                    782:                CALL DLASET( 'A', M, 1, ZERO, ZERO, A(1,p), LDA )
                    783:                SVA(p) = ZERO
                    784:             END IF
                    785:  700     CONTINUE
                    786:       END IF
                    787: *
                    788: *     Preconditioning using QR factorization with pivoting
                    789: *
                    790:       IF ( ROWPIV ) THEN
                    791: *        Optional row permutation (Bjoerck row pivoting):
                    792: *        A result by Cox and Higham shows that the Bjoerck's
                    793: *        row pivoting combined with standard column pivoting
                    794: *        has similar effect as Powell-Reid complete pivoting.
                    795: *        The ell-infinity norms of A are made nonincreasing.
                    796:          DO 1952 p = 1, M - 1
                    797:             q = IDAMAX( M-p+1, WORK(M+N+p), 1 ) + p - 1
                    798:             IWORK(2*N+p) = q
                    799:             IF ( p .NE. q ) THEN
                    800:                TEMP1       = WORK(M+N+p)
                    801:                WORK(M+N+p) = WORK(M+N+q)
                    802:                WORK(M+N+q) = TEMP1
                    803:             END IF
                    804:  1952    CONTINUE
                    805:          CALL DLASWP( N, A, LDA, 1, M-1, IWORK(2*N+1), 1 )
                    806:       END IF
                    807: *
                    808: *     End of the preparation phase (scaling, optional sorting and
                    809: *     transposing, optional flushing of small columns).
                    810: *
                    811: *     Preconditioning
                    812: *
                    813: *     If the full SVD is needed, the right singular vectors are computed
                    814: *     from a matrix equation, and for that we need theoretical analysis
                    815: *     of the Businger-Golub pivoting. So we use DGEQP3 as the first RR QRF.
                    816: *     In all other cases the first RR QRF can be chosen by other criteria
                    817: *     (eg speed by replacing global with restricted window pivoting, such
                    818: *     as in SGEQPX from TOMS # 782). Good results will be obtained using
                    819: *     SGEQPX with properly (!) chosen numerical parameters.
                    820: *     Any improvement of DGEQP3 improves overal performance of DGEJSV.
                    821: *
                    822: *     A * P1 = Q1 * [ R1^t 0]^t:
                    823:       DO 1963 p = 1, N
                    824: *        .. all columns are free columns
                    825:          IWORK(p) = 0
                    826:  1963 CONTINUE
                    827:       CALL DGEQP3( M,N,A,LDA, IWORK,WORK, WORK(N+1),LWORK-N, IERR )
                    828: *
                    829: *     The upper triangular matrix R1 from the first QRF is inspected for
                    830: *     rank deficiency and possibilities for deflation, or possible
                    831: *     ill-conditioning. Depending on the user specified flag L2RANK,
                    832: *     the procedure explores possibilities to reduce the numerical
                    833: *     rank by inspecting the computed upper triangular factor. If
                    834: *     L2RANK or L2ABER are up, then DGEJSV will compute the SVD of
                    835: *     A + dA, where ||dA|| <= f(M,N)*EPSLN.
                    836: *
                    837:       NR = 1
                    838:       IF ( L2ABER ) THEN
                    839: *        Standard absolute error bound suffices. All sigma_i with
                    840: *        sigma_i < N*EPSLN*||A|| are flushed to zero. This is an
                    841: *        agressive enforcement of lower numerical rank by introducing a
                    842: *        backward error of the order of N*EPSLN*||A||.
                    843:          TEMP1 = DSQRT(DBLE(N))*EPSLN
                    844:          DO 3001 p = 2, N
                    845:             IF ( DABS(A(p,p)) .GE. (TEMP1*DABS(A(1,1))) ) THEN
                    846:                NR = NR + 1
                    847:             ELSE
                    848:                GO TO 3002
                    849:             END IF
                    850:  3001    CONTINUE
                    851:  3002    CONTINUE
                    852:       ELSE IF ( L2RANK ) THEN
                    853: *        .. similarly as above, only slightly more gentle (less agressive).
                    854: *        Sudden drop on the diagonal of R1 is used as the criterion for
                    855: *        close-to-rank-defficient.
                    856:          TEMP1 = DSQRT(SFMIN)
                    857:          DO 3401 p = 2, N
                    858:             IF ( ( DABS(A(p,p)) .LT. (EPSLN*DABS(A(p-1,p-1))) ) .OR.
1.6     ! bertrand  859:      $           ( DABS(A(p,p)) .LT. SMALL ) .OR.
        !           860:      $           ( L2KILL .AND. (DABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3402
1.1       bertrand  861:             NR = NR + 1
                    862:  3401    CONTINUE
                    863:  3402    CONTINUE
                    864: *
                    865:       ELSE
                    866: *        The goal is high relative accuracy. However, if the matrix
                    867: *        has high scaled condition number the relative accuracy is in
                    868: *        general not feasible. Later on, a condition number estimator
                    869: *        will be deployed to estimate the scaled condition number.
                    870: *        Here we just remove the underflowed part of the triangular
                    871: *        factor. This prevents the situation in which the code is
                    872: *        working hard to get the accuracy not warranted by the data.
                    873:          TEMP1  = DSQRT(SFMIN)
                    874:          DO 3301 p = 2, N
                    875:             IF ( ( DABS(A(p,p)) .LT. SMALL ) .OR.
1.6     ! bertrand  876:      $          ( L2KILL .AND. (DABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3302
1.1       bertrand  877:             NR = NR + 1
                    878:  3301    CONTINUE
                    879:  3302    CONTINUE
                    880: *
                    881:       END IF
                    882: *
                    883:       ALMORT = .FALSE.
                    884:       IF ( NR .EQ. N ) THEN
                    885:          MAXPRJ = ONE
                    886:          DO 3051 p = 2, N
                    887:             TEMP1  = DABS(A(p,p)) / SVA(IWORK(p))
                    888:             MAXPRJ = DMIN1( MAXPRJ, TEMP1 )
                    889:  3051    CONTINUE
                    890:          IF ( MAXPRJ**2 .GE. ONE - DBLE(N)*EPSLN ) ALMORT = .TRUE.
