dgejsv.c

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/* dgejsv.f -- translated by f2c (version 20061008).
   You must link the resulting object file with libf2c:
        on Microsoft Windows system, link with libf2c.lib;
        on Linux or Unix systems, link with .../path/to/libf2c.a -lm
        or, if you install libf2c.a in a standard place, with -lf2c -lm
        -- in that order, at the end of the command line, as in
                cc *.o -lf2c -lm
        Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,

                http://www.netlib.org/f2c/libf2c.zip
*/

#include "f2c.h"
#include "blaswrap.h"

/* Table of constant values */

static integer c__1 = 1;
static doublereal c_b34 = 0.;
static doublereal c_b35 = 1.;
static integer c__0 = 0;
static integer c_n1 = -1;

/* Subroutine */ int dgejsv_(char *joba, char *jobu, char *jobv, char *jobr, 
        char *jobt, char *jobp, integer *m, integer *n, doublereal *a, 
        integer *lda, doublereal *sva, doublereal *u, integer *ldu, 
        doublereal *v, integer *ldv, doublereal *work, integer *lwork, 
        integer *iwork, integer *info)
{
    /* System generated locals */
    integer a_dim1, a_offset, u_dim1, u_offset, v_dim1, v_offset, i__1, i__2, 
            i__3, i__4, i__5, i__6, i__7, i__8, i__9, i__10;
    doublereal d__1, d__2, d__3, d__4;

    /* Builtin functions */
    double sqrt(doublereal), log(doublereal), d_sign(doublereal *, doublereal 
            *);
    integer i_dnnt(doublereal *);

    /* Local variables */
    integer p, q, n1, nr;
    doublereal big, xsc, big1;
    logical defr;
    doublereal aapp, aaqq;
    logical kill;
    integer ierr;
    extern doublereal dnrm2_(integer *, doublereal *, integer *);
    doublereal temp1;
    logical jracc;
    extern /* Subroutine */ int dscal_(integer *, doublereal *, doublereal *, 
            integer *);
    extern logical lsame_(char *, char *);
    doublereal small, entra, sfmin;
    logical lsvec;
    extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, 
            doublereal *, integer *), dswap_(integer *, doublereal *, integer 
            *, doublereal *, integer *);
    doublereal epsln;
    logical rsvec;
    extern /* Subroutine */ int dtrsm_(char *, char *, char *, char *, 
            integer *, integer *, doublereal *, doublereal *, integer *, 
            doublereal *, integer *);
    logical l2aber;
    extern /* Subroutine */ int dgeqp3_(integer *, integer *, doublereal *, 
            integer *, integer *, doublereal *, doublereal *, integer *, 
            integer *);
    doublereal condr1, condr2, uscal1, uscal2;
    logical l2kill, l2rank, l2tran, l2pert;
    extern doublereal dlamch_(char *);
    extern /* Subroutine */ int dgelqf_(integer *, integer *, doublereal *, 
            integer *, doublereal *, doublereal *, integer *, integer *);
    extern integer idamax_(integer *, doublereal *, integer *);
    doublereal scalem;
    extern /* Subroutine */ int dlascl_(char *, integer *, integer *, 
            doublereal *, doublereal *, integer *, integer *, doublereal *, 
            integer *, integer *);
    doublereal sconda;
    logical goscal;
    doublereal aatmin;
    extern /* Subroutine */ int dgeqrf_(integer *, integer *, doublereal *, 
            integer *, doublereal *, doublereal *, integer *, integer *);
    doublereal aatmax;
    extern /* Subroutine */ int dlacpy_(char *, integer *, integer *, 
            doublereal *, integer *, doublereal *, integer *), 
            dlaset_(char *, integer *, integer *, doublereal *, doublereal *, 
            doublereal *, integer *), xerbla_(char *, integer *);
    logical noscal;
    extern /* Subroutine */ int dpocon_(char *, integer *, doublereal *, 
            integer *, doublereal *, doublereal *, doublereal *, integer *, 
            integer *), dgesvj_(char *, char *, char *, integer *, 
            integer *, doublereal *, integer *, doublereal *, integer *, 
            doublereal *, integer *, doublereal *, integer *, integer *), dlassq_(integer *, doublereal *, integer 
            *, doublereal *, doublereal *), dlaswp_(integer *, doublereal *, 
            integer *, integer *, integer *, integer *, integer *);
    doublereal entrat;
    logical almort;
    extern /* Subroutine */ int dorgqr_(integer *, integer *, integer *, 
            doublereal *, integer *, doublereal *, doublereal *, integer *, 
            integer *), dormlq_(char *, char *, integer *, integer *, integer 
            *, doublereal *, integer *, doublereal *, doublereal *, integer *, 
             doublereal *, integer *, integer *);
    doublereal maxprj;
    logical errest;
    extern /* Subroutine */ int dormqr_(char *, char *, integer *, integer *, 
            integer *, doublereal *, integer *, doublereal *, doublereal *, 
            integer *, doublereal *, integer *, integer *);
    logical transp, rowpiv;
    doublereal cond_ok__;
    integer warning, numrank;


/*  -- LAPACK routine (version 3.2)                                    -- */

/*  -- Contributed by Zlatko Drmac of the University of Zagreb and     -- */
/*  -- Kresimir Veselic of the Fernuniversitaet Hagen                  -- */
/*  -- November 2008                                                   -- */

/*  -- LAPACK is a software package provided by Univ. of Tennessee,    -- */
/*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */

/* This routine is also part of SIGMA (version 1.23, October 23. 2008.) */
/* SIGMA is a library of algorithms for highly accurate algorithms for */
/* computation of SVD, PSVD, QSVD, (H,K)-SVD, and for solution of the */
/* eigenvalue problems Hx = lambda M x, H M x = lambda x with H, M > 0. */

/*     -#- Scalar Arguments -#- */


/*     -#- Array Arguments -#- */

/*     .. */

/*  Purpose */
/*  ~~~~~~~ */
/*  DGEJSV computes the singular value decomposition (SVD) of a real M-by-N */
/*  matrix [A], where M >= N. The SVD of [A] is written as */

/*               [A] = [U] * [SIGMA] * [V]^t, */

/*  where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N */
/*  diagonal elements, [U] is an M-by-N (or M-by-M) orthonormal matrix, and */
/*  [V] is an N-by-N orthogonal matrix. The diagonal elements of [SIGMA] are */
/*  the singular values of [A]. The columns of [U] and [V] are the left and */
/*  the right singular vectors of [A], respectively. The matrices [U] and [V] */
/*  are computed and stored in the arrays U and V, respectively. The diagonal */
/*  of [SIGMA] is computed and stored in the array SVA. */

/*  Further details */
/*  ~~~~~~~~~~~~~~~ */
/*  DGEJSV implements a preconditioned Jacobi SVD algorithm. It uses SGEQP3, */
/*  SGEQRF, and SGELQF as preprocessors and preconditioners. Optionally, an */
/*  additional row pivoting can be used as a preprocessor, which in some */
/*  cases results in much higher accuracy. An example is matrix A with the */
/*  structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned */
/*  diagonal matrices and C is well-conditioned matrix. In that case, complete */
/*  pivoting in the first QR factorizations provides accuracy dependent on the */
/*  condition number of C, and independent of D1, D2. Such higher accuracy is */
/*  not completely understood theoretically, but it works well in practice. */
/*  Further, if A can be written as A = B*D, with well-conditioned B and some */
/*  diagonal D, then the high accuracy is guaranteed, both theoretically and */
/*  in software, independent of D. For more details see [1], [2]. */
/*     The computational range for the singular values can be the full range */
/*  ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS */
/*  & LAPACK routines called by DGEJSV are implemented to work in that range. */
/*  If that is not the case, then the restriction for safe computation with */
/*  the singular values in the range of normalized IEEE numbers is that the */
/*  spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not */
/*  overflow. This code (DGEJSV) is best used in this restricted range, */
/*  meaning that singular values of magnitude below ||A||_2 / SLAMCH('O') are */
/*  returned as zeros. See JOBR for details on this. */
/*     Further, this implementation is somewhat slower than the one described */
/*  in [1,2] due to replacement of some non-LAPACK components, and because */
/*  the choice of some tuning parameters in the iterative part (DGESVJ) is */
/*  left to the implementer on a particular machine. */
/*     The rank revealing QR factorization (in this code: SGEQP3) should be */
/*  implemented as in [3]. We have a new version of SGEQP3 under development */
/*  that is more robust than the current one in LAPACK, with a cleaner cut in */
/*  rank defficient cases. It will be available in the SIGMA library [4]. */
/*  If M is much larger than N, it is obvious that the inital QRF with */
/*  column pivoting can be preprocessed by the QRF without pivoting. That */
/*  well known trick is not used in DGEJSV because in some cases heavy row */
/*  weighting can be treated with complete pivoting. The overhead in cases */
/*  M much larger than N is then only due to pivoting, but the benefits in */
/*  terms of accuracy have prevailed. The implementer/user can incorporate */
/*  this extra QRF step easily. The implementer can also improve data movement */
/*  (matrix transpose, matrix copy, matrix transposed copy) - this */
/*  implementation of DGEJSV uses only the simplest, naive data movement. */

/*  Contributors */
/*  ~~~~~~~~~~~~ */
/*  Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany) */

/*  References */
/*  ~~~~~~~~~~ */
/* [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I. */
/*     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342. */
/*     LAPACK Working note 169. */
/* [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II. */
/*     SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362. */
/*     LAPACK Working note 170. */
/* [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR */
/*     factorization software - a case study. */
/*     ACM Trans. Math. Softw. Vol. 35, No 2 (2008), pp. 1-28. */
/*     LAPACK Working note 176. */
/* [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV, */
/*     QSVD, (H,K)-SVD computations. */
/*     Department of Mathematics, University of Zagreb, 2008. */

/*  Bugs, examples and comments */
/*  ~~~~~~~~~~~~~~~~~~~~~~~~~~~ */
/*  Please report all bugs and send interesting examples and/or comments to */
/*  drmac@math.hr. Thank you. */

