sgsvj0.c

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/* sgsvj0.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 integer c__0 = 0;
static real c_b42 = 1.f;

/* Subroutine */ int sgsvj0_(char *jobv, integer *m, integer *n, real *a, 
        integer *lda, real *d__, real *sva, integer *mv, real *v, integer *
        ldv, real *eps, real *sfmin, real *tol, integer *nsweep, real *work, 
        integer *lwork, integer *info)
{
    /* System generated locals */
    integer a_dim1, a_offset, v_dim1, v_offset, i__1, i__2, i__3, i__4, i__5, 
            i__6;
    real r__1, r__2;

    /* Builtin functions */
    double sqrt(doublereal), r_sign(real *, real *);

    /* Local variables */
    real bigtheta;
    integer pskipped, i__, p, q;
    real t, rootsfmin, cs, sn;
    integer ir1, jbc;
    real big;
    integer kbl, igl, ibr, jgl, nbl, mvl;
    real aapp, aapq, aaqq;
    integer ierr;
    extern doublereal sdot_(integer *, real *, integer *, real *, integer *);
    real aapp0, temp1;
    extern doublereal snrm2_(integer *, real *, integer *);
    real apoaq, aqoap;
    extern logical lsame_(char *, char *);
    real theta, small, fastr[5];
    logical applv, rsvec;
    extern /* Subroutine */ int scopy_(integer *, real *, integer *, real *, 
            integer *);
    logical rotok;
    extern /* Subroutine */ int sswap_(integer *, real *, integer *, real *, 
            integer *), saxpy_(integer *, real *, real *, integer *, real *, 
            integer *), srotm_(integer *, real *, integer *, real *, integer *
, real *), xerbla_(char *, integer *);
    integer ijblsk, swband;
    extern /* Subroutine */ int slascl_(char *, integer *, integer *, real *, 
            real *, integer *, integer *, real *, integer *, integer *);
    extern integer isamax_(integer *, real *, integer *);
    integer blskip;
    real mxaapq, thsign;
    extern /* Subroutine */ int slassq_(integer *, real *, integer *, real *, 
            real *);
    real mxsinj;
    integer emptsw, notrot, iswrot, lkahead;
    real rootbig, rooteps;
    integer rowskip;
    real roottol;


/*  -- 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 */
/*  ~~~~~~~ */
/*  SGSVJ0 is called from SGESVJ as a pre-processor and that is its main */
/*  purpose. It applies Jacobi rotations in the same way as SGESVJ does, but */
/*  it does not check convergence (stopping criterion). Few tuning */
/*  parameters (marked by [TP]) are available for the implementer. */

/*  Further details */
/*  ~~~~~~~~~~~~~~~ */
/*  SGSVJ0 is used just to enable SGESVJ to call a simplified version of */
/*  itself to work on a submatrix of the original matrix. */

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

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

/*  Arguments */
/*  ~~~~~~~~~ */

/*  JOBV    (input) CHARACTER*1 */
/*          Specifies whether the output from this procedure is used */
/*          to compute the matrix V: */
/*          = 'V': the product of the Jacobi rotations is accumulated */
/*                 by postmulyiplying the N-by-N array V. */
/*                (See the description of V.) */
/*          = 'A': the product of the Jacobi rotations is accumulated */
/*                 by postmulyiplying the MV-by-N array V. */
/*                (See the descriptions of MV and V.) */
/*          = 'N': the Jacobi rotations are not accumulated. */

/*  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/output) REAL array, dimension (LDA,N) */
/*          On entry, M-by-N matrix A, such that A*diag(D) represents */
/*          the input matrix. */
/*          On exit, */
/*          A_onexit * D_onexit represents the input matrix A*diag(D) */
/*          post-multiplied by a sequence of Jacobi rotations, where the */
/*          rotation threshold and the total number of sweeps are given in */
/*          TOL and NSWEEP, respectively. */
/*          (See the descriptions of D, TOL and NSWEEP.) */

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

/*  D       (input/workspace/output) REAL array, dimension (N) */
/*          The array D accumulates the scaling factors from the fast scaled */
/*          Jacobi rotations. */
/*          On entry, A*diag(D) represents the input matrix. */
/*          On exit, A_onexit*diag(D_onexit) represents the input matrix */
/*          post-multiplied by a sequence of Jacobi rotations, where the */
/*          rotation threshold and the total number of sweeps are given in */
/*          TOL and NSWEEP, respectively. */
/*          (See the descriptions of A, TOL and NSWEEP.) */

/*  SVA     (input/workspace/output) REAL array, dimension (N) */
/*          On entry, SVA contains the Euclidean norms of the columns of */
/*          the matrix A*diag(D). */
/*          On exit, SVA contains the Euclidean norms of the columns of */
/*          the matrix onexit*diag(D_onexit). */

/*  MV      (input) INTEGER */
/*          If JOBV .EQ. 'A', then MV rows of V are post-multipled by a */
/*                           sequence of Jacobi rotations. */
/*          If JOBV = 'N',   then MV is not referenced. */

