zbdsqr.c

Go to the documentation of this file.

/* zbdsqr.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 doublereal c_b15 = -.125;
static integer c__1 = 1;
static doublereal c_b49 = 1.;
static doublereal c_b72 = -1.;

/* Subroutine */ int zbdsqr_(char *uplo, integer *n, integer *ncvt, integer *
        nru, integer *ncc, doublereal *d__, doublereal *e, doublecomplex *vt, 
        integer *ldvt, doublecomplex *u, integer *ldu, doublecomplex *c__, 
        integer *ldc, doublereal *rwork, integer *info)
{
    /* System generated locals */
    integer c_dim1, c_offset, u_dim1, u_offset, vt_dim1, vt_offset, i__1, 
            i__2;
    doublereal d__1, d__2, d__3, d__4;

    /* Builtin functions */
    double pow_dd(doublereal *, doublereal *), sqrt(doublereal), d_sign(
            doublereal *, doublereal *);

    /* Local variables */
    doublereal f, g, h__;
    integer i__, j, m;
    doublereal r__, cs;
    integer ll;
    doublereal sn, mu;
    integer nm1, nm12, nm13, lll;
    doublereal eps, sll, tol, abse;
    integer idir;
    doublereal abss;
    integer oldm;
    doublereal cosl;
    integer isub, iter;
    doublereal unfl, sinl, cosr, smin, smax, sinr;
    extern /* Subroutine */ int dlas2_(doublereal *, doublereal *, doublereal 
            *, doublereal *, doublereal *);
    extern logical lsame_(char *, char *);
    doublereal oldcs;
    integer oldll;
    doublereal shift, sigmn, oldsn;
    integer maxit;
    doublereal sminl, sigmx;
    logical lower;
    extern /* Subroutine */ int zlasr_(char *, char *, char *, integer *, 
            integer *, doublereal *, doublereal *, doublecomplex *, integer *), zdrot_(integer *, doublecomplex *, 
            integer *, doublecomplex *, integer *, doublereal *, doublereal *)
            , zswap_(integer *, doublecomplex *, integer *, doublecomplex *, 
            integer *), dlasq1_(integer *, doublereal *, doublereal *, 
            doublereal *, integer *), dlasv2_(doublereal *, doublereal *, 
            doublereal *, doublereal *, doublereal *, doublereal *, 
            doublereal *, doublereal *, doublereal *);
    extern doublereal dlamch_(char *);
    extern /* Subroutine */ int dlartg_(doublereal *, doublereal *, 
            doublereal *, doublereal *, doublereal *), xerbla_(char *, 
            integer *), zdscal_(integer *, doublereal *, 
            doublecomplex *, integer *);
    doublereal sminoa, thresh;
    logical rotate;
    doublereal tolmul;


/*  -- LAPACK routine (version 3.2) -- */
/*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/*     November 2006 */

/*     .. Scalar Arguments .. */
/*     .. */
/*     .. Array Arguments .. */
/*     .. */

/*  Purpose */
/*  ======= */

/*  ZBDSQR computes the singular values and, optionally, the right and/or */
/*  left singular vectors from the singular value decomposition (SVD) of */
/*  a real N-by-N (upper or lower) bidiagonal matrix B using the implicit */
/*  zero-shift QR algorithm.  The SVD of B has the form */

/*     B = Q * S * P**H */

/*  where S is the diagonal matrix of singular values, Q is an orthogonal */
/*  matrix of left singular vectors, and P is an orthogonal matrix of */
/*  right singular vectors.  If left singular vectors are requested, this */
/*  subroutine actually returns U*Q instead of Q, and, if right singular */
/*  vectors are requested, this subroutine returns P**H*VT instead of */
/*  P**H, for given complex input matrices U and VT.  When U and VT are */
/*  the unitary matrices that reduce a general matrix A to bidiagonal */
/*  form: A = U*B*VT, as computed by ZGEBRD, then */