/*  Arguments */
/*  ~~~~~~~~~ */
/* ............................................................................ */
/* . JOBA   (input) CHARACTER*1 */
/* .        Specifies the level of accuracy: */
/* .      = 'C': This option works well (high relative accuracy) if A = B * D, */
/* .             with well-conditioned B and arbitrary diagonal matrix D. */
/* .             The accuracy cannot be spoiled by COLUMN scaling. The */
/* .             accuracy of the computed output depends on the condition of */
/* .             B, and the procedure aims at the best theoretical accuracy. */
/* .             The relative error max_{i=1:N}|d sigma_i| / sigma_i is */
/* .             bounded by f(M,N)*epsilon* cond(B), independent of D. */
/* .             The input matrix is preprocessed with the QRF with column */
/* .             pivoting. This initial preprocessing and preconditioning by */
/* .             a rank revealing QR factorization is common for all values of */
/* .             JOBA. Additional actions are specified as follows: */
/* .      = 'E': Computation as with 'C' with an additional estimate of the */
/* .             condition number of B. It provides a realistic error bound. */
/* .      = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings */
/* .             D1, D2, and well-conditioned matrix C, this option gives */
/* .             higher accuracy than the 'C' option. If the structure of the */
/* .             input matrix is not known, and relative accuracy is */
/* .             desirable, then this option is advisable. The input matrix A */
/* .             is preprocessed with QR factorization with FULL (row and */
/* .             column) pivoting. */
/* .      = 'G'  Computation as with 'F' with an additional estimate of the */
/* .             condition number of B, where A=D*B. If A has heavily weighted */
/* .             rows, then using this condition number gives too pessimistic */
/* .             error bound. */
/* .      = 'A': Small singular values are the noise and the matrix is treated */
/* .             as numerically rank defficient. The error in the computed */
/* .             singular values is bounded by f(m,n)*epsilon*||A||. */
/* .             The computed SVD A = U * S * V^t restores A up to */
/* .             f(m,n)*epsilon*||A||. */
/* .             This gives the procedure the licence to discard (set to zero) */
/* .             all singular values below N*epsilon*||A||. */
/* .      = 'R': Similar as in 'A'. Rank revealing property of the initial */
/* .             QR factorization is used do reveal (using triangular factor) */
/* .             a gap sigma_{r+1} < epsilon * sigma_r in which case the */
/* .             numerical RANK is declared to be r. The SVD is computed with */
/* .             absolute error bounds, but more accurately than with 'A'. */
/* . */
/* . JOBU   (input) CHARACTER*1 */
/* .        Specifies whether to compute the columns of U: */
/* .      = 'U': N columns of U are returned in the array U. */
/* .      = 'F': full set of M left sing. vectors is returned in the array U. */
/* .      = 'W': U may be used as workspace of length M*N. See the description */
/* .             of U. */
/* .      = 'N': U is not computed. */
/* . */
/* . JOBV   (input) CHARACTER*1 */
/* .        Specifies whether to compute the matrix V: */
/* .      = 'V': N columns of V are returned in the array V; Jacobi rotations */
/* .             are not explicitly accumulated. */
/* .      = 'J': N columns of V are returned in the array V, but they are */
/* .             computed as the product of Jacobi rotations. This option is */
/* .             allowed only if JOBU .NE. 'N', i.e. in computing the full SVD. */
/* .      = 'W': V may be used as workspace of length N*N. See the description */
/* .             of V. */
/* .      = 'N': V is not computed. */
/* . */
/* . JOBR   (input) CHARACTER*1 */
/* .        Specifies the RANGE for the singular values. Issues the licence to */
/* .        set to zero small positive singular values if they are outside */
/* .        specified range. If A .NE. 0 is scaled so that the largest singular */
/* .        value of c*A is around DSQRT(BIG), BIG=SLAMCH('O'), then JOBR issues */
/* .        the licence to kill columns of A whose norm in c*A is less than */
/* .        DSQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN, */
/* .        where SFMIN=SLAMCH('S'), EPSLN=SLAMCH('E'). */
/* .      = 'N': Do not kill small columns of c*A. This option assumes that */
/* .             BLAS and QR factorizations and triangular solvers are */
/* .             implemented to work in that range. If the condition of A */
/* .             is greater than BIG, use DGESVJ. */
/* .      = 'R': RESTRICTED range for sigma(c*A) is [DSQRT(SFMIN), DSQRT(BIG)] */
/* .             (roughly, as described above). This option is recommended. */
/* .                                            ~~~~~~~~~~~~~~~~~~~~~~~~~~~ */
/* .        For computing the singular values in the FULL range [SFMIN,BIG] */
/* .        use DGESVJ. */
/* . */
/* . JOBT   (input) CHARACTER*1 */
/* .        If the matrix is square then the procedure may determine to use */
/* .        transposed A if A^t seems to be better with respect to convergence. */
/* .        If the matrix is not square, JOBT is ignored. This is subject to */
/* .        changes in the future. */
/* .        The decision is based on two values of entropy over the adjoint */
/* .        orbit of A^t * A. See the descriptions of WORK(6) and WORK(7). */
/* .      = 'T': transpose if entropy test indicates possibly faster */
/* .        convergence of Jacobi process if A^t is taken as input. If A is */
/* .        replaced with A^t, then the row pivoting is included automatically. */
/* .      = 'N': do not speculate. */
/* .        This option can be used to compute only the singular values, or the */
/* .        full SVD (U, SIGMA and V). For only one set of singular vectors */
/* .        (U or V), the caller should provide both U and V, as one of the */
/* .        matrices is used as workspace if the matrix A is transposed. */
/* .        The implementer can easily remove this constraint and make the */
/* .        code more complicated. See the descriptions of U and V. */
/* . */
/* . JOBP   (input) CHARACTER*1 */
/* .        Issues the licence to introduce structured perturbations to drown */
/* .        denormalized numbers. This licence should be active if the */
/* .        denormals are poorly implemented, causing slow computation, */
/* .        especially in cases of fast convergence (!). For details see [1,2]. */
/* .        For the sake of simplicity, this perturbations are included only */
/* .        when the full SVD or only the singular values are requested. The */
/* .        implementer/user can easily add the perturbation for the cases of */
/* .        computing one set of singular vectors. */
/* .      = 'P': introduce perturbation */
/* .      = 'N': do not perturb */
/* ............................................................................ */

/*  M      (input) INTEGER */
/*         The number of rows of the input matrix A.  M >= 0. */

/*  N      (input) INTEGER */
/*         The number of columns of the input matrix A. M >= N >= 0. */

/*  A       (input/workspace) REAL array, dimension (LDA,N) */
/*          On entry, the M-by-N matrix A. */

/*  LDA     (input) INTEGER */
/*          The leading dimension of the array A.  LDA >= max(1,M). */

/*  SVA     (workspace/output) REAL array, dimension (N) */
/*          On exit, */
/*          - For WORK(1)/WORK(2) = ONE: The singular values of A. During the */
/*            computation SVA contains Euclidean column norms of the */
/*            iterated matrices in the array A. */
/*          - For WORK(1) .NE. WORK(2): The singular values of A are */
/*            (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if */
/*            sigma_max(A) overflows or if small singular values have been */
/*            saved from underflow by scaling the input matrix A. */
/*          - If JOBR='R' then some of the singular values may be returned */
/*            as exact zeros obtained by "set to zero" because they are */
/*            below the numerical rank threshold or are denormalized numbers. */

/*  U       (workspace/output) REAL array, dimension ( LDU, N ) */
/*          If JOBU = 'U', then U contains on exit the M-by-N matrix of */
/*                         the left singular vectors. */
/*          If JOBU = 'F', then U contains on exit the M-by-M matrix of */
/*                         the left singular vectors, including an ONB */
/*                         of the orthogonal complement of the Range(A). */
/*          If JOBU = 'W'  .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N), */
/*                         then U is used as workspace if the procedure */
/*                         replaces A with A^t. In that case, [V] is computed */
/*                         in U as left singular vectors of A^t and then */
/*                         copied back to the V array. This 'W' option is just */
/*                         a reminder to the caller that in this case U is */
/*                         reserved as workspace of length N*N. */
/*          If JOBU = 'N'  U is not referenced. */

/* LDU      (input) INTEGER */
/*          The leading dimension of the array U,  LDU >= 1. */
/*          IF  JOBU = 'U' or 'F' or 'W',  then LDU >= M. */

/*  V       (workspace/output) REAL array, dimension ( LDV, N ) */
/*          If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of */
/*                         the right singular vectors; */
/*          If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N), */
/*                         then V is used as workspace if the pprocedure */
/*                         replaces A with A^t. In that case, [U] is computed */
/*                         in V as right singular vectors of A^t and then */
/*                         copied back to the U array. This 'W' option is just */
/*                         a reminder to the caller that in this case V is */
/*                         reserved as workspace of length N*N. */
/*          If JOBV = 'N'  V is not referenced. */

/*  LDV     (input) INTEGER */
/*          The leading dimension of the array V,  LDV >= 1. */
/*          If JOBV = 'V' or 'J' or 'W', then LDV >= N. */

/*  WORK    (workspace/output) REAL array, dimension at least LWORK. */
/*          On exit, */
/*          WORK(1) = SCALE = WORK(2) / WORK(1) is the scaling factor such */
/*                    that SCALE*SVA(1:N) are the computed singular values */
/*                    of A. (See the description of SVA().) */
/*          WORK(2) = See the description of WORK(1). */
/*          WORK(3) = SCONDA is an estimate for the condition number of */
/*                    column equilibrated A. (If JOBA .EQ. 'E' or 'G') */
/*                    SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1). */
/*                    It is computed using DPOCON. It holds */
/*                    N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA */
/*                    where R is the triangular factor from the QRF of A. */
/*                    However, if R is truncated and the numerical rank is */
/*                    determined to be strictly smaller than N, SCONDA is */
/*                    returned as -1, thus indicating that the smallest */
/*                    singular values might be lost. */

/*          If full SVD is needed, the following two condition numbers are */
/*          useful for the analysis of the algorithm. They are provied for */
/*          a developer/implementer who is familiar with the details of */
/*          the method. */

/*          WORK(4) = an estimate of the scaled condition number of the */
/*                    triangular factor in the first QR factorization. */
/*          WORK(5) = an estimate of the scaled condition number of the */
/*                    triangular factor in the second QR factorization. */
/*          The following two parameters are computed if JOBT .EQ. 'T'. */
/*          They are provided for a developer/implementer who is familiar */
/*          with the details of the method. */

/*          WORK(6) = the entropy of A^t*A :: this is the Shannon entropy */
/*                    of diag(A^t*A) / Trace(A^t*A) taken as point in the */
/*                    probability simplex. */
/*          WORK(7) = the entropy of A*A^t. */

/*  LWORK   (input) INTEGER */
/*          Length of WORK to confirm proper allocation of work space. */
/*          LWORK depends on the job: */

/*          If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and */
/*            -> .. no scaled condition estimate required ( JOBE.EQ.'N'): */
/*               LWORK >= max(2*M+N,4*N+1,7). This is the minimal requirement. */
/*               For optimal performance (blocked code) the optimal value */
/*               is LWORK >= max(2*M+N,3*N+(N+1)*NB,7). Here NB is the optimal */
/*               block size for xGEQP3/xGEQRF. */
/*            -> .. an estimate of the scaled condition number of A is */
/*               required (JOBA='E', 'G'). In this case, LWORK is the maximum */
/*               of the above and N*N+4*N, i.e. LWORK >= max(2*M+N,N*N+4N,7). */

/*          If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'), */
/*            -> the minimal requirement is LWORK >= max(2*N+M,7). */
/*            -> For optimal performance, LWORK >= max(2*N+M,2*N+N*NB,7), */
/*               where NB is the optimal block size. */

/*          If SIGMA and the left singular vectors are needed */
/*            -> the minimal requirement is LWORK >= max(2*N+M,7). */
/*            -> For optimal performance, LWORK >= max(2*N+M,2*N+N*NB,7), */
/*               where NB is the optimal block size. */

/*          If full SVD is needed ( JOBU.EQ.'U' or 'F', JOBV.EQ.'V' ) and */
/*            -> .. the singular vectors are computed without explicit */
/*               accumulation of the Jacobi rotations, LWORK >= 6*N+2*N*N */
/*            -> .. in the iterative part, the Jacobi rotations are */
/*               explicitly accumulated (option, see the description of JOBV), */
/*               then the minimal requirement is LWORK >= max(M+3*N+N*N,7). */
/*               For better performance, if NB is the optimal block size, */
/*               LWORK >= max(3*N+N*N+M,3*N+N*N+N*NB,7). */

/*  IWORK   (workspace/output) INTEGER array, dimension M+3*N. */
/*          On exit, */
/*          IWORK(1) = the numerical rank determined after the initial */
/*                     QR factorization with pivoting. See the descriptions */
/*                     of JOBA and JOBR. */
/*          IWORK(2) = the number of the computed nonzero singular values */
/*          IWORK(3) = if nonzero, a warning message: */
/*                     If IWORK(3).EQ.1 then some of the column norms of A */
/*                     were denormalized floats. The requested high accuracy */
/*                     is not warranted by the data. */