                    891:       END IF
                    892: *
                    893: *
                    894:       SCONDA = - ONE
                    895:       CONDR1 = - ONE
                    896:       CONDR2 = - ONE
                    897: *
                    898:       IF ( ERREST ) THEN
                    899:          IF ( N .EQ. NR ) THEN
                    900:             IF ( RSVEC ) THEN
                    901: *              .. V is available as workspace
                    902:                CALL DLACPY( 'U', N, N, A, LDA, V, LDV )
                    903:                DO 3053 p = 1, N
                    904:                   TEMP1 = SVA(IWORK(p))
                    905:                   CALL DSCAL( p, ONE/TEMP1, V(1,p), 1 )
                    906:  3053          CONTINUE
                    907:                CALL DPOCON( 'U', N, V, LDV, ONE, TEMP1,
1.6     ! bertrand  908:      $              WORK(N+1), IWORK(2*N+M+1), IERR )
1.1       bertrand  909:             ELSE IF ( LSVEC ) THEN
                    910: *              .. U is available as workspace
                    911:                CALL DLACPY( 'U', N, N, A, LDA, U, LDU )
                    912:                DO 3054 p = 1, N
                    913:                   TEMP1 = SVA(IWORK(p))
                    914:                   CALL DSCAL( p, ONE/TEMP1, U(1,p), 1 )
                    915:  3054          CONTINUE
                    916:                CALL DPOCON( 'U', N, U, LDU, ONE, TEMP1,
1.6     ! bertrand  917:      $              WORK(N+1), IWORK(2*N+M+1), IERR )
1.1       bertrand  918:             ELSE
                    919:                CALL DLACPY( 'U', N, N, A, LDA, WORK(N+1), N )
                    920:                DO 3052 p = 1, N
                    921:                   TEMP1 = SVA(IWORK(p))
                    922:                   CALL DSCAL( p, ONE/TEMP1, WORK(N+(p-1)*N+1), 1 )
                    923:  3052          CONTINUE
                    924: *           .. the columns of R are scaled to have unit Euclidean lengths.
                    925:                CALL DPOCON( 'U', N, WORK(N+1), N, ONE, TEMP1,
1.6     ! bertrand  926:      $              WORK(N+N*N+1), IWORK(2*N+M+1), IERR )
1.1       bertrand  927:             END IF
                    928:             SCONDA = ONE / DSQRT(TEMP1)
                    929: *           SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1).
                    930: *           N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
                    931:          ELSE
                    932:             SCONDA = - ONE
                    933:          END IF
                    934:       END IF
                    935: *
                    936:       L2PERT = L2PERT .AND. ( DABS( A(1,1)/A(NR,NR) ) .GT. DSQRT(BIG1) )
                    937: *     If there is no violent scaling, artificial perturbation is not needed.
                    938: *
                    939: *     Phase 3:
                    940: *
                    941:       IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN
                    942: *
                    943: *         Singular Values only
                    944: *
                    945: *         .. transpose A(1:NR,1:N)
                    946:          DO 1946 p = 1, MIN0( N-1, NR )
                    947:             CALL DCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 )
                    948:  1946    CONTINUE
                    949: *
                    950: *        The following two DO-loops introduce small relative perturbation
                    951: *        into the strict upper triangle of the lower triangular matrix.
                    952: *        Small entries below the main diagonal are also changed.
                    953: *        This modification is useful if the computing environment does not
                    954: *        provide/allow FLUSH TO ZERO underflow, for it prevents many
                    955: *        annoying denormalized numbers in case of strongly scaled matrices.
                    956: *        The perturbation is structured so that it does not introduce any
                    957: *        new perturbation of the singular values, and it does not destroy
                    958: *        the job done by the preconditioner.
                    959: *        The licence for this perturbation is in the variable L2PERT, which
                    960: *        should be .FALSE. if FLUSH TO ZERO underflow is active.
                    961: *
                    962:          IF ( .NOT. ALMORT ) THEN
                    963: *
                    964:             IF ( L2PERT ) THEN
                    965: *              XSC = DSQRT(SMALL)
                    966:                XSC = EPSLN / DBLE(N)
                    967:                DO 4947 q = 1, NR
                    968:                   TEMP1 = XSC*DABS(A(q,q))
                    969:                   DO 4949 p = 1, N
                    970:                      IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) )
1.6     ! bertrand  971:      $                    .OR. ( p .LT. q ) )
        !           972:      $                     A(p,q) = DSIGN( TEMP1, A(p,q) )
1.1       bertrand  973:  4949             CONTINUE
                    974:  4947          CONTINUE
                    975:             ELSE
                    976:                CALL DLASET( 'U', NR-1,NR-1, ZERO,ZERO, A(1,2),LDA )
                    977:             END IF
                    978: *
                    979: *            .. second preconditioning using the QR factorization
                    980: *
                    981:             CALL DGEQRF( N,NR, A,LDA, WORK, WORK(N+1),LWORK-N, IERR )
                    982: *
                    983: *           .. and transpose upper to lower triangular
                    984:             DO 1948 p = 1, NR - 1
                    985:                CALL DCOPY( NR-p, A(p,p+1), LDA, A(p+1,p), 1 )
                    986:  1948       CONTINUE
                    987: *
                    988:          END IF
                    989: *
                    990: *           Row-cyclic Jacobi SVD algorithm with column pivoting
                    991: *
                    992: *           .. again some perturbation (a "background noise") is added
                    993: *           to drown denormals
                    994:             IF ( L2PERT ) THEN
                    995: *              XSC = DSQRT(SMALL)
                    996:                XSC = EPSLN / DBLE(N)
                    997:                DO 1947 q = 1, NR
                    998:                   TEMP1 = XSC*DABS(A(q,q))
                    999:                   DO 1949 p = 1, NR
                   1000:                      IF ( ( (p.GT.q) .AND. (DABS(A(p,q)).LE.TEMP1) )
1.6     ! bertrand 1001:      $                       .OR. ( p .LT. q ) )
        !          1002:      $                   A(p,q) = DSIGN( TEMP1, A(p,q) )
1.1       bertrand 1003:  1949             CONTINUE
                   1004:  1947          CONTINUE
                   1005:             ELSE
                   1006:                CALL DLASET( 'U', NR-1, NR-1, ZERO, ZERO, A(1,2), LDA )
                   1007:             END IF
                   1008: *
                   1009: *           .. and one-sided Jacobi rotations are started on a lower
                   1010: *           triangular matrix (plus perturbation which is ignored in
                   1011: *           the part which destroys triangular form (confusing?!))