/*  V       (input/output) REAL array, dimension (LDV,N) */
/*          If JOBV .EQ. 'V' then N rows of V are post-multipled by a */
/*                           sequence of Jacobi rotations. */
/*          If JOBV .EQ. 'A' then MV rows of V are post-multipled by a */
/*                           sequence of Jacobi rotations. */
/*          If JOBV = 'N',   then V is not referenced. */

/*  LDV     (input) INTEGER */
/*          The leading dimension of the array V,  LDV >= 1. */
/*          If JOBV = 'V', LDV .GE. N. */
/*          If JOBV = 'A', LDV .GE. MV. */

/*  EPS     (input) INTEGER */
/*          EPS = SLAMCH('Epsilon') */

/*  SFMIN   (input) INTEGER */
/*          SFMIN = SLAMCH('Safe Minimum') */

/*  TOL     (input) REAL */
/*          TOL is the threshold for Jacobi rotations. For a pair */
/*          A(:,p), A(:,q) of pivot columns, the Jacobi rotation is */
/*          applied only if ABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL. */

/*  NSWEEP  (input) INTEGER */
/*          NSWEEP is the number of sweeps of Jacobi rotations to be */
/*          performed. */

/*  WORK    (workspace) REAL array, dimension LWORK. */

/*  LWORK   (input) INTEGER */
/*          LWORK is the dimension of WORK. LWORK .GE. M. */

/*  INFO    (output) INTEGER */
/*          = 0 : successful exit. */
/*          < 0 : if INFO = -i, then the i-th argument had an illegal value */

/*     Local Parameters */
/*     Local Scalars */
/*     Local Arrays */

/*     Intrinsic Functions */

/*     External Functions */

/*     External Subroutines */

/*     ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~| */

    /* Parameter adjustments */
    --sva;
    --d__;
    a_dim1 = *lda;
    a_offset = 1 + a_dim1;
    a -= a_offset;
    v_dim1 = *ldv;
    v_offset = 1 + v_dim1;
    v -= v_offset;
    --work;

    /* Function Body */
    applv = lsame_(jobv, "A");
    rsvec = lsame_(jobv, "V");
    if (! (rsvec || applv || lsame_(jobv, "N"))) {
        *info = -1;
    } else if (*m < 0) {
        *info = -2;
    } else if (*n < 0 || *n > *m) {
        *info = -3;
    } else if (*lda < *m) {
        *info = -5;
    } else if (*mv < 0) {
        *info = -8;
    } else if (*ldv < *m) {
        *info = -10;
    } else if (*tol <= *eps) {
        *info = -13;
    } else if (*nsweep < 0) {
        *info = -14;
    } else if (*lwork < *m) {
        *info = -16;
    } else {
        *info = 0;
    }

/*     #:( */
    if (*info != 0) {
        i__1 = -(*info);
        xerbla_("SGSVJ0", &i__1);
        return 0;
    }

    if (rsvec) {
        mvl = *n;
    } else if (applv) {
        mvl = *mv;
    }
    rsvec = rsvec || applv;
    rooteps = sqrt(*eps);
    rootsfmin = sqrt(*sfmin);
    small = *sfmin / *eps;
    big = 1.f / *sfmin;
    rootbig = 1.f / rootsfmin;
    bigtheta = 1.f / rooteps;
    roottol = sqrt(*tol);


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

    emptsw = *n * (*n - 1) / 2;
    notrot = 0;
    fastr[0] = 0.f;

/*     -#- Row-cyclic pivot strategy with de Rijk's pivoting -#- */

    swband = 0;
/* [TP] SWBAND is a tuning parameter. It is meaningful and effective */
/*     if SGESVJ is used as a computational routine in the preconditioned */
/*     Jacobi SVD algorithm SGESVJ. For sweeps i=1:SWBAND the procedure */
/*     ...... */
    kbl = min(8,*n);
/* [TP] KBL is a tuning parameter that defines the tile size in the */
/*     tiling of the p-q loops of pivot pairs. In general, an optimal */
/*     value of KBL depends on the matrix dimensions and on the */
/*     parameters of the computer's memory. */

    nbl = *n / kbl;
    if (nbl * kbl != *n) {
        ++nbl;
    }
/* Computing 2nd power */
    i__1 = kbl;
    blskip = i__1 * i__1 + 1;
/* [TP] BLKSKIP is a tuning parameter that depends on SWBAND and KBL. */
    rowskip = min(5,kbl);
/* [TP] ROWSKIP is a tuning parameter. */
    lkahead = 1;
/* [TP] LKAHEAD is a tuning parameter. */
    swband = 0;
    pskipped = 0;

    i__1 = *nsweep;
    for (i__ = 1; i__ <= i__1; ++i__) {
/*     .. go go go ... */

        mxaapq = 0.f;
        mxsinj = 0.f;
        iswrot = 0;

        notrot = 0;
        pskipped = 0;

        i__2 = nbl;
        for (ibr = 1; ibr <= i__2; ++ibr) {
            igl = (ibr - 1) * kbl + 1;

/* Computing MIN */
            i__4 = lkahead, i__5 = nbl - ibr;
            i__3 = min(i__4,i__5);
            for (ir1 = 0; ir1 <= i__3; ++ir1) {

                igl += ir1 * kbl;