/*  INFO    (output) INTEGER */
/*           < 0  : if INFO = -i, then the i-th argument had an illegal value. */
/*           = 0 :  successfull exit; */
/*           > 0 :  DGEJSV  did not converge in the maximal allowed number */
/*                  of sweeps. The computed values may be inaccurate. */

/* ............................................................................ */

/*     Local Parameters: */


/*     Local Scalars: */


/*     Intrinsic Functions: */


/*     External Functions: */


/*     External Subroutines ( BLAS, LAPACK ): */



/* ............................................................................ */

/*     Test the input arguments */

    /* Parameter adjustments */
    --sva;
    a_dim1 = *lda;
    a_offset = 1 + a_dim1;
    a -= a_offset;
    u_dim1 = *ldu;
    u_offset = 1 + u_dim1;
    u -= u_offset;
    v_dim1 = *ldv;
    v_offset = 1 + v_dim1;
    v -= v_offset;
    --work;
    --iwork;

    /* Function Body */
    lsvec = lsame_(jobu, "U") || lsame_(jobu, "F");
    jracc = lsame_(jobv, "J");
    rsvec = lsame_(jobv, "V") || jracc;
    rowpiv = lsame_(joba, "F") || lsame_(joba, "G");
    l2rank = lsame_(joba, "R");
    l2aber = lsame_(joba, "A");
    errest = lsame_(joba, "E") || lsame_(joba, "G");
    l2tran = lsame_(jobt, "T");
    l2kill = lsame_(jobr, "R");
    defr = lsame_(jobr, "N");
    l2pert = lsame_(jobp, "P");

    if (! (rowpiv || l2rank || l2aber || errest || lsame_(joba, "C"))) {
        *info = -1;
    } else if (! (lsvec || lsame_(jobu, "N") || lsame_(
            jobu, "W"))) {
        *info = -2;
    } else if (! (rsvec || lsame_(jobv, "N") || lsame_(
            jobv, "W")) || jracc && ! lsvec) {
        *info = -3;
    } else if (! (l2kill || defr)) {
        *info = -4;
    } else if (! (l2tran || lsame_(jobt, "N"))) {
        *info = -5;
    } else if (! (l2pert || lsame_(jobp, "N"))) {
        *info = -6;
    } else if (*m < 0) {
        *info = -7;
    } else if (*n < 0 || *n > *m) {
        *info = -8;
    } else if (*lda < *m) {
        *info = -10;
    } else if (lsvec && *ldu < *m) {
        *info = -13;
    } else if (rsvec && *ldv < *n) {
        *info = -14;
    } else /* if(complicated condition) */ {
/* Computing MAX */
        i__1 = 7, i__2 = (*n << 2) + 1, i__1 = max(i__1,i__2), i__2 = (*m << 
                1) + *n;
/* Computing MAX */
        i__3 = 7, i__4 = (*n << 2) + *n * *n, i__3 = max(i__3,i__4), i__4 = (*
                m << 1) + *n;
/* Computing MAX */
        i__5 = 7, i__6 = (*n << 1) + *m;
/* Computing MAX */
        i__7 = 7, i__8 = (*n << 1) + *m;
/* Computing MAX */
        i__9 = 7, i__10 = *m + *n * 3 + *n * *n;
        if (! (lsvec || rsvec || errest) && *lwork < max(i__1,i__2) || ! (
                lsvec || lsvec) && errest && *lwork < max(i__3,i__4) || lsvec 
                && ! rsvec && *lwork < max(i__5,i__6) || rsvec && ! lsvec && *
                lwork < max(i__7,i__8) || lsvec && rsvec && ! jracc && *lwork 
                < *n * 6 + (*n << 1) * *n || lsvec && rsvec && jracc && *
                lwork < max(i__9,i__10)) {
            *info = -17;
        } else {
/*        #:) */
            *info = 0;
        }
    }

    if (*info != 0) {
/*       #:( */
        i__1 = -(*info);
        xerbla_("DGEJSV", &i__1);
    }

/*     Quick return for void matrix (Y3K safe) */
/* #:) */
    if (*m == 0 || *n == 0) {
        return 0;
    }

/*     Determine whether the matrix U should be M x N or M x M */

    if (lsvec) {
        n1 = *n;
        if (lsame_(jobu, "F")) {
            n1 = *m;
        }
    }

/*     Set numerical parameters */

/* !    NOTE: Make sure DLAMCH() does not fail on the target architecture. */

    epsln = dlamch_("Epsilon");
    sfmin = dlamch_("SafeMinimum");
    small = sfmin / epsln;
    big = dlamch_("O");
/*     BIG   = ONE / SFMIN */

/*     Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N */

/* (!)  If necessary, scale SVA() to protect the largest norm from */
/*     overflow. It is possible that this scaling pushes the smallest */
/*     column norm left from the underflow threshold (extreme case). */

    scalem = 1. / sqrt((doublereal) (*m) * (doublereal) (*n));
    noscal = TRUE_;
    goscal = TRUE_;
    i__1 = *n;
    for (p = 1; p <= i__1; ++p) {
        aapp = 0.;
        aaqq = 0.;
        dlassq_(m, &a[p * a_dim1 + 1], &c__1, &aapp, &aaqq);
        if (aapp > big) {
            *info = -9;
            i__2 = -(*info);
            xerbla_("DGEJSV", &i__2);
            return 0;
        }
        aaqq = sqrt(aaqq);
        if (aapp < big / aaqq && noscal) {
            sva[p] = aapp * aaqq;
        } else {
            noscal = FALSE_;
            sva[p] = aapp * (aaqq * scalem);
            if (goscal) {
                goscal = FALSE_;
                i__2 = p - 1;
                dscal_(&i__2, &scalem, &sva[1], &c__1);
            }
        }
/* L1874: */
    }

    if (noscal) {
        scalem = 1.;
    }

    aapp = 0.;
    aaqq = big;
    i__1 = *n;
    for (p = 1; p <= i__1; ++p) {
/* Computing MAX */
        d__1 = aapp, d__2 = sva[p];
        aapp = max(d__1,d__2);
        if (sva[p] != 0.) {
/* Computing MIN */
            d__1 = aaqq, d__2 = sva[p];
            aaqq = min(d__1,d__2);
        }
/* L4781: */
    }

/*     Quick return for zero M x N matrix */
/* #:) */
    if (aapp == 0.) {
        if (lsvec) {
            dlaset_("G", m, &n1, &c_b34, &c_b35, &u[u_offset], ldu)
                    ;
        }
        if (rsvec) {
            dlaset_("G", n, n, &c_b34, &c_b35, &v[v_offset], ldv);
        }
        work[1] = 1.;
        work[2] = 1.;
        if (errest) {
            work[3] = 1.;
        }
        if (lsvec && rsvec) {
            work[4] = 1.;
            work[5] = 1.;
        }
        if (l2tran) {
            work[6] = 0.;
            work[7] = 0.;
        }
        iwork[1] = 0;
        iwork[2] = 0;
        return 0;
    }

/*     Issue warning if denormalized column norms detected. Override the */
/*     high relative accuracy request. Issue licence to kill columns */
/*     (set them to zero) whose norm is less than sigma_max / BIG (roughly). */
/* #:( */
    warning = 0;
    if (aaqq <= sfmin) {
        l2rank = TRUE_;
        l2kill = TRUE_;
        warning = 1;
    }

/*     Quick return for one-column matrix */
/* #:) */
    if (*n == 1) {

        if (lsvec) {
            dlascl_("G", &c__0, &c__0, &sva[1], &scalem, m, &c__1, &a[a_dim1 
                    + 1], lda, &ierr);
            dlacpy_("A", m, &c__1, &a[a_offset], lda, &u[u_offset], ldu);
/*           computing all M left singular vectors of the M x 1 matrix */
            if (n1 != *n) {
                i__1 = *lwork - *n;
                dgeqrf_(m, n, &u[u_offset], ldu, &work[1], &work[*n + 1], &
                        i__1, &ierr);
                i__1 = *lwork - *n;
                dorgqr_(m, &n1, &c__1, &u[u_offset], ldu, &work[1], &work[*n 
                        + 1], &i__1, &ierr);
                dcopy_(m, &a[a_dim1 + 1], &c__1, &u[u_dim1 + 1], &c__1);
            }
        }
        if (rsvec) {
            v[v_dim1 + 1] = 1.;
        }
        if (sva[1] < big * scalem) {
            sva[1] /= scalem;
            scalem = 1.;
        }
        work[1] = 1. / scalem;
        work[2] = 1.;
        if (sva[1] != 0.) {
            iwork[1] = 1;
            if (sva[1] / scalem >= sfmin) {
                iwork[2] = 1;
            } else {
                iwork[2] = 0;
            }
        } else {
            iwork[1] = 0;
            iwork[2] = 0;
        }
        if (errest) {
            work[3] = 1.;
        }
        if (lsvec && rsvec) {
            work[4] = 1.;
            work[5] = 1.;
        }
        if (l2tran) {
            work[6] = 0.;
            work[7] = 0.;
        }
        return 0;

    }

    transp = FALSE_;
    l2tran = l2tran && *m == *n;

    aatmax = -1.;
    aatmin = big;
    if (rowpiv || l2tran) {

/*     Compute the row norms, needed to determine row pivoting sequence */
/*     (in the case of heavily row weighted A, row pivoting is strongly */
/*     advised) and to collect information needed to compare the */
/*     structures of A * A^t and A^t * A (in the case L2TRAN.EQ..TRUE.). */

        if (l2tran) {
            i__1 = *m;
            for (p = 1; p <= i__1; ++p) {
                xsc = 0.;
                temp1 = 0.;
                dlassq_(n, &a[p + a_dim1], lda, &xsc, &temp1);
/*              DLASSQ gets both the ell_2 and the ell_infinity norm */
/*              in one pass through the vector */
                work[*m + *n + p] = xsc * scalem;
                work[*n + p] = xsc * (scalem * sqrt(temp1));
/* Computing MAX */
                d__1 = aatmax, d__2 = work[*n + p];
                aatmax = max(d__1,d__2);
                if (work[*n + p] != 0.) {
/* Computing MIN */
                    d__1 = aatmin, d__2 = work[*n + p];
                    aatmin = min(d__1,d__2);
                }
/* L1950: */
            }
        } else {
            i__1 = *m;
            for (p = 1; p <= i__1; ++p) {
                work[*m + *n + p] = scalem * (d__1 = a[p + idamax_(n, &a[p + 
                        a_dim1], lda) * a_dim1], abs(d__1));
/* Computing MAX */
                d__1 = aatmax, d__2 = work[*m + *n + p];
                aatmax = max(d__1,d__2);
/* Computing MIN */
                d__1 = aatmin, d__2 = work[*m + *n + p];
                aatmin = min(d__1,d__2);
/* L1904: */
            }
        }

    }

/*     For square matrix A try to determine whether A^t  would be  better */
/*     input for the preconditioned Jacobi SVD, with faster convergence. */
/*     The decision is based on an O(N) function of the vector of column */
/*     and row norms of A, based on the Shannon entropy. This should give */
/*     the right choice in most cases when the difference actually matters. */
/*     It may fail and pick the slower converging side. */

    entra = 0.;
    entrat = 0.;
    if (l2tran) {

        xsc = 0.;
        temp1 = 0.;
        dlassq_(n, &sva[1], &c__1, &xsc, &temp1);
        temp1 = 1. / temp1;

        entra = 0.;
        i__1 = *n;
        for (p = 1; p <= i__1; ++p) {
/* Computing 2nd power */
            d__1 = sva[p] / xsc;
            big1 = d__1 * d__1 * temp1;
            if (big1 != 0.) {
                entra += big1 * log(big1);
            }
/* L1113: */
        }
        entra = -entra / log((doublereal) (*n));