/*     A = (U*Q) * S * (P**H*VT) */

/*  is the SVD of A.  Optionally, the subroutine may also compute Q**H*C */
/*  for a given complex input matrix C. */

/*  See "Computing  Small Singular Values of Bidiagonal Matrices With */
/*  Guaranteed High Relative Accuracy," by J. Demmel and W. Kahan, */
/*  LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11, */
/*  no. 5, pp. 873-912, Sept 1990) and */
/*  "Accurate singular values and differential qd algorithms," by */
/*  B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics */
/*  Department, University of California at Berkeley, July 1992 */
/*  for a detailed description of the algorithm. */

/*  Arguments */
/*  ========= */

/*  UPLO    (input) CHARACTER*1 */
/*          = 'U':  B is upper bidiagonal; */
/*          = 'L':  B is lower bidiagonal. */

/*  N       (input) INTEGER */
/*          The order of the matrix B.  N >= 0. */

/*  NCVT    (input) INTEGER */
/*          The number of columns of the matrix VT. NCVT >= 0. */

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

/*  NCC     (input) INTEGER */
/*          The number of columns of the matrix C. NCC >= 0. */

/*  D       (input/output) DOUBLE PRECISION array, dimension (N) */
/*          On entry, the n diagonal elements of the bidiagonal matrix B. */
/*          On exit, if INFO=0, the singular values of B in decreasing */
/*          order. */

/*  E       (input/output) DOUBLE PRECISION array, dimension (N-1) */
/*          On entry, the N-1 offdiagonal elements of the bidiagonal */
/*          matrix B. */
/*          On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E */
/*          will contain the diagonal and superdiagonal elements of a */
/*          bidiagonal matrix orthogonally equivalent to the one given */
/*          as input. */

/*  VT      (input/output) COMPLEX*16 array, dimension (LDVT, NCVT) */
/*          On entry, an N-by-NCVT matrix VT. */
/*          On exit, VT is overwritten by P**H * VT. */
/*          Not referenced if NCVT = 0. */

/*  LDVT    (input) INTEGER */
/*          The leading dimension of the array VT. */
/*          LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0. */

/*  U       (input/output) COMPLEX*16 array, dimension (LDU, N) */
/*          On entry, an NRU-by-N matrix U. */
/*          On exit, U is overwritten by U * Q. */
/*          Not referenced if NRU = 0. */

/*  LDU     (input) INTEGER */
/*          The leading dimension of the array U.  LDU >= max(1,NRU). */

/*  C       (input/output) COMPLEX*16 array, dimension (LDC, NCC) */
/*          On entry, an N-by-NCC matrix C. */
/*          On exit, C is overwritten by Q**H * C. */
/*          Not referenced if NCC = 0. */

/*  LDC     (input) INTEGER */
/*          The leading dimension of the array C. */
/*          LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0. */

/*  RWORK   (workspace) DOUBLE PRECISION array, dimension (2*N) */
/*          if NCVT = NRU = NCC = 0, (max(1, 4*N-4)) otherwise */

/*  INFO    (output) INTEGER */
/*          = 0:  successful exit */
/*          < 0:  If INFO = -i, the i-th argument had an illegal value */
/*          > 0:  the algorithm did not converge; D and E contain the */
/*                elements of a bidiagonal matrix which is orthogonally */
/*                similar to the input matrix B;  if INFO = i, i */
/*                elements of E have not converged to zero. */

/*  Internal Parameters */
/*  =================== */

/*  TOLMUL  DOUBLE PRECISION, default = max(10,min(100,EPS**(-1/8))) */
/*          TOLMUL controls the convergence criterion of the QR loop. */
/*          If it is positive, TOLMUL*EPS is the desired relative */
/*             precision in the computed singular values. */
/*          If it is negative, abs(TOLMUL*EPS*sigma_max) is the */
/*             desired absolute accuracy in the computed singular */
/*             values (corresponds to relative accuracy */
/*             abs(TOLMUL*EPS) in the largest singular value. */
/*          abs(TOLMUL) should be between 1 and 1/EPS, and preferably */
/*             between 10 (for fast convergence) and .1/EPS */
/*             (for there to be some accuracy in the results). */
/*          Default is to lose at either one eighth or 2 of the */
/*             available decimal digits in each computed singular value */
/*             (whichever is smaller). */