/* Computing MIN */
                i__5 = igl + kbl - 1, i__6 = *n - 1;
                i__4 = min(i__5,i__6);
                for (p = igl; p <= i__4; ++p) {
/*     .. de Rijk's pivoting */
                    i__5 = *n - p + 1;
                    q = isamax_(&i__5, &sva[p], &c__1) + p - 1;
                    if (p != q) {
                        sswap_(m, &a[p * a_dim1 + 1], &c__1, &a[q * a_dim1 + 
                                1], &c__1);
                        if (rsvec) {
                            sswap_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[q * 
                                    v_dim1 + 1], &c__1);
                        }
                        temp1 = sva[p];
                        sva[p] = sva[q];
                        sva[q] = temp1;
                        temp1 = d__[p];
                        d__[p] = d__[q];
                        d__[q] = temp1;
                    }

                    if (ir1 == 0) {

/*        Column norms are periodically updated by explicit */
/*        norm computation. */
/*        Caveat: */
/*        Some BLAS implementations compute SNRM2(M,A(1,p),1) */
/*        as SQRT(SDOT(M,A(1,p),1,A(1,p),1)), which may result in */
/*        overflow for ||A(:,p)||_2 > SQRT(overflow_threshold), and */
/*        undeflow for ||A(:,p)||_2 < SQRT(underflow_threshold). */
/*        Hence, SNRM2 cannot be trusted, not even in the case when */
/*        the true norm is far from the under(over)flow boundaries. */
/*        If properly implemented SNRM2 is available, the IF-THEN-ELSE */
/*        below should read "AAPP = SNRM2( M, A(1,p), 1 ) * D(p)". */

                        if (sva[p] < rootbig && sva[p] > rootsfmin) {
                            sva[p] = snrm2_(m, &a[p * a_dim1 + 1], &c__1) * 
                                    d__[p];
                        } else {
                            temp1 = 0.f;
                            aapp = 0.f;
                            slassq_(m, &a[p * a_dim1 + 1], &c__1, &temp1, &
                                    aapp);
                            sva[p] = temp1 * sqrt(aapp) * d__[p];
                        }
                        aapp = sva[p];
                    } else {
                        aapp = sva[p];
                    }

                    if (aapp > 0.f) {

                        pskipped = 0;

/* Computing MIN */
                        i__6 = igl + kbl - 1;
                        i__5 = min(i__6,*n);
                        for (q = p + 1; q <= i__5; ++q) {

                            aaqq = sva[q];
                            if (aaqq > 0.f) {

                                aapp0 = aapp;
                                if (aaqq >= 1.f) {
                                    rotok = small * aapp <= aaqq;
                                    if (aapp < big / aaqq) {
                                        aapq = sdot_(m, &a[p * a_dim1 + 1], &
                                                c__1, &a[q * a_dim1 + 1], &
                                                c__1) * d__[p] * d__[q] / 
                                                aaqq / aapp;
                                    } else {
                                        scopy_(m, &a[p * a_dim1 + 1], &c__1, &
                                                work[1], &c__1);
                                        slascl_("G", &c__0, &c__0, &aapp, &
                                                d__[p], m, &c__1, &work[1], 
                                                lda, &ierr);
                                        aapq = sdot_(m, &work[1], &c__1, &a[q 
                                                * a_dim1 + 1], &c__1) * d__[q]
                                                 / aaqq;
                                    }
                                } else {
                                    rotok = aapp <= aaqq / small;
                                    if (aapp > small / aaqq) {
                                        aapq = sdot_(m, &a[p * a_dim1 + 1], &
                                                c__1, &a[q * a_dim1 + 1], &
                                                c__1) * d__[p] * d__[q] / 
                                                aaqq / aapp;
                                    } else {
                                        scopy_(m, &a[q * a_dim1 + 1], &c__1, &
                                                work[1], &c__1);
                                        slascl_("G", &c__0, &c__0, &aaqq, &
                                                d__[q], m, &c__1, &work[1], 
                                                lda, &ierr);
                                        aapq = sdot_(m, &work[1], &c__1, &a[p 
                                                * a_dim1 + 1], &c__1) * d__[p]
                                                 / aapp;
                                    }
                                }

/* Computing MAX */
                                r__1 = mxaapq, r__2 = dabs(aapq);
                                mxaapq = dmax(r__1,r__2);

/*        TO rotate or NOT to rotate, THAT is the question ... */

                                if (dabs(aapq) > *tol) {

/*           .. rotate */
/*           ROTATED = ROTATED + ONE */

                                    if (ir1 == 0) {
                                        notrot = 0;
                                        pskipped = 0;
                                        ++iswrot;
                                    }

                                    if (rotok) {

                                        aqoap = aaqq / aapp;
                                        apoaq = aapp / aaqq;
                                        theta = (r__1 = aqoap - apoaq, dabs(
                                                r__1)) * -.5f / aapq;

