Esempio n. 1
0
/* Subroutine */ int dlalsd_(char *uplo, integer *smlsiz, integer *n, integer 
	*nrhs, doublereal *d__, doublereal *e, doublereal *b, integer *ldb, 
	doublereal *rcond, integer *rank, doublereal *work, integer *iwork, 
	integer *info)
{
    /* System generated locals */
    integer b_dim1, b_offset, i__1, i__2;
    doublereal d__1;

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

    /* Local variables */
    integer c__, i__, j, k;
    doublereal r__;
    integer s, u, z__;
    doublereal cs;
    integer bx;
    doublereal sn;
    integer st, vt, nm1, st1;
    doublereal eps;
    integer iwk;
    doublereal tol;
    integer difl, difr;
    doublereal rcnd;
    integer perm, nsub;
    extern /* Subroutine */ int drot_(integer *, doublereal *, integer *, 
	    doublereal *, integer *, doublereal *, doublereal *);
    integer nlvl, sqre, bxst;
    extern /* Subroutine */ int dgemm_(char *, char *, integer *, integer *, 
	    integer *, doublereal *, doublereal *, integer *, doublereal *, 
	    integer *, doublereal *, doublereal *, integer *),
	     dcopy_(integer *, doublereal *, integer *, doublereal *, integer 
	    *);
    integer poles, sizei, nsize, nwork, icmpq1, icmpq2;
    extern doublereal dlamch_(char *);
    extern /* Subroutine */ int dlasda_(integer *, integer *, integer *, 
	    integer *, doublereal *, doublereal *, doublereal *, integer *, 
	    doublereal *, integer *, doublereal *, doublereal *, doublereal *, 
	     doublereal *, integer *, integer *, integer *, integer *, 
	    doublereal *, doublereal *, doublereal *, doublereal *, integer *, 
	     integer *), dlalsa_(integer *, integer *, integer *, integer *, 
	    doublereal *, integer *, doublereal *, integer *, doublereal *, 
	    integer *, doublereal *, integer *, doublereal *, doublereal *, 
	    doublereal *, doublereal *, integer *, integer *, integer *, 
	    integer *, doublereal *, doublereal *, doublereal *, doublereal *, 
	     integer *, integer *), dlascl_(char *, integer *, integer *, 
	    doublereal *, doublereal *, integer *, integer *, doublereal *, 
	    integer *, integer *);
    extern integer idamax_(integer *, doublereal *, integer *);
    extern /* Subroutine */ int dlasdq_(char *, integer *, integer *, integer 
	    *, integer *, integer *, doublereal *, doublereal *, doublereal *, 
	     integer *, doublereal *, integer *, doublereal *, integer *, 
	    doublereal *, integer *), dlacpy_(char *, integer *, 
	    integer *, doublereal *, integer *, doublereal *, integer *), dlartg_(doublereal *, doublereal *, doublereal *, 
	    doublereal *, doublereal *), dlaset_(char *, integer *, integer *, 
	     doublereal *, doublereal *, doublereal *, integer *), 
	    xerbla_(char *, integer *);
    integer givcol;
    extern doublereal dlanst_(char *, integer *, doublereal *, doublereal *);
    extern /* Subroutine */ int dlasrt_(char *, integer *, doublereal *, 
	    integer *);
    doublereal orgnrm;
    integer givnum, givptr, smlszp;


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

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

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

/*  DLALSD uses the singular value decomposition of A to solve the least */
/*  squares problem of finding X to minimize the Euclidean norm of each */
/*  column of A*X-B, where A is N-by-N upper bidiagonal, and X and B */
/*  are N-by-NRHS. The solution X overwrites B. */

/*  The singular values of A smaller than RCOND times the largest */
/*  singular value are treated as zero in solving the least squares */
/*  problem; in this case a minimum norm solution is returned. */
/*  The actual singular values are returned in D in ascending order. */

/*  This code makes very mild assumptions about floating point */
/*  arithmetic. It will work on machines with a guard digit in */
/*  add/subtract, or on those binary machines without guard digits */
/*  which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2. */
/*  It could conceivably fail on hexadecimal or decimal machines */
/*  without guard digits, but we know of none. */

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

/*  UPLO   (input) CHARACTER*1 */
/*         = 'U': D and E define an upper bidiagonal matrix. */
/*         = 'L': D and E define a  lower bidiagonal matrix. */

/*  SMLSIZ (input) INTEGER */
/*         The maximum size of the subproblems at the bottom of the */
/*         computation tree. */

/*  N      (input) INTEGER */
/*         The dimension of the  bidiagonal matrix.  N >= 0. */

/*  NRHS   (input) INTEGER */
/*         The number of columns of B. NRHS must be at least 1. */

/*  D      (input/output) DOUBLE PRECISION array, dimension (N) */
/*         On entry D contains the main diagonal of the bidiagonal */
/*         matrix. On exit, if INFO = 0, D contains its singular values. */

/*  E      (input/output) DOUBLE PRECISION array, dimension (N-1) */
/*         Contains the super-diagonal entries of the bidiagonal matrix. */
/*         On exit, E has been destroyed. */

/*  B      (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS) */
/*         On input, B contains the right hand sides of the least */
/*         squares problem. On output, B contains the solution X. */

/*  LDB    (input) INTEGER */
/*         The leading dimension of B in the calling subprogram. */
/*         LDB must be at least max(1,N). */

/*  RCOND  (input) DOUBLE PRECISION */
/*         The singular values of A less than or equal to RCOND times */
/*         the largest singular value are treated as zero in solving */
/*         the least squares problem. If RCOND is negative, */
/*         machine precision is used instead. */
/*         For example, if diag(S)*X=B were the least squares problem, */
/*         where diag(S) is a diagonal matrix of singular values, the */
/*         solution would be X(i) = B(i) / S(i) if S(i) is greater than */
/*         RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to */
/*         RCOND*max(S). */

/*  RANK   (output) INTEGER */
/*         The number of singular values of A greater than RCOND times */
/*         the largest singular value. */

/*  WORK   (workspace) DOUBLE PRECISION array, dimension at least */
/*         (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2), */
/*         where NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1). */

/*  IWORK  (workspace) INTEGER array, dimension at least */
/*         (3*N*NLVL + 11*N) */

/*  INFO   (output) INTEGER */
/*         = 0:  successful exit. */
/*         < 0:  if INFO = -i, the i-th argument had an illegal value. */
/*         > 0:  The algorithm failed to compute an singular value while */
/*               working on the submatrix lying in rows and columns */
/*               INFO/(N+1) through MOD(INFO,N+1). */

/*  Further Details */
/*  =============== */

/*  Based on contributions by */
/*     Ming Gu and Ren-Cang Li, Computer Science Division, University of */
/*       California at Berkeley, USA */
/*     Osni Marques, LBNL/NERSC, USA */

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

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

/*     Test the input parameters. */

    /* Parameter adjustments */
    --d__;
    --e;
    b_dim1 = *ldb;
    b_offset = 1 + b_dim1;
    b -= b_offset;
    --work;
    --iwork;

    /* Function Body */
    *info = 0;

    if (*n < 0) {
	*info = -3;
    } else if (*nrhs < 1) {
	*info = -4;
    } else if (*ldb < 1 || *ldb < *n) {
	*info = -8;
    }
    if (*info != 0) {
	i__1 = -(*info);
	xerbla_("DLALSD", &i__1);
	return 0;
    }

    eps = dlamch_("Epsilon");

/*     Set up the tolerance. */

    if (*rcond <= 0. || *rcond >= 1.) {
	rcnd = eps;
    } else {
	rcnd = *rcond;
    }

    *rank = 0;

/*     Quick return if possible. */

    if (*n == 0) {
	return 0;
    } else if (*n == 1) {
	if (d__[1] == 0.) {
	    dlaset_("A", &c__1, nrhs, &c_b6, &c_b6, &b[b_offset], ldb);
	} else {
	    *rank = 1;
	    dlascl_("G", &c__0, &c__0, &d__[1], &c_b11, &c__1, nrhs, &b[
		    b_offset], ldb, info);
	    d__[1] = abs(d__[1]);
	}
	return 0;
    }

/*     Rotate the matrix if it is lower bidiagonal. */

    if (*(unsigned char *)uplo == 'L') {
	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];
	    if (*nrhs == 1) {
		drot_(&c__1, &b[i__ + b_dim1], &c__1, &b[i__ + 1 + b_dim1], &
			c__1, &cs, &sn);
	    } else {
		work[(i__ << 1) - 1] = cs;
		work[i__ * 2] = sn;
	    }
/* L10: */
	}
	if (*nrhs > 1) {
	    i__1 = *nrhs;
	    for (i__ = 1; i__ <= i__1; ++i__) {
		i__2 = *n - 1;
		for (j = 1; j <= i__2; ++j) {
		    cs = work[(j << 1) - 1];
		    sn = work[j * 2];
		    drot_(&c__1, &b[j + i__ * b_dim1], &c__1, &b[j + 1 + i__ *
			     b_dim1], &c__1, &cs, &sn);
/* L20: */
		}
/* L30: */
	    }
	}
    }

/*     Scale. */

    nm1 = *n - 1;
    orgnrm = dlanst_("M", n, &d__[1], &e[1]);
    if (orgnrm == 0.) {
	dlaset_("A", n, nrhs, &c_b6, &c_b6, &b[b_offset], ldb);
	return 0;
    }

    dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, &c__1, &d__[1], n, info);
    dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, &nm1, &c__1, &e[1], &nm1, 
	    info);