/*  MAXITR  INTEGER, default = 6 */
/*          MAXITR controls the maximum number of passes of the */
/*          algorithm through its inner loop. The algorithms stops */
/*          (and so fails to converge) if the number of passes */
/*          through the inner loop exceeds MAXITR*N**2. */

/*  ===================================================================== */

/*     .. Parameters .. */
/*     .. */
/*     .. Local Scalars .. */
/*     .. */
/*     .. External Functions .. */
/*     .. */
/*     .. External Subroutines .. */
/*     .. */
/*     .. Intrinsic Functions .. */
/*     .. */
/*     .. Executable Statements .. */

/*     Test the input parameters. */

    /* Parameter adjustments */
    --d__;
    --e;
    vt_dim1 = *ldvt;
    vt_offset = 1 + vt_dim1;
    vt -= vt_offset;
    u_dim1 = *ldu;
    u_offset = 1 + u_dim1;
    u -= u_offset;
    c_dim1 = *ldc;
    c_offset = 1 + c_dim1;
    c__ -= c_offset;
    --rwork;

    /* Function Body */
    *info = 0;
    lower = lsame_(uplo, "L");
    if (! lsame_(uplo, "U") && ! lower) {
        *info = -1;
    } else if (*n < 0) {
        *info = -2;
    } else if (*ncvt < 0) {
        *info = -3;
    } else if (*nru < 0) {
        *info = -4;
    } else if (*ncc < 0) {
        *info = -5;
    } else if (*ncvt == 0 && *ldvt < 1 || *ncvt > 0 && *ldvt < max(1,*n)) {
        *info = -9;
    } else if (*ldu < max(1,*nru)) {
        *info = -11;
    } else if (*ncc == 0 && *ldc < 1 || *ncc > 0 && *ldc < max(1,*n)) {
        *info = -13;
    }
    if (*info != 0) {
        i__1 = -(*info);
        xerbla_("ZBDSQR", &i__1);
        return 0;
    }
    if (*n == 0) {
        return 0;
    }
    if (*n == 1) {
        goto L160;
    }

/*     ROTATE is true if any singular vectors desired, false otherwise */

    rotate = *ncvt > 0 || *nru > 0 || *ncc > 0;

/*     If no singular vectors desired, use qd algorithm */

    if (! rotate) {
        dlasq1_(n, &d__[1], &e[1], &rwork[1], info);
        return 0;
    }

    nm1 = *n - 1;
    nm12 = nm1 + nm1;
    nm13 = nm12 + nm1;
    idir = 0;

/*     Get machine constants */

    eps = dlamch_("Epsilon");
    unfl = dlamch_("Safe minimum");

/*     If matrix lower bidiagonal, rotate to be upper bidiagonal */
/*     by applying Givens rotations on the left */

    if (lower) {
        i__1 = *n - 1;
        for (i__ = 1; i__ <= i__1; ++i__) {
            dlartg_(&d__[i__], &e[i__], &cs, &sn, &r__);
            d__[i__] = r__;
            e[i__] = sn * d__[i__ + 1];
            d__[i__ + 1] = cs * d__[i__ + 1];
            rwork[i__] = cs;
            rwork[nm1 + i__] = sn;
/* L10: */
        }

/*        Update singular vectors if desired */

        if (*nru > 0) {
            zlasr_("R", "V", "F", nru, n, &rwork[1], &rwork[*n], &u[u_offset], 
                     ldu);
        }
        if (*ncc > 0) {
            zlasr_("L", "V", "F", n, ncc, &rwork[1], &rwork[*n], &c__[
                    c_offset], ldc);
        }
    }

