dlasq1.c

#include "clapack.h"

/* Table of constant values */

static integer c__1 = 1;
static integer c__2 = 2;
static integer c__0 = 0;

/* Subroutine */ int dlasq1_(integer *n, doublereal *d__, doublereal *e, 
	doublereal *work, integer *info)
{
    /* System generated locals */
    integer i__1, i__2;
    doublereal d__1, d__2, d__3;

    /* Builtin functions */
    double sqrt(doublereal);

    /* Local variables */
    integer i__;
    doublereal eps;
    extern /* Subroutine */ int dlas2_(doublereal *, doublereal *, doublereal 
	    *, doublereal *, doublereal *);
    doublereal scale;
    integer iinfo;
    doublereal sigmn;
    extern /* Subroutine */ int dcopy_(integer *, doublereal *, integer *, 
	    doublereal *, integer *);
    doublereal sigmx;
    extern /* Subroutine */ int dlasq2_(integer *, doublereal *, integer *);
    extern doublereal dlamch_(char *);
    extern /* Subroutine */ int dlascl_(char *, integer *, integer *, 
	    doublereal *, doublereal *, integer *, integer *, doublereal *, 
	    integer *, integer *);
    doublereal safmin;
    extern /* Subroutine */ int xerbla_(char *, integer *), dlasrt_(
	    char *, integer *, doublereal *, integer *);


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

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

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

/*  DLASQ1 computes the singular values of a real N-by-N bidiagonal */
/*  matrix with diagonal D and off-diagonal E. The singular values */
/*  are computed to high relative accuracy, in the absence of */
/*  denormalization, underflow and overflow. The algorithm was first */
/*  presented in */

/*  "Accurate singular values and differential qd algorithms" by K. V. */
/*  Fernando and B. N. Parlett, Numer. Math., Vol-67, No. 2, pp. 191-230, */
/*  1994, */

/*  and the present implementation is described in "An implementation of */
/*  the dqds Algorithm (Positive Case)", LAPACK Working Note. */

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

/*  N     (input) INTEGER */
/*        The number of rows and columns in the matrix. N >= 0. */

/*  D     (input/output) DOUBLE PRECISION array, dimension (N) */
/*        On entry, D contains the diagonal elements of the */
/*        bidiagonal matrix whose SVD is desired. On normal exit, */
/*        D contains the singular values in decreasing order. */

/*  E     (input/output) DOUBLE PRECISION array, dimension (N) */
/*        On entry, elements E(1:N-1) contain the off-diagonal elements */
/*        of the bidiagonal matrix whose SVD is desired. */
/*        On exit, E is overwritten. */

/*  WORK  (workspace) DOUBLE PRECISION array, dimension (4*N) */

/*  INFO  (output) INTEGER */
/*        = 0: successful exit */
/*        < 0: if INFO = -i, the i-th argument had an illegal value */
/*        > 0: the algorithm failed */
/*             = 1, a split was marked by a positive value in E */
/*             = 2, current block of Z not diagonalized after 30*N */
/*                  iterations (in inner while loop) */
/*             = 3, termination criterion of outer while loop not met */
/*                  (program created more than N unreduced blocks) */

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

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

    /* Parameter adjustments */
    --work;
    --e;
    --d__;

    /* Function Body */
    *info = 0;
    if (*n < 0) {
	*info = -2;
	i__1 = -(*info);
	xerbla_("DLASQ1", &i__1);
	return 0;
    } else if (*n == 0) {
	return 0;
    } else if (*n == 1) {
	d__[1] = abs(d__[1]);
	return 0;
    } else if (*n == 2) {
	dlas2_(&d__[1], &e[1], &d__[2], &sigmn, &sigmx);
	d__[1] = sigmx;
	d__[2] = sigmn;
	return 0;
    }

/*     Estimate the largest singular value. */

    sigmx = 0.;
    i__1 = *n - 1;
    for (i__ = 1; i__ <= i__1; ++i__) {
	d__[i__] = (d__1 = d__[i__], abs(d__1));
/* Computing MAX */
	d__2 = sigmx, d__3 = (d__1 = e[i__], abs(d__1));
	sigmx = max(d__2,d__3);
/* L10: */
    }
    d__[*n] = (d__1 = d__[*n], abs(d__1));

/*     Early return if SIGMX is zero (matrix is already diagonal). */

    if (sigmx == 0.) {
	dlasrt_("D", n, &d__[1], &iinfo);
	return 0;
    }

    i__1 = *n;
    for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing MAX */
	d__1 = sigmx, d__2 = d__[i__];
	sigmx = max(d__1,d__2);
/* L20: */
    }

/*     Copy D and E into WORK (in the Z format) and scale (squaring the */
/*     input data makes scaling by a power of the radix pointless). */

    eps = dlamch_("Precision");
    safmin = dlamch_("Safe minimum");
    scale = sqrt(eps / safmin);
    dcopy_(n, &d__[1], &c__1, &work[1], &c__2);
    i__1 = *n - 1;
    dcopy_(&i__1, &e[1], &c__1, &work[2], &c__2);
    i__1 = (*n << 1) - 1;
    i__2 = (*n << 1) - 1;
    dlascl_("G", &c__0, &c__0, &sigmx, &scale, &i__1, &c__1, &work[1], &i__2, 
	    &iinfo);

/*     Compute the q's and e's. */

    i__1 = (*n << 1) - 1;
    for (i__ = 1; i__ <= i__1; ++i__) {
/* Computing 2nd power */
	d__1 = work[i__];
	work[i__] = d__1 * d__1;
/* L30: */
    }
    work[*n * 2] = 0.;

    dlasq2_(n, &work[1], info);

    if (*info == 0) {
	i__1 = *n;
	for (i__ = 1; i__ <= i__1; ++i__) {
	    d__[i__] = sqrt(work[i__]);
/* L40: */
	}
	dlascl_("G", &c__0, &c__0, &scale, &sigmx, n, &c__1, &d__[1], n, &
		iinfo);
    }

    return 0;

/*     End of DLASQ1 */

} /* dlasq1_ */