/*     If N is smaller than the minimum divide size SMLSIZ, then solve */
/*     the problem with another solver. */

    if (*n <= *smlsiz) {
	nwork = *n * *n + 1;
	dlaset_("A", n, n, &c_b6, &c_b11, &work[1], n);
	dlasdq_("U", &c__0, n, n, &c__0, nrhs, &d__[1], &e[1], &work[1], n, &
		work[1], n, &b[b_offset], ldb, &work[nwork], info);
	if (*info != 0) {
	    return 0;
	}
	tol = rcnd * (d__1 = d__[idamax_(n, &d__[1], &c__1)], abs(d__1));
	i__1 = *n;
	for (i__ = 1; i__ <= i__1; ++i__) {
	    if (d__[i__] <= tol) {
		dlaset_("A", &c__1, nrhs, &c_b6, &c_b6, &b[i__ + b_dim1], ldb);
	    } else {
		dlascl_("G", &c__0, &c__0, &d__[i__], &c_b11, &c__1, nrhs, &b[
			i__ + b_dim1], ldb, info);
		++(*rank);
	    }
/* L40: */
	}
	dgemm_("T", "N", n, nrhs, n, &c_b11, &work[1], n, &b[b_offset], ldb, &
		c_b6, &work[nwork], n);
	dlacpy_("A", n, nrhs, &work[nwork], n, &b[b_offset], ldb);

/*        Unscale. */

	dlascl_("G", &c__0, &c__0, &c_b11, &orgnrm, n, &c__1, &d__[1], n, 
		info);
	dlasrt_("D", n, &d__[1], info);
	dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, nrhs, &b[b_offset], 
		ldb, info);

	return 0;
    }

/*     Book-keeping and setting up some constants. */

    nlvl = (integer) (log((doublereal) (*n) / (doublereal) (*smlsiz + 1)) / 
	    log(2.)) + 1;

    smlszp = *smlsiz + 1;

    u = 1;
    vt = *smlsiz * *n + 1;
    difl = vt + smlszp * *n;
    difr = difl + nlvl * *n;
    z__ = difr + (nlvl * *n << 1);
    c__ = z__ + nlvl * *n;
    s = c__ + *n;
    poles = s + *n;
    givnum = poles + (nlvl << 1) * *n;
    bx = givnum + (nlvl << 1) * *n;
    nwork = bx + *n * *nrhs;

    sizei = *n + 1;
    k = sizei + *n;
    givptr = k + *n;
    perm = givptr + *n;
    givcol = perm + nlvl * *n;
    iwk = givcol + (nlvl * *n << 1);

    st = 1;
    sqre = 0;
    icmpq1 = 1;
    icmpq2 = 0;
    nsub = 0;

    i__1 = *n;
    for (i__ = 1; i__ <= i__1; ++i__) {
	if ((d__1 = d__[i__], abs(d__1)) < eps) {
	    d__[i__] = d_sign(&eps, &d__[i__]);
	}
/* L50: */
    }

    i__1 = nm1;
    for (i__ = 1; i__ <= i__1; ++i__) {
	if ((d__1 = e[i__], abs(d__1)) < eps || i__ == nm1) {
	    ++nsub;
	    iwork[nsub] = st;

/*           Subproblem found. First determine its size and then */
/*           apply divide and conquer on it. */

	    if (i__ < nm1) {

/*              A subproblem with E(I) small for I < NM1. */

		nsize = i__ - st + 1;
		iwork[sizei + nsub - 1] = nsize;
	    } else if ((d__1 = e[i__], abs(d__1)) >= eps) {

/*              A subproblem with E(NM1) not too small but I = NM1. */

		nsize = *n - st + 1;
		iwork[sizei + nsub - 1] = nsize;
	    } else {

/*              A subproblem with E(NM1) small. This implies an */
/*              1-by-1 subproblem at D(N), which is not solved */
/*              explicitly. */

		nsize = i__ - st + 1;
		iwork[sizei + nsub - 1] = nsize;
		++nsub;
		iwork[nsub] = *n;
		iwork[sizei + nsub - 1] = 1;
		dcopy_(nrhs, &b[*n + b_dim1], ldb, &work[bx + nm1], n);
	    }
	    st1 = st - 1;
	    if (nsize == 1) {

/*              This is a 1-by-1 subproblem and is not solved */
/*              explicitly. */

		dcopy_(nrhs, &b[st + b_dim1], ldb, &work[bx + st1], n);
	    } else if (nsize <= *smlsiz) {

/*              This is a small subproblem and is solved by DLASDQ. */

		dlaset_("A", &nsize, &nsize, &c_b6, &c_b11, &work[vt + st1], 
			n);
		dlasdq_("U", &c__0, &nsize, &nsize, &c__0, nrhs, &d__[st], &e[
			st], &work[vt + st1], n, &work[nwork], n, &b[st + 
			b_dim1], ldb, &work[nwork], info);
		if (*info != 0) {
		    return 0;
		}
		dlacpy_("A", &nsize, nrhs, &b[st + b_dim1], ldb, &work[bx + 
			st1], n);
	    } else {

/*              A large problem. Solve it using divide and conquer. */

		dlasda_(&icmpq1, smlsiz, &nsize, &sqre, &d__[st], &e[st], &
			work[u + st1], n, &work[vt + st1], &iwork[k + st1], &
			work[difl + st1], &work[difr + st1], &work[z__ + st1], 
			 &work[poles + st1], &iwork[givptr + st1], &iwork[
			givcol + st1], n, &iwork[perm + st1], &work[givnum + 
			st1], &work[c__ + st1], &work[s + st1], &work[nwork], 
			&iwork[iwk], info);
		if (*info != 0) {
		    return 0;
		}
		bxst = bx + st1;
		dlalsa_(&icmpq2, smlsiz, &nsize, nrhs, &b[st + b_dim1], ldb, &
			work[bxst], n, &work[u + st1], n, &work[vt + st1], &
			iwork[k + st1], &work[difl + st1], &work[difr + st1], 
			&work[z__ + st1], &work[poles + st1], &iwork[givptr + 
			st1], &iwork[givcol + st1], n, &iwork[perm + st1], &
			work[givnum + st1], &work[c__ + st1], &work[s + st1], 
			&work[nwork], &iwork[iwk], info);
		if (*info != 0) {
		    return 0;
		}
	    }
	    st = i__ + 1;
	}
/* L60: */
    }

/*     Apply the singular values and treat the tiny ones as zero. */

    tol = rcnd * (d__1 = d__[idamax_(n, &d__[1], &c__1)], abs(d__1));

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

/*        Some of the elements in D can be negative because 1-by-1 */
/*        subproblems were not solved explicitly. */

	if ((d__1 = d__[i__], abs(d__1)) <= tol) {
	    dlaset_("A", &c__1, nrhs, &c_b6, &c_b6, &work[bx + i__ - 1], n);
	} else {
	    ++(*rank);
	    dlascl_("G", &c__0, &c__0, &d__[i__], &c_b11, &c__1, nrhs, &work[
		    bx + i__ - 1], n, info);
	}
	d__[i__] = (d__1 = d__[i__], abs(d__1));
/* L70: */
    }

/*     Now apply back the right singular vectors. */

    icmpq2 = 1;
    i__1 = nsub;
    for (i__ = 1; i__ <= i__1; ++i__) {
	st = iwork[i__];
	st1 = st - 1;
	nsize = iwork[sizei + i__ - 1];
	bxst = bx + st1;
	if (nsize == 1) {
	    dcopy_(nrhs, &work[bxst], n, &b[st + b_dim1], ldb);
	} else if (nsize <= *smlsiz) {
	    dgemm_("T", "N", &nsize, nrhs, &nsize, &c_b11, &work[vt + st1], n, 
		     &work[bxst], n, &c_b6, &b[st + b_dim1], ldb);
	} else {
	    dlalsa_(&icmpq2, smlsiz, &nsize, nrhs, &work[bxst], n, &b[st + 
		    b_dim1], ldb, &work[u + st1], n, &work[vt + st1], &iwork[
		    k + st1], &work[difl + st1], &work[difr + st1], &work[z__ 
		    + st1], &work[poles + st1], &iwork[givptr + st1], &iwork[
		    givcol + st1], n, &iwork[perm + st1], &work[givnum + st1], 
		     &work[c__ + st1], &work[s + st1], &work[nwork], &iwork[
		    iwk], info);
	    if (*info != 0) {
		return 0;
	    }
	}
/* L80: */
    }

/*     Unscale and sort the singular values. */

    dlascl_("G", &c__0, &c__0, &c_b11, &orgnrm, n, &c__1, &d__[1], n, info);
    dlasrt_("D", n, &d__[1], info);
    dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, nrhs, &b[b_offset], ldb, 
	    info);

    return 0;

/*     End of DLALSD */

} /* dlalsd_ */
Esempio n. 2
0
/* Subroutine */ int dbdsdc_(char *uplo, char *compq, integer *n, doublereal *
	d__, doublereal *e, doublereal *u, integer *ldu, doublereal *vt, 
	integer *ldvt, doublereal *q, integer *iq, doublereal *work, integer *
	iwork, integer *info)
{
    /* System generated locals */
    integer u_dim1, u_offset, vt_dim1, vt_offset, i__1, i__2;
    doublereal d__1;