                   1012: *
                   1013:             CALL DGESVJ( 'L', 'NoU', 'NoV', NR, NR, A, LDA, SVA,
1.6     ! bertrand 1014:      $                      N, V, LDV, WORK, LWORK, INFO )
1.1       bertrand 1015: *
                   1016:             SCALEM  = WORK(1)
                   1017:             NUMRANK = IDNINT(WORK(2))
                   1018: *
                   1019: *
                   1020:       ELSE IF ( RSVEC .AND. ( .NOT. LSVEC ) ) THEN
                   1021: *
                   1022: *        -> Singular Values and Right Singular Vectors <-
                   1023: *
                   1024:          IF ( ALMORT ) THEN
                   1025: *
                   1026: *           .. in this case NR equals N
                   1027:             DO 1998 p = 1, NR
                   1028:                CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
                   1029:  1998       CONTINUE
                   1030:             CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
                   1031: *
                   1032:             CALL DGESVJ( 'L','U','N', N, NR, V,LDV, SVA, NR, A,LDA,
1.6     ! bertrand 1033:      $                  WORK, LWORK, INFO )
1.1       bertrand 1034:             SCALEM  = WORK(1)
                   1035:             NUMRANK = IDNINT(WORK(2))
                   1036: 
                   1037:          ELSE
                   1038: *
                   1039: *        .. two more QR factorizations ( one QRF is not enough, two require
                   1040: *        accumulated product of Jacobi rotations, three are perfect )
                   1041: *
                   1042:             CALL DLASET( 'Lower', NR-1, NR-1, ZERO, ZERO, A(2,1), LDA )
                   1043:             CALL DGELQF( NR, N, A, LDA, WORK, WORK(N+1), LWORK-N, IERR)
                   1044:             CALL DLACPY( 'Lower', NR, NR, A, LDA, V, LDV )
                   1045:             CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
                   1046:             CALL DGEQRF( NR, NR, V, LDV, WORK(N+1), WORK(2*N+1),
1.6     ! bertrand 1047:      $                   LWORK-2*N, IERR )
1.1       bertrand 1048:             DO 8998 p = 1, NR
                   1049:                CALL DCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )
                   1050:  8998       CONTINUE
                   1051:             CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
                   1052: *
                   1053:             CALL DGESVJ( 'Lower', 'U','N', NR, NR, V,LDV, SVA, NR, U,
1.6     ! bertrand 1054:      $                  LDU, WORK(N+1), LWORK, INFO )
1.1       bertrand 1055:             SCALEM  = WORK(N+1)
                   1056:             NUMRANK = IDNINT(WORK(N+2))
                   1057:             IF ( NR .LT. N ) THEN
                   1058:                CALL DLASET( 'A',N-NR, NR, ZERO,ZERO, V(NR+1,1),   LDV )
                   1059:                CALL DLASET( 'A',NR, N-NR, ZERO,ZERO, V(1,NR+1),   LDV )
                   1060:                CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE, V(NR+1,NR+1), LDV )
                   1061:             END IF
                   1062: *
                   1063:          CALL DORMLQ( 'Left', 'Transpose', N, N, NR, A, LDA, WORK,
1.6     ! bertrand 1064:      $               V, LDV, WORK(N+1), LWORK-N, IERR )
1.1       bertrand 1065: *
                   1066:          END IF
                   1067: *
                   1068:          DO 8991 p = 1, N
                   1069:             CALL DCOPY( N, V(p,1), LDV, A(IWORK(p),1), LDA )
                   1070:  8991    CONTINUE
                   1071:          CALL DLACPY( 'All', N, N, A, LDA, V, LDV )
                   1072: *
                   1073:          IF ( TRANSP ) THEN
                   1074:             CALL DLACPY( 'All', N, N, V, LDV, U, LDU )
                   1075:          END IF
                   1076: *
                   1077:       ELSE IF ( LSVEC .AND. ( .NOT. RSVEC ) ) THEN
                   1078: *
                   1079: *        .. Singular Values and Left Singular Vectors                 ..
                   1080: *
                   1081: *        .. second preconditioning step to avoid need to accumulate
                   1082: *        Jacobi rotations in the Jacobi iterations.
                   1083:          DO 1965 p = 1, NR
                   1084:             CALL DCOPY( N-p+1, A(p,p), LDA, U(p,p), 1 )
                   1085:  1965    CONTINUE
                   1086:          CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )
                   1087: *
                   1088:          CALL DGEQRF( N, NR, U, LDU, WORK(N+1), WORK(2*N+1),
1.6     ! bertrand 1089:      $              LWORK-2*N, IERR )
1.1       bertrand 1090: *
                   1091:          DO 1967 p = 1, NR - 1
                   1092:             CALL DCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )
                   1093:  1967    CONTINUE
                   1094:          CALL DLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )
                   1095: *
                   1096:          CALL DGESVJ( 'Lower', 'U', 'N', NR,NR, U, LDU, SVA, NR, A,
1.6     ! bertrand 1097:      $        LDA, WORK(N+1), LWORK-N, INFO )
1.1       bertrand 1098:          SCALEM  = WORK(N+1)
                   1099:          NUMRANK = IDNINT(WORK(N+2))
                   1100: *
                   1101:          IF ( NR .LT. M ) THEN
                   1102:             CALL DLASET( 'A',  M-NR, NR,ZERO, ZERO, U(NR+1,1), LDU )
                   1103:             IF ( NR .LT. N1 ) THEN
                   1104:                CALL DLASET( 'A',NR, N1-NR, ZERO, ZERO, U(1,NR+1), LDU )
                   1105:                CALL DLASET( 'A',M-NR,N1-NR,ZERO,ONE,U(NR+1,NR+1), LDU )
                   1106:             END IF
                   1107:          END IF
                   1108: *
                   1109:          CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,
1.6     ! bertrand 1110:      $               LDU, WORK(N+1), LWORK-N, IERR )
1.1       bertrand 1111: *
                   1112:          IF ( ROWPIV )
1.6     ! bertrand 1113:      $       CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
1.1       bertrand 1114: *
                   1115:          DO 1974 p = 1, N1
                   1116:             XSC = ONE / DNRM2( M, U(1,p), 1 )
                   1117:             CALL DSCAL( M, XSC, U(1,p), 1 )
                   1118:  1974    CONTINUE
                   1119: *
                   1120:          IF ( TRANSP ) THEN
                   1121:             CALL DLACPY( 'All', N, N, U, LDU, V, LDV )
                   1122:          END IF
                   1123: *
                   1124:       ELSE
                   1125: *
                   1126: *        .. Full SVD ..
