dsyequb.c

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/* dsyequb.f -- translated by f2c (version 20061008).
   You must link the resulting object file with libf2c:
        on Microsoft Windows system, link with libf2c.lib;
        on Linux or Unix systems, link with .../path/to/libf2c.a -lm
        or, if you install libf2c.a in a standard place, with -lf2c -lm
        -- in that order, at the end of the command line, as in
                cc *.o -lf2c -lm
        Source for libf2c is in /netlib/f2c/libf2c.zip, e.g.,

                http://www.netlib.org/f2c/libf2c.zip
*/

#include "f2c.h"
#include "blaswrap.h"

/* Table of constant values */

static integer c__1 = 1;

/* Subroutine */ int dsyequb_(char *uplo, integer *n, doublereal *a, integer *
        lda, doublereal *s, doublereal *scond, doublereal *amax, doublereal *
        work, integer *info)
{
    /* System generated locals */
    integer a_dim1, a_offset, i__1, i__2;
    doublereal d__1, d__2, d__3;

    /* Builtin functions */
    double sqrt(doublereal), log(doublereal), pow_di(doublereal *, integer *);

    /* Local variables */
    doublereal d__;
    integer i__, j;
    doublereal t, u, c0, c1, c2, si;
    logical up;
    doublereal avg, std, tol, base;
    integer iter;
    doublereal smin, smax, scale;
    extern logical lsame_(char *, char *);
    doublereal sumsq;
    extern doublereal dlamch_(char *);
    extern /* Subroutine */ int xerbla_(char *, integer *);
    doublereal bignum;
    extern /* Subroutine */ int dlassq_(integer *, doublereal *, integer *, 
            doublereal *, doublereal *);
    doublereal smlnum;


/*     -- LAPACK routine (version 3.2)                                 -- */
/*     -- Contributed by James Demmel, Deaglan Halligan, Yozo Hida and -- */
/*     -- Jason Riedy of Univ. of California Berkeley.                 -- */
/*     -- November 2008                                                -- */

/*     -- LAPACK is a software package provided by Univ. of Tennessee, -- */
/*     -- Univ. of California Berkeley and NAG Ltd.                    -- */

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

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

/*  DSYEQUB computes row and column scalings intended to equilibrate a */
/*  symmetric matrix A and reduce its condition number */
/*  (with respect to the two-norm).  S contains the scale factors, */
/*  S(i) = 1/sqrt(A(i,i)), chosen so that the scaled matrix B with */
/*  elements B(i,j) = S(i)*A(i,j)*S(j) has ones on the diagonal.  This */
/*  choice of S puts the condition number of B within a factor N of the */
/*  smallest possible condition number over all possible diagonal */
/*  scalings. */

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

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

/*  A       (input) DOUBLE PRECISION array, dimension (LDA,N) */
/*          The N-by-N symmetric matrix whose scaling */
/*          factors are to be computed.  Only the diagonal elements of A */
/*          are referenced. */

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

/*  S       (output) DOUBLE PRECISION array, dimension (N) */
/*          If INFO = 0, S contains the scale factors for A. */

/*  SCOND   (output) DOUBLE PRECISION */
/*          If INFO = 0, S contains the ratio of the smallest S(i) to */
/*          the largest S(i).  If SCOND >= 0.1 and AMAX is neither too */
/*          large nor too small, it is not worth scaling by S. */

/*  AMAX    (output) DOUBLE PRECISION */
/*          Absolute value of largest matrix element.  If AMAX is very */
/*          close to overflow or very close to underflow, the matrix */
/*          should be scaled. */
/*  INFO    (output) INTEGER */
/*          = 0:  successful exit */
/*          < 0:  if INFO = -i, the i-th argument had an illegal value */
/*          > 0:  if INFO = i, the i-th diagonal element is nonpositive. */

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

/*  Reference: Livne, O.E. and Golub, G.H., "Scaling by Binormalization", */
/*  Numerical Algorithms, vol. 35, no. 1, pp. 97-120, January 2004. */
/*  DOI 10.1023/B:NUMA.0000016606.32820.69 */
/*  Tech report version: http://ruready.utah.edu/archive/papers/bin.pdf */

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

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

/*     Test input parameters. */

    /* Parameter adjustments */
    a_dim1 = *lda;
    a_offset = 1 + a_dim1;
    a -= a_offset;
    --s;
    --work;

    /* Function Body */
    *info = 0;
    if (! (lsame_(uplo, "U") || lsame_(uplo, "L"))) {
        *info = -1;
    } else if (*n < 0) {
        *info = -2;
    } else if (*lda < max(1,*n)) {
        *info = -4;
    }
    if (*info != 0) {
        i__1 = -(*info);
        xerbla_("DSYEQUB", &i__1);
        return 0;
    }
    up = lsame_(uplo, "U");
    *amax = 0.;