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

    /* Local variables */
    integer i__, j, k;
    doublereal p, r__;
    integer z__, ic, ii, kk;
    doublereal cs;
    integer is, iu;
    doublereal sn;
    integer nm1;
    doublereal eps;
    integer ivt, difl, difr, ierr, perm, mlvl, sqre;
    extern logical lsame_(char *, char *);
    extern /* Subroutine */ int dlasr_(char *, char *, char *, integer *, 
	    integer *, doublereal *, doublereal *, doublereal *, integer *), dcopy_(integer *, doublereal *, integer *
, doublereal *, integer *), dswap_(integer *, doublereal *, 
	    integer *, doublereal *, integer *);
    integer poles, iuplo, nsize, start;
    extern /* Subroutine */ int dlasd0_(integer *, integer *, doublereal *, 
	    doublereal *, doublereal *, integer *, doublereal *, integer *, 
	    integer *, integer *, doublereal *, integer *);
    extern doublereal dlamch_(char *);
    extern /* Subroutine */ int dlasda_(integer *, integer *, integer *, 
	    integer *, doublereal *, doublereal *, doublereal *, integer *, 
	    doublereal *, integer *, doublereal *, doublereal *, doublereal *, 
	     doublereal *, integer *, integer *, integer *, integer *, 
	    doublereal *, doublereal *, doublereal *, doublereal *, integer *, 
	     integer *), dlascl_(char *, integer *, integer *, doublereal *, 
	    doublereal *, integer *, integer *, doublereal *, integer *, 
	    integer *), dlasdq_(char *, integer *, integer *, integer 
	    *, integer *, integer *, doublereal *, doublereal *, doublereal *, 
	     integer *, doublereal *, integer *, doublereal *, integer *, 
	    doublereal *, integer *), dlaset_(char *, integer *, 
	    integer *, doublereal *, doublereal *, doublereal *, integer *), dlartg_(doublereal *, doublereal *, doublereal *, 
	    doublereal *, doublereal *);
    extern integer ilaenv_(integer *, char *, char *, integer *, integer *, 
	    integer *, integer *);
    extern /* Subroutine */ int xerbla_(char *, integer *);
    integer givcol;
    extern doublereal dlanst_(char *, integer *, doublereal *, doublereal *);
    integer icompq;
    doublereal orgnrm;
    integer givnum, givptr, qstart, smlsiz, wstart, smlszp;


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

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

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

/*  DBDSDC computes the singular value decomposition (SVD) of a real */
/*  N-by-N (upper or lower) bidiagonal matrix B:  B = U * S * VT, */
/*  using a divide and conquer method, where S is a diagonal matrix */
/*  with non-negative diagonal elements (the singular values of B), and */
/*  U and VT are orthogonal matrices of left and right singular vectors, */
/*  respectively. DBDSDC can be used to compute all singular values, */
/*  and optionally, singular vectors or singular vectors in compact form. */

/*  This code makes very mild assumptions about floating point */
/*  arithmetic. It will work on machines with a guard digit in */
/*  add/subtract, or on those binary machines without guard digits */
/*  which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. */
/*  It could conceivably fail on hexadecimal or decimal machines */
/*  without guard digits, but we know of none.  See DLASD3 for details. */

/*  The code currently calls DLASDQ if singular values only are desired. */
/*  However, it can be slightly modified to compute singular values */
/*  using the divide and conquer method. */

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

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

/*  COMPQ   (input) CHARACTER*1 */
/*          Specifies whether singular vectors are to be computed */
/*          as follows: */
/*          = 'N':  Compute singular values only; */
/*          = 'P':  Compute singular values and compute singular */
/*                  vectors in compact form; */
/*          = 'I':  Compute singular values and singular vectors. */

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

/*  E       (input/output) DOUBLE PRECISION array, dimension (N-1) */
/*          On entry, the elements of E contain the offdiagonal */
/*          elements of the bidiagonal matrix whose SVD is desired. */
/*          On exit, E has been destroyed. */

/*  U       (output) DOUBLE PRECISION array, dimension (LDU,N) */
/*          If  COMPQ = 'I', then: */
/*             On exit, if INFO = 0, U contains the left singular vectors */
/*             of the bidiagonal matrix. */
/*          For other values of COMPQ, U is not referenced. */

/*  LDU     (input) INTEGER */
/*          The leading dimension of the array U.  LDU >= 1. */
/*          If singular vectors are desired, then LDU >= max( 1, N ). */

/*  VT      (output) DOUBLE PRECISION array, dimension (LDVT,N) */
/*          If  COMPQ = 'I', then: */
/*             On exit, if INFO = 0, VT' contains the right singular */
/*             vectors of the bidiagonal matrix. */
/*          For other values of COMPQ, VT is not referenced. */

/*  LDVT    (input) INTEGER */
/*          The leading dimension of the array VT.  LDVT >= 1. */
/*          If singular vectors are desired, then LDVT >= max( 1, N ). */

/*  Q       (output) DOUBLE PRECISION array, dimension (LDQ) */
/*          If  COMPQ = 'P', then: */
/*             On exit, if INFO = 0, Q and IQ contain the left */
/*             and right singular vectors in a compact form, */
/*             requiring O(N log N) space instead of 2*N**2. */
/*             In particular, Q contains all the DOUBLE PRECISION data in */
/*             LDQ >= N*(11 + 2*SMLSIZ + 8*INT(LOG_2(N/(SMLSIZ+1)))) */
/*             words of memory, where SMLSIZ is returned by ILAENV and */
/*             is equal to the maximum size of the subproblems at the */
/*             bottom of the computation tree (usually about 25). */
/*          For other values of COMPQ, Q is not referenced. */

/*  IQ      (output) INTEGER array, dimension (LDIQ) */
/*          If  COMPQ = 'P', then: */
/*             On exit, if INFO = 0, Q and IQ contain the left */
/*             and right singular vectors in a compact form, */
/*             requiring O(N log N) space instead of 2*N**2. */
/*             In particular, IQ contains all INTEGER data in */
/*             LDIQ >= N*(3 + 3*INT(LOG_2(N/(SMLSIZ+1)))) */
/*             words of memory, where SMLSIZ is returned by ILAENV and */
/*             is equal to the maximum size of the subproblems at the */
/*             bottom of the computation tree (usually about 25). */
/*          For other values of COMPQ, IQ is not referenced. */

/*  WORK    (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) */
/*          If COMPQ = 'N' then LWORK >= (4 * N). */
/*          If COMPQ = 'P' then LWORK >= (6 * N). */
/*          If COMPQ = 'I' then LWORK >= (3 * N**2 + 4 * N). */

/*  IWORK   (workspace) INTEGER array, dimension (8*N) */

/*  INFO    (output) INTEGER */
/*          = 0:  successful exit. */
/*          < 0:  if INFO = -i, the i-th argument had an illegal value. */
/*          > 0:  The algorithm failed to compute an singular value. */
/*                The update process of divide and conquer failed. */

/*  Further Details */
/*  =============== */

/*  Based on contributions by */
/*     Ming Gu and Huan Ren, Computer Science Division, University of */
/*     California at Berkeley, USA */

/*  ===================================================================== */
/*  Changed dimension statement in comment describing E from (N) to */
/*  (N-1).  Sven, 17 Feb 05. */
/*  ===================================================================== */

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

/*     Test the input parameters. */

    /* Parameter adjustments */
    --d__;
    --e;
    u_dim1 = *ldu;
    u_offset = 1 + u_dim1;
    u -= u_offset;
    vt_dim1 = *ldvt;
    vt_offset = 1 + vt_dim1;
    vt -= vt_offset;
    --q;
    --iq;
    --work;
    --iwork;

    /* Function Body */
    *info = 0;

    iuplo = 0;
    if (lsame_(uplo, "U")) {
	iuplo = 1;
    }
    if (lsame_(uplo, "L")) {
	iuplo = 2;
    }
    if (lsame_(compq, "N")) {
	icompq = 0;
    } else if (lsame_(compq, "P")) {
	icompq = 1;
    } else if (lsame_(compq, "I")) {
	icompq = 2;
    } else {
	icompq = -1;
    }
    if (iuplo == 0) {
	*info = -1;
    } else if (icompq < 0) {
	*info = -2;
    } else if (*n < 0) {
	*info = -3;
    } else if (*ldu < 1 || icompq == 2 && *ldu < *n) {
	*info = -7;
    } else if (*ldvt < 1 || icompq == 2 && *ldvt < *n) {
	*info = -9;
    }
    if (*info != 0) {
	i__1 = -(*info);
	xerbla_("DBDSDC", &i__1);
	return 0;
    }

/*     Quick return if possible */

    if (*n == 0) {
	return 0;
    }
    smlsiz = ilaenv_(&c__9, "DBDSDC", " ", &c__0, &c__0, &c__0, &c__0);
    if (*n == 1) {
	if (icompq == 1) {
	    q[1] = d_sign(&c_b15, &d__[1]);
	    q[smlsiz * *n + 1] = 1.;
	} else if (icompq == 2) {
	    u[u_dim1 + 1] = d_sign(&c_b15, &d__[1]);
	    vt[vt_dim1 + 1] = 1.;
	}
	d__[1] = abs(d__[1]);
	return 0;
    }
    nm1 = *n - 1;

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

    wstart = 1;
    qstart = 3;
    if (icompq == 1) {
	dcopy_(n, &d__[1], &c__1, &q[1], &c__1);
	i__1 = *n - 1;
	dcopy_(&i__1, &e[1], &c__1, &q[*n + 1], &c__1);
    }
    if (iuplo == 2) {
	qstart = 5;
	wstart = (*n << 1) - 1;
	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];
	    if (icompq == 1) {
		q[i__ + (*n << 1)] = cs;
		q[i__ + *n * 3] = sn;
	    } else if (icompq == 2) {
		work[i__] = cs;
		work[nm1 + i__] = -sn;
	    }
/* L10: */
	}
    }

/*     If ICOMPQ = 0, use DLASDQ to compute the singular values. */

    if (icompq == 0) {
	dlasdq_("U", &c__0, n, &c__0, &c__0, &c__0, &d__[1], &e[1], &vt[
		vt_offset], ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[
		wstart], info);
	goto L40;
    }