                   1127: *
                   1128:          IF ( .NOT. JRACC ) THEN
                   1129: *
                   1130:          IF ( .NOT. ALMORT ) THEN
                   1131: *
                   1132: *           Second Preconditioning Step (QRF [with pivoting])
                   1133: *           Note that the composition of TRANSPOSE, QRF and TRANSPOSE is
                   1134: *           equivalent to an LQF CALL. Since in many libraries the QRF
                   1135: *           seems to be better optimized than the LQF, we do explicit
                   1136: *           transpose and use the QRF. This is subject to changes in an
                   1137: *           optimized implementation of DGEJSV.
                   1138: *
                   1139:             DO 1968 p = 1, NR
                   1140:                CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
                   1141:  1968       CONTINUE
                   1142: *
                   1143: *           .. the following two loops perturb small entries to avoid
                   1144: *           denormals in the second QR factorization, where they are
                   1145: *           as good as zeros. This is done to avoid painfully slow
                   1146: *           computation with denormals. The relative size of the perturbation
                   1147: *           is a parameter that can be changed by the implementer.
                   1148: *           This perturbation device will be obsolete on machines with
                   1149: *           properly implemented arithmetic.
                   1150: *           To switch it off, set L2PERT=.FALSE. To remove it from  the
                   1151: *           code, remove the action under L2PERT=.TRUE., leave the ELSE part.
                   1152: *           The following two loops should be blocked and fused with the
                   1153: *           transposed copy above.
                   1154: *
                   1155:             IF ( L2PERT ) THEN
                   1156:                XSC = DSQRT(SMALL)
                   1157:                DO 2969 q = 1, NR
                   1158:                   TEMP1 = XSC*DABS( V(q,q) )
                   1159:                   DO 2968 p = 1, N
                   1160:                      IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 )
1.6     ! bertrand 1161:      $                   .OR. ( p .LT. q ) )
        !          1162:      $                   V(p,q) = DSIGN( TEMP1, V(p,q) )
        !          1163:                      IF ( p .LT. q ) V(p,q) = - V(p,q)
1.1       bertrand 1164:  2968             CONTINUE
                   1165:  2969          CONTINUE
                   1166:             ELSE
                   1167:                CALL DLASET( 'U', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
                   1168:             END IF
                   1169: *
                   1170: *           Estimate the row scaled condition number of R1
                   1171: *           (If R1 is rectangular, N > NR, then the condition number
                   1172: *           of the leading NR x NR submatrix is estimated.)
                   1173: *
                   1174:             CALL DLACPY( 'L', NR, NR, V, LDV, WORK(2*N+1), NR )
                   1175:             DO 3950 p = 1, NR
                   1176:                TEMP1 = DNRM2(NR-p+1,WORK(2*N+(p-1)*NR+p),1)
                   1177:                CALL DSCAL(NR-p+1,ONE/TEMP1,WORK(2*N+(p-1)*NR+p),1)
                   1178:  3950       CONTINUE
                   1179:             CALL DPOCON('Lower',NR,WORK(2*N+1),NR,ONE,TEMP1,
1.6     ! bertrand 1180:      $                   WORK(2*N+NR*NR+1),IWORK(M+2*N+1),IERR)
1.1       bertrand 1181:             CONDR1 = ONE / DSQRT(TEMP1)
                   1182: *           .. here need a second oppinion on the condition number
                   1183: *           .. then assume worst case scenario
                   1184: *           R1 is OK for inverse <=> CONDR1 .LT. DBLE(N)
                   1185: *           more conservative    <=> CONDR1 .LT. DSQRT(DBLE(N))
                   1186: *
                   1187:             COND_OK = DSQRT(DBLE(NR))
                   1188: *[TP]       COND_OK is a tuning parameter.
                   1189: 
                   1190:             IF ( CONDR1 .LT. COND_OK ) THEN
                   1191: *              .. the second QRF without pivoting. Note: in an optimized
                   1192: *              implementation, this QRF should be implemented as the QRF
                   1193: *              of a lower triangular matrix.
                   1194: *              R1^t = Q2 * R2
                   1195:                CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
1.6     ! bertrand 1196:      $              LWORK-2*N, IERR )
1.1       bertrand 1197: *
                   1198:                IF ( L2PERT ) THEN
                   1199:                   XSC = DSQRT(SMALL)/EPSLN
                   1200:                   DO 3959 p = 2, NR
                   1201:                      DO 3958 q = 1, p - 1
                   1202:                         TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q)))
                   1203:                         IF ( DABS(V(q,p)) .LE. TEMP1 )
1.6     ! bertrand 1204:      $                     V(q,p) = DSIGN( TEMP1, V(q,p) )
1.1       bertrand 1205:  3958                CONTINUE
                   1206:  3959             CONTINUE
                   1207:                END IF
                   1208: *
                   1209:                IF ( NR .NE. N )
1.6     ! bertrand 1210:      $         CALL DLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N )
1.1       bertrand 1211: *              .. save ...
                   1212: *
                   1213: *           .. this transposed copy should be better than naive
                   1214:                DO 1969 p = 1, NR - 1
                   1215:                   CALL DCOPY( NR-p, V(p,p+1), LDV, V(p+1,p), 1 )
                   1216:  1969          CONTINUE
                   1217: *
                   1218:                CONDR2 = CONDR1
                   1219: *
                   1220:             ELSE
                   1221: *
                   1222: *              .. ill-conditioned case: second QRF with pivoting
                   1223: *              Note that windowed pivoting would be equaly good
                   1224: *              numerically, and more run-time efficient. So, in
                   1225: *              an optimal implementation, the next call to DGEQP3
                   1226: *              should be replaced with eg. CALL SGEQPX (ACM TOMS #782)
                   1227: *              with properly (carefully) chosen parameters.
