dlasd7.c

#include "clapack.h"

/* Table of constant values */

static integer c__1 = 1;

/* Subroutine */ int dlasd7_(integer *icompq, integer *nl, integer *nr, 
	integer *sqre, integer *k, doublereal *d__, doublereal *z__, 
	doublereal *zw, doublereal *vf, doublereal *vfw, doublereal *vl, 
	doublereal *vlw, doublereal *alpha, doublereal *beta, doublereal *
	dsigma, integer *idx, integer *idxp, integer *idxq, integer *perm, 
	integer *givptr, integer *givcol, integer *ldgcol, doublereal *givnum, 
	 integer *ldgnum, doublereal *c__, doublereal *s, integer *info)
{
    /* System generated locals */
    integer givcol_dim1, givcol_offset, givnum_dim1, givnum_offset, i__1;
    doublereal d__1, d__2;

    /* Local variables */
    integer i__, j, m, n, k2;
    doublereal z1;
    integer jp;
    doublereal eps, tau, tol;
    integer nlp1, nlp2, idxi, idxj;
    extern /* Subroutine */ int drot_(integer *, doublereal *, integer *, 
	    doublereal *, integer *, doublereal *, doublereal *);
    integer idxjp;
    extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, 
	    doublereal *, integer *);
    integer jprev;
    extern doublereal dlapy2_(doublereal *, doublereal *), dlamch_(char *);
    extern /* Subroutine */ int dlamrg_(integer *, integer *, doublereal *, 
	    integer *, integer *, integer *), xerbla_(char *, integer *);
    doublereal hlftol;


/*  -- LAPACK auxiliary routine (version 3.1) -- */
/*     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. */
/*     November 2006 */

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

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

/*  DLASD7 merges the two sets of singular values together into a single */
/*  sorted set. Then it tries to deflate the size of the problem. There */
/*  are two ways in which deflation can occur:  when two or more singular */
/*  values are close together or if there is a tiny entry in the Z */
/*  vector. For each such occurrence the order of the related */
/*  secular equation problem is reduced by one. */

/*  DLASD7 is called from DLASD6. */

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

/*  ICOMPQ  (input) INTEGER */
/*          Specifies whether singular vectors are to be computed */
/*          in compact form, as follows: */
/*          = 0: Compute singular values only. */
/*          = 1: Compute singular vectors of upper */
/*               bidiagonal matrix in compact form. */

/*  NL     (input) INTEGER */
/*         The row dimension of the upper block. NL >= 1. */

/*  NR     (input) INTEGER */
/*         The row dimension of the lower block. NR >= 1. */

/*  SQRE   (input) INTEGER */
/*         = 0: the lower block is an NR-by-NR square matrix. */
/*         = 1: the lower block is an NR-by-(NR+1) rectangular matrix. */

/*         The bidiagonal matrix has */
/*         N = NL + NR + 1 rows and */
/*         M = N + SQRE >= N columns. */

/*  K      (output) INTEGER */
/*         Contains the dimension of the non-deflated matrix, this is */
/*         the order of the related secular equation. 1 <= K <=N. */

/*  D      (input/output) DOUBLE PRECISION array, dimension ( N ) */
/*         On entry D contains the singular values of the two submatrices */
/*         to be combined. On exit D contains the trailing (N-K) updated */
/*         singular values (those which were deflated) sorted into */
/*         increasing order. */

/*  Z      (output) DOUBLE PRECISION array, dimension ( M ) */
/*         On exit Z contains the updating row vector in the secular */
/*         equation. */

/*  ZW     (workspace) DOUBLE PRECISION array, dimension ( M ) */
/*         Workspace for Z. */

/*  VF     (input/output) DOUBLE PRECISION array, dimension ( M ) */
/*         On entry, VF(1:NL+1) contains the first components of all */
/*         right singular vectors of the upper block; and VF(NL+2:M) */
/*         contains the first components of all right singular vectors */
/*         of the lower block. On exit, VF contains the first components */
/*         of all right singular vectors of the bidiagonal matrix. */

/*  VFW    (workspace) DOUBLE PRECISION array, dimension ( M ) */
/*         Workspace for VF. */