/*     If N is smaller than the minimum divide size SMLSIZ, then solve */
/*     the problem with another solver. */

    if (*n <= smlsiz) {
	if (icompq == 2) {
	    dlaset_("A", n, n, &c_b29, &c_b15, &u[u_offset], ldu);
	    dlaset_("A", n, n, &c_b29, &c_b15, &vt[vt_offset], ldvt);
	    dlasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &vt[vt_offset]
, ldvt, &u[u_offset], ldu, &u[u_offset], ldu, &work[
		    wstart], info);
	} else if (icompq == 1) {
	    iu = 1;
	    ivt = iu + *n;
	    dlaset_("A", n, n, &c_b29, &c_b15, &q[iu + (qstart - 1) * *n], n);
	    dlaset_("A", n, n, &c_b29, &c_b15, &q[ivt + (qstart - 1) * *n], n);
	    dlasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &q[ivt + (
		    qstart - 1) * *n], n, &q[iu + (qstart - 1) * *n], n, &q[
		    iu + (qstart - 1) * *n], n, &work[wstart], info);
	}
	goto L40;
    }

    if (icompq == 2) {
	dlaset_("A", n, n, &c_b29, &c_b15, &u[u_offset], ldu);
	dlaset_("A", n, n, &c_b29, &c_b15, &vt[vt_offset], ldvt);
    }

/*     Scale. */

    orgnrm = dlanst_("M", n, &d__[1], &e[1]);
    if (orgnrm == 0.) {
	return 0;
    }
    dlascl_("G", &c__0, &c__0, &orgnrm, &c_b15, n, &c__1, &d__[1], n, &ierr);
    dlascl_("G", &c__0, &c__0, &orgnrm, &c_b15, &nm1, &c__1, &e[1], &nm1, &
	    ierr);

    eps = dlamch_("Epsilon");

    mlvl = (integer) (log((doublereal) (*n) / (doublereal) (smlsiz + 1)) / 
	    log(2.)) + 1;
    smlszp = smlsiz + 1;

    if (icompq == 1) {
	iu = 1;
	ivt = smlsiz + 1;
	difl = ivt + smlszp;
	difr = difl + mlvl;
	z__ = difr + (mlvl << 1);
	ic = z__ + mlvl;
	is = ic + 1;
	poles = is + 1;
	givnum = poles + (mlvl << 1);

	k = 1;
	givptr = 2;
	perm = 3;
	givcol = perm + mlvl;
    }

    i__1 = *n;
    for (i__ = 1; i__ <= i__1; ++i__) {
	if ((d__1 = d__[i__], abs(d__1)) < eps) {
	    d__[i__] = d_sign(&eps, &d__[i__]);
	}
/* L20: */
    }

    start = 1;
    sqre = 0;

    i__1 = nm1;
    for (i__ = 1; i__ <= i__1; ++i__) {
	if ((d__1 = e[i__], abs(d__1)) < eps || i__ == nm1) {

/*        Subproblem found. First determine its size and then */
/*        apply divide and conquer on it. */

	    if (i__ < nm1) {

/*        A subproblem with E(I) small for I < NM1. */

		nsize = i__ - start + 1;
	    } else if ((d__1 = e[i__], abs(d__1)) >= eps) {

/*        A subproblem with E(NM1) not too small but I = NM1. */

		nsize = *n - start + 1;
	    } else {

/*        A subproblem with E(NM1) small. This implies an */
/*        1-by-1 subproblem at D(N). Solve this 1-by-1 problem */
/*        first. */

		nsize = i__ - start + 1;
		if (icompq == 2) {
		    u[*n + *n * u_dim1] = d_sign(&c_b15, &d__[*n]);
		    vt[*n + *n * vt_dim1] = 1.;
		} else if (icompq == 1) {
		    q[*n + (qstart - 1) * *n] = d_sign(&c_b15, &d__[*n]);
		    q[*n + (smlsiz + qstart - 1) * *n] = 1.;
		}
		d__[*n] = (d__1 = d__[*n], abs(d__1));
	    }
	    if (icompq == 2) {
		dlasd0_(&nsize, &sqre, &d__[start], &e[start], &u[start + 
			start * u_dim1], ldu, &vt[start + start * vt_dim1], 
			ldvt, &smlsiz, &iwork[1], &work[wstart], info);
	    } else {
		dlasda_(&icompq, &smlsiz, &nsize, &sqre, &d__[start], &e[
			start], &q[start + (iu + qstart - 2) * *n], n, &q[
			start + (ivt + qstart - 2) * *n], &iq[start + k * *n], 
			 &q[start + (difl + qstart - 2) * *n], &q[start + (
			difr + qstart - 2) * *n], &q[start + (z__ + qstart - 
			2) * *n], &q[start + (poles + qstart - 2) * *n], &iq[
			start + givptr * *n], &iq[start + givcol * *n], n, &
			iq[start + perm * *n], &q[start + (givnum + qstart - 
			2) * *n], &q[start + (ic + qstart - 2) * *n], &q[
			start + (is + qstart - 2) * *n], &work[wstart], &
			iwork[1], info);
		if (*info != 0) {
		    return 0;
		}
	    }
	    start = i__ + 1;
	}
/* L30: */
    }

/*     Unscale */

    dlascl_("G", &c__0, &c__0, &c_b15, &orgnrm, n, &c__1, &d__[1], n, &ierr);
L40:

/*     Use Selection Sort to minimize swaps of singular vectors */

    i__1 = *n;
    for (ii = 2; ii <= i__1; ++ii) {
	i__ = ii - 1;
	kk = i__;
	p = d__[i__];
	i__2 = *n;
	for (j = ii; j <= i__2; ++j) {
	    if (d__[j] > p) {
		kk = j;
		p = d__[j];
	    }
/* L50: */
	}
	if (kk != i__) {
	    d__[kk] = d__[i__];
	    d__[i__] = p;
	    if (icompq == 1) {
		iq[i__] = kk;
	    } else if (icompq == 2) {
		dswap_(n, &u[i__ * u_dim1 + 1], &c__1, &u[kk * u_dim1 + 1], &
			c__1);
		dswap_(n, &vt[i__ + vt_dim1], ldvt, &vt[kk + vt_dim1], ldvt);
	    }
	} else if (icompq == 1) {
	    iq[i__] = i__;
	}
/* L60: */
    }

/*     If ICOMPQ = 1, use IQ(N,1) as the indicator for UPLO */

    if (icompq == 1) {
	if (iuplo == 1) {
	    iq[*n] = 1;
	} else {
	    iq[*n] = 0;
	}
    }

/*     If B is lower bidiagonal, update U by those Givens rotations */
/*     which rotated B to be upper bidiagonal */

    if (iuplo == 2 && icompq == 2) {
	dlasr_("L", "V", "B", n, n, &work[1], &work[*n], &u[u_offset], ldu);
    }

    return 0;

/*     End of DBDSDC */

} /* dbdsdc_ */
Esempio n. 3
0
/* Subroutine */ int zlalsd_(char *uplo, integer *smlsiz, integer *n, integer 
	*nrhs, doublereal *d__, doublereal *e, doublecomplex *b, integer *ldb,
	 doublereal *rcond, integer *rank, doublecomplex *work, doublereal *
	rwork, integer *iwork, integer *info)
{
    /* System generated locals */
    integer b_dim1, b_offset, i__1, i__2, i__3, i__4, i__5, i__6;
    doublereal d__1;
    doublecomplex z__1;

    /* Builtin functions */
    double d_imag(doublecomplex *), log(doublereal), d_sign(doublereal *, 
	    doublereal *);

    /* Local variables */
    static integer difl, difr, jcol, irwb, perm, nsub, nlvl, sqre, bxst, jrow,
	     irwu, c__, i__, j, k;
    static doublereal r__;
    static integer s, u, jimag;
    extern /* Subroutine */ int dgemm_(char *, char *, integer *, integer *, 
	    integer *, doublereal *, doublereal *, integer *, doublereal *, 
	    integer *, doublereal *, doublereal *, integer *);
    static integer z__, jreal, irwib, poles, sizei, irwrb, nsize;
    extern /* Subroutine */ int zdrot_(integer *, doublecomplex *, integer *, 
	    doublecomplex *, integer *, doublereal *, doublereal *), zcopy_(
	    integer *, doublecomplex *, integer *, doublecomplex *, integer *)
	    ;
    static integer irwvt, icmpq1, icmpq2;
    static doublereal cs;
    extern doublereal dlamch_(char *);
    extern /* Subroutine */ int dlasda_(integer *, integer *, integer *, 
	    integer *, doublereal *, doublereal *, doublereal *, integer *, 
	    doublereal *, integer *, doublereal *, doublereal *, doublereal *,
	     doublereal *, integer *, integer *, integer *, integer *, 
	    doublereal *, doublereal *, doublereal *, doublereal *, integer *,
	     integer *);
    static integer bx;
    static doublereal sn;
    extern /* Subroutine */ int dlascl_(char *, integer *, integer *, 
	    doublereal *, doublereal *, integer *, integer *, doublereal *, 
	    integer *, integer *);
    extern integer idamax_(integer *, doublereal *, integer *);
    static integer st;
    extern /* Subroutine */ int dlasdq_(char *, integer *, integer *, integer 
	    *, integer *, integer *, doublereal *, doublereal *, doublereal *,
	     integer *, doublereal *, integer *, doublereal *, integer *, 
	    doublereal *, integer *);
    static integer vt;
    extern /* Subroutine */ int dlaset_(char *, integer *, integer *, 
	    doublereal *, doublereal *, doublereal *, integer *), 
	    dlartg_(doublereal *, doublereal *, doublereal *, doublereal *, 
	    doublereal *), xerbla_(char *, integer *);
    static integer givcol;
    extern doublereal dlanst_(char *, integer *, doublereal *, doublereal *);
    extern /* Subroutine */ int zlalsa_(integer *, integer *, integer *, 
	    integer *, doublecomplex *, integer *, doublecomplex *, integer *,
	     doublereal *, integer *, doublereal *, integer *, doublereal *, 
	    doublereal *, doublereal *, doublereal *, integer *, integer *, 
	    integer *, integer *, doublereal *, doublereal *, doublereal *, 
	    doublereal *, integer *, integer *), zlascl_(char *, integer *, 
	    integer *, doublereal *, doublereal *, integer *, integer *, 
	    doublecomplex *, integer *, integer *), dlasrt_(char *, 
	    integer *, doublereal *, integer *), zlacpy_(char *, 
	    integer *, integer *, doublecomplex *, integer *, doublecomplex *,
	     integer *), zlaset_(char *, integer *, integer *, 
	    doublecomplex *, doublecomplex *, doublecomplex *, integer *);
    static doublereal orgnrm;
    static integer givnum, givptr, nm1, nrwork, irwwrk, smlszp, st1;
    static doublereal eps;
    static integer iwk;
    static doublereal tol;


#define b_subscr(a_1,a_2) (a_2)*b_dim1 + a_1
#define b_ref(a_1,a_2) b[b_subscr(a_1,a_2)]


/*  -- LAPACK routine (version 3.0) --   
       Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd.,   
       Courant Institute, Argonne National Lab, and Rice University   
       October 31, 1999   


    Purpose   
    =======   

    ZLALSD uses the singular value decomposition of A to solve the least   
    squares problem of finding X to minimize the Euclidean norm of each   
    column of A*X-B, where A is N-by-N upper bidiagonal, and X and B   
    are N-by-NRHS. The solution X overwrites B.   