                   1228: *
                   1229: *              R1^t * P2 = Q2 * R2
                   1230:                DO 3003 p = 1, NR
                   1231:                   IWORK(N+p) = 0
                   1232:  3003          CONTINUE
                   1233:                CALL DGEQP3( N, NR, V, LDV, IWORK(N+1), WORK(N+1),
1.6     ! bertrand 1234:      $                  WORK(2*N+1), LWORK-2*N, IERR )
1.1       bertrand 1235: **               CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
1.6     ! bertrand 1236: **     $              LWORK-2*N, IERR )
1.1       bertrand 1237:                IF ( L2PERT ) THEN
                   1238:                   XSC = DSQRT(SMALL)
                   1239:                   DO 3969 p = 2, NR
                   1240:                      DO 3968 q = 1, p - 1
                   1241:                         TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q)))
                   1242:                         IF ( DABS(V(q,p)) .LE. TEMP1 )
1.6     ! bertrand 1243:      $                     V(q,p) = DSIGN( TEMP1, V(q,p) )
1.1       bertrand 1244:  3968                CONTINUE
                   1245:  3969             CONTINUE
                   1246:                END IF
                   1247: *
                   1248:                CALL DLACPY( 'A', N, NR, V, LDV, WORK(2*N+1), N )
                   1249: *
                   1250:                IF ( L2PERT ) THEN
                   1251:                   XSC = DSQRT(SMALL)
                   1252:                   DO 8970 p = 2, NR
                   1253:                      DO 8971 q = 1, p - 1
                   1254:                         TEMP1 = XSC * DMIN1(DABS(V(p,p)),DABS(V(q,q)))
                   1255:                         V(p,q) = - DSIGN( TEMP1, V(q,p) )
                   1256:  8971                CONTINUE
                   1257:  8970             CONTINUE
                   1258:                ELSE
                   1259:                   CALL DLASET( 'L',NR-1,NR-1,ZERO,ZERO,V(2,1),LDV )
                   1260:                END IF
                   1261: *              Now, compute R2 = L3 * Q3, the LQ factorization.
                   1262:                CALL DGELQF( NR, NR, V, LDV, WORK(2*N+N*NR+1),
1.6     ! bertrand 1263:      $               WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, IERR )
1.1       bertrand 1264: *              .. and estimate the condition number
                   1265:                CALL DLACPY( 'L',NR,NR,V,LDV,WORK(2*N+N*NR+NR+1),NR )
                   1266:                DO 4950 p = 1, NR
                   1267:                   TEMP1 = DNRM2( p, WORK(2*N+N*NR+NR+p), NR )
                   1268:                   CALL DSCAL( p, ONE/TEMP1, WORK(2*N+N*NR+NR+p), NR )
                   1269:  4950          CONTINUE
                   1270:                CALL DPOCON( 'L',NR,WORK(2*N+N*NR+NR+1),NR,ONE,TEMP1,
1.6     ! bertrand 1271:      $              WORK(2*N+N*NR+NR+NR*NR+1),IWORK(M+2*N+1),IERR )
1.1       bertrand 1272:                CONDR2 = ONE / DSQRT(TEMP1)
                   1273: *
                   1274:                IF ( CONDR2 .GE. COND_OK ) THEN
                   1275: *                 .. save the Householder vectors used for Q3
                   1276: *                 (this overwrittes the copy of R2, as it will not be
                   1277: *                 needed in this branch, but it does not overwritte the
                   1278: *                 Huseholder vectors of Q2.).
                   1279:                   CALL DLACPY( 'U', NR, NR, V, LDV, WORK(2*N+1), N )
                   1280: *                 .. and the rest of the information on Q3 is in
                   1281: *                 WORK(2*N+N*NR+1:2*N+N*NR+N)
                   1282:                END IF
                   1283: *
                   1284:             END IF
                   1285: *
                   1286:             IF ( L2PERT ) THEN
                   1287:                XSC = DSQRT(SMALL)
                   1288:                DO 4968 q = 2, NR
                   1289:                   TEMP1 = XSC * V(q,q)
                   1290:                   DO 4969 p = 1, q - 1
                   1291: *                    V(p,q) = - DSIGN( TEMP1, V(q,p) )
                   1292:                      V(p,q) = - DSIGN( TEMP1, V(p,q) )
                   1293:  4969             CONTINUE
                   1294:  4968          CONTINUE
                   1295:             ELSE
                   1296:                CALL DLASET( 'U', NR-1,NR-1, ZERO,ZERO, V(1,2), LDV )
                   1297:             END IF
                   1298: *
                   1299: *        Second preconditioning finished; continue with Jacobi SVD
                   1300: *        The input matrix is lower trinagular.
                   1301: *
                   1302: *        Recover the right singular vectors as solution of a well
                   1303: *        conditioned triangular matrix equation.
                   1304: *
                   1305:             IF ( CONDR1 .LT. COND_OK ) THEN
                   1306: *
                   1307:                CALL DGESVJ( 'L','U','N',NR,NR,V,LDV,SVA,NR,U,
1.6     ! bertrand 1308:      $              LDU,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,INFO )
1.1       bertrand 1309:                SCALEM  = WORK(2*N+N*NR+NR+1)
                   1310:                NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))
                   1311:                DO 3970 p = 1, NR
                   1312:                   CALL DCOPY( NR, V(1,p), 1, U(1,p), 1 )
                   1313:                   CALL DSCAL( NR, SVA(p),    V(1,p), 1 )
                   1314:  3970          CONTINUE
                   1315: 
                   1316: *        .. pick the right matrix equation and solve it
                   1317: *
1.6     ! bertrand 1318:                IF ( NR .EQ. N ) THEN
1.1       bertrand 1319: * :))             .. best case, R1 is inverted. The solution of this matrix
                   1320: *                 equation is Q2*V2 = the product of the Jacobi rotations
                   1321: *                 used in DGESVJ, premultiplied with the orthogonal matrix
                   1322: *                 from the second QR factorization.
                   1323:                   CALL DTRSM( 'L','U','N','N', NR,NR,ONE, A,LDA, V,LDV )
                   1324:                ELSE
                   1325: *                 .. R1 is well conditioned, but non-square. Transpose(R2)
                   1326: *                 is inverted to get the product of the Jacobi rotations
                   1327: *                 used in DGESVJ. The Q-factor from the second QR
                   1328: *                 factorization is then built in explicitly.