/*  VL     (input/output) DOUBLE PRECISION array, dimension ( M ) */
/*         On entry, VL(1:NL+1) contains the  last components of all */
/*         right singular vectors of the upper block; and VL(NL+2:M) */
/*         contains the last components of all right singular vectors */
/*         of the lower block. On exit, VL contains the last components */
/*         of all right singular vectors of the bidiagonal matrix. */

/*  VLW    (workspace) DOUBLE PRECISION array, dimension ( M ) */
/*         Workspace for VL. */

/*  ALPHA  (input) DOUBLE PRECISION */
/*         Contains the diagonal element associated with the added row. */

/*  BETA   (input) DOUBLE PRECISION */
/*         Contains the off-diagonal element associated with the added */
/*         row. */

/*  DSIGMA (output) DOUBLE PRECISION array, dimension ( N ) */
/*         Contains a copy of the diagonal elements (K-1 singular values */
/*         and one zero) in the secular equation. */

/*  IDX    (workspace) INTEGER array, dimension ( N ) */
/*         This will contain the permutation used to sort the contents of */
/*         D into ascending order. */

/*  IDXP   (workspace) INTEGER array, dimension ( N ) */
/*         This will contain the permutation used to place deflated */
/*         values of D at the end of the array. On output IDXP(2:K) */
/*         points to the nondeflated D-values and IDXP(K+1:N) */
/*         points to the deflated singular values. */

/*  IDXQ   (input) INTEGER array, dimension ( N ) */
/*         This contains the permutation which separately sorts the two */
/*         sub-problems in D into ascending order.  Note that entries in */
/*         the first half of this permutation must first be moved one */
/*         position backward; and entries in the second half */
/*         must first have NL+1 added to their values. */

/*  PERM   (output) INTEGER array, dimension ( N ) */
/*         The permutations (from deflation and sorting) to be applied */
/*         to each singular block. Not referenced if ICOMPQ = 0. */

/*  GIVPTR (output) INTEGER */
/*         The number of Givens rotations which took place in this */
/*         subproblem. Not referenced if ICOMPQ = 0. */

/*  GIVCOL (output) INTEGER array, dimension ( LDGCOL, 2 ) */
/*         Each pair of numbers indicates a pair of columns to take place */
/*         in a Givens rotation. Not referenced if ICOMPQ = 0. */

/*  LDGCOL (input) INTEGER */
/*         The leading dimension of GIVCOL, must be at least N. */

/*  GIVNUM (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 ) */
/*         Each number indicates the C or S value to be used in the */
/*         corresponding Givens rotation. Not referenced if ICOMPQ = 0. */

/*  LDGNUM (input) INTEGER */
/*         The leading dimension of GIVNUM, must be at least N. */

/*  C      (output) DOUBLE PRECISION */
/*         C contains garbage if SQRE =0 and the C-value of a Givens */
/*         rotation related to the right null space if SQRE = 1. */

/*  S      (output) DOUBLE PRECISION */
/*         S contains garbage if SQRE =0 and the S-value of a Givens */
/*         rotation related to the right null space if SQRE = 1. */

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

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

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

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

/*     .. Parameters .. */
/*     .. */
/*     .. Local Scalars .. */

/*     .. */
/*     .. External Subroutines .. */
/*     .. */
/*     .. External Functions .. */
/*     .. */
/*     .. Intrinsic Functions .. */
/*     .. */
/*     .. Executable Statements .. */

/*     Test the input parameters. */

    /* Parameter adjustments */
    --d__;
    --z__;
    --zw;
    --vf;
    --vfw;
    --vl;
    --vlw;
    --dsigma;
    --idx;
    --idxp;
    --idxq;
    --perm;
    givcol_dim1 = *ldgcol;
    givcol_offset = 1 + givcol_dim1;
    givcol -= givcol_offset;
    givnum_dim1 = *ldgnum;
    givnum_offset = 1 + givnum_dim1;
    givnum -= givnum_offset;