    The singular values of A smaller than RCOND times the largest   
    singular value are treated as zero in solving the least squares   
    problem; in this case a minimum norm solution is returned.   
    The actual singular values are returned in D in ascending order.   

    This code makes very mild assumptions about floating point   
    arithmetic. It will work on machines with a guard digit in   
    add/subtract, or on those binary machines without guard digits   
    which subtract like the Cray XMP, Cray YMP, Cray C 90, or Cray 2.   
    It could conceivably fail on hexadecimal or decimal machines   
    without guard digits, but we know of none.   

    Arguments   
    =========   

    UPLO   (input) CHARACTER*1   
           = 'U': D and E define an upper bidiagonal matrix.   
           = 'L': D and E define a  lower bidiagonal matrix.   

    SMLSIZ (input) INTEGER   
           The maximum size of the subproblems at the bottom of the   
           computation tree.   

    N      (input) INTEGER   
           The dimension of the  bidiagonal matrix.  N >= 0.   

    NRHS   (input) INTEGER   
           The number of columns of B. NRHS must be at least 1.   

    D      (input/output) DOUBLE PRECISION array, dimension (N)   
           On entry D contains the main diagonal of the bidiagonal   
           matrix. On exit, if INFO = 0, D contains its singular values.   

    E      (input) DOUBLE PRECISION array, dimension (N-1)   
           Contains the super-diagonal entries of the bidiagonal matrix.   
           On exit, E has been destroyed.   

    B      (input/output) COMPLEX*16 array, dimension (LDB,NRHS)   
           On input, B contains the right hand sides of the least   
           squares problem. On output, B contains the solution X.   

    LDB    (input) INTEGER   
           The leading dimension of B in the calling subprogram.   
           LDB must be at least max(1,N).   

    RCOND  (input) DOUBLE PRECISION   
           The singular values of A less than or equal to RCOND times   
           the largest singular value are treated as zero in solving   
           the least squares problem. If RCOND is negative,   
           machine precision is used instead.   
           For example, if diag(S)*X=B were the least squares problem,   
           where diag(S) is a diagonal matrix of singular values, the   
           solution would be X(i) = B(i) / S(i) if S(i) is greater than   
           RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to   
           RCOND*max(S).   

    RANK   (output) INTEGER   
           The number of singular values of A greater than RCOND times   
           the largest singular value.   

    WORK   (workspace) COMPLEX*16 array, dimension at least   
           (N * NRHS).   

    RWORK  (workspace) DOUBLE PRECISION array, dimension at least   
           (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + (SMLSIZ+1)**2),   
           where   
           NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )   

    IWORK  (workspace) INTEGER array, dimension at least   
           (3*N*NLVL + 11*N).   

    INFO   (output) INTEGER   
           = 0:  successful exit.   
           < 0:  if INFO = -i, the i-th argument had an illegal value.   
           > 0:  The algorithm failed to compute an singular value while   
                 working on the submatrix lying in rows and columns   
                 INFO/(N+1) through MOD(INFO,N+1).   

    Further Details   
    ===============   

    Based on contributions by   
       Ming Gu and Ren-Cang Li, Computer Science Division, University of   
         California at Berkeley, USA   
       Osni Marques, LBNL/NERSC, USA   

    =====================================================================   


       Test the input parameters.   

       Parameter adjustments */
    --d__;
    --e;
    b_dim1 = *ldb;
    b_offset = 1 + b_dim1 * 1;
    b -= b_offset;
    --work;
    --rwork;
    --iwork;

    /* Function Body */
    *info = 0;

    if (*n < 0) {
	*info = -3;
    } else if (*nrhs < 1) {
	*info = -4;
    } else if (*ldb < 1 || *ldb < *n) {
	*info = -8;
    }
    if (*info != 0) {
	i__1 = -(*info);
	xerbla_("ZLALSD", &i__1);
	return 0;
    }

    eps = dlamch_("Epsilon");

/*     Set up the tolerance. */

    if (*rcond <= 0. || *rcond >= 1.) {
	*rcond = eps;
    }

    *rank = 0;

/*     Quick return if possible. */

    if (*n == 0) {
	return 0;
    } else if (*n == 1) {
	if (d__[1] == 0.) {
	    zlaset_("A", &c__1, nrhs, &c_b1, &c_b1, &b[b_offset], ldb);
	} else {
	    *rank = 1;
	    zlascl_("G", &c__0, &c__0, &d__[1], &c_b10, &c__1, nrhs, &b[
		    b_offset], ldb, info);
	    d__[1] = abs(d__[1]);
	}
	return 0;
    }

/*     Rotate the matrix if it is lower bidiagonal. */

    if (*(unsigned char *)uplo == 'L') {
	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];
	    if (*nrhs == 1) {
		zdrot_(&c__1, &b_ref(i__, 1), &c__1, &b_ref(i__ + 1, 1), &
			c__1, &cs, &sn);
	    } else {
		rwork[(i__ << 1) - 1] = cs;
		rwork[i__ * 2] = sn;
	    }
/* L10: */
	}
	if (*nrhs > 1) {
	    i__1 = *nrhs;
	    for (i__ = 1; i__ <= i__1; ++i__) {
		i__2 = *n - 1;
		for (j = 1; j <= i__2; ++j) {
		    cs = rwork[(j << 1) - 1];
		    sn = rwork[j * 2];
		    zdrot_(&c__1, &b_ref(j, i__), &c__1, &b_ref(j + 1, i__), &
			    c__1, &cs, &sn);
/* L20: */
		}
/* L30: */
	    }
	}
    }

/*     Scale. */

    nm1 = *n - 1;
    orgnrm = dlanst_("M", n, &d__[1], &e[1]);
    if (orgnrm == 0.) {
	zlaset_("A", n, nrhs, &c_b1, &c_b1, &b[b_offset], ldb);
	return 0;
    }

    dlascl_("G", &c__0, &c__0, &orgnrm, &c_b10, n, &c__1, &d__[1], n, info);
    dlascl_("G", &c__0, &c__0, &orgnrm, &c_b10, &nm1, &c__1, &e[1], &nm1, 
	    info);

/*     If N is smaller than the minimum divide size SMLSIZ, then solve   
       the problem with another solver. */

    if (*n <= *smlsiz) {
	irwu = 1;
	irwvt = irwu + *n * *n;
	irwwrk = irwvt + *n * *n;
	irwrb = irwwrk;
	irwib = irwrb + *n * *nrhs;
	irwb = irwib + *n * *nrhs;
	dlaset_("A", n, n, &c_b35, &c_b10, &rwork[irwu], n);
	dlaset_("A", n, n, &c_b35, &c_b10, &rwork[irwvt], n);
	dlasdq_("U", &c__0, n, n, n, &c__0, &d__[1], &e[1], &rwork[irwvt], n, 
		&rwork[irwu], n, &rwork[irwwrk], &c__1, &rwork[irwwrk], info);
	if (*info != 0) {
	    return 0;
	}

/*        In the real version, B is passed to DLASDQ and multiplied   
          internally by Q'. Here B is complex and that product is   
          computed below in two steps (real and imaginary parts). */

	j = irwb - 1;
	i__1 = *nrhs;
	for (jcol = 1; jcol <= i__1; ++jcol) {
	    i__2 = *n;
	    for (jrow = 1; jrow <= i__2; ++jrow) {
		++j;
		i__3 = b_subscr(jrow, jcol);
		rwork[j] = b[i__3].r;
/* L40: */
	    }
/* L50: */
	}
	dgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwu], n, &rwork[irwb], n,
		 &c_b35, &rwork[irwrb], n);
	j = irwb - 1;
	i__1 = *nrhs;
	for (jcol = 1; jcol <= i__1; ++jcol) {
	    i__2 = *n;
	    for (jrow = 1; jrow <= i__2; ++jrow) {
		++j;
		rwork[j] = d_imag(&b_ref(jrow, jcol));
/* L60: */
	    }
/* L70: */
	}
	dgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwu], n, &rwork[irwb], n,
		 &c_b35, &rwork[irwib], n);
	jreal = irwrb - 1;
	jimag = irwib - 1;
	i__1 = *nrhs;
	for (jcol = 1; jcol <= i__1; ++jcol) {
	    i__2 = *n;
	    for (jrow = 1; jrow <= i__2; ++jrow) {
		++jreal;
		++jimag;
		i__3 = b_subscr(jrow, jcol);
		i__4 = jreal;
		i__5 = jimag;
		z__1.r = rwork[i__4], z__1.i = rwork[i__5];
		b[i__3].r = z__1.r, b[i__3].i = z__1.i;
/* L80: */
	    }
/* L90: */
	}

	tol = *rcond * (d__1 = d__[idamax_(n, &d__[1], &c__1)], abs(d__1));
	i__1 = *n;
	for (i__ = 1; i__ <= i__1; ++i__) {
	    if (d__[i__] <= tol) {
		zlaset_("A", &c__1, nrhs, &c_b1, &c_b1, &b_ref(i__, 1), ldb);
	    } else {
		zlascl_("G", &c__0, &c__0, &d__[i__], &c_b10, &c__1, nrhs, &
			b_ref(i__, 1), ldb, info);
		++(*rank);
	    }
/* L100: */
	}

/*        Since B is complex, the following call to DGEMM is performed   
          in two steps (real and imaginary parts). That is for V * B   
          (in the real version of the code V' is stored in WORK).   