/*     Compute singular values to relative accuracy TOL */
/*     (By setting TOL to be negative, algorithm will compute */
/*     singular values to absolute accuracy ABS(TOL)*norm(input matrix)) */

/* Computing MAX */
/* Computing MIN */
    d__3 = 100., d__4 = pow_dd(&eps, &c_b15);
    d__1 = 10., d__2 = min(d__3,d__4);
    tolmul = max(d__1,d__2);
    tol = tolmul * eps;

/*     Compute approximate maximum, minimum singular values */

    smax = 0.;
    i__1 = *n;
    for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing MAX */
        d__2 = smax, d__3 = (d__1 = d__[i__], abs(d__1));
        smax = max(d__2,d__3);
/* L20: */
    }
    i__1 = *n - 1;
    for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing MAX */
        d__2 = smax, d__3 = (d__1 = e[i__], abs(d__1));
        smax = max(d__2,d__3);
/* L30: */
    }
    sminl = 0.;
    if (tol >= 0.) {

/*        Relative accuracy desired */

        sminoa = abs(d__[1]);
        if (sminoa == 0.) {
            goto L50;
        }
        mu = sminoa;
        i__1 = *n;
        for (i__ = 2; i__ <= i__1; ++i__) {
            mu = (d__2 = d__[i__], abs(d__2)) * (mu / (mu + (d__1 = e[i__ - 1]
                    , abs(d__1))));
            sminoa = min(sminoa,mu);
            if (sminoa == 0.) {
                goto L50;
            }
/* L40: */
        }
L50:
        sminoa /= sqrt((doublereal) (*n));
/* Computing MAX */
        d__1 = tol * sminoa, d__2 = *n * 6 * *n * unfl;
        thresh = max(d__1,d__2);
    } else {

/*        Absolute accuracy desired */

/* Computing MAX */
        d__1 = abs(tol) * smax, d__2 = *n * 6 * *n * unfl;
        thresh = max(d__1,d__2);
    }

/*     Prepare for main iteration loop for the singular values */
/*     (MAXIT is the maximum number of passes through the inner */
/*     loop permitted before nonconvergence signalled.) */

    maxit = *n * 6 * *n;
    iter = 0;
    oldll = -1;
    oldm = -1;

/*     M points to last element of unconverged part of matrix */

    m = *n;

/*     Begin main iteration loop */

L60:

/*     Check for convergence or exceeding iteration count */

    if (m <= 1) {
        goto L160;
    }
    if (iter > maxit) {
        goto L200;
    }

/*     Find diagonal block of matrix to work on */

    if (tol < 0. && (d__1 = d__[m], abs(d__1)) <= thresh) {
        d__[m] = 0.;
    }
    smax = (d__1 = d__[m], abs(d__1));
    smin = smax;
    i__1 = m - 1;
    for (lll = 1; lll <= i__1; ++lll) {
        ll = m - lll;
        abss = (d__1 = d__[ll], abs(d__1));
        abse = (d__1 = e[ll], abs(d__1));
        if (tol < 0. && abss <= thresh) {
            d__[ll] = 0.;
        }
        if (abse <= thresh) {
            goto L80;
        }
        smin = min(smin,abss);
/* Computing MAX */
        d__1 = max(smax,abss);
        smax = max(d__1,abse);
/* L70: */
    }
    ll = 0;
    goto L90;
L80:
    e[ll] = 0.;

/*     Matrix splits since E(LL) = 0 */

    if (ll == m - 1) {

/*        Convergence of bottom singular value, return to top of loop */

        --m;
        goto L60;
    }
L90:
    ++ll;

/*     E(LL) through E(M-1) are nonzero, E(LL-1) is zero */

    if (ll == m - 1) {

/*        2 by 2 block, handle separately */

        dlasv2_(&d__[m - 1], &e[m - 1], &d__[m], &sigmn, &sigmx, &sinr, &cosr, 
                 &sinl, &cosl);
        d__[m - 1] = sigmx;
        e[m - 1] = 0.;
        d__[m] = sigmn;