    /* Function Body */
    *info = 0;
    n = *nl + *nr + 1;
    m = n + *sqre;

    if (*icompq < 0 || *icompq > 1) {
	*info = -1;
    } else if (*nl < 1) {
	*info = -2;
    } else if (*nr < 1) {
	*info = -3;
    } else if (*sqre < 0 || *sqre > 1) {
	*info = -4;
    } else if (*ldgcol < n) {
	*info = -22;
    } else if (*ldgnum < n) {
	*info = -24;
    }
    if (*info != 0) {
	i__1 = -(*info);
	xerbla_("DLASD7", &i__1);
	return 0;
    }

    nlp1 = *nl + 1;
    nlp2 = *nl + 2;
    if (*icompq == 1) {
	*givptr = 0;
    }

/*     Generate the first part of the vector Z and move the singular */
/*     values in the first part of D one position backward. */

    z1 = *alpha * vl[nlp1];
    vl[nlp1] = 0.;
    tau = vf[nlp1];
    for (i__ = *nl; i__ >= 1; --i__) {
	z__[i__ + 1] = *alpha * vl[i__];
	vl[i__] = 0.;
	vf[i__ + 1] = vf[i__];
	d__[i__ + 1] = d__[i__];
	idxq[i__ + 1] = idxq[i__] + 1;
/* L10: */
    }
    vf[1] = tau;

/*     Generate the second part of the vector Z. */

    i__1 = m;
    for (i__ = nlp2; i__ <= i__1; ++i__) {
	z__[i__] = *beta * vf[i__];
	vf[i__] = 0.;
/* L20: */
    }

/*     Sort the singular values into increasing order */

    i__1 = n;
    for (i__ = nlp2; i__ <= i__1; ++i__) {
	idxq[i__] += nlp1;
/* L30: */
    }

/*     DSIGMA, IDXC, IDXC, and ZW are used as storage space. */

    i__1 = n;
    for (i__ = 2; i__ <= i__1; ++i__) {
	dsigma[i__] = d__[idxq[i__]];
	zw[i__] = z__[idxq[i__]];
	vfw[i__] = vf[idxq[i__]];
	vlw[i__] = vl[idxq[i__]];
/* L40: */
    }

    dlamrg_(nl, nr, &dsigma[2], &c__1, &c__1, &idx[2]);

    i__1 = n;
    for (i__ = 2; i__ <= i__1; ++i__) {
	idxi = idx[i__] + 1;
	d__[i__] = dsigma[idxi];
	z__[i__] = zw[idxi];
	vf[i__] = vfw[idxi];
	vl[i__] = vlw[idxi];
/* L50: */
    }

/*     Calculate the allowable deflation tolerence */

    eps = dlamch_("Epsilon");
/* Computing MAX */
    d__1 = abs(*alpha), d__2 = abs(*beta);
    tol = max(d__1,d__2);
/* Computing MAX */
    d__2 = (d__1 = d__[n], abs(d__1));
    tol = eps * 64. * max(d__2,tol);

/*     There are 2 kinds of deflation -- first a value in the z-vector */
/*     is small, second two (or more) singular values are very close */
/*     together (their difference is small). */

/*     If the value in the z-vector is small, we simply permute the */
/*     array so that the corresponding singular value is moved to the */
/*     end. */

/*     If two values in the D-vector are close, we perform a two-sided */
/*     rotation designed to make one of the corresponding z-vector */
/*     entries zero, and then permute the array so that the deflated */
/*     singular value is moved to the end. */

/*     If there are multiple singular values then the problem deflates. */
/*     Here the number of equal singular values are found.  As each equal */
/*     singular value is found, an elementary reflector is computed to */
/*     rotate the corresponding singular subspace so that the */
/*     corresponding components of Z are zero in this new basis. */

    *k = 1;
    k2 = n + 1;
    i__1 = n;
    for (j = 2; j <= i__1; ++j) {
	if ((d__1 = z__[j], abs(d__1)) <= tol) {

/*           Deflate due to small z component. */

	    --k2;
	    idxp[k2] = j;
	    if (j == n) {
		goto L100;
	    }
	} else {
	    jprev = j;
	    goto L70;
	}
/* L60: */
    }
L70:
    j = jprev;
L80:
    ++j;
    if (j > n) {
	goto L90;
    }
    if ((d__1 = z__[j], abs(d__1)) <= tol) {