          CALL DGEMM( 'T', 'N', N, NRHS, N, ONE, WORK, N, B, LDB, ZERO,   
      $               WORK( NWORK ), N ) */

	j = irwb - 1;
	i__1 = *nrhs;
	for (jcol = 1; jcol <= i__1; ++jcol) {
	    i__2 = *n;
	    for (jrow = 1; jrow <= i__2; ++jrow) {
		++j;
		i__3 = b_subscr(jrow, jcol);
		rwork[j] = b[i__3].r;
/* L110: */
	    }
/* L120: */
	}
	dgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwvt], n, &rwork[irwb], 
		n, &c_b35, &rwork[irwrb], n);
	j = irwb - 1;
	i__1 = *nrhs;
	for (jcol = 1; jcol <= i__1; ++jcol) {
	    i__2 = *n;
	    for (jrow = 1; jrow <= i__2; ++jrow) {
		++j;
		rwork[j] = d_imag(&b_ref(jrow, jcol));
/* L130: */
	    }
/* L140: */
	}
	dgemm_("T", "N", n, nrhs, n, &c_b10, &rwork[irwvt], n, &rwork[irwb], 
		n, &c_b35, &rwork[irwib], n);
	jreal = irwrb - 1;
	jimag = irwib - 1;
	i__1 = *nrhs;
	for (jcol = 1; jcol <= i__1; ++jcol) {
	    i__2 = *n;
	    for (jrow = 1; jrow <= i__2; ++jrow) {
		++jreal;
		++jimag;
		i__3 = b_subscr(jrow, jcol);
		i__4 = jreal;
		i__5 = jimag;
		z__1.r = rwork[i__4], z__1.i = rwork[i__5];
		b[i__3].r = z__1.r, b[i__3].i = z__1.i;
/* L150: */
	    }
/* L160: */
	}

/*        Unscale. */

	dlascl_("G", &c__0, &c__0, &c_b10, &orgnrm, n, &c__1, &d__[1], n, 
		info);
	dlasrt_("D", n, &d__[1], info);
	zlascl_("G", &c__0, &c__0, &orgnrm, &c_b10, n, nrhs, &b[b_offset], 
		ldb, info);

	return 0;
    }

/*     Book-keeping and setting up some constants. */

    nlvl = (integer) (log((doublereal) (*n) / (doublereal) (*smlsiz + 1)) / 
	    log(2.)) + 1;

    smlszp = *smlsiz + 1;

    u = 1;
    vt = *smlsiz * *n + 1;
    difl = vt + smlszp * *n;
    difr = difl + nlvl * *n;
    z__ = difr + (nlvl * *n << 1);
    c__ = z__ + nlvl * *n;
    s = c__ + *n;
    poles = s + *n;
    givnum = poles + (nlvl << 1) * *n;
    nrwork = givnum + (nlvl << 1) * *n;
    bx = 1;

    irwrb = nrwork;
    irwib = irwrb + *smlsiz * *nrhs;
    irwb = irwib + *smlsiz * *nrhs;

    sizei = *n + 1;
    k = sizei + *n;
    givptr = k + *n;
    perm = givptr + *n;
    givcol = perm + nlvl * *n;
    iwk = givcol + (nlvl * *n << 1);

    st = 1;
    sqre = 0;
    icmpq1 = 1;
    icmpq2 = 0;
    nsub = 0;

    i__1 = *n;
    for (i__ = 1; i__ <= i__1; ++i__) {
	if ((d__1 = d__[i__], abs(d__1)) < eps) {
	    d__[i__] = d_sign(&eps, &d__[i__]);
	}
/* L170: */
    }

    i__1 = nm1;
    for (i__ = 1; i__ <= i__1; ++i__) {
	if ((d__1 = e[i__], abs(d__1)) < eps || i__ == nm1) {
	    ++nsub;
	    iwork[nsub] = st;

/*           Subproblem found. First determine its size and then   
             apply divide and conquer on it. */

	    if (i__ < nm1) {

/*              A subproblem with E(I) small for I < NM1. */

		nsize = i__ - st + 1;
		iwork[sizei + nsub - 1] = nsize;
	    } else if ((d__1 = e[i__], abs(d__1)) >= eps) {

/*              A subproblem with E(NM1) not too small but I = NM1. */

		nsize = *n - st + 1;
		iwork[sizei + nsub - 1] = nsize;
	    } else {

/*              A subproblem with E(NM1) small. This implies an   
                1-by-1 subproblem at D(N), which is not solved   
                explicitly. */

		nsize = i__ - st + 1;
		iwork[sizei + nsub - 1] = nsize;
		++nsub;
		iwork[nsub] = *n;
		iwork[sizei + nsub - 1] = 1;
		zcopy_(nrhs, &b_ref(*n, 1), ldb, &work[bx + nm1], n);
	    }
	    st1 = st - 1;
	    if (nsize == 1) {

/*              This is a 1-by-1 subproblem and is not solved   
                explicitly. */

		zcopy_(nrhs, &b_ref(st, 1), ldb, &work[bx + st1], n);
	    } else if (nsize <= *smlsiz) {

/*              This is a small subproblem and is solved by DLASDQ. */

		dlaset_("A", &nsize, &nsize, &c_b35, &c_b10, &rwork[vt + st1],
			 n);
		dlaset_("A", &nsize, &nsize, &c_b35, &c_b10, &rwork[u + st1], 
			n);
		dlasdq_("U", &c__0, &nsize, &nsize, &nsize, &c__0, &d__[st], &
			e[st], &rwork[vt + st1], n, &rwork[u + st1], n, &
			rwork[nrwork], &c__1, &rwork[nrwork], info)
			;
		if (*info != 0) {
		    return 0;
		}

/*              In the real version, B is passed to DLASDQ and multiplied   
                internally by Q'. Here B is complex and that product is   
                computed below in two steps (real and imaginary parts). */

		j = irwb - 1;
		i__2 = *nrhs;
		for (jcol = 1; jcol <= i__2; ++jcol) {
		    i__3 = st + nsize - 1;
		    for (jrow = st; jrow <= i__3; ++jrow) {
			++j;
			i__4 = b_subscr(jrow, jcol);
			rwork[j] = b[i__4].r;
/* L180: */
		    }
/* L190: */
		}
		dgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[u + st1]
			, n, &rwork[irwb], &nsize, &c_b35, &rwork[irwrb], &
			nsize);
		j = irwb - 1;
		i__2 = *nrhs;
		for (jcol = 1; jcol <= i__2; ++jcol) {
		    i__3 = st + nsize - 1;
		    for (jrow = st; jrow <= i__3; ++jrow) {
			++j;
			rwork[j] = d_imag(&b_ref(jrow, jcol));
/* L200: */
		    }
/* L210: */
		}
		dgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[u + st1]
			, n, &rwork[irwb], &nsize, &c_b35, &rwork[irwib], &
			nsize);
		jreal = irwrb - 1;
		jimag = irwib - 1;
		i__2 = *nrhs;
		for (jcol = 1; jcol <= i__2; ++jcol) {
		    i__3 = st + nsize - 1;
		    for (jrow = st; jrow <= i__3; ++jrow) {
			++jreal;
			++jimag;
			i__4 = b_subscr(jrow, jcol);
			i__5 = jreal;
			i__6 = jimag;
			z__1.r = rwork[i__5], z__1.i = rwork[i__6];
			b[i__4].r = z__1.r, b[i__4].i = z__1.i;
/* L220: */
		    }
/* L230: */
		}

		zlacpy_("A", &nsize, nrhs, &b_ref(st, 1), ldb, &work[bx + st1]
			, n);
	    } else {

/*              A large problem. Solve it using divide and conquer. */

		dlasda_(&icmpq1, smlsiz, &nsize, &sqre, &d__[st], &e[st], &
			rwork[u + st1], n, &rwork[vt + st1], &iwork[k + st1], 
			&rwork[difl + st1], &rwork[difr + st1], &rwork[z__ + 
			st1], &rwork[poles + st1], &iwork[givptr + st1], &
			iwork[givcol + st1], n, &iwork[perm + st1], &rwork[
			givnum + st1], &rwork[c__ + st1], &rwork[s + st1], &
			rwork[nrwork], &iwork[iwk], info);
		if (*info != 0) {
		    return 0;
		}
		bxst = bx + st1;
		zlalsa_(&icmpq2, smlsiz, &nsize, nrhs, &b_ref(st, 1), ldb, &
			work[bxst], n, &rwork[u + st1], n, &rwork[vt + st1], &
			iwork[k + st1], &rwork[difl + st1], &rwork[difr + st1]
			, &rwork[z__ + st1], &rwork[poles + st1], &iwork[
			givptr + st1], &iwork[givcol + st1], n, &iwork[perm + 
			st1], &rwork[givnum + st1], &rwork[c__ + st1], &rwork[
			s + st1], &rwork[nrwork], &iwork[iwk], info);
		if (*info != 0) {
		    return 0;
		}
	    }
	    st = i__ + 1;
	}
/* L240: */
    }