                                        if (dabs(theta) > bigtheta) {

                                            t = .5f / theta;
                                            fastr[2] = t * d__[p] / d__[q];
                                            fastr[3] = -t * d__[q] / d__[p];
                                            srotm_(m, &a[p * a_dim1 + 1], &
                                                    c__1, &a[q * a_dim1 + 1], 
                                                    &c__1, fastr);
                                            if (rsvec) {
                          srotm_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[q * 
                                  v_dim1 + 1], &c__1, fastr);
                                            }
/* Computing MAX */
                                            r__1 = 0.f, r__2 = t * apoaq * 
                                                    aapq + 1.f;
                                            sva[q] = aaqq * sqrt((dmax(r__1,
                                                    r__2)));
                                            aapp *= sqrt(1.f - t * aqoap * 
                                                    aapq);
/* Computing MAX */
                                            r__1 = mxsinj, r__2 = dabs(t);
                                            mxsinj = dmax(r__1,r__2);

                                        } else {

/*                 .. choose correct signum for THETA and rotate */

                                            thsign = -r_sign(&c_b42, &aapq);
                                            t = 1.f / (theta + thsign * sqrt(
                                                    theta * theta + 1.f));
                                            cs = sqrt(1.f / (t * t + 1.f));
                                            sn = t * cs;

/* Computing MAX */
                                            r__1 = mxsinj, r__2 = dabs(sn);
                                            mxsinj = dmax(r__1,r__2);
/* Computing MAX */
                                            r__1 = 0.f, r__2 = t * apoaq * 
                                                    aapq + 1.f;
                                            sva[q] = aaqq * sqrt((dmax(r__1,
                                                    r__2)));
/* Computing MAX */
                                            r__1 = 0.f, r__2 = 1.f - t * 
                                                    aqoap * aapq;
                                            aapp *= sqrt((dmax(r__1,r__2)));

                                            apoaq = d__[p] / d__[q];
                                            aqoap = d__[q] / d__[p];
                                            if (d__[p] >= 1.f) {
                          if (d__[q] >= 1.f) {
                              fastr[2] = t * apoaq;
                              fastr[3] = -t * aqoap;
                              d__[p] *= cs;
                              d__[q] *= cs;
                              srotm_(m, &a[p * a_dim1 + 1], &c__1, &a[q * 
                                      a_dim1 + 1], &c__1, fastr);
                              if (rsvec) {
                                  srotm_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[
                                          q * v_dim1 + 1], &c__1, fastr);
                              }
                          } else {
                              r__1 = -t * aqoap;
                              saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, &a[
                                      p * a_dim1 + 1], &c__1);
                              r__1 = cs * sn * apoaq;
                              saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, &a[
                                      q * a_dim1 + 1], &c__1);
                              d__[p] *= cs;
                              d__[q] /= cs;
                              if (rsvec) {
                                  r__1 = -t * aqoap;
                                  saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], &
                                          c__1, &v[p * v_dim1 + 1], &c__1);
                                  r__1 = cs * sn * apoaq;
                                  saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], &
                                          c__1, &v[q * v_dim1 + 1], &c__1);
                              }
                          }
                                            } else {
                          if (d__[q] >= 1.f) {
                              r__1 = t * apoaq;
                              saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, &a[
                                      q * a_dim1 + 1], &c__1);
                              r__1 = -cs * sn * aqoap;
                              saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, &a[
                                      p * a_dim1 + 1], &c__1);
                              d__[p] /= cs;
                              d__[q] *= cs;
                              if (rsvec) {
                                  r__1 = t * apoaq;
                                  saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], &
                                          c__1, &v[q * v_dim1 + 1], &c__1);
                                  r__1 = -cs * sn * aqoap;
                                  saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], &
                                          c__1, &v[p * v_dim1 + 1], &c__1);
                              }
                          } else {
                              if (d__[p] >= d__[q]) {
                                  r__1 = -t * aqoap;
                                  saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, 
                                          &a[p * a_dim1 + 1], &c__1);
                                  r__1 = cs * sn * apoaq;
                                  saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, 
                                          &a[q * a_dim1 + 1], &c__1);
                                  d__[p] *= cs;
                                  d__[q] /= cs;
                                  if (rsvec) {
                                      r__1 = -t * aqoap;
                                      saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], 
                                              &c__1, &v[p * v_dim1 + 1], &
                                              c__1);
                                      r__1 = cs * sn * apoaq;
                                      saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], 
                                              &c__1, &v[q * v_dim1 + 1], &
                                              c__1);
                                  }
                              } else {
                                  r__1 = t * apoaq;
                                  saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, 
                                          &a[q * a_dim1 + 1], &c__1);
                                  r__1 = -cs * sn * aqoap;
                                  saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, 
                                          &a[p * a_dim1 + 1], &c__1);
                                  d__[p] /= cs;
                                  d__[q] *= cs;
                                  if (rsvec) {
                                      r__1 = t * apoaq;
                                      saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], 
                                              &c__1, &v[q * v_dim1 + 1], &
                                              c__1);
                                      r__1 = -cs * sn * aqoap;
                                      saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], 
                                              &c__1, &v[p * v_dim1 + 1], &
                                              c__1);
                                  }
                              }
                          }
                                            }
                                        }