/*        Now, SVA().^2/Trace(A^t * A) is a point in the probability simplex. */
/*        It is derived from the diagonal of  A^t * A.  Do the same with the */
/*        diagonal of A * A^t, compute the entropy of the corresponding */
/*        probability distribution. Note that A * A^t and A^t * A have the */
/*        same trace. */

        entrat = 0.;
        i__1 = *n + *m;
        for (p = *n + 1; p <= i__1; ++p) {
/* Computing 2nd power */
            d__1 = work[p] / xsc;
            big1 = d__1 * d__1 * temp1;
            if (big1 != 0.) {
                entrat += big1 * log(big1);
            }
/* L1114: */
        }
        entrat = -entrat / log((doublereal) (*m));

/*        Analyze the entropies and decide A or A^t. Smaller entropy */
/*        usually means better input for the algorithm. */

        transp = entrat < entra;

/*        If A^t is better than A, transpose A. */

        if (transp) {
/*           In an optimal implementation, this trivial transpose */
/*           should be replaced with faster transpose. */
            i__1 = *n - 1;
            for (p = 1; p <= i__1; ++p) {
                i__2 = *n;
                for (q = p + 1; q <= i__2; ++q) {
                    temp1 = a[q + p * a_dim1];
                    a[q + p * a_dim1] = a[p + q * a_dim1];
                    a[p + q * a_dim1] = temp1;
/* L1116: */
                }
/* L1115: */
            }
            i__1 = *n;
            for (p = 1; p <= i__1; ++p) {
                work[*m + *n + p] = sva[p];
                sva[p] = work[*n + p];
/* L1117: */
            }
            temp1 = aapp;
            aapp = aatmax;
            aatmax = temp1;
            temp1 = aaqq;
            aaqq = aatmin;
            aatmin = temp1;
            kill = lsvec;
            lsvec = rsvec;
            rsvec = kill;

            rowpiv = TRUE_;
        }

    }
/*     END IF L2TRAN */

/*     Scale the matrix so that its maximal singular value remains less */
/*     than DSQRT(BIG) -- the matrix is scaled so that its maximal column */
/*     has Euclidean norm equal to DSQRT(BIG/N). The only reason to keep */
/*     DSQRT(BIG) instead of BIG is the fact that DGEJSV uses LAPACK and */
/*     BLAS routines that, in some implementations, are not capable of */
/*     working in the full interval [SFMIN,BIG] and that they may provoke */
/*     overflows in the intermediate results. If the singular values spread */
/*     from SFMIN to BIG, then DGESVJ will compute them. So, in that case, */
/*     one should use DGESVJ instead of DGEJSV. */

    big1 = sqrt(big);
    temp1 = sqrt(big / (doublereal) (*n));

    dlascl_("G", &c__0, &c__0, &aapp, &temp1, n, &c__1, &sva[1], n, &ierr);
    if (aaqq > aapp * sfmin) {
        aaqq = aaqq / aapp * temp1;
    } else {
        aaqq = aaqq * temp1 / aapp;
    }
    temp1 *= scalem;
    dlascl_("G", &c__0, &c__0, &aapp, &temp1, m, n, &a[a_offset], lda, &ierr);

/*     To undo scaling at the end of this procedure, multiply the */
/*     computed singular values with USCAL2 / USCAL1. */

    uscal1 = temp1;
    uscal2 = aapp;

    if (l2kill) {
/*        L2KILL enforces computation of nonzero singular values in */
/*        the restricted range of condition number of the initial A, */
/*        sigma_max(A) / sigma_min(A) approx. DSQRT(BIG)/DSQRT(SFMIN). */
        xsc = sqrt(sfmin);
    } else {
        xsc = small;

/*        Now, if the condition number of A is too big, */
/*        sigma_max(A) / sigma_min(A) .GT. DSQRT(BIG/N) * EPSLN / SFMIN, */
/*        as a precaution measure, the full SVD is computed using DGESVJ */
/*        with accumulated Jacobi rotations. This provides numerically */
/*        more robust computation, at the cost of slightly increased run */
/*        time. Depending on the concrete implementation of BLAS and LAPACK */
/*        (i.e. how they behave in presence of extreme ill-conditioning) the */
/*        implementor may decide to remove this switch. */
        if (aaqq < sqrt(sfmin) && lsvec && rsvec) {
            jracc = TRUE_;
        }

    }
    if (aaqq < xsc) {
        i__1 = *n;
        for (p = 1; p <= i__1; ++p) {
            if (sva[p] < xsc) {
                dlaset_("A", m, &c__1, &c_b34, &c_b34, &a[p * a_dim1 + 1], 
                        lda);
                sva[p] = 0.;
            }
/* L700: */
        }
    }

/*     Preconditioning using QR factorization with pivoting */

    if (rowpiv) {
/*        Optional row permutation (Bjoerck row pivoting): */
/*        A result by Cox and Higham shows that the Bjoerck's */
/*        row pivoting combined with standard column pivoting */
/*        has similar effect as Powell-Reid complete pivoting. */
/*        The ell-infinity norms of A are made nonincreasing. */
        i__1 = *m - 1;
        for (p = 1; p <= i__1; ++p) {
            i__2 = *m - p + 1;
            q = idamax_(&i__2, &work[*m + *n + p], &c__1) + p - 1;
            iwork[(*n << 1) + p] = q;
            if (p != q) {
                temp1 = work[*m + *n + p];
                work[*m + *n + p] = work[*m + *n + q];
                work[*m + *n + q] = temp1;
            }
/* L1952: */
        }
        i__1 = *m - 1;
        dlaswp_(n, &a[a_offset], lda, &c__1, &i__1, &iwork[(*n << 1) + 1], &
                c__1);
    }

/*     End of the preparation phase (scaling, optional sorting and */
/*     transposing, optional flushing of small columns). */

/*     Preconditioning */

/*     If the full SVD is needed, the right singular vectors are computed */
/*     from a matrix equation, and for that we need theoretical analysis */
/*     of the Businger-Golub pivoting. So we use DGEQP3 as the first RR QRF. */
/*     In all other cases the first RR QRF can be chosen by other criteria */
/*     (eg speed by replacing global with restricted window pivoting, such */
/*     as in SGEQPX from TOMS # 782). Good results will be obtained using */
/*     SGEQPX with properly (!) chosen numerical parameters. */
/*     Any improvement of DGEQP3 improves overal performance of DGEJSV. */

/*     A * P1 = Q1 * [ R1^t 0]^t: */
    i__1 = *n;
    for (p = 1; p <= i__1; ++p) {
/*        .. all columns are free columns */
        iwork[p] = 0;
/* L1963: */
    }
    i__1 = *lwork - *n;
    dgeqp3_(m, n, &a[a_offset], lda, &iwork[1], &work[1], &work[*n + 1], &
            i__1, &ierr);

/*     The upper triangular matrix R1 from the first QRF is inspected for */
/*     rank deficiency and possibilities for deflation, or possible */
/*     ill-conditioning. Depending on the user specified flag L2RANK, */
/*     the procedure explores possibilities to reduce the numerical */
/*     rank by inspecting the computed upper triangular factor. If */
/*     L2RANK or L2ABER are up, then DGEJSV will compute the SVD of */
/*     A + dA, where ||dA|| <= f(M,N)*EPSLN. */

    nr = 1;
    if (l2aber) {
/*        Standard absolute error bound suffices. All sigma_i with */
/*        sigma_i < N*EPSLN*||A|| are flushed to zero. This is an */
/*        agressive enforcement of lower numerical rank by introducing a */
/*        backward error of the order of N*EPSLN*||A||. */
        temp1 = sqrt((doublereal) (*n)) * epsln;
        i__1 = *n;
        for (p = 2; p <= i__1; ++p) {
            if ((d__2 = a[p + p * a_dim1], abs(d__2)) >= temp1 * (d__1 = a[
                    a_dim1 + 1], abs(d__1))) {
                ++nr;
            } else {
                goto L3002;
            }
/* L3001: */
        }
L3002:
        ;
    } else if (l2rank) {
/*        .. similarly as above, only slightly more gentle (less agressive). */
/*        Sudden drop on the diagonal of R1 is used as the criterion for */
/*        close-to-rank-defficient. */
        temp1 = sqrt(sfmin);
        i__1 = *n;
        for (p = 2; p <= i__1; ++p) {
            if ((d__2 = a[p + p * a_dim1], abs(d__2)) < epsln * (d__1 = a[p - 
                    1 + (p - 1) * a_dim1], abs(d__1)) || (d__3 = a[p + p * 
                    a_dim1], abs(d__3)) < small || l2kill && (d__4 = a[p + p *
                     a_dim1], abs(d__4)) < temp1) {
                goto L3402;
            }
            ++nr;
/* L3401: */
        }
L3402:

        ;
    } else {
/*        The goal is high relative accuracy. However, if the matrix */
/*        has high scaled condition number the relative accuracy is in */
/*        general not feasible. Later on, a condition number estimator */
/*        will be deployed to estimate the scaled condition number. */
/*        Here we just remove the underflowed part of the triangular */
/*        factor. This prevents the situation in which the code is */
/*        working hard to get the accuracy not warranted by the data. */
        temp1 = sqrt(sfmin);
        i__1 = *n;
        for (p = 2; p <= i__1; ++p) {
            if ((d__1 = a[p + p * a_dim1], abs(d__1)) < small || l2kill && (
                    d__2 = a[p + p * a_dim1], abs(d__2)) < temp1) {
                goto L3302;
            }
            ++nr;
/* L3301: */
        }
L3302:

        ;
    }

    almort = FALSE_;
    if (nr == *n) {
        maxprj = 1.;
        i__1 = *n;
        for (p = 2; p <= i__1; ++p) {
            temp1 = (d__1 = a[p + p * a_dim1], abs(d__1)) / sva[iwork[p]];
            maxprj = min(maxprj,temp1);
/* L3051: */
        }
/* Computing 2nd power */
        d__1 = maxprj;
        if (d__1 * d__1 >= 1. - (doublereal) (*n) * epsln) {
            almort = TRUE_;
        }
    }


    sconda = -1.;
    condr1 = -1.;
    condr2 = -1.;

    if (errest) {
        if (*n == nr) {
            if (rsvec) {
/*              .. V is available as workspace */
                dlacpy_("U", n, n, &a[a_offset], lda, &v[v_offset], ldv);
                i__1 = *n;
                for (p = 1; p <= i__1; ++p) {
                    temp1 = sva[iwork[p]];
                    d__1 = 1. / temp1;
                    dscal_(&p, &d__1, &v[p * v_dim1 + 1], &c__1);
/* L3053: */
                }
                dpocon_("U", n, &v[v_offset], ldv, &c_b35, &temp1, &work[*n + 
                        1], &iwork[(*n << 1) + *m + 1], &ierr);
            } else if (lsvec) {
/*              .. U is available as workspace */
                dlacpy_("U", n, n, &a[a_offset], lda, &u[u_offset], ldu);
                i__1 = *n;
                for (p = 1; p <= i__1; ++p) {
                    temp1 = sva[iwork[p]];
                    d__1 = 1. / temp1;
                    dscal_(&p, &d__1, &u[p * u_dim1 + 1], &c__1);
/* L3054: */
                }
                dpocon_("U", n, &u[u_offset], ldu, &c_b35, &temp1, &work[*n + 
                        1], &iwork[(*n << 1) + *m + 1], &ierr);
            } else {
                dlacpy_("U", n, n, &a[a_offset], lda, &work[*n + 1], n);
                i__1 = *n;
                for (p = 1; p <= i__1; ++p) {
                    temp1 = sva[iwork[p]];
                    d__1 = 1. / temp1;
                    dscal_(&p, &d__1, &work[*n + (p - 1) * *n + 1], &c__1);
/* L3052: */
                }
/*           .. the columns of R are scaled to have unit Euclidean lengths. */
                dpocon_("U", n, &work[*n + 1], n, &c_b35, &temp1, &work[*n + *
                        n * *n + 1], &iwork[(*n << 1) + *m + 1], &ierr);
            }
            sconda = 1. / sqrt(temp1);
/*           SCONDA is an estimate of DSQRT(||(R^t * R)^(-1)||_1). */
/*           N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA */
        } else {
            sconda = -1.;
        }
    }

    l2pert = l2pert && (d__1 = a[a_dim1 + 1] / a[nr + nr * a_dim1], abs(d__1))
             > sqrt(big1);
/*     If there is no violent scaling, artificial perturbation is not needed. */