/*        Compute singular vectors, if desired */

        if (*ncvt > 0) {
            zdrot_(ncvt, &vt[m - 1 + vt_dim1], ldvt, &vt[m + vt_dim1], ldvt, &
                    cosr, &sinr);
        }
        if (*nru > 0) {
            zdrot_(nru, &u[(m - 1) * u_dim1 + 1], &c__1, &u[m * u_dim1 + 1], &
                    c__1, &cosl, &sinl);
        }
        if (*ncc > 0) {
            zdrot_(ncc, &c__[m - 1 + c_dim1], ldc, &c__[m + c_dim1], ldc, &
                    cosl, &sinl);
        }
        m += -2;
        goto L60;
    }

/*     If working on new submatrix, choose shift direction */
/*     (from larger end diagonal element towards smaller) */

    if (ll > oldm || m < oldll) {
        if ((d__1 = d__[ll], abs(d__1)) >= (d__2 = d__[m], abs(d__2))) {

/*           Chase bulge from top (big end) to bottom (small end) */

            idir = 1;
        } else {

/*           Chase bulge from bottom (big end) to top (small end) */

            idir = 2;
        }
    }

/*     Apply convergence tests */

    if (idir == 1) {

/*        Run convergence test in forward direction */
/*        First apply standard test to bottom of matrix */

        if ((d__2 = e[m - 1], abs(d__2)) <= abs(tol) * (d__1 = d__[m], abs(
                d__1)) || tol < 0. && (d__3 = e[m - 1], abs(d__3)) <= thresh) 
                {
            e[m - 1] = 0.;
            goto L60;
        }

        if (tol >= 0.) {

/*           If relative accuracy desired, */
/*           apply convergence criterion forward */

            mu = (d__1 = d__[ll], abs(d__1));
            sminl = mu;
            i__1 = m - 1;
            for (lll = ll; lll <= i__1; ++lll) {
                if ((d__1 = e[lll], abs(d__1)) <= tol * mu) {
                    e[lll] = 0.;
                    goto L60;
                }
                mu = (d__2 = d__[lll + 1], abs(d__2)) * (mu / (mu + (d__1 = e[
                        lll], abs(d__1))));
                sminl = min(sminl,mu);
/* L100: */
            }
        }

    } else {

/*        Run convergence test in backward direction */
/*        First apply standard test to top of matrix */

        if ((d__2 = e[ll], abs(d__2)) <= abs(tol) * (d__1 = d__[ll], abs(d__1)
                ) || tol < 0. && (d__3 = e[ll], abs(d__3)) <= thresh) {
            e[ll] = 0.;
            goto L60;
        }

        if (tol >= 0.) {

/*           If relative accuracy desired, */
/*           apply convergence criterion backward */

            mu = (d__1 = d__[m], abs(d__1));
            sminl = mu;
            i__1 = ll;
            for (lll = m - 1; lll >= i__1; --lll) {
                if ((d__1 = e[lll], abs(d__1)) <= tol * mu) {
                    e[lll] = 0.;
                    goto L60;
                }
                mu = (d__2 = d__[lll], abs(d__2)) * (mu / (mu + (d__1 = e[lll]
                        , abs(d__1))));
                sminl = min(sminl,mu);
/* L110: */
            }
        }
    }
    oldll = ll;
    oldm = m;

/*     Compute shift.  First, test if shifting would ruin relative */
/*     accuracy, and if so set the shift to zero. */

/* Computing MAX */
    d__1 = eps, d__2 = tol * .01;
    if (tol >= 0. && *n * tol * (sminl / smax) <= max(d__1,d__2)) {