/*        Deflate due to small z component. */

	--k2;
	idxp[k2] = j;
    } else {

/*        Check if singular values are close enough to allow deflation. */

	if ((d__1 = d__[j] - d__[jprev], abs(d__1)) <= tol) {

/*           Deflation is possible. */

	    *s = z__[jprev];
	    *c__ = z__[j];

/*           Find sqrt(a**2+b**2) without overflow or */
/*           destructive underflow. */

	    tau = dlapy2_(c__, s);
	    z__[j] = tau;
	    z__[jprev] = 0.;
	    *c__ /= tau;
	    *s = -(*s) / tau;

/*           Record the appropriate Givens rotation */

	    if (*icompq == 1) {
		++(*givptr);
		idxjp = idxq[idx[jprev] + 1];
		idxj = idxq[idx[j] + 1];
		if (idxjp <= nlp1) {
		    --idxjp;
		}
		if (idxj <= nlp1) {
		    --idxj;
		}
		givcol[*givptr + (givcol_dim1 << 1)] = idxjp;
		givcol[*givptr + givcol_dim1] = idxj;
		givnum[*givptr + (givnum_dim1 << 1)] = *c__;
		givnum[*givptr + givnum_dim1] = *s;
	    }
	    drot_(&c__1, &vf[jprev], &c__1, &vf[j], &c__1, c__, s);
	    drot_(&c__1, &vl[jprev], &c__1, &vl[j], &c__1, c__, s);
	    --k2;
	    idxp[k2] = jprev;
	    jprev = j;
	} else {
	    ++(*k);
	    zw[*k] = z__[jprev];
	    dsigma[*k] = d__[jprev];
	    idxp[*k] = jprev;
	    jprev = j;
	}
    }
    goto L80;
L90:

/*     Record the last singular value. */

    ++(*k);
    zw[*k] = z__[jprev];
    dsigma[*k] = d__[jprev];
    idxp[*k] = jprev;

L100:

/*     Sort the singular values into DSIGMA. The singular values which */
/*     were not deflated go into the first K slots of DSIGMA, except */
/*     that DSIGMA(1) is treated separately. */

    i__1 = n;
    for (j = 2; j <= i__1; ++j) {
	jp = idxp[j];
	dsigma[j] = d__[jp];
	vfw[j] = vf[jp];
	vlw[j] = vl[jp];
/* L110: */
    }
    if (*icompq == 1) {
	i__1 = n;
	for (j = 2; j <= i__1; ++j) {
	    jp = idxp[j];
	    perm[j] = idxq[idx[jp] + 1];
	    if (perm[j] <= nlp1) {
		--perm[j];
	    }
/* L120: */
	}
    }

/*     The deflated singular values go back into the last N - K slots of */
/*     D. */

    i__1 = n - *k;
    dcopy_(&i__1, &dsigma[*k + 1], &c__1, &d__[*k + 1], &c__1);

/*     Determine DSIGMA(1), DSIGMA(2), Z(1), VF(1), VL(1), VF(M), and */
/*     VL(M). */

    dsigma[1] = 0.;
    hlftol = tol / 2.;
    if (abs(dsigma[2]) <= hlftol) {
	dsigma[2] = hlftol;
    }
    if (m > n) {
	z__[1] = dlapy2_(&z1, &z__[m]);
	if (z__[1] <= tol) {
	    *c__ = 1.;
	    *s = 0.;
	    z__[1] = tol;
	} else {
	    *c__ = z1 / z__[1];
	    *s = -z__[m] / z__[1];
	}
	drot_(&c__1, &vf[m], &c__1, &vf[1], &c__1, c__, s);
	drot_(&c__1, &vl[m], &c__1, &vl[1], &c__1, c__, s);
    } else {
	if (abs(z1) <= tol) {
	    z__[1] = tol;
	} else {
	    z__[1] = z1;
	}
    }

/*     Restore Z, VF, and VL. */

    i__1 = *k - 1;
    dcopy_(&i__1, &zw[2], &c__1, &z__[2], &c__1);
    i__1 = n - 1;
    dcopy_(&i__1, &vfw[2], &c__1, &vf[2], &c__1);
    i__1 = n - 1;
    dcopy_(&i__1, &vlw[2], &c__1, &vl[2], &c__1);

    return 0;

/*     End of DLASD7 */

} /* dlasd7_ */