                                    } else {
/*              .. have to use modified Gram-Schmidt like transformation */
                                        scopy_(m, &a[p * a_dim1 + 1], &c__1, &
                                                work[1], &c__1);
                                        slascl_("G", &c__0, &c__0, &aapp, &
                                                c_b42, m, &c__1, &work[1], 
                                                lda, &ierr);
                                        slascl_("G", &c__0, &c__0, &aaqq, &
                                                c_b42, m, &c__1, &a[q * 
                                                a_dim1 + 1], lda, &ierr);
                                        temp1 = -aapq * d__[p] / d__[q];
                                        saxpy_(m, &temp1, &work[1], &c__1, &a[
                                                q * a_dim1 + 1], &c__1);
                                        slascl_("G", &c__0, &c__0, &c_b42, &
                                                aaqq, m, &c__1, &a[q * a_dim1 
                                                + 1], lda, &ierr);
/* Computing MAX */
                                        r__1 = 0.f, r__2 = 1.f - aapq * aapq;
                                        sva[q] = aaqq * sqrt((dmax(r__1,r__2))
                                                );
                                        mxsinj = dmax(mxsinj,*sfmin);
                                    }
/*           END IF ROTOK THEN ... ELSE */

/*           In the case of cancellation in updating SVA(q), SVA(p) */
/*           recompute SVA(q), SVA(p). */
/* Computing 2nd power */
                                    r__1 = sva[q] / aaqq;
                                    if (r__1 * r__1 <= rooteps) {
                                        if (aaqq < rootbig && aaqq > 
                                                rootsfmin) {
                                            sva[q] = snrm2_(m, &a[q * a_dim1 
                                                    + 1], &c__1) * d__[q];
                                        } else {
                                            t = 0.f;
                                            aaqq = 0.f;
                                            slassq_(m, &a[q * a_dim1 + 1], &
                                                    c__1, &t, &aaqq);
                                            sva[q] = t * sqrt(aaqq) * d__[q];
                                        }
                                    }
                                    if (aapp / aapp0 <= rooteps) {
                                        if (aapp < rootbig && aapp > 
                                                rootsfmin) {
                                            aapp = snrm2_(m, &a[p * a_dim1 + 
                                                    1], &c__1) * d__[p];
                                        } else {
                                            t = 0.f;
                                            aapp = 0.f;
                                            slassq_(m, &a[p * a_dim1 + 1], &
                                                    c__1, &t, &aapp);
                                            aapp = t * sqrt(aapp) * d__[p];
                                        }
                                        sva[p] = aapp;
                                    }

                                } else {
/*        A(:,p) and A(:,q) already numerically orthogonal */
                                    if (ir1 == 0) {
                                        ++notrot;
                                    }
                                    ++pskipped;
                                }
                            } else {
/*        A(:,q) is zero column */
                                if (ir1 == 0) {
                                    ++notrot;
                                }
                                ++pskipped;
                            }

                            if (i__ <= swband && pskipped > rowskip) {
                                if (ir1 == 0) {
                                    aapp = -aapp;
                                }
                                notrot = 0;
                                goto L2103;
                            }

/* L2002: */
                        }
/*     END q-LOOP */

L2103:
/*     bailed out of q-loop */
                        sva[p] = aapp;
                    } else {
                        sva[p] = aapp;
                        if (ir1 == 0 && aapp == 0.f) {
/* Computing MIN */
                            i__5 = igl + kbl - 1;
                            notrot = notrot + min(i__5,*n) - p;
                        }
                    }

/* L2001: */
                }
/*     end of the p-loop */
/*     end of doing the block ( ibr, ibr ) */
/* L1002: */
            }
/*     end of ir1-loop */

/* ........................................................ */
/* ... go to the off diagonal blocks */

            igl = (ibr - 1) * kbl + 1;

            i__3 = nbl;
            for (jbc = ibr + 1; jbc <= i__3; ++jbc) {

                jgl = (jbc - 1) * kbl + 1;

/*        doing the block at ( ibr, jbc ) */

                ijblsk = 0;
/* Computing MIN */
                i__5 = igl + kbl - 1;
                i__4 = min(i__5,*n);
                for (p = igl; p <= i__4; ++p) {

                    aapp = sva[p];

                    if (aapp > 0.f) {

                        pskipped = 0;

/* Computing MIN */
                        i__6 = jgl + kbl - 1;
                        i__5 = min(i__6,*n);
                        for (q = jgl; q <= i__5; ++q) {

                            aaqq = sva[q];

                            if (aaqq > 0.f) {
                                aapp0 = aapp;