/*        Use a zero shift to avoid loss of relative accuracy */

        shift = 0.;
    } else {

/*        Compute the shift from 2-by-2 block at end of matrix */

        if (idir == 1) {
            sll = (d__1 = d__[ll], abs(d__1));
            dlas2_(&d__[m - 1], &e[m - 1], &d__[m], &shift, &r__);
        } else {
            sll = (d__1 = d__[m], abs(d__1));
            dlas2_(&d__[ll], &e[ll], &d__[ll + 1], &shift, &r__);
        }

/*        Test if shift negligible, and if so set to zero */

        if (sll > 0.) {
/* Computing 2nd power */
            d__1 = shift / sll;
            if (d__1 * d__1 < eps) {
                shift = 0.;
            }
        }
    }

/*     Increment iteration count */

    iter = iter + m - ll;

/*     If SHIFT = 0, do simplified QR iteration */

    if (shift == 0.) {
        if (idir == 1) {

/*           Chase bulge from top to bottom */
/*           Save cosines and sines for later singular vector updates */

            cs = 1.;
            oldcs = 1.;
            i__1 = m - 1;
            for (i__ = ll; i__ <= i__1; ++i__) {
                d__1 = d__[i__] * cs;
                dlartg_(&d__1, &e[i__], &cs, &sn, &r__);
                if (i__ > ll) {
                    e[i__ - 1] = oldsn * r__;
                }
                d__1 = oldcs * r__;
                d__2 = d__[i__ + 1] * sn;
                dlartg_(&d__1, &d__2, &oldcs, &oldsn, &d__[i__]);
                rwork[i__ - ll + 1] = cs;
                rwork[i__ - ll + 1 + nm1] = sn;
                rwork[i__ - ll + 1 + nm12] = oldcs;
                rwork[i__ - ll + 1 + nm13] = oldsn;
/* L120: */
            }
            h__ = d__[m] * cs;
            d__[m] = h__ * oldcs;
            e[m - 1] = h__ * oldsn;

/*           Update singular vectors */

            if (*ncvt > 0) {
                i__1 = m - ll + 1;
                zlasr_("L", "V", "F", &i__1, ncvt, &rwork[1], &rwork[*n], &vt[
                        ll + vt_dim1], ldvt);
            }
            if (*nru > 0) {
                i__1 = m - ll + 1;
                zlasr_("R", "V", "F", nru, &i__1, &rwork[nm12 + 1], &rwork[
                        nm13 + 1], &u[ll * u_dim1 + 1], ldu);
            }
            if (*ncc > 0) {
                i__1 = m - ll + 1;
                zlasr_("L", "V", "F", &i__1, ncc, &rwork[nm12 + 1], &rwork[
                        nm13 + 1], &c__[ll + c_dim1], ldc);
            }

/*           Test convergence */

            if ((d__1 = e[m - 1], abs(d__1)) <= thresh) {
                e[m - 1] = 0.;
            }

        } else {

/*           Chase bulge from bottom to top */
/*           Save cosines and sines for later singular vector updates */

            cs = 1.;
            oldcs = 1.;
            i__1 = ll + 1;
            for (i__ = m; i__ >= i__1; --i__) {
                d__1 = d__[i__] * cs;
                dlartg_(&d__1, &e[i__ - 1], &cs, &sn, &r__);
                if (i__ < m) {
                    e[i__] = oldsn * r__;
                }
                d__1 = oldcs * r__;
                d__2 = d__[i__ - 1] * sn;
                dlartg_(&d__1, &d__2, &oldcs, &oldsn, &d__[i__]);
                rwork[i__ - ll] = cs;
                rwork[i__ - ll + nm1] = -sn;
                rwork[i__ - ll + nm12] = oldcs;
                rwork[i__ - ll + nm13] = -oldsn;
/* L130: */
            }
            h__ = d__[ll] * cs;
            d__[ll] = h__ * oldcs;
            e[ll] = h__ * oldsn;