                   1329:                   CALL DTRSM('L','U','T','N',NR,NR,ONE,WORK(2*N+1),
1.6     ! bertrand 1330:      $                 N,V,LDV)
1.1       bertrand 1331:                   IF ( NR .LT. N ) THEN
                   1332:                     CALL DLASET('A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV)
                   1333:                     CALL DLASET('A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV)
                   1334:                     CALL DLASET('A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV)
                   1335:                   END IF
                   1336:                   CALL DORMQR('L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
1.6     ! bertrand 1337:      $                 V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR)
1.1       bertrand 1338:                END IF
                   1339: *
                   1340:             ELSE IF ( CONDR2 .LT. COND_OK ) THEN
                   1341: *
                   1342: * :)           .. the input matrix A is very likely a relative of
                   1343: *              the Kahan matrix :)
                   1344: *              The matrix R2 is inverted. The solution of the matrix equation
                   1345: *              is Q3^T*V3 = the product of the Jacobi rotations (appplied to
                   1346: *              the lower triangular L3 from the LQ factorization of
                   1347: *              R2=L3*Q3), pre-multiplied with the transposed Q3.
                   1348:                CALL DGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR, U,
1.6     ! bertrand 1349:      $              LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )
1.1       bertrand 1350:                SCALEM  = WORK(2*N+N*NR+NR+1)
                   1351:                NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))
                   1352:                DO 3870 p = 1, NR
                   1353:                   CALL DCOPY( NR, V(1,p), 1, U(1,p), 1 )
                   1354:                   CALL DSCAL( NR, SVA(p),    U(1,p), 1 )
                   1355:  3870          CONTINUE
                   1356:                CALL DTRSM('L','U','N','N',NR,NR,ONE,WORK(2*N+1),N,U,LDU)
                   1357: *              .. apply the permutation from the second QR factorization
                   1358:                DO 873 q = 1, NR
                   1359:                   DO 872 p = 1, NR
                   1360:                      WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
                   1361:  872              CONTINUE
                   1362:                   DO 874 p = 1, NR
                   1363:                      U(p,q) = WORK(2*N+N*NR+NR+p)
                   1364:  874              CONTINUE
                   1365:  873           CONTINUE
                   1366:                IF ( NR .LT. N ) THEN
                   1367:                   CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )
                   1368:                   CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )
                   1369:                   CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )
                   1370:                END IF
                   1371:                CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
1.6     ! bertrand 1372:      $              V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
1.1       bertrand 1373:             ELSE
                   1374: *              Last line of defense.
                   1375: * #:(          This is a rather pathological case: no scaled condition
                   1376: *              improvement after two pivoted QR factorizations. Other
                   1377: *              possibility is that the rank revealing QR factorization
                   1378: *              or the condition estimator has failed, or the COND_OK
                   1379: *              is set very close to ONE (which is unnecessary). Normally,
                   1380: *              this branch should never be executed, but in rare cases of
                   1381: *              failure of the RRQR or condition estimator, the last line of
                   1382: *              defense ensures that DGEJSV completes the task.
                   1383: *              Compute the full SVD of L3 using DGESVJ with explicit
                   1384: *              accumulation of Jacobi rotations.
                   1385:                CALL DGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR, U,
1.6     ! bertrand 1386:      $              LDU, WORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, INFO )
1.1       bertrand 1387:                SCALEM  = WORK(2*N+N*NR+NR+1)
                   1388:                NUMRANK = IDNINT(WORK(2*N+N*NR+NR+2))
                   1389:                IF ( NR .LT. N ) THEN
                   1390:                   CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )
                   1391:                   CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )
                   1392:                   CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )
                   1393:                END IF
                   1394:                CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
1.6     ! bertrand 1395:      $              V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
1.1       bertrand 1396: *
                   1397:                CALL DORMLQ( 'L', 'T', NR, NR, NR, WORK(2*N+1), N,
1.6     ! bertrand 1398:      $              WORK(2*N+N*NR+1), U, LDU, WORK(2*N+N*NR+NR+1),
        !          1399:      $              LWORK-2*N-N*NR-NR, IERR )
1.1       bertrand 1400:                DO 773 q = 1, NR
                   1401:                   DO 772 p = 1, NR
                   1402:                      WORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
                   1403:  772              CONTINUE
                   1404:                   DO 774 p = 1, NR
                   1405:                      U(p,q) = WORK(2*N+N*NR+NR+p)
                   1406:  774              CONTINUE
                   1407:  773           CONTINUE
                   1408: *
                   1409:             END IF
                   1410: *
                   1411: *           Permute the rows of V using the (column) permutation from the
                   1412: *           first QRF. Also, scale the columns to make them unit in
                   1413: *           Euclidean norm. This applies to all cases.
                   1414: *
                   1415:             TEMP1 = DSQRT(DBLE(N)) * EPSLN
                   1416:             DO 1972 q = 1, N
                   1417:                DO 972 p = 1, N
                   1418:                   WORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
                   1419:   972          CONTINUE
                   1420:                DO 973 p = 1, N
                   1421:                   V(p,q) = WORK(2*N+N*NR+NR+p)
                   1422:   973          CONTINUE
                   1423:                XSC = ONE / DNRM2( N, V(1,q), 1 )
                   1424:                IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
1.6     ! bertrand 1425:      $           CALL DSCAL( N, XSC, V(1,q), 1 )
1.1       bertrand 1426:  1972       CONTINUE
                   1427: *           At this moment, V contains the right singular vectors of A.
                   1428: *           Next, assemble the left singular vector matrix U (M x N).
                   1429:             IF ( NR .LT. M ) THEN
                   1430:                CALL DLASET( 'A', M-NR, NR, ZERO, ZERO, U(NR+1,1), LDU )
                   1431:                IF ( NR .LT. N1 ) THEN
                   1432:                   CALL DLASET('A',NR,N1-NR,ZERO,ZERO,U(1,NR+1),LDU)
                   1433:                   CALL DLASET('A',M-NR,N1-NR,ZERO,ONE,U(NR+1,NR+1),LDU)
                   1434:                END IF
                   1435:             END IF
                   1436: *
                   1437: *           The Q matrix from the first QRF is built into the left singular
                   1438: *           matrix U. This applies to all cases.
                   1439: *
                   1440:             CALL DORMQR( 'Left', 'No_Tr', M, N1, N, A, LDA, WORK, U,
1.6     ! bertrand 1441:      $           LDU, WORK(N+1), LWORK-N, IERR )
1.1       bertrand 1442: 
                   1443: *           The columns of U are normalized. The cost is O(M*N) flops.