/*     -#- M x 2 Jacobi SVD -#- */

/*        -#- Safe Gram matrix computation -#- */

                                if (aaqq >= 1.f) {
                                    if (aapp >= aaqq) {
                                        rotok = small * aapp <= aaqq;
                                    } else {
                                        rotok = small * aaqq <= aapp;
                                    }
                                    if (aapp < big / aaqq) {
                                        aapq = sdot_(m, &a[p * a_dim1 + 1], &
                                                c__1, &a[q * a_dim1 + 1], &
                                                c__1) * d__[p] * d__[q] / 
                                                aaqq / aapp;
                                    } else {
                                        scopy_(m, &a[p * a_dim1 + 1], &c__1, &
                                                work[1], &c__1);
                                        slascl_("G", &c__0, &c__0, &aapp, &
                                                d__[p], m, &c__1, &work[1], 
                                                lda, &ierr);
                                        aapq = sdot_(m, &work[1], &c__1, &a[q 
                                                * a_dim1 + 1], &c__1) * d__[q]
                                                 / aaqq;
                                    }
                                } else {
                                    if (aapp >= aaqq) {
                                        rotok = aapp <= aaqq / small;
                                    } else {
                                        rotok = aaqq <= aapp / small;
                                    }
                                    if (aapp > small / aaqq) {
                                        aapq = sdot_(m, &a[p * a_dim1 + 1], &
                                                c__1, &a[q * a_dim1 + 1], &
                                                c__1) * d__[p] * d__[q] / 
                                                aaqq / aapp;
                                    } else {
                                        scopy_(m, &a[q * a_dim1 + 1], &c__1, &
                                                work[1], &c__1);
                                        slascl_("G", &c__0, &c__0, &aaqq, &
                                                d__[q], m, &c__1, &work[1], 
                                                lda, &ierr);
                                        aapq = sdot_(m, &work[1], &c__1, &a[p 
                                                * a_dim1 + 1], &c__1) * d__[p]
                                                 / aapp;
                                    }
                                }

/* Computing MAX */
                                r__1 = mxaapq, r__2 = dabs(aapq);
                                mxaapq = dmax(r__1,r__2);

/*        TO rotate or NOT to rotate, THAT is the question ... */

                                if (dabs(aapq) > *tol) {
                                    notrot = 0;
/*           ROTATED  = ROTATED + 1 */
                                    pskipped = 0;
                                    ++iswrot;

                                    if (rotok) {

                                        aqoap = aaqq / aapp;
                                        apoaq = aapp / aaqq;
                                        theta = (r__1 = aqoap - apoaq, dabs(
                                                r__1)) * -.5f / aapq;
                                        if (aaqq > aapp0) {
                                            theta = -theta;
                                        }

                                        if (dabs(theta) > bigtheta) {
                                            t = .5f / theta;
                                            fastr[2] = t * d__[p] / d__[q];
                                            fastr[3] = -t * d__[q] / d__[p];
                                            srotm_(m, &a[p * a_dim1 + 1], &
                                                    c__1, &a[q * a_dim1 + 1], 
                                                    &c__1, fastr);
                                            if (rsvec) {
                          srotm_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[q * 
                                  v_dim1 + 1], &c__1, fastr);
                                            }
/* Computing MAX */
                                            r__1 = 0.f, r__2 = t * apoaq * 
                                                    aapq + 1.f;
                                            sva[q] = aaqq * sqrt((dmax(r__1,
                                                    r__2)));
/* Computing MAX */
                                            r__1 = 0.f, r__2 = 1.f - t * 
                                                    aqoap * aapq;
                                            aapp *= sqrt((dmax(r__1,r__2)));
/* Computing MAX */
                                            r__1 = mxsinj, r__2 = dabs(t);
                                            mxsinj = dmax(r__1,r__2);
                                        } else {

/*                 .. choose correct signum for THETA and rotate */

                                            thsign = -r_sign(&c_b42, &aapq);
                                            if (aaqq > aapp0) {
                          thsign = -thsign;
                                            }
                                            t = 1.f / (theta + thsign * sqrt(
                                                    theta * theta + 1.f));
                                            cs = sqrt(1.f / (t * t + 1.f));
                                            sn = t * cs;
/* Computing MAX */
                                            r__1 = mxsinj, r__2 = dabs(sn);
                                            mxsinj = dmax(r__1,r__2);
/* Computing MAX */
                                            r__1 = 0.f, r__2 = t * apoaq * 
                                                    aapq + 1.f;
                                            sva[q] = aaqq * sqrt((dmax(r__1,
                                                    r__2)));
                                            aapp *= sqrt(1.f - t * aqoap * 
                                                    aapq);

                                            apoaq = d__[p] / d__[q];
                                            aqoap = d__[q] / d__[p];
                                            if (d__[p] >= 1.f) {