/*           Update singular vectors */

            if (*ncvt > 0) {
                i__1 = m - ll + 1;
                zlasr_("L", "V", "B", &i__1, ncvt, &rwork[nm12 + 1], &rwork[
                        nm13 + 1], &vt[ll + vt_dim1], ldvt);
            }
            if (*nru > 0) {
                i__1 = m - ll + 1;
                zlasr_("R", "V", "B", nru, &i__1, &rwork[1], &rwork[*n], &u[
                        ll * u_dim1 + 1], ldu);
            }
            if (*ncc > 0) {
                i__1 = m - ll + 1;
                zlasr_("L", "V", "B", &i__1, ncc, &rwork[1], &rwork[*n], &c__[
                        ll + c_dim1], ldc);
            }

/*           Test convergence */

            if ((d__1 = e[ll], abs(d__1)) <= thresh) {
                e[ll] = 0.;
            }
        }
    } else {

/*        Use nonzero shift */

        if (idir == 1) {

/*           Chase bulge from top to bottom */
/*           Save cosines and sines for later singular vector updates */

            f = ((d__1 = d__[ll], abs(d__1)) - shift) * (d_sign(&c_b49, &d__[
                    ll]) + shift / d__[ll]);
            g = e[ll];
            i__1 = m - 1;
            for (i__ = ll; i__ <= i__1; ++i__) {
                dlartg_(&f, &g, &cosr, &sinr, &r__);
                if (i__ > ll) {
                    e[i__ - 1] = r__;
                }
                f = cosr * d__[i__] + sinr * e[i__];
                e[i__] = cosr * e[i__] - sinr * d__[i__];
                g = sinr * d__[i__ + 1];
                d__[i__ + 1] = cosr * d__[i__ + 1];
                dlartg_(&f, &g, &cosl, &sinl, &r__);
                d__[i__] = r__;
                f = cosl * e[i__] + sinl * d__[i__ + 1];
                d__[i__ + 1] = cosl * d__[i__ + 1] - sinl * e[i__];
                if (i__ < m - 1) {
                    g = sinl * e[i__ + 1];
                    e[i__ + 1] = cosl * e[i__ + 1];
                }
                rwork[i__ - ll + 1] = cosr;
                rwork[i__ - ll + 1 + nm1] = sinr;
                rwork[i__ - ll + 1 + nm12] = cosl;
                rwork[i__ - ll + 1 + nm13] = sinl;
/* L140: */
            }
            e[m - 1] = f;

/*           Update singular vectors */

            if (*ncvt > 0) {
                i__1 = m - ll + 1;
                zlasr_("L", "V", "F", &i__1, ncvt, &rwork[1], &rwork[*n], &vt[
                        ll + vt_dim1], ldvt);
            }
            if (*nru > 0) {
                i__1 = m - ll + 1;
                zlasr_("R", "V", "F", nru, &i__1, &rwork[nm12 + 1], &rwork[
                        nm13 + 1], &u[ll * u_dim1 + 1], ldu);
            }
            if (*ncc > 0) {
                i__1 = m - ll + 1;
                zlasr_("L", "V", "F", &i__1, ncc, &rwork[nm12 + 1], &rwork[
                        nm13 + 1], &c__[ll + c_dim1], ldc);
            }

/*           Test convergence */

            if ((d__1 = e[m - 1], abs(d__1)) <= thresh) {
                e[m - 1] = 0.;
            }

        } else {

/*           Chase bulge from bottom to top */
/*           Save cosines and sines for later singular vector updates */