                   1444:             TEMP1 = DSQRT(DBLE(M)) * EPSLN
                   1445:             DO 1973 p = 1, NR
                   1446:                XSC = ONE / DNRM2( M, U(1,p), 1 )
                   1447:                IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
1.6     ! bertrand 1448:      $          CALL DSCAL( M, XSC, U(1,p), 1 )
1.1       bertrand 1449:  1973       CONTINUE
                   1450: *
                   1451: *           If the initial QRF is computed with row pivoting, the left
                   1452: *           singular vectors must be adjusted.
                   1453: *
                   1454:             IF ( ROWPIV )
1.6     ! bertrand 1455:      $          CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
1.1       bertrand 1456: *
                   1457:          ELSE
                   1458: *
                   1459: *        .. the initial matrix A has almost orthogonal columns and
                   1460: *        the second QRF is not needed
                   1461: *
                   1462:             CALL DLACPY( 'Upper', N, N, A, LDA, WORK(N+1), N )
                   1463:             IF ( L2PERT ) THEN
                   1464:                XSC = DSQRT(SMALL)
                   1465:                DO 5970 p = 2, N
                   1466:                   TEMP1 = XSC * WORK( N + (p-1)*N + p )
                   1467:                   DO 5971 q = 1, p - 1
                   1468:                      WORK(N+(q-1)*N+p)=-DSIGN(TEMP1,WORK(N+(p-1)*N+q))
                   1469:  5971             CONTINUE
                   1470:  5970          CONTINUE
                   1471:             ELSE
                   1472:                CALL DLASET( 'Lower',N-1,N-1,ZERO,ZERO,WORK(N+2),N )
                   1473:             END IF
                   1474: *
                   1475:             CALL DGESVJ( 'Upper', 'U', 'N', N, N, WORK(N+1), N, SVA,
1.6     ! bertrand 1476:      $           N, U, LDU, WORK(N+N*N+1), LWORK-N-N*N, INFO )
1.1       bertrand 1477: *
                   1478:             SCALEM  = WORK(N+N*N+1)
                   1479:             NUMRANK = IDNINT(WORK(N+N*N+2))
                   1480:             DO 6970 p = 1, N
                   1481:                CALL DCOPY( N, WORK(N+(p-1)*N+1), 1, U(1,p), 1 )
                   1482:                CALL DSCAL( N, SVA(p), WORK(N+(p-1)*N+1), 1 )
                   1483:  6970       CONTINUE
                   1484: *
                   1485:             CALL DTRSM( 'Left', 'Upper', 'NoTrans', 'No UD', N, N,
1.6     ! bertrand 1486:      $           ONE, A, LDA, WORK(N+1), N )
1.1       bertrand 1487:             DO 6972 p = 1, N
                   1488:                CALL DCOPY( N, WORK(N+p), N, V(IWORK(p),1), LDV )
                   1489:  6972       CONTINUE
                   1490:             TEMP1 = DSQRT(DBLE(N))*EPSLN
                   1491:             DO 6971 p = 1, N
                   1492:                XSC = ONE / DNRM2( N, V(1,p), 1 )
                   1493:                IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
1.6     ! bertrand 1494:      $            CALL DSCAL( N, XSC, V(1,p), 1 )
1.1       bertrand 1495:  6971       CONTINUE
                   1496: *
                   1497: *           Assemble the left singular vector matrix U (M x N).
                   1498: *
                   1499:             IF ( N .LT. M ) THEN
1.4       bertrand 1500:                CALL DLASET( 'A',  M-N, N, ZERO, ZERO, U(N+1,1), LDU )
1.1       bertrand 1501:                IF ( N .LT. N1 ) THEN
                   1502:                   CALL DLASET( 'A',N,  N1-N, ZERO, ZERO,  U(1,N+1),LDU )
1.4       bertrand 1503:                   CALL DLASET( 'A',M-N,N1-N, ZERO, ONE,U(N+1,N+1),LDU )
1.1       bertrand 1504:                END IF
                   1505:             END IF
                   1506:             CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,
1.6     ! bertrand 1507:      $           LDU, WORK(N+1), LWORK-N, IERR )
1.1       bertrand 1508:             TEMP1 = DSQRT(DBLE(M))*EPSLN
                   1509:             DO 6973 p = 1, N1
                   1510:                XSC = ONE / DNRM2( M, U(1,p), 1 )
                   1511:                IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
1.6     ! bertrand 1512:      $            CALL DSCAL( M, XSC, U(1,p), 1 )
1.1       bertrand 1513:  6973       CONTINUE
                   1514: *
                   1515:             IF ( ROWPIV )
1.6     ! bertrand 1516:      $         CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
1.1       bertrand 1517: *
                   1518:          END IF
                   1519: *
                   1520: *        end of the  >> almost orthogonal case <<  in the full SVD
                   1521: *
                   1522:          ELSE
                   1523: *
                   1524: *        This branch deploys a preconditioned Jacobi SVD with explicitly
                   1525: *        accumulated rotations. It is included as optional, mainly for
                   1526: *        experimental purposes. It does perfom well, and can also be used.
                   1527: *        In this implementation, this branch will be automatically activated
                   1528: *        if the  condition number sigma_max(A) / sigma_min(A) is predicted
                   1529: *        to be greater than the overflow threshold. This is because the
                   1530: *        a posteriori computation of the singular vectors assumes robust
                   1531: *        implementation of BLAS and some LAPACK procedures, capable of working
                   1532: *        in presence of extreme values. Since that is not always the case, ...