/*     Apply the singular values and treat the tiny ones as zero. */

    tol = *rcond * (d__1 = d__[idamax_(n, &d__[1], &c__1)], abs(d__1));

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

/*        Some of the elements in D can be negative because 1-by-1   
          subproblems were not solved explicitly. */

	if ((d__1 = d__[i__], abs(d__1)) <= tol) {
	    zlaset_("A", &c__1, nrhs, &c_b1, &c_b1, &work[bx + i__ - 1], n);
	} else {
	    ++(*rank);
	    zlascl_("G", &c__0, &c__0, &d__[i__], &c_b10, &c__1, nrhs, &work[
		    bx + i__ - 1], n, info);
	}
	d__[i__] = (d__1 = d__[i__], abs(d__1));
/* L250: */
    }

/*     Now apply back the right singular vectors. */

    icmpq2 = 1;
    i__1 = nsub;
    for (i__ = 1; i__ <= i__1; ++i__) {
	st = iwork[i__];
	st1 = st - 1;
	nsize = iwork[sizei + i__ - 1];
	bxst = bx + st1;
	if (nsize == 1) {
	    zcopy_(nrhs, &work[bxst], n, &b_ref(st, 1), ldb);
	} else if (nsize <= *smlsiz) {

/*           Since B and BX are complex, the following call to DGEMM   
             is performed in two steps (real and imaginary parts).   

             CALL DGEMM( 'T', 'N', NSIZE, NRHS, NSIZE, ONE,   
      $                  RWORK( VT+ST1 ), N, RWORK( BXST ), N, ZERO,   
      $                  B( ST, 1 ), LDB ) */

	    j = bxst - *n - 1;
	    jreal = irwb - 1;
	    i__2 = *nrhs;
	    for (jcol = 1; jcol <= i__2; ++jcol) {
		j += *n;
		i__3 = nsize;
		for (jrow = 1; jrow <= i__3; ++jrow) {
		    ++jreal;
		    i__4 = j + jrow;
		    rwork[jreal] = work[i__4].r;
/* L260: */
		}
/* L270: */
	    }
	    dgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[vt + st1], 
		    n, &rwork[irwb], &nsize, &c_b35, &rwork[irwrb], &nsize);
	    j = bxst - *n - 1;
	    jimag = irwb - 1;
	    i__2 = *nrhs;
	    for (jcol = 1; jcol <= i__2; ++jcol) {
		j += *n;
		i__3 = nsize;
		for (jrow = 1; jrow <= i__3; ++jrow) {
		    ++jimag;
		    rwork[jimag] = d_imag(&work[j + jrow]);
/* L280: */
		}
/* L290: */
	    }
	    dgemm_("T", "N", &nsize, nrhs, &nsize, &c_b10, &rwork[vt + st1], 
		    n, &rwork[irwb], &nsize, &c_b35, &rwork[irwib], &nsize);
	    jreal = irwrb - 1;
	    jimag = irwib - 1;
	    i__2 = *nrhs;
	    for (jcol = 1; jcol <= i__2; ++jcol) {
		i__3 = st + nsize - 1;
		for (jrow = st; jrow <= i__3; ++jrow) {
		    ++jreal;
		    ++jimag;
		    i__4 = b_subscr(jrow, jcol);
		    i__5 = jreal;
		    i__6 = jimag;
		    z__1.r = rwork[i__5], z__1.i = rwork[i__6];
		    b[i__4].r = z__1.r, b[i__4].i = z__1.i;
/* L300: */
		}
/* L310: */
	    }
	} else {
	    zlalsa_(&icmpq2, smlsiz, &nsize, nrhs, &work[bxst], n, &b_ref(st, 
		    1), ldb, &rwork[u + st1], n, &rwork[vt + st1], &iwork[k + 
		    st1], &rwork[difl + st1], &rwork[difr + st1], &rwork[z__ 
		    + st1], &rwork[poles + st1], &iwork[givptr + st1], &iwork[
		    givcol + st1], n, &iwork[perm + st1], &rwork[givnum + st1]
		    , &rwork[c__ + st1], &rwork[s + st1], &rwork[nrwork], &
		    iwork[iwk], info);
	    if (*info != 0) {
		return 0;
	    }
	}
/* L320: */
    }

/*     Unscale and sort the singular values. */

    dlascl_("G", &c__0, &c__0, &c_b10, &orgnrm, n, &c__1, &d__[1], n, info);
    dlasrt_("D", n, &d__[1], info);
    zlascl_("G", &c__0, &c__0, &orgnrm, &c_b10, n, nrhs, &b[b_offset], ldb, 
	    info);

    return 0;