            f = ((d__1 = d__[m], abs(d__1)) - shift) * (d_sign(&c_b49, &d__[m]
                    ) + shift / d__[m]);
            g = e[m - 1];
            i__1 = ll + 1;
            for (i__ = m; i__ >= i__1; --i__) {
                dlartg_(&f, &g, &cosr, &sinr, &r__);
                if (i__ < m) {
                    e[i__] = r__;
                }
                f = cosr * d__[i__] + sinr * e[i__ - 1];
                e[i__ - 1] = cosr * e[i__ - 1] - sinr * d__[i__];
                g = sinr * d__[i__ - 1];
                d__[i__ - 1] = cosr * d__[i__ - 1];
                dlartg_(&f, &g, &cosl, &sinl, &r__);
                d__[i__] = r__;
                f = cosl * e[i__ - 1] + sinl * d__[i__ - 1];
                d__[i__ - 1] = cosl * d__[i__ - 1] - sinl * e[i__ - 1];
                if (i__ > ll + 1) {
                    g = sinl * e[i__ - 2];
                    e[i__ - 2] = cosl * e[i__ - 2];
                }
                rwork[i__ - ll] = cosr;
                rwork[i__ - ll + nm1] = -sinr;
                rwork[i__ - ll + nm12] = cosl;
                rwork[i__ - ll + nm13] = -sinl;
/* L150: */
            }
            e[ll] = f;

/*           Test convergence */

            if ((d__1 = e[ll], abs(d__1)) <= thresh) {
                e[ll] = 0.;
            }

/*           Update singular vectors if desired */

            if (*ncvt > 0) {
                i__1 = m - ll + 1;
                zlasr_("L", "V", "B", &i__1, ncvt, &rwork[nm12 + 1], &rwork[
                        nm13 + 1], &vt[ll + vt_dim1], ldvt);
            }
            if (*nru > 0) {
                i__1 = m - ll + 1;
                zlasr_("R", "V", "B", nru, &i__1, &rwork[1], &rwork[*n], &u[
                        ll * u_dim1 + 1], ldu);
            }
            if (*ncc > 0) {
                i__1 = m - ll + 1;
                zlasr_("L", "V", "B", &i__1, ncc, &rwork[1], &rwork[*n], &c__[
                        ll + c_dim1], ldc);
            }
        }
    }

/*     QR iteration finished, go back and check convergence */

    goto L60;

/*     All singular values converged, so make them positive */

L160:
    i__1 = *n;
    for (i__ = 1; i__ <= i__1; ++i__) {
        if (d__[i__] < 0.) {
            d__[i__] = -d__[i__];

/*           Change sign of singular vectors, if desired */

            if (*ncvt > 0) {
                zdscal_(ncvt, &c_b72, &vt[i__ + vt_dim1], ldvt);
            }
        }
/* L170: */
    }

/*     Sort the singular values into decreasing order (insertion sort on */
/*     singular values, but only one transposition per singular vector) */

    i__1 = *n - 1;
    for (i__ = 1; i__ <= i__1; ++i__) {

/*        Scan for smallest D(I) */

        isub = 1;
        smin = d__[1];
        i__2 = *n + 1 - i__;
        for (j = 2; j <= i__2; ++j) {
            if (d__[j] <= smin) {
                isub = j;
                smin = d__[j];
            }
/* L180: */
        }
        if (isub != *n + 1 - i__) {

/*           Swap singular values and vectors */

            d__[isub] = d__[*n + 1 - i__];
            d__[*n + 1 - i__] = smin;
            if (*ncvt > 0) {
                zswap_(ncvt, &vt[isub + vt_dim1], ldvt, &vt[*n + 1 - i__ + 
                        vt_dim1], ldvt);
            }
            if (*nru > 0) {
                zswap_(nru, &u[isub * u_dim1 + 1], &c__1, &u[(*n + 1 - i__) * 
                        u_dim1 + 1], &c__1);
            }
            if (*ncc > 0) {
                zswap_(ncc, &c__[isub + c_dim1], ldc, &c__[*n + 1 - i__ + 
                        c_dim1], ldc);
            }
        }
/* L190: */
    }
    goto L220;

/*     Maximum number of iterations exceeded, failure to converge */

L200:
    *info = 0;
    i__1 = *n - 1;
    for (i__ = 1; i__ <= i__1; ++i__) {
        if (e[i__] != 0.) {
            ++(*info);
        }
/* L210: */
    }
L220:
    return 0;

/*     End of ZBDSQR */

} /* zbdsqr_ */