                          if (d__[q] >= 1.f) {
                              fastr[2] = t * apoaq;
                              fastr[3] = -t * aqoap;
                              d__[p] *= cs;
                              d__[q] *= cs;
                              srotm_(m, &a[p * a_dim1 + 1], &c__1, &a[q * 
                                      a_dim1 + 1], &c__1, fastr);
                              if (rsvec) {
                                  srotm_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[
                                          q * v_dim1 + 1], &c__1, fastr);
                              }
                          } else {
                              r__1 = -t * aqoap;
                              saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, &a[
                                      p * a_dim1 + 1], &c__1);
                              r__1 = cs * sn * apoaq;
                              saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, &a[
                                      q * a_dim1 + 1], &c__1);
                              if (rsvec) {
                                  r__1 = -t * aqoap;
                                  saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], &
                                          c__1, &v[p * v_dim1 + 1], &c__1);
                                  r__1 = cs * sn * apoaq;
                                  saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], &
                                          c__1, &v[q * v_dim1 + 1], &c__1);
                              }
                              d__[p] *= cs;
                              d__[q] /= cs;
                          }
                                            } else {
                          if (d__[q] >= 1.f) {
                              r__1 = t * apoaq;
                              saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, &a[
                                      q * a_dim1 + 1], &c__1);
                              r__1 = -cs * sn * aqoap;
                              saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, &a[
                                      p * a_dim1 + 1], &c__1);
                              if (rsvec) {
                                  r__1 = t * apoaq;
                                  saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], &
                                          c__1, &v[q * v_dim1 + 1], &c__1);
                                  r__1 = -cs * sn * aqoap;
                                  saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], &
                                          c__1, &v[p * v_dim1 + 1], &c__1);
                              }
                              d__[p] /= cs;
                              d__[q] *= cs;
                          } else {
                              if (d__[p] >= d__[q]) {
                                  r__1 = -t * aqoap;
                                  saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, 
                                          &a[p * a_dim1 + 1], &c__1);
                                  r__1 = cs * sn * apoaq;
                                  saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, 
                                          &a[q * a_dim1 + 1], &c__1);
                                  d__[p] *= cs;
                                  d__[q] /= cs;
                                  if (rsvec) {
                                      r__1 = -t * aqoap;
                                      saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], 
                                              &c__1, &v[p * v_dim1 + 1], &
                                              c__1);
                                      r__1 = cs * sn * apoaq;
                                      saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], 
                                              &c__1, &v[q * v_dim1 + 1], &
                                              c__1);
                                  }
                              } else {
                                  r__1 = t * apoaq;
                                  saxpy_(m, &r__1, &a[p * a_dim1 + 1], &c__1, 
                                          &a[q * a_dim1 + 1], &c__1);
                                  r__1 = -cs * sn * aqoap;
                                  saxpy_(m, &r__1, &a[q * a_dim1 + 1], &c__1, 
                                          &a[p * a_dim1 + 1], &c__1);
                                  d__[p] /= cs;
                                  d__[q] *= cs;
                                  if (rsvec) {
                                      r__1 = t * apoaq;
                                      saxpy_(&mvl, &r__1, &v[p * v_dim1 + 1], 
                                              &c__1, &v[q * v_dim1 + 1], &
                                              c__1);
                                      r__1 = -cs * sn * aqoap;
                                      saxpy_(&mvl, &r__1, &v[q * v_dim1 + 1], 
                                              &c__1, &v[p * v_dim1 + 1], &
                                              c__1);
                                  }
                              }
                          }
                                            }
                                        }

                                    } else {
                                        if (aapp > aaqq) {
                                            scopy_(m, &a[p * a_dim1 + 1], &
                                                    c__1, &work[1], &c__1);
                                            slascl_("G", &c__0, &c__0, &aapp, 
                                                    &c_b42, m, &c__1, &work[1]
, lda, &ierr);
                                            slascl_("G", &c__0, &c__0, &aaqq, 
                                                    &c_b42, m, &c__1, &a[q * 
                                                    a_dim1 + 1], lda, &ierr);
                                            temp1 = -aapq * d__[p] / d__[q];
                                            saxpy_(m, &temp1, &work[1], &c__1, 
                                                     &a[q * a_dim1 + 1], &
                                                    c__1);
                                            slascl_("G", &c__0, &c__0, &c_b42, 
                                                     &aaqq, m, &c__1, &a[q * 
                                                    a_dim1 + 1], lda, &ierr);
/* Computing MAX */
                                            r__1 = 0.f, r__2 = 1.f - aapq * 
                                                    aapq;
                                            sva[q] = aaqq * sqrt((dmax(r__1,
                                                    r__2)));
                                            mxsinj = dmax(mxsinj,*sfmin);
                                        } else {
                                            scopy_(m, &a[q * a_dim1 + 1], &
                                                    c__1, &work[1], &c__1);
                                            slascl_("G", &c__0, &c__0, &aaqq, 
                                                    &c_b42, m, &c__1, &work[1]
, lda, &ierr);
                                            slascl_("G", &c__0, &c__0, &aapp, 
                                                    &c_b42, m, &c__1, &a[p * 
                                                    a_dim1 + 1], lda, &ierr);
                                            temp1 = -aapq * d__[q] / d__[p];
                                            saxpy_(m, &temp1, &work[1], &c__1, 
                                                     &a[p * a_dim1 + 1], &
                                                    c__1);
                                            slascl_("G", &c__0, &c__0, &c_b42, 
                                                     &aapp, m, &c__1, &a[p * 
                                                    a_dim1 + 1], lda, &ierr);
/* Computing MAX */
                                            r__1 = 0.f, r__2 = 1.f - aapq * 
                                                    aapq;
                                            sva[p] = aapp * sqrt((dmax(r__1,
                                                    r__2)));
                                            mxsinj = dmax(mxsinj,*sfmin);
                                        }
                                    }
/*           END IF ROTOK THEN ... ELSE */