/*     Phase 3: */

    if (! (rsvec || lsvec)) {

/*         Singular Values only */

/*         .. transpose A(1:NR,1:N) */
/* Computing MIN */
        i__2 = *n - 1;
        i__1 = min(i__2,nr);
        for (p = 1; p <= i__1; ++p) {
            i__2 = *n - p;
            dcopy_(&i__2, &a[p + (p + 1) * a_dim1], lda, &a[p + 1 + p * 
                    a_dim1], &c__1);
/* L1946: */
        }

/*        The following two DO-loops introduce small relative perturbation */
/*        into the strict upper triangle of the lower triangular matrix. */
/*        Small entries below the main diagonal are also changed. */
/*        This modification is useful if the computing environment does not */
/*        provide/allow FLUSH TO ZERO underflow, for it prevents many */
/*        annoying denormalized numbers in case of strongly scaled matrices. */
/*        The perturbation is structured so that it does not introduce any */
/*        new perturbation of the singular values, and it does not destroy */
/*        the job done by the preconditioner. */
/*        The licence for this perturbation is in the variable L2PERT, which */
/*        should be .FALSE. if FLUSH TO ZERO underflow is active. */

        if (! almort) {

            if (l2pert) {
/*              XSC = DSQRT(SMALL) */
                xsc = epsln / (doublereal) (*n);
                i__1 = nr;
                for (q = 1; q <= i__1; ++q) {
                    temp1 = xsc * (d__1 = a[q + q * a_dim1], abs(d__1));
                    i__2 = *n;
                    for (p = 1; p <= i__2; ++p) {
                        if (p > q && (d__1 = a[p + q * a_dim1], abs(d__1)) <= 
                                temp1 || p < q) {
                            a[p + q * a_dim1] = d_sign(&temp1, &a[p + q * 
                                    a_dim1]);
                        }
/* L4949: */
                    }
/* L4947: */
                }
            } else {
                i__1 = nr - 1;
                i__2 = nr - 1;
                dlaset_("U", &i__1, &i__2, &c_b34, &c_b34, &a[(a_dim1 << 1) + 
                        1], lda);
            }

/*            .. second preconditioning using the QR factorization */

            i__1 = *lwork - *n;
            dgeqrf_(n, &nr, &a[a_offset], lda, &work[1], &work[*n + 1], &i__1, 
                     &ierr);

/*           .. and transpose upper to lower triangular */
            i__1 = nr - 1;
            for (p = 1; p <= i__1; ++p) {
                i__2 = nr - p;
                dcopy_(&i__2, &a[p + (p + 1) * a_dim1], lda, &a[p + 1 + p * 
                        a_dim1], &c__1);
/* L1948: */
            }

        }

/*           Row-cyclic Jacobi SVD algorithm with column pivoting */

/*           .. again some perturbation (a "background noise") is added */
/*           to drown denormals */
        if (l2pert) {
/*              XSC = DSQRT(SMALL) */
            xsc = epsln / (doublereal) (*n);
            i__1 = nr;
            for (q = 1; q <= i__1; ++q) {
                temp1 = xsc * (d__1 = a[q + q * a_dim1], abs(d__1));
                i__2 = nr;
                for (p = 1; p <= i__2; ++p) {
                    if (p > q && (d__1 = a[p + q * a_dim1], abs(d__1)) <= 
                            temp1 || p < q) {
                        a[p + q * a_dim1] = d_sign(&temp1, &a[p + q * a_dim1])
                                ;
                    }
/* L1949: */
                }
/* L1947: */
            }
        } else {
            i__1 = nr - 1;
            i__2 = nr - 1;
            dlaset_("U", &i__1, &i__2, &c_b34, &c_b34, &a[(a_dim1 << 1) + 1], 
                    lda);
        }

/*           .. and one-sided Jacobi rotations are started on a lower */
/*           triangular matrix (plus perturbation which is ignored in */
/*           the part which destroys triangular form (confusing?!)) */

        dgesvj_("L", "NoU", "NoV", &nr, &nr, &a[a_offset], lda, &sva[1], n, &
                v[v_offset], ldv, &work[1], lwork, info);

        scalem = work[1];
        numrank = i_dnnt(&work[2]);


    } else if (rsvec && ! lsvec) {

/*        -> Singular Values and Right Singular Vectors <- */

        if (almort) {

/*           .. in this case NR equals N */
            i__1 = nr;
            for (p = 1; p <= i__1; ++p) {
                i__2 = *n - p + 1;
                dcopy_(&i__2, &a[p + p * a_dim1], lda, &v[p + p * v_dim1], &
                        c__1);
/* L1998: */
            }
            i__1 = nr - 1;
            i__2 = nr - 1;
            dlaset_("Upper", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 1) + 
                    1], ldv);

            dgesvj_("L", "U", "N", n, &nr, &v[v_offset], ldv, &sva[1], &nr, &
                    a[a_offset], lda, &work[1], lwork, info);
            scalem = work[1];
            numrank = i_dnnt(&work[2]);
        } else {

/*        .. two more QR factorizations ( one QRF is not enough, two require */
/*        accumulated product of Jacobi rotations, three are perfect ) */

            i__1 = nr - 1;
            i__2 = nr - 1;
            dlaset_("Lower", &i__1, &i__2, &c_b34, &c_b34, &a[a_dim1 + 2], 
                    lda);
            i__1 = *lwork - *n;
            dgelqf_(&nr, n, &a[a_offset], lda, &work[1], &work[*n + 1], &i__1, 
                     &ierr);
            dlacpy_("Lower", &nr, &nr, &a[a_offset], lda, &v[v_offset], ldv);
            i__1 = nr - 1;
            i__2 = nr - 1;
            dlaset_("Upper", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 1) + 
                    1], ldv);
            i__1 = *lwork - (*n << 1);
            dgeqrf_(&nr, &nr, &v[v_offset], ldv, &work[*n + 1], &work[(*n << 
                    1) + 1], &i__1, &ierr);
            i__1 = nr;
            for (p = 1; p <= i__1; ++p) {
                i__2 = nr - p + 1;
                dcopy_(&i__2, &v[p + p * v_dim1], ldv, &v[p + p * v_dim1], &
                        c__1);
/* L8998: */
            }
            i__1 = nr - 1;
            i__2 = nr - 1;
            dlaset_("Upper", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 1) + 
                    1], ldv);

            dgesvj_("Lower", "U", "N", &nr, &nr, &v[v_offset], ldv, &sva[1], &
                    nr, &u[u_offset], ldu, &work[*n + 1], lwork, info);
            scalem = work[*n + 1];
            numrank = i_dnnt(&work[*n + 2]);
            if (nr < *n) {
                i__1 = *n - nr;
                dlaset_("A", &i__1, &nr, &c_b34, &c_b34, &v[nr + 1 + v_dim1], 
                        ldv);
                i__1 = *n - nr;
                dlaset_("A", &nr, &i__1, &c_b34, &c_b34, &v[(nr + 1) * v_dim1 
                        + 1], ldv);
                i__1 = *n - nr;
                i__2 = *n - nr;
                dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &v[nr + 1 + (nr + 
                        1) * v_dim1], ldv);
            }

            i__1 = *lwork - *n;
            dormlq_("Left", "Transpose", n, n, &nr, &a[a_offset], lda, &work[
                    1], &v[v_offset], ldv, &work[*n + 1], &i__1, &ierr);

        }

        i__1 = *n;
        for (p = 1; p <= i__1; ++p) {
            dcopy_(n, &v[p + v_dim1], ldv, &a[iwork[p] + a_dim1], lda);
/* L8991: */
        }
        dlacpy_("All", n, n, &a[a_offset], lda, &v[v_offset], ldv);

        if (transp) {
            dlacpy_("All", n, n, &v[v_offset], ldv, &u[u_offset], ldu);
        }

    } else if (lsvec && ! rsvec) {

/*        -#- Singular Values and Left Singular Vectors                 -#- */

/*        .. second preconditioning step to avoid need to accumulate */
/*        Jacobi rotations in the Jacobi iterations. */
        i__1 = nr;
        for (p = 1; p <= i__1; ++p) {
            i__2 = *n - p + 1;
            dcopy_(&i__2, &a[p + p * a_dim1], lda, &u[p + p * u_dim1], &c__1);
/* L1965: */
        }
        i__1 = nr - 1;
        i__2 = nr - 1;
        dlaset_("Upper", &i__1, &i__2, &c_b34, &c_b34, &u[(u_dim1 << 1) + 1], 
                ldu);

        i__1 = *lwork - (*n << 1);
        dgeqrf_(n, &nr, &u[u_offset], ldu, &work[*n + 1], &work[(*n << 1) + 1]
, &i__1, &ierr);

        i__1 = nr - 1;
        for (p = 1; p <= i__1; ++p) {
            i__2 = nr - p;
            dcopy_(&i__2, &u[p + (p + 1) * u_dim1], ldu, &u[p + 1 + p * 
                    u_dim1], &c__1);
/* L1967: */
        }
        i__1 = nr - 1;
        i__2 = nr - 1;
        dlaset_("Upper", &i__1, &i__2, &c_b34, &c_b34, &u[(u_dim1 << 1) + 1], 
                ldu);

        i__1 = *lwork - *n;
        dgesvj_("Lower", "U", "N", &nr, &nr, &u[u_offset], ldu, &sva[1], &nr, 
                &a[a_offset], lda, &work[*n + 1], &i__1, info);
        scalem = work[*n + 1];
        numrank = i_dnnt(&work[*n + 2]);

        if (nr < *m) {
            i__1 = *m - nr;
            dlaset_("A", &i__1, &nr, &c_b34, &c_b34, &u[nr + 1 + u_dim1], ldu);
            if (nr < n1) {
                i__1 = n1 - nr;
                dlaset_("A", &nr, &i__1, &c_b34, &c_b34, &u[(nr + 1) * u_dim1 
                        + 1], ldu);
                i__1 = *m - nr;
                i__2 = n1 - nr;
                dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &u[nr + 1 + (nr + 
                        1) * u_dim1], ldu);
            }
        }

        i__1 = *lwork - *n;
        dormqr_("Left", "No Tr", m, &n1, n, &a[a_offset], lda, &work[1], &u[
                u_offset], ldu, &work[*n + 1], &i__1, &ierr);

        if (rowpiv) {
            i__1 = *m - 1;
            dlaswp_(&n1, &u[u_offset], ldu, &c__1, &i__1, &iwork[(*n << 1) + 
                    1], &c_n1);
        }

        i__1 = n1;
        for (p = 1; p <= i__1; ++p) {
            xsc = 1. / dnrm2_(m, &u[p * u_dim1 + 1], &c__1);
            dscal_(m, &xsc, &u[p * u_dim1 + 1], &c__1);
/* L1974: */
        }

        if (transp) {
            dlacpy_("All", n, n, &u[u_offset], ldu, &v[v_offset], ldv);
        }