                   1533: *
                   1534:          DO 7968 p = 1, NR
                   1535:             CALL DCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
                   1536:  7968    CONTINUE
                   1537: *
                   1538:          IF ( L2PERT ) THEN
                   1539:             XSC = DSQRT(SMALL/EPSLN)
                   1540:             DO 5969 q = 1, NR
                   1541:                TEMP1 = XSC*DABS( V(q,q) )
                   1542:                DO 5968 p = 1, N
                   1543:                   IF ( ( p .GT. q ) .AND. ( DABS(V(p,q)) .LE. TEMP1 )
1.6     ! bertrand 1544:      $                .OR. ( p .LT. q ) )
        !          1545:      $                V(p,q) = DSIGN( TEMP1, V(p,q) )
        !          1546:                   IF ( p .LT. q ) V(p,q) = - V(p,q)
1.1       bertrand 1547:  5968          CONTINUE
                   1548:  5969       CONTINUE
                   1549:          ELSE
                   1550:             CALL DLASET( 'U', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
                   1551:          END IF
                   1552: 
                   1553:          CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1),
1.6     ! bertrand 1554:      $        LWORK-2*N, IERR )
1.1       bertrand 1555:          CALL DLACPY( 'L', N, NR, V, LDV, WORK(2*N+1), N )
                   1556: *
                   1557:          DO 7969 p = 1, NR
                   1558:             CALL DCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 )
                   1559:  7969    CONTINUE
                   1560: 
                   1561:          IF ( L2PERT ) THEN
                   1562:             XSC = DSQRT(SMALL/EPSLN)
                   1563:             DO 9970 q = 2, NR
                   1564:                DO 9971 p = 1, q - 1
                   1565:                   TEMP1 = XSC * DMIN1(DABS(U(p,p)),DABS(U(q,q)))
                   1566:                   U(p,q) = - DSIGN( TEMP1, U(q,p) )
                   1567:  9971          CONTINUE
                   1568:  9970       CONTINUE
                   1569:          ELSE
                   1570:             CALL DLASET('U', NR-1, NR-1, ZERO, ZERO, U(1,2), LDU )
                   1571:          END IF
                   1572: 
                   1573:          CALL DGESVJ( 'G', 'U', 'V', NR, NR, U, LDU, SVA,
1.6     ! bertrand 1574:      $        N, V, LDV, WORK(2*N+N*NR+1), LWORK-2*N-N*NR, INFO )
1.1       bertrand 1575:          SCALEM  = WORK(2*N+N*NR+1)
                   1576:          NUMRANK = IDNINT(WORK(2*N+N*NR+2))
                   1577: 
                   1578:          IF ( NR .LT. N ) THEN
                   1579:             CALL DLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )
                   1580:             CALL DLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )
                   1581:             CALL DLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )
                   1582:          END IF
                   1583: 
                   1584:          CALL DORMQR( 'L','N',N,N,NR,WORK(2*N+1),N,WORK(N+1),
1.6     ! bertrand 1585:      $        V,LDV,WORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
1.1       bertrand 1586: *
                   1587: *           Permute the rows of V using the (column) permutation from the
                   1588: *           first QRF. Also, scale the columns to make them unit in
                   1589: *           Euclidean norm. This applies to all cases.
                   1590: *
                   1591:             TEMP1 = DSQRT(DBLE(N)) * EPSLN
                   1592:             DO 7972 q = 1, N
                   1593:                DO 8972 p = 1, N
                   1594:                   WORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
                   1595:  8972          CONTINUE
                   1596:                DO 8973 p = 1, N
                   1597:                   V(p,q) = WORK(2*N+N*NR+NR+p)
                   1598:  8973          CONTINUE
                   1599:                XSC = ONE / DNRM2( N, V(1,q), 1 )
                   1600:                IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
1.6     ! bertrand 1601:      $           CALL DSCAL( N, XSC, V(1,q), 1 )
1.1       bertrand 1602:  7972       CONTINUE
                   1603: *
                   1604: *           At this moment, V contains the right singular vectors of A.
                   1605: *           Next, assemble the left singular vector matrix U (M x N).
                   1606: *
1.4       bertrand 1607:          IF ( NR .LT. M ) THEN
                   1608:             CALL DLASET( 'A',  M-NR, NR, ZERO, ZERO, U(NR+1,1), LDU )
                   1609:             IF ( NR .LT. N1 ) THEN
                   1610:                CALL DLASET( 'A',NR,  N1-NR, ZERO, ZERO,  U(1,NR+1),LDU )
                   1611:                CALL DLASET( 'A',M-NR,N1-NR, ZERO, ONE,U(NR+1,NR+1),LDU )
1.1       bertrand 1612:             END IF
                   1613:          END IF
                   1614: *
                   1615:          CALL DORMQR( 'Left', 'No Tr', M, N1, N, A, LDA, WORK, U,
1.6     ! bertrand 1616:      $        LDU, WORK(N+1), LWORK-N, IERR )
1.1       bertrand 1617: *
                   1618:             IF ( ROWPIV )
1.6     ! bertrand 1619:      $         CALL DLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
1.1       bertrand 1620: *
                   1621: *
                   1622:          END IF
                   1623:          IF ( TRANSP ) THEN
                   1624: *           .. swap U and V because the procedure worked on A^t
                   1625:             DO 6974 p = 1, N
                   1626:                CALL DSWAP( N, U(1,p), 1, V(1,p), 1 )
                   1627:  6974       CONTINUE
                   1628:          END IF
                   1629: *
                   1630:       END IF
                   1631: *     end of the full SVD
                   1632: *
                   1633: *     Undo scaling, if necessary (and possible)
                   1634: *
                   1635:       IF ( USCAL2 .LE. (BIG/SVA(1))*USCAL1 ) THEN
                   1636:          CALL DLASCL( 'G', 0, 0, USCAL1, USCAL2, NR, 1, SVA, N, IERR )
                   1637:          USCAL1 = ONE
                   1638:          USCAL2 = ONE
                   1639:       END IF
                   1640: *
                   1641:       IF ( NR .LT. N ) THEN
                   1642:          DO 3004 p = NR+1, N
                   1643:             SVA(p) = ZERO
                   1644:  3004    CONTINUE
                   1645:       END IF
                   1646: *
                   1647:       WORK(1) = USCAL2 * SCALEM
                   1648:       WORK(2) = USCAL1
                   1649:       IF ( ERREST ) WORK(3) = SCONDA
                   1650:       IF ( LSVEC .AND. RSVEC ) THEN
                   1651:          WORK(4) = CONDR1
                   1652:          WORK(5) = CONDR2
                   1653:       END IF
                   1654:       IF ( L2TRAN ) THEN
                   1655:          WORK(6) = ENTRA
                   1656:          WORK(7) = ENTRAT
                   1657:       END IF
                   1658: *
                   1659:       IWORK(1) = NR
                   1660:       IWORK(2) = NUMRANK
                   1661:       IWORK(3) = WARNING
                   1662: *
                   1663:       RETURN
                   1664: *     ..
                   1665: *     .. END OF DGEJSV
                   1666: *     ..
                   1667:       END
                   1668: *

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