/*           In the case of cancellation in updating SVA(q) */
/*           .. recompute SVA(q) */
/* Computing 2nd power */
                                    r__1 = sva[q] / aaqq;
                                    if (r__1 * r__1 <= rooteps) {
                                        if (aaqq < rootbig && aaqq > 
                                                rootsfmin) {
                                            sva[q] = snrm2_(m, &a[q * a_dim1 
                                                    + 1], &c__1) * d__[q];
                                        } else {
                                            t = 0.f;
                                            aaqq = 0.f;
                                            slassq_(m, &a[q * a_dim1 + 1], &
                                                    c__1, &t, &aaqq);
                                            sva[q] = t * sqrt(aaqq) * d__[q];
                                        }
                                    }
/* Computing 2nd power */
                                    r__1 = aapp / aapp0;
                                    if (r__1 * r__1 <= rooteps) {
                                        if (aapp < rootbig && aapp > 
                                                rootsfmin) {
                                            aapp = snrm2_(m, &a[p * a_dim1 + 
                                                    1], &c__1) * d__[p];
                                        } else {
                                            t = 0.f;
                                            aapp = 0.f;
                                            slassq_(m, &a[p * a_dim1 + 1], &
                                                    c__1, &t, &aapp);
                                            aapp = t * sqrt(aapp) * d__[p];
                                        }
                                        sva[p] = aapp;
                                    }
/*              end of OK rotation */
                                } else {
                                    ++notrot;
                                    ++pskipped;
                                    ++ijblsk;
                                }
                            } else {
                                ++notrot;
                                ++pskipped;
                                ++ijblsk;
                            }

                            if (i__ <= swband && ijblsk >= blskip) {
                                sva[p] = aapp;
                                notrot = 0;
                                goto L2011;
                            }
                            if (i__ <= swband && pskipped > rowskip) {
                                aapp = -aapp;
                                notrot = 0;
                                goto L2203;
                            }

/* L2200: */
                        }
/*        end of the q-loop */
L2203:

                        sva[p] = aapp;

                    } else {
                        if (aapp == 0.f) {
/* Computing MIN */
                            i__5 = jgl + kbl - 1;
                            notrot = notrot + min(i__5,*n) - jgl + 1;
                        }
                        if (aapp < 0.f) {
                            notrot = 0;
                        }
                    }
/* L2100: */
                }
/*     end of the p-loop */
/* L2010: */
            }
/*     end of the jbc-loop */
L2011:
/* 2011 bailed out of the jbc-loop */
/* Computing MIN */
            i__4 = igl + kbl - 1;
            i__3 = min(i__4,*n);
            for (p = igl; p <= i__3; ++p) {
                sva[p] = (r__1 = sva[p], dabs(r__1));
/* L2012: */
            }

/* L2000: */
        }
/* 2000 :: end of the ibr-loop */

/*     .. update SVA(N) */
        if (sva[*n] < rootbig && sva[*n] > rootsfmin) {
            sva[*n] = snrm2_(m, &a[*n * a_dim1 + 1], &c__1) * d__[*n];
        } else {
            t = 0.f;
            aapp = 0.f;
            slassq_(m, &a[*n * a_dim1 + 1], &c__1, &t, &aapp);
            sva[*n] = t * sqrt(aapp) * d__[*n];
        }

/*     Additional steering devices */

        if (i__ < swband && (mxaapq <= roottol || iswrot <= *n)) {
            swband = i__;
        }

        if (i__ > swband + 1 && mxaapq < (real) (*n) * *tol && (real) (*n) * 
                mxaapq * mxsinj < *tol) {
            goto L1994;
        }

        if (notrot >= emptsw) {
            goto L1994;
        }
/* L1993: */
    }
/*     end i=1:NSWEEP loop */
/* #:) Reaching this point means that the procedure has comleted the given */
/*     number of iterations. */
    *info = *nsweep - 1;
    goto L1995;
L1994:
/* #:) Reaching this point means that during the i-th sweep all pivots were */
/*     below the given tolerance, causing early exit. */

    *info = 0;
/* #:) INFO = 0 confirms successful iterations. */
L1995:

/*     Sort the vector D. */
    i__1 = *n - 1;
    for (p = 1; p <= i__1; ++p) {
        i__2 = *n - p + 1;
        q = isamax_(&i__2, &sva[p], &c__1) + p - 1;
        if (p != q) {
            temp1 = sva[p];
            sva[p] = sva[q];
            sva[q] = temp1;
            temp1 = d__[p];
            d__[p] = d__[q];
            d__[q] = temp1;
            sswap_(m, &a[p * a_dim1 + 1], &c__1, &a[q * a_dim1 + 1], &c__1);
            if (rsvec) {
                sswap_(&mvl, &v[p * v_dim1 + 1], &c__1, &v[q * v_dim1 + 1], &
                        c__1);
            }
        }
/* L5991: */
    }

    return 0;
/*     .. */
/*     .. END OF SGSVJ0 */
/*     .. */
} /* sgsvj0_ */