    } else {

/*        -#- Full SVD -#- */

        if (! jracc) {

            if (! almort) {

/*           Second Preconditioning Step (QRF [with pivoting]) */
/*           Note that the composition of TRANSPOSE, QRF and TRANSPOSE is */
/*           equivalent to an LQF CALL. Since in many libraries the QRF */
/*           seems to be better optimized than the LQF, we do explicit */
/*           transpose and use the QRF. This is subject to changes in an */
/*           optimized implementation of DGEJSV. */

                i__1 = nr;
                for (p = 1; p <= i__1; ++p) {
                    i__2 = *n - p + 1;
                    dcopy_(&i__2, &a[p + p * a_dim1], lda, &v[p + p * v_dim1], 
                             &c__1);
/* L1968: */
                }

/*           .. the following two loops perturb small entries to avoid */
/*           denormals in the second QR factorization, where they are */
/*           as good as zeros. This is done to avoid painfully slow */
/*           computation with denormals. The relative size of the perturbation */
/*           is a parameter that can be changed by the implementer. */
/*           This perturbation device will be obsolete on machines with */
/*           properly implemented arithmetic. */
/*           To switch it off, set L2PERT=.FALSE. To remove it from  the */
/*           code, remove the action under L2PERT=.TRUE., leave the ELSE part. */
/*           The following two loops should be blocked and fused with the */
/*           transposed copy above. */

                if (l2pert) {
                    xsc = sqrt(small);
                    i__1 = nr;
                    for (q = 1; q <= i__1; ++q) {
                        temp1 = xsc * (d__1 = v[q + q * v_dim1], abs(d__1));
                        i__2 = *n;
                        for (p = 1; p <= i__2; ++p) {
                            if (p > q && (d__1 = v[p + q * v_dim1], abs(d__1))
                                     <= temp1 || p < q) {
                                v[p + q * v_dim1] = d_sign(&temp1, &v[p + q * 
                                        v_dim1]);
                            }
                            if (p < q) {
                                v[p + q * v_dim1] = -v[p + q * v_dim1];
                            }
/* L2968: */
                        }
/* L2969: */
                    }
                } else {
                    i__1 = nr - 1;
                    i__2 = nr - 1;
                    dlaset_("U", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 
                            1) + 1], ldv);
                }

/*           Estimate the row scaled condition number of R1 */
/*           (If R1 is rectangular, N > NR, then the condition number */
/*           of the leading NR x NR submatrix is estimated.) */

                dlacpy_("L", &nr, &nr, &v[v_offset], ldv, &work[(*n << 1) + 1]
, &nr);
                i__1 = nr;
                for (p = 1; p <= i__1; ++p) {
                    i__2 = nr - p + 1;
                    temp1 = dnrm2_(&i__2, &work[(*n << 1) + (p - 1) * nr + p], 
                             &c__1);
                    i__2 = nr - p + 1;
                    d__1 = 1. / temp1;
                    dscal_(&i__2, &d__1, &work[(*n << 1) + (p - 1) * nr + p], 
                            &c__1);
/* L3950: */
                }
                dpocon_("Lower", &nr, &work[(*n << 1) + 1], &nr, &c_b35, &
                        temp1, &work[(*n << 1) + nr * nr + 1], &iwork[*m + (*
                        n << 1) + 1], &ierr);
                condr1 = 1. / sqrt(temp1);
/*           .. here need a second oppinion on the condition number */
/*           .. then assume worst case scenario */
/*           R1 is OK for inverse <=> CONDR1 .LT. DBLE(N) */
/*           more conservative    <=> CONDR1 .LT. DSQRT(DBLE(N)) */

                cond_ok__ = sqrt((doublereal) nr);
/* [TP]       COND_OK is a tuning parameter. */
                if (condr1 < cond_ok__) {
/*              .. the second QRF without pivoting. Note: in an optimized */
/*              implementation, this QRF should be implemented as the QRF */
/*              of a lower triangular matrix. */
/*              R1^t = Q2 * R2 */
                    i__1 = *lwork - (*n << 1);
                    dgeqrf_(n, &nr, &v[v_offset], ldv, &work[*n + 1], &work[(*
                            n << 1) + 1], &i__1, &ierr);

                    if (l2pert) {
                        xsc = sqrt(small) / epsln;
                        i__1 = nr;
                        for (p = 2; p <= i__1; ++p) {
                            i__2 = p - 1;
                            for (q = 1; q <= i__2; ++q) {
/* Computing MIN */
                                d__3 = (d__1 = v[p + p * v_dim1], abs(d__1)), 
                                        d__4 = (d__2 = v[q + q * v_dim1], abs(
                                        d__2));
                                temp1 = xsc * min(d__3,d__4);
                                if ((d__1 = v[q + p * v_dim1], abs(d__1)) <= 
                                        temp1) {
                                    v[q + p * v_dim1] = d_sign(&temp1, &v[q + 
                                            p * v_dim1]);
                                }
/* L3958: */
                            }
/* L3959: */
                        }
                    }

                    if (nr != *n) {
                        dlacpy_("A", n, &nr, &v[v_offset], ldv, &work[(*n << 
                                1) + 1], n);
                    }
/*              .. save ... */

/*           .. this transposed copy should be better than naive */
                    i__1 = nr - 1;
                    for (p = 1; p <= i__1; ++p) {
                        i__2 = nr - p;
                        dcopy_(&i__2, &v[p + (p + 1) * v_dim1], ldv, &v[p + 1 
                                + p * v_dim1], &c__1);
/* L1969: */
                    }

                    condr2 = condr1;

                } else {

/*              .. ill-conditioned case: second QRF with pivoting */
/*              Note that windowed pivoting would be equaly good */
/*              numerically, and more run-time efficient. So, in */
/*              an optimal implementation, the next call to DGEQP3 */
/*              should be replaced with eg. CALL SGEQPX (ACM TOMS #782) */
/*              with properly (carefully) chosen parameters. */

/*              R1^t * P2 = Q2 * R2 */
                    i__1 = nr;
                    for (p = 1; p <= i__1; ++p) {
                        iwork[*n + p] = 0;
/* L3003: */
                    }
                    i__1 = *lwork - (*n << 1);
                    dgeqp3_(n, &nr, &v[v_offset], ldv, &iwork[*n + 1], &work[*
                            n + 1], &work[(*n << 1) + 1], &i__1, &ierr);
/* *               CALL DGEQRF( N, NR, V, LDV, WORK(N+1), WORK(2*N+1), */
/* *     &              LWORK-2*N, IERR ) */
                    if (l2pert) {
                        xsc = sqrt(small);
                        i__1 = nr;
                        for (p = 2; p <= i__1; ++p) {
                            i__2 = p - 1;
                            for (q = 1; q <= i__2; ++q) {
/* Computing MIN */
                                d__3 = (d__1 = v[p + p * v_dim1], abs(d__1)), 
                                        d__4 = (d__2 = v[q + q * v_dim1], abs(
                                        d__2));
                                temp1 = xsc * min(d__3,d__4);
                                if ((d__1 = v[q + p * v_dim1], abs(d__1)) <= 
                                        temp1) {
                                    v[q + p * v_dim1] = d_sign(&temp1, &v[q + 
                                            p * v_dim1]);
                                }
/* L3968: */
                            }
/* L3969: */
                        }
                    }

                    dlacpy_("A", n, &nr, &v[v_offset], ldv, &work[(*n << 1) + 
                            1], n);

                    if (l2pert) {
                        xsc = sqrt(small);
                        i__1 = nr;
                        for (p = 2; p <= i__1; ++p) {
                            i__2 = p - 1;
                            for (q = 1; q <= i__2; ++q) {
/* Computing MIN */
                                d__3 = (d__1 = v[p + p * v_dim1], abs(d__1)), 
                                        d__4 = (d__2 = v[q + q * v_dim1], abs(
                                        d__2));
                                temp1 = xsc * min(d__3,d__4);
                                v[p + q * v_dim1] = -d_sign(&temp1, &v[q + p *
                                         v_dim1]);
/* L8971: */
                            }
/* L8970: */
                        }
                    } else {
                        i__1 = nr - 1;
                        i__2 = nr - 1;
                        dlaset_("L", &i__1, &i__2, &c_b34, &c_b34, &v[v_dim1 
                                + 2], ldv);
                    }
/*              Now, compute R2 = L3 * Q3, the LQ factorization. */
                    i__1 = *lwork - (*n << 1) - *n * nr - nr;
                    dgelqf_(&nr, &nr, &v[v_offset], ldv, &work[(*n << 1) + *n 
                            * nr + 1], &work[(*n << 1) + *n * nr + nr + 1], &
                            i__1, &ierr);
/*              .. and estimate the condition number */
                    dlacpy_("L", &nr, &nr, &v[v_offset], ldv, &work[(*n << 1) 
                            + *n * nr + nr + 1], &nr);
                    i__1 = nr;
                    for (p = 1; p <= i__1; ++p) {
                        temp1 = dnrm2_(&p, &work[(*n << 1) + *n * nr + nr + p]
, &nr);
                        d__1 = 1. / temp1;
                        dscal_(&p, &d__1, &work[(*n << 1) + *n * nr + nr + p], 
                                 &nr);
/* L4950: */
                    }
                    dpocon_("L", &nr, &work[(*n << 1) + *n * nr + nr + 1], &
                            nr, &c_b35, &temp1, &work[(*n << 1) + *n * nr + 
                            nr + nr * nr + 1], &iwork[*m + (*n << 1) + 1], &
                            ierr);
                    condr2 = 1. / sqrt(temp1);

                    if (condr2 >= cond_ok__) {
/*                 .. save the Householder vectors used for Q3 */
/*                 (this overwrittes the copy of R2, as it will not be */
/*                 needed in this branch, but it does not overwritte the */
/*                 Huseholder vectors of Q2.). */
                        dlacpy_("U", &nr, &nr, &v[v_offset], ldv, &work[(*n <<
                                 1) + 1], n);
/*                 .. and the rest of the information on Q3 is in */
/*                 WORK(2*N+N*NR+1:2*N+N*NR+N) */
                    }

                }

                if (l2pert) {
                    xsc = sqrt(small);
                    i__1 = nr;
                    for (q = 2; q <= i__1; ++q) {
                        temp1 = xsc * v[q + q * v_dim1];
                        i__2 = q - 1;
                        for (p = 1; p <= i__2; ++p) {
/*                    V(p,q) = - DSIGN( TEMP1, V(q,p) ) */
                            v[p + q * v_dim1] = -d_sign(&temp1, &v[p + q * 
                                    v_dim1]);
/* L4969: */
                        }
/* L4968: */
                    }
                } else {
                    i__1 = nr - 1;
                    i__2 = nr - 1;
                    dlaset_("U", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 
                            1) + 1], ldv);
                }

/*        Second preconditioning finished; continue with Jacobi SVD */
/*        The input matrix is lower trinagular. */