/*     End of ZLALSD */

} /* zlalsd_ */
Esempio n. 4
0
/* Subroutine */
int dlalsd_(char *uplo, integer *smlsiz, integer *n, integer *nrhs, doublereal *d__, doublereal *e, doublereal *b, integer *ldb, doublereal *rcond, integer *rank, doublereal *work, integer *iwork, integer *info)
{
    /* System generated locals */
    integer b_dim1, b_offset, i__1, i__2;
    doublereal d__1;
    /* Builtin functions */
    double log(doublereal), d_sign(doublereal *, doublereal *);
    /* Local variables */
    integer c__, i__, j, k;
    doublereal r__;
    integer s, u, z__;
    doublereal cs;
    integer bx;
    doublereal sn;
    integer st, vt, nm1, st1;
    doublereal eps;
    integer iwk;
    doublereal tol;
    integer difl, difr;
    doublereal rcnd;
    integer perm, nsub;
    extern /* Subroutine */
    int drot_(integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *);
    integer nlvl, sqre, bxst;
    extern /* Subroutine */
    int dgemm_(char *, char *, integer *, integer *, integer *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, integer *), dcopy_(integer *, doublereal *, integer *, doublereal *, integer *);
    integer poles, sizei, nsize, nwork, icmpq1, icmpq2;
    extern doublereal dlamch_(char *);
    extern /* Subroutine */
    int dlasda_(integer *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *), dlalsa_(integer *, integer *, integer *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, doublereal *, integer *, integer *), dlascl_(char *, integer *, integer *, doublereal *, doublereal *, integer *, integer *, doublereal *, integer *, integer *);
    extern integer idamax_(integer *, doublereal *, integer *);
    extern /* Subroutine */
    int dlasdq_(char *, integer *, integer *, integer *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *, doublereal *, integer *), dlacpy_(char *, integer *, integer *, doublereal *, integer *, doublereal *, integer *), dlartg_(doublereal *, doublereal *, doublereal *, doublereal *, doublereal *), dlaset_(char *, integer *, integer *, doublereal *, doublereal *, doublereal *, integer *), xerbla_(char *, integer *);
    integer givcol;
    extern doublereal dlanst_(char *, integer *, doublereal *, doublereal *);
    extern /* Subroutine */
    int dlasrt_(char *, integer *, doublereal *, integer *);
    doublereal orgnrm;
    integer givnum, givptr, smlszp;
    /* -- LAPACK computational routine (version 3.4.2) -- */
    /* -- LAPACK is a software package provided by Univ. of Tennessee, -- */
    /* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- */
    /* September 2012 */
    /* .. Scalar Arguments .. */
    /* .. */
    /* .. Array Arguments .. */
    /* .. */
    /* ===================================================================== */
    /* .. Parameters .. */
    /* .. */
    /* .. Local Scalars .. */
    /* .. */
    /* .. External Functions .. */
    /* .. */
    /* .. External Subroutines .. */
    /* .. */
    /* .. Intrinsic Functions .. */
    /* .. */
    /* .. Executable Statements .. */
    /* Test the input parameters. */
    /* Parameter adjustments */
    --d__;
    --e;
    b_dim1 = *ldb;
    b_offset = 1 + b_dim1;
    b -= b_offset;
    --work;
    --iwork;
    /* Function Body */
    *info = 0;
    if (*n < 0)
    {
        *info = -3;
    }
    else if (*nrhs < 1)
    {
        *info = -4;
    }
    else if (*ldb < 1 || *ldb < *n)
    {
        *info = -8;
    }
    if (*info != 0)
    {
        i__1 = -(*info);
        xerbla_("DLALSD", &i__1);
        return 0;
    }
    eps = dlamch_("Epsilon");
    /* Set up the tolerance. */
    if (*rcond <= 0. || *rcond >= 1.)
    {
        rcnd = eps;
    }
    else
    {
        rcnd = *rcond;
    }
    *rank = 0;
    /* Quick return if possible. */
    if (*n == 0)
    {
        return 0;
    }
    else if (*n == 1)
    {
        if (d__[1] == 0.)
        {
            dlaset_("A", &c__1, nrhs, &c_b6, &c_b6, &b[b_offset], ldb);
        }
        else
        {
            *rank = 1;
            dlascl_("G", &c__0, &c__0, &d__[1], &c_b11, &c__1, nrhs, &b[ b_offset], ldb, info);
            d__[1] = abs(d__[1]);
        }
        return 0;
    }
    /* Rotate the matrix if it is lower bidiagonal. */
    if (*(unsigned char *)uplo == 'L')
    {
        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];
            if (*nrhs == 1)
            {
                drot_(&c__1, &b[i__ + b_dim1], &c__1, &b[i__ + 1 + b_dim1], & c__1, &cs, &sn);
            }
            else
            {
                work[(i__ << 1) - 1] = cs;
                work[i__ * 2] = sn;
            }
            /* L10: */
        }
        if (*nrhs > 1)
        {
            i__1 = *nrhs;
            for (i__ = 1;
                    i__ <= i__1;
                    ++i__)
            {
                i__2 = *n - 1;
                for (j = 1;
                        j <= i__2;
                        ++j)
                {
                    cs = work[(j << 1) - 1];
                    sn = work[j * 2];
                    drot_(&c__1, &b[j + i__ * b_dim1], &c__1, &b[j + 1 + i__ * b_dim1], &c__1, &cs, &sn);
                    /* L20: */
                }
                /* L30: */
            }
        }
    }
    /* Scale. */
    nm1 = *n - 1;
    orgnrm = dlanst_("M", n, &d__[1], &e[1]);
    if (orgnrm == 0.)
    {
        dlaset_("A", n, nrhs, &c_b6, &c_b6, &b[b_offset], ldb);
        return 0;
    }
    dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, &c__1, &d__[1], n, info);
    dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, &nm1, &c__1, &e[1], &nm1, info);
    /* If N is smaller than the minimum divide size SMLSIZ, then solve */
    /* the problem with another solver. */
    if (*n <= *smlsiz)
    {
        nwork = *n * *n + 1;
        dlaset_("A", n, n, &c_b6, &c_b11, &work[1], n);
        dlasdq_("U", &c__0, n, n, &c__0, nrhs, &d__[1], &e[1], &work[1], n, & work[1], n, &b[b_offset], ldb, &work[nwork], info);
        if (*info != 0)
        {
            return 0;
        }
        tol = rcnd * (d__1 = d__[idamax_(n, &d__[1], &c__1)], abs(d__1));
        i__1 = *n;
        for (i__ = 1;
                i__ <= i__1;
                ++i__)
        {
            if (d__[i__] <= tol)
            {
                dlaset_("A", &c__1, nrhs, &c_b6, &c_b6, &b[i__ + b_dim1], ldb);
            }
            else
            {
                dlascl_("G", &c__0, &c__0, &d__[i__], &c_b11, &c__1, nrhs, &b[ i__ + b_dim1], ldb, info);
                ++(*rank);
            }
            /* L40: */
        }
        dgemm_("T", "N", n, nrhs, n, &c_b11, &work[1], n, &b[b_offset], ldb, & c_b6, &work[nwork], n);
        dlacpy_("A", n, nrhs, &work[nwork], n, &b[b_offset], ldb);
        /* Unscale. */
        dlascl_("G", &c__0, &c__0, &c_b11, &orgnrm, n, &c__1, &d__[1], n, info);
        dlasrt_("D", n, &d__[1], info);
        dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, nrhs, &b[b_offset], ldb, info);
        return 0;
    }
    /* Book-keeping and setting up some constants. */
    nlvl = (integer) (log((doublereal) (*n) / (doublereal) (*smlsiz + 1)) / log(2.)) + 1;
    smlszp = *smlsiz + 1;
    u = 1;
    vt = *smlsiz * *n + 1;
    difl = vt + smlszp * *n;
    difr = difl + nlvl * *n;
    z__ = difr + (nlvl * *n << 1);
    c__ = z__ + nlvl * *n;
    s = c__ + *n;
    poles = s + *n;
    givnum = poles + (nlvl << 1) * *n;
    bx = givnum + (nlvl << 1) * *n;
    nwork = bx + *n * *nrhs;
    sizei = *n + 1;
    k = sizei + *n;
    givptr = k + *n;
    perm = givptr + *n;
    givcol = perm + nlvl * *n;
    iwk = givcol + (nlvl * *n << 1);
    st = 1;
    sqre = 0;
    icmpq1 = 1;
    icmpq2 = 0;
    nsub = 0;
    i__1 = *n;
    for (i__ = 1;
            i__ <= i__1;
            ++i__)
    {
        if ((d__1 = d__[i__], abs(d__1)) < eps)
        {
            d__[i__] = d_sign(&eps, &d__[i__]);
        }
        /* L50: */
    }
    i__1 = nm1;
    for (i__ = 1;
            i__ <= i__1;
            ++i__)
    {
        if ((d__1 = e[i__], abs(d__1)) < eps || i__ == nm1)
        {
            ++nsub;
            iwork[nsub] = st;
            /* Subproblem found. First determine its size and then */
            /* apply divide and conquer on it. */
            if (i__ < nm1)
            {
                /* A subproblem with E(I) small for I < NM1. */
                nsize = i__ - st + 1;
                iwork[sizei + nsub - 1] = nsize;
            }
            else if ((d__1 = e[i__], abs(d__1)) >= eps)
            {
                /* A subproblem with E(NM1) not too small but I = NM1. */
                nsize = *n - st + 1;
                iwork[sizei + nsub - 1] = nsize;
            }
            else
            {
                /* A subproblem with E(NM1) small. This implies an */
                /* 1-by-1 subproblem at D(N), which is not solved */
                /* explicitly. */
                nsize = i__ - st + 1;
                iwork[sizei + nsub - 1] = nsize;
                ++nsub;
                iwork[nsub] = *n;
                iwork[sizei + nsub - 1] = 1;
                dcopy_(nrhs, &b[*n + b_dim1], ldb, &work[bx + nm1], n);
            }
            st1 = st - 1;
            if (nsize == 1)
            {
                /* This is a 1-by-1 subproblem and is not solved */
                /* explicitly. */
                dcopy_(nrhs, &b[st + b_dim1], ldb, &work[bx + st1], n);
            }
            else if (nsize <= *smlsiz)
            {
                /* This is a small subproblem and is solved by DLASDQ. */
                dlaset_("A", &nsize, &nsize, &c_b6, &c_b11, &work[vt + st1], n);
                dlasdq_("U", &c__0, &nsize, &nsize, &c__0, nrhs, &d__[st], &e[ st], &work[vt + st1], n, &work[nwork], n, &b[st + b_dim1], ldb, &work[nwork], info);
                if (*info != 0)
                {
                    return 0;
                }
                dlacpy_("A", &nsize, nrhs, &b[st + b_dim1], ldb, &work[bx + st1], n);
            }
            else
            {
                /* A large problem. Solve it using divide and conquer. */
                dlasda_(&icmpq1, smlsiz, &nsize, &sqre, &d__[st], &e[st], & work[u + st1], n, &work[vt + st1], &iwork[k + st1], & work[difl + st1], &work[difr + st1], &work[z__ + st1], &work[poles + st1], &iwork[givptr + st1], &iwork[ givcol + st1], n, &iwork[perm + st1], &work[givnum + st1], &work[c__ + st1], &work[s + st1], &work[nwork], &iwork[iwk], info);
                if (*info != 0)
                {
                    return 0;
                }
                bxst = bx + st1;
                dlalsa_(&icmpq2, smlsiz, &nsize, nrhs, &b[st + b_dim1], ldb, & work[bxst], n, &work[u + st1], n, &work[vt + st1], & iwork[k + st1], &work[difl + st1], &work[difr + st1], &work[z__ + st1], &work[poles + st1], &iwork[givptr + st1], &iwork[givcol + st1], n, &iwork[perm + st1], & work[givnum + st1], &work[c__ + st1], &work[s + st1], &work[nwork], &iwork[iwk], info);
                if (*info != 0)
                {
                    return 0;
                }
            }
            st = i__ + 1;
        }
        /* L60: */
    }
    /* Apply the singular values and treat the tiny ones as zero. */
    tol = rcnd * (d__1 = d__[idamax_(n, &d__[1], &c__1)], abs(d__1));
    i__1 = *n;
    for (i__ = 1;
            i__ <= i__1;
            ++i__)
    {
        /* Some of the elements in D can be negative because 1-by-1 */
        /* subproblems were not solved explicitly. */
        if ((d__1 = d__[i__], abs(d__1)) <= tol)
        {
            dlaset_("A", &c__1, nrhs, &c_b6, &c_b6, &work[bx + i__ - 1], n);
        }
        else
        {
            ++(*rank);
            dlascl_("G", &c__0, &c__0, &d__[i__], &c_b11, &c__1, nrhs, &work[ bx + i__ - 1], n, info);
        }
        d__[i__] = (d__1 = d__[i__], abs(d__1));
        /* L70: */
    }
    /* Now apply back the right singular vectors. */
    icmpq2 = 1;
    i__1 = nsub;
    for (i__ = 1;
            i__ <= i__1;
            ++i__)
    {
        st = iwork[i__];
        st1 = st - 1;
        nsize = iwork[sizei + i__ - 1];
        bxst = bx + st1;
        if (nsize == 1)
        {
            dcopy_(nrhs, &work[bxst], n, &b[st + b_dim1], ldb);
        }
        else if (nsize <= *smlsiz)
        {
            dgemm_("T", "N", &nsize, nrhs, &nsize, &c_b11, &work[vt + st1], n, &work[bxst], n, &c_b6, &b[st + b_dim1], ldb);
        }
        else
        {
            dlalsa_(&icmpq2, smlsiz, &nsize, nrhs, &work[bxst], n, &b[st + b_dim1], ldb, &work[u + st1], n, &work[vt + st1], &iwork[ k + st1], &work[difl + st1], &work[difr + st1], &work[z__ + st1], &work[poles + st1], &iwork[givptr + st1], &iwork[ givcol + st1], n, &iwork[perm + st1], &work[givnum + st1], &work[c__ + st1], &work[s + st1], &work[nwork], &iwork[ iwk], info);
            if (*info != 0)
            {
                return 0;
            }
        }
        /* L80: */
    }
    /* Unscale and sort the singular values. */
    dlascl_("G", &c__0, &c__0, &c_b11, &orgnrm, n, &c__1, &d__[1], n, info);
    dlasrt_("D", n, &d__[1], info);
    dlascl_("G", &c__0, &c__0, &orgnrm, &c_b11, n, nrhs, &b[b_offset], ldb, info);
    return 0;
    /* End of DLALSD */
}