/*        Recover the right singular vectors as solution of a well */
/*        conditioned triangular matrix equation. */

                if (condr1 < cond_ok__) {

                    i__1 = *lwork - (*n << 1) - *n * nr - nr;
                    dgesvj_("L", "U", "N", &nr, &nr, &v[v_offset], ldv, &sva[
                            1], &nr, &u[u_offset], ldu, &work[(*n << 1) + *n *
                             nr + nr + 1], &i__1, info);
                    scalem = work[(*n << 1) + *n * nr + nr + 1];
                    numrank = i_dnnt(&work[(*n << 1) + *n * nr + nr + 2]);
                    i__1 = nr;
                    for (p = 1; p <= i__1; ++p) {
                        dcopy_(&nr, &v[p * v_dim1 + 1], &c__1, &u[p * u_dim1 
                                + 1], &c__1);
                        dscal_(&nr, &sva[p], &v[p * v_dim1 + 1], &c__1);
/* L3970: */
                    }
/*        .. pick the right matrix equation and solve it */

                    if (nr == *n) {
/* :))             .. best case, R1 is inverted. The solution of this matrix */
/*                 equation is Q2*V2 = the product of the Jacobi rotations */
/*                 used in DGESVJ, premultiplied with the orthogonal matrix */
/*                 from the second QR factorization. */
                        dtrsm_("L", "U", "N", "N", &nr, &nr, &c_b35, &a[
                                a_offset], lda, &v[v_offset], ldv);
                    } else {
/*                 .. R1 is well conditioned, but non-square. Transpose(R2) */
/*                 is inverted to get the product of the Jacobi rotations */
/*                 used in DGESVJ. The Q-factor from the second QR */
/*                 factorization is then built in explicitly. */
                        dtrsm_("L", "U", "T", "N", &nr, &nr, &c_b35, &work[(*
                                n << 1) + 1], n, &v[v_offset], ldv);
                        if (nr < *n) {
                            i__1 = *n - nr;
                            dlaset_("A", &i__1, &nr, &c_b34, &c_b34, &v[nr + 
                                    1 + v_dim1], ldv);
                            i__1 = *n - nr;
                            dlaset_("A", &nr, &i__1, &c_b34, &c_b34, &v[(nr + 
                                    1) * v_dim1 + 1], ldv);
                            i__1 = *n - nr;
                            i__2 = *n - nr;
                            dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &v[nr 
                                    + 1 + (nr + 1) * v_dim1], ldv);
                        }
                        i__1 = *lwork - (*n << 1) - *n * nr - nr;
                        dormqr_("L", "N", n, n, &nr, &work[(*n << 1) + 1], n, 
                                &work[*n + 1], &v[v_offset], ldv, &work[(*n <<
                                 1) + *n * nr + nr + 1], &i__1, &ierr);
                    }

                } else if (condr2 < cond_ok__) {

/* :)           .. the input matrix A is very likely a relative of */
/*              the Kahan matrix :) */
/*              The matrix R2 is inverted. The solution of the matrix equation */
/*              is Q3^T*V3 = the product of the Jacobi rotations (appplied to */
/*              the lower triangular L3 from the LQ factorization of */
/*              R2=L3*Q3), pre-multiplied with the transposed Q3. */
                    i__1 = *lwork - (*n << 1) - *n * nr - nr;
                    dgesvj_("L", "U", "N", &nr, &nr, &v[v_offset], ldv, &sva[
                            1], &nr, &u[u_offset], ldu, &work[(*n << 1) + *n *
                             nr + nr + 1], &i__1, info);
                    scalem = work[(*n << 1) + *n * nr + nr + 1];
                    numrank = i_dnnt(&work[(*n << 1) + *n * nr + nr + 2]);
                    i__1 = nr;
                    for (p = 1; p <= i__1; ++p) {
                        dcopy_(&nr, &v[p * v_dim1 + 1], &c__1, &u[p * u_dim1 
                                + 1], &c__1);
                        dscal_(&nr, &sva[p], &u[p * u_dim1 + 1], &c__1);
/* L3870: */
                    }
                    dtrsm_("L", "U", "N", "N", &nr, &nr, &c_b35, &work[(*n << 
                            1) + 1], n, &u[u_offset], ldu);
/*              .. apply the permutation from the second QR factorization */
                    i__1 = nr;
                    for (q = 1; q <= i__1; ++q) {
                        i__2 = nr;
                        for (p = 1; p <= i__2; ++p) {
                            work[(*n << 1) + *n * nr + nr + iwork[*n + p]] = 
                                    u[p + q * u_dim1];
/* L872: */
                        }
                        i__2 = nr;
                        for (p = 1; p <= i__2; ++p) {
                            u[p + q * u_dim1] = work[(*n << 1) + *n * nr + nr 
                                    + p];
/* L874: */
                        }
/* L873: */
                    }
                    if (nr < *n) {
                        i__1 = *n - nr;
                        dlaset_("A", &i__1, &nr, &c_b34, &c_b34, &v[nr + 1 + 
                                v_dim1], ldv);
                        i__1 = *n - nr;
                        dlaset_("A", &nr, &i__1, &c_b34, &c_b34, &v[(nr + 1) *
                                 v_dim1 + 1], ldv);
                        i__1 = *n - nr;
                        i__2 = *n - nr;
                        dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &v[nr + 1 
                                + (nr + 1) * v_dim1], ldv);
                    }
                    i__1 = *lwork - (*n << 1) - *n * nr - nr;
                    dormqr_("L", "N", n, n, &nr, &work[(*n << 1) + 1], n, &
                            work[*n + 1], &v[v_offset], ldv, &work[(*n << 1) 
                            + *n * nr + nr + 1], &i__1, &ierr);
                } else {
/*              Last line of defense. */
/* #:(          This is a rather pathological case: no scaled condition */
/*              improvement after two pivoted QR factorizations. Other */
/*              possibility is that the rank revealing QR factorization */
/*              or the condition estimator has failed, or the COND_OK */
/*              is set very close to ONE (which is unnecessary). Normally, */
/*              this branch should never be executed, but in rare cases of */
/*              failure of the RRQR or condition estimator, the last line of */
/*              defense ensures that DGEJSV completes the task. */
/*              Compute the full SVD of L3 using DGESVJ with explicit */
/*              accumulation of Jacobi rotations. */
                    i__1 = *lwork - (*n << 1) - *n * nr - nr;
                    dgesvj_("L", "U", "V", &nr, &nr, &v[v_offset], ldv, &sva[
                            1], &nr, &u[u_offset], ldu, &work[(*n << 1) + *n *
                             nr + nr + 1], &i__1, info);
                    scalem = work[(*n << 1) + *n * nr + nr + 1];
                    numrank = i_dnnt(&work[(*n << 1) + *n * nr + nr + 2]);
                    if (nr < *n) {
                        i__1 = *n - nr;
                        dlaset_("A", &i__1, &nr, &c_b34, &c_b34, &v[nr + 1 + 
                                v_dim1], ldv);
                        i__1 = *n - nr;
                        dlaset_("A", &nr, &i__1, &c_b34, &c_b34, &v[(nr + 1) *
                                 v_dim1 + 1], ldv);
                        i__1 = *n - nr;
                        i__2 = *n - nr;
                        dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &v[nr + 1 
                                + (nr + 1) * v_dim1], ldv);
                    }
                    i__1 = *lwork - (*n << 1) - *n * nr - nr;
                    dormqr_("L", "N", n, n, &nr, &work[(*n << 1) + 1], n, &
                            work[*n + 1], &v[v_offset], ldv, &work[(*n << 1) 
                            + *n * nr + nr + 1], &i__1, &ierr);

                    i__1 = *lwork - (*n << 1) - *n * nr - nr;
                    dormlq_("L", "T", &nr, &nr, &nr, &work[(*n << 1) + 1], n, 
                            &work[(*n << 1) + *n * nr + 1], &u[u_offset], ldu, 
                             &work[(*n << 1) + *n * nr + nr + 1], &i__1, &
                            ierr);
                    i__1 = nr;
                    for (q = 1; q <= i__1; ++q) {
                        i__2 = nr;
                        for (p = 1; p <= i__2; ++p) {
                            work[(*n << 1) + *n * nr + nr + iwork[*n + p]] = 
                                    u[p + q * u_dim1];
/* L772: */
                        }
                        i__2 = nr;
                        for (p = 1; p <= i__2; ++p) {
                            u[p + q * u_dim1] = work[(*n << 1) + *n * nr + nr 
                                    + p];
/* L774: */
                        }
/* L773: */
                    }

                }

/*           Permute the rows of V using the (column) permutation from the */
/*           first QRF. Also, scale the columns to make them unit in */
/*           Euclidean norm. This applies to all cases. */

                temp1 = sqrt((doublereal) (*n)) * epsln;
                i__1 = *n;
                for (q = 1; q <= i__1; ++q) {
                    i__2 = *n;
                    for (p = 1; p <= i__2; ++p) {
                        work[(*n << 1) + *n * nr + nr + iwork[p]] = v[p + q * 
                                v_dim1];
/* L972: */
                    }
                    i__2 = *n;
                    for (p = 1; p <= i__2; ++p) {
                        v[p + q * v_dim1] = work[(*n << 1) + *n * nr + nr + p]
                                ;
/* L973: */
                    }
                    xsc = 1. / dnrm2_(n, &v[q * v_dim1 + 1], &c__1);
                    if (xsc < 1. - temp1 || xsc > temp1 + 1.) {
                        dscal_(n, &xsc, &v[q * v_dim1 + 1], &c__1);
                    }
/* L1972: */
                }
/*           At this moment, V contains the right singular vectors of A. */
/*           Next, assemble the left singular vector matrix U (M x N). */
                if (nr < *m) {
                    i__1 = *m - nr;
                    dlaset_("A", &i__1, &nr, &c_b34, &c_b34, &u[nr + 1 + 
                            u_dim1], ldu);
                    if (nr < n1) {
                        i__1 = n1 - nr;
                        dlaset_("A", &nr, &i__1, &c_b34, &c_b34, &u[(nr + 1) *
                                 u_dim1 + 1], ldu);
                        i__1 = *m - nr;
                        i__2 = n1 - nr;
                        dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &u[nr + 1 
                                + (nr + 1) * u_dim1], ldu);
                    }
                }

/*           The Q matrix from the first QRF is built into the left singular */
/*           matrix U. This applies to all cases. */

                i__1 = *lwork - *n;
                dormqr_("Left", "No_Tr", m, &n1, n, &a[a_offset], lda, &work[
                        1], &u[u_offset], ldu, &work[*n + 1], &i__1, &ierr);
/*           The columns of U are normalized. The cost is O(M*N) flops. */
                temp1 = sqrt((doublereal) (*m)) * epsln;
                i__1 = nr;
                for (p = 1; p <= i__1; ++p) {
                    xsc = 1. / dnrm2_(m, &u[p * u_dim1 + 1], &c__1);
                    if (xsc < 1. - temp1 || xsc > temp1 + 1.) {
                        dscal_(m, &xsc, &u[p * u_dim1 + 1], &c__1);
                    }
/* L1973: */
                }

/*           If the initial QRF is computed with row pivoting, the left */
/*           singular vectors must be adjusted. */

                if (rowpiv) {
                    i__1 = *m - 1;
                    dlaswp_(&n1, &u[u_offset], ldu, &c__1, &i__1, &iwork[(*n 
                            << 1) + 1], &c_n1);
                }

            } else {

/*        .. the initial matrix A has almost orthogonal columns and */
/*        the second QRF is not needed */

                dlacpy_("Upper", n, n, &a[a_offset], lda, &work[*n + 1], n);
                if (l2pert) {
                    xsc = sqrt(small);
                    i__1 = *n;
                    for (p = 2; p <= i__1; ++p) {
                        temp1 = xsc * work[*n + (p - 1) * *n + p];
                        i__2 = p - 1;
                        for (q = 1; q <= i__2; ++q) {
                            work[*n + (q - 1) * *n + p] = -d_sign(&temp1, &
                                    work[*n + (p - 1) * *n + q]);
/* L5971: */
                        }
/* L5970: */
                    }
                } else {
                    i__1 = *n - 1;
                    i__2 = *n - 1;
                    dlaset_("Lower", &i__1, &i__2, &c_b34, &c_b34, &work[*n + 
                            2], n);
                }

                i__1 = *lwork - *n - *n * *n;
                dgesvj_("Upper", "U", "N", n, n, &work[*n + 1], n, &sva[1], n, 
                         &u[u_offset], ldu, &work[*n + *n * *n + 1], &i__1, 
                        info);

                scalem = work[*n + *n * *n + 1];
                numrank = i_dnnt(&work[*n + *n * *n + 2]);
                i__1 = *n;
                for (p = 1; p <= i__1; ++p) {
                    dcopy_(n, &work[*n + (p - 1) * *n + 1], &c__1, &u[p * 
                            u_dim1 + 1], &c__1);
                    dscal_(n, &sva[p], &work[*n + (p - 1) * *n + 1], &c__1);
/* L6970: */
                }

                dtrsm_("Left", "Upper", "NoTrans", "No UD", n, n, &c_b35, &a[
                        a_offset], lda, &work[*n + 1], n);
                i__1 = *n;
                for (p = 1; p <= i__1; ++p) {
                    dcopy_(n, &work[*n + p], n, &v[iwork[p] + v_dim1], ldv);
/* L6972: */
                }
                temp1 = sqrt((doublereal) (*n)) * epsln;
                i__1 = *n;
                for (p = 1; p <= i__1; ++p) {
                    xsc = 1. / dnrm2_(n, &v[p * v_dim1 + 1], &c__1);
                    if (xsc < 1. - temp1 || xsc > temp1 + 1.) {
                        dscal_(n, &xsc, &v[p * v_dim1 + 1], &c__1);
                    }
/* L6971: */
                }

/*           Assemble the left singular vector matrix U (M x N). */

                if (*n < *m) {
                    i__1 = *m - *n;
                    dlaset_("A", &i__1, n, &c_b34, &c_b34, &u[nr + 1 + u_dim1]
, ldu);
                    if (*n < n1) {
                        i__1 = n1 - *n;
                        dlaset_("A", n, &i__1, &c_b34, &c_b34, &u[(*n + 1) * 
                                u_dim1 + 1], ldu);
                        i__1 = *m - *n;
                        i__2 = n1 - *n;
                        dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &u[nr + 1 
                                + (*n + 1) * u_dim1], ldu);
                    }
                }
                i__1 = *lwork - *n;
                dormqr_("Left", "No Tr", m, &n1, n, &a[a_offset], lda, &work[
                        1], &u[u_offset], ldu, &work[*n + 1], &i__1, &ierr);
                temp1 = sqrt((doublereal) (*m)) * epsln;
                i__1 = n1;
                for (p = 1; p <= i__1; ++p) {
                    xsc = 1. / dnrm2_(m, &u[p * u_dim1 + 1], &c__1);
                    if (xsc < 1. - temp1 || xsc > temp1 + 1.) {
                        dscal_(m, &xsc, &u[p * u_dim1 + 1], &c__1);
                    }
/* L6973: */
                }

                if (rowpiv) {
                    i__1 = *m - 1;
                    dlaswp_(&n1, &u[u_offset], ldu, &c__1, &i__1, &iwork[(*n 
                            << 1) + 1], &c_n1);
                }

            }

/*        end of the  >> almost orthogonal case <<  in the full SVD */

        } else {

/*        This branch deploys a preconditioned Jacobi SVD with explicitly */
/*        accumulated rotations. It is included as optional, mainly for */
/*        experimental purposes. It does perfom well, and can also be used. */
/*        In this implementation, this branch will be automatically activated */
/*        if the  condition number sigma_max(A) / sigma_min(A) is predicted */
/*        to be greater than the overflow threshold. This is because the */
/*        a posteriori computation of the singular vectors assumes robust */
/*        implementation of BLAS and some LAPACK procedures, capable of working */
/*        in presence of extreme values. Since that is not always the case, ... */

            i__1 = nr;
            for (p = 1; p <= i__1; ++p) {
                i__2 = *n - p + 1;
                dcopy_(&i__2, &a[p + p * a_dim1], lda, &v[p + p * v_dim1], &
                        c__1);
/* L7968: */
            }

            if (l2pert) {
                xsc = sqrt(small / epsln);
                i__1 = nr;
                for (q = 1; q <= i__1; ++q) {
                    temp1 = xsc * (d__1 = v[q + q * v_dim1], abs(d__1));
                    i__2 = *n;
                    for (p = 1; p <= i__2; ++p) {
                        if (p > q && (d__1 = v[p + q * v_dim1], abs(d__1)) <= 
                                temp1 || p < q) {
                            v[p + q * v_dim1] = d_sign(&temp1, &v[p + q * 
                                    v_dim1]);
                        }
                        if (p < q) {
                            v[p + q * v_dim1] = -v[p + q * v_dim1];
                        }
/* L5968: */
                    }
/* L5969: */
                }
            } else {
                i__1 = nr - 1;
                i__2 = nr - 1;
                dlaset_("U", &i__1, &i__2, &c_b34, &c_b34, &v[(v_dim1 << 1) + 
                        1], ldv);
            }
            i__1 = *lwork - (*n << 1);
            dgeqrf_(n, &nr, &v[v_offset], ldv, &work[*n + 1], &work[(*n << 1) 
                    + 1], &i__1, &ierr);
            dlacpy_("L", n, &nr, &v[v_offset], ldv, &work[(*n << 1) + 1], n);

            i__1 = nr;
            for (p = 1; p <= i__1; ++p) {
                i__2 = nr - p + 1;
                dcopy_(&i__2, &v[p + p * v_dim1], ldv, &u[p + p * u_dim1], &
                        c__1);
/* L7969: */
            }
            if (l2pert) {
                xsc = sqrt(small / epsln);
                i__1 = nr;
                for (q = 2; q <= i__1; ++q) {
                    i__2 = q - 1;
                    for (p = 1; p <= i__2; ++p) {
/* Computing MIN */
                        d__3 = (d__1 = u[p + p * u_dim1], abs(d__1)), d__4 = (
                                d__2 = u[q + q * u_dim1], abs(d__2));
                        temp1 = xsc * min(d__3,d__4);
                        u[p + q * u_dim1] = -d_sign(&temp1, &u[q + p * u_dim1]
                                );
/* L9971: */
                    }
/* L9970: */
                }
            } else {
                i__1 = nr - 1;
                i__2 = nr - 1;
                dlaset_("U", &i__1, &i__2, &c_b34, &c_b34, &u[(u_dim1 << 1) + 
                        1], ldu);
            }
            i__1 = *lwork - (*n << 1) - *n * nr;
            dgesvj_("G", "U", "V", &nr, &nr, &u[u_offset], ldu, &sva[1], n, &
                    v[v_offset], ldv, &work[(*n << 1) + *n * nr + 1], &i__1, 
                    info);
            scalem = work[(*n << 1) + *n * nr + 1];
            numrank = i_dnnt(&work[(*n << 1) + *n * nr + 2]);
            if (nr < *n) {
                i__1 = *n - nr;
                dlaset_("A", &i__1, &nr, &c_b34, &c_b34, &v[nr + 1 + v_dim1], 
                        ldv);
                i__1 = *n - nr;
                dlaset_("A", &nr, &i__1, &c_b34, &c_b34, &v[(nr + 1) * v_dim1 
                        + 1], ldv);
                i__1 = *n - nr;
                i__2 = *n - nr;
                dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &v[nr + 1 + (nr + 
                        1) * v_dim1], ldv);
            }
            i__1 = *lwork - (*n << 1) - *n * nr - nr;
            dormqr_("L", "N", n, n, &nr, &work[(*n << 1) + 1], n, &work[*n + 
                    1], &v[v_offset], ldv, &work[(*n << 1) + *n * nr + nr + 1]
, &i__1, &ierr);

/*           Permute the rows of V using the (column) permutation from the */
/*           first QRF. Also, scale the columns to make them unit in */
/*           Euclidean norm. This applies to all cases. */

            temp1 = sqrt((doublereal) (*n)) * epsln;
            i__1 = *n;
            for (q = 1; q <= i__1; ++q) {
                i__2 = *n;
                for (p = 1; p <= i__2; ++p) {
                    work[(*n << 1) + *n * nr + nr + iwork[p]] = v[p + q * 
                            v_dim1];
/* L8972: */
                }
                i__2 = *n;
                for (p = 1; p <= i__2; ++p) {
                    v[p + q * v_dim1] = work[(*n << 1) + *n * nr + nr + p];
/* L8973: */
                }
                xsc = 1. / dnrm2_(n, &v[q * v_dim1 + 1], &c__1);
                if (xsc < 1. - temp1 || xsc > temp1 + 1.) {
                    dscal_(n, &xsc, &v[q * v_dim1 + 1], &c__1);
                }
/* L7972: */
            }

/*           At this moment, V contains the right singular vectors of A. */
/*           Next, assemble the left singular vector matrix U (M x N). */

            if (*n < *m) {
                i__1 = *m - *n;
                dlaset_("A", &i__1, n, &c_b34, &c_b34, &u[nr + 1 + u_dim1], 
                        ldu);
                if (*n < n1) {
                    i__1 = n1 - *n;
                    dlaset_("A", n, &i__1, &c_b34, &c_b34, &u[(*n + 1) * 
                            u_dim1 + 1], ldu);
                    i__1 = *m - *n;
                    i__2 = n1 - *n;
                    dlaset_("A", &i__1, &i__2, &c_b34, &c_b35, &u[nr + 1 + (*
                            n + 1) * u_dim1], ldu);
                }
            }

            i__1 = *lwork - *n;
            dormqr_("Left", "No Tr", m, &n1, n, &a[a_offset], lda, &work[1], &
                    u[u_offset], ldu, &work[*n + 1], &i__1, &ierr);

            if (rowpiv) {
                i__1 = *m - 1;
                dlaswp_(&n1, &u[u_offset], ldu, &c__1, &i__1, &iwork[(*n << 1)
                         + 1], &c_n1);
            }


        }
        if (transp) {
/*           .. swap U and V because the procedure worked on A^t */
            i__1 = *n;
            for (p = 1; p <= i__1; ++p) {
                dswap_(n, &u[p * u_dim1 + 1], &c__1, &v[p * v_dim1 + 1], &
                        c__1);
/* L6974: */
            }
        }

    }
/*     end of the full SVD */

/*     Undo scaling, if necessary (and possible) */

    if (uscal2 <= big / sva[1] * uscal1) {
        dlascl_("G", &c__0, &c__0, &uscal1, &uscal2, &nr, &c__1, &sva[1], n, &
                ierr);
        uscal1 = 1.;
        uscal2 = 1.;
    }

    if (nr < *n) {
        i__1 = *n;
        for (p = nr + 1; p <= i__1; ++p) {
            sva[p] = 0.;
/* L3004: */
        }
    }

    work[1] = uscal2 * scalem;
    work[2] = uscal1;
    if (errest) {
        work[3] = sconda;
    }
    if (lsvec && rsvec) {
        work[4] = condr1;
        work[5] = condr2;
    }
    if (l2tran) {
        work[6] = entra;
        work[7] = entrat;
    }

    iwork[1] = nr;
    iwork[2] = numrank;
    iwork[3] = warning;

    return 0;
/*     .. */
/*     .. END OF DGEJSV */
/*     .. */
} /* dgejsv_ */