/*     Quick return if possible. */

    if (*n == 0) {
        *scond = 1.;
        return 0;
    }
    i__1 = *n;
    for (i__ = 1; i__ <= i__1; ++i__) {
        s[i__] = 0.;
    }
    *amax = 0.;
    if (up) {
        i__1 = *n;
        for (j = 1; j <= i__1; ++j) {
            i__2 = j - 1;
            for (i__ = 1; i__ <= i__2; ++i__) {
/* Computing MAX */
                d__2 = s[i__], d__3 = (d__1 = a[i__ + j * a_dim1], abs(d__1));
                s[i__] = max(d__2,d__3);
/* Computing MAX */
                d__2 = s[j], d__3 = (d__1 = a[i__ + j * a_dim1], abs(d__1));
                s[j] = max(d__2,d__3);
/* Computing MAX */
                d__2 = *amax, d__3 = (d__1 = a[i__ + j * a_dim1], abs(d__1));
                *amax = max(d__2,d__3);
            }
/* Computing MAX */
            d__2 = s[j], d__3 = (d__1 = a[j + j * a_dim1], abs(d__1));
            s[j] = max(d__2,d__3);
/* Computing MAX */
            d__2 = *amax, d__3 = (d__1 = a[j + j * a_dim1], abs(d__1));
            *amax = max(d__2,d__3);
        }
    } else {
        i__1 = *n;
        for (j = 1; j <= i__1; ++j) {
/* Computing MAX */
            d__2 = s[j], d__3 = (d__1 = a[j + j * a_dim1], abs(d__1));
            s[j] = max(d__2,d__3);
/* Computing MAX */
            d__2 = *amax, d__3 = (d__1 = a[j + j * a_dim1], abs(d__1));
            *amax = max(d__2,d__3);
            i__2 = *n;
            for (i__ = j + 1; i__ <= i__2; ++i__) {
/* Computing MAX */
                d__2 = s[i__], d__3 = (d__1 = a[i__ + j * a_dim1], abs(d__1));
                s[i__] = max(d__2,d__3);
/* Computing MAX */
                d__2 = s[j], d__3 = (d__1 = a[i__ + j * a_dim1], abs(d__1));
                s[j] = max(d__2,d__3);
/* Computing MAX */
                d__2 = *amax, d__3 = (d__1 = a[i__ + j * a_dim1], abs(d__1));
                *amax = max(d__2,d__3);
            }
        }
    }
    i__1 = *n;
    for (j = 1; j <= i__1; ++j) {
        s[j] = 1. / s[j];
    }
    tol = 1. / sqrt(*n * 2.);
    for (iter = 1; iter <= 100; ++iter) {
        scale = 0.;
        sumsq = 0.;
/*       BETA = |A|S */
        i__1 = *n;
        for (i__ = 1; i__ <= i__1; ++i__) {
            work[i__] = 0.;
        }
        if (up) {
            i__1 = *n;
            for (j = 1; j <= i__1; ++j) {
                i__2 = j - 1;
                for (i__ = 1; i__ <= i__2; ++i__) {
                    t = (d__1 = a[i__ + j * a_dim1], abs(d__1));
                    work[i__] += (d__1 = a[i__ + j * a_dim1], abs(d__1)) * s[
                            j];
                    work[j] += (d__1 = a[i__ + j * a_dim1], abs(d__1)) * s[
                            i__];
                }
                work[j] += (d__1 = a[j + j * a_dim1], abs(d__1)) * s[j];
            }
        } else {
            i__1 = *n;
            for (j = 1; j <= i__1; ++j) {
                work[j] += (d__1 = a[j + j * a_dim1], abs(d__1)) * s[j];
                i__2 = *n;
                for (i__ = j + 1; i__ <= i__2; ++i__) {
                    t = (d__1 = a[i__ + j * a_dim1], abs(d__1));
                    work[i__] += (d__1 = a[i__ + j * a_dim1], abs(d__1)) * s[
                            j];
                    work[j] += (d__1 = a[i__ + j * a_dim1], abs(d__1)) * s[
                            i__];
                }
            }
        }
/*       avg = s^T beta / n */
        avg = 0.;
        i__1 = *n;
        for (i__ = 1; i__ <= i__1; ++i__) {
            avg += s[i__] * work[i__];
        }
        avg /= *n;
        std = 0.;
        i__1 = *n * 3;
        for (i__ = (*n << 1) + 1; i__ <= i__1; ++i__) {
            work[i__] = s[i__ - (*n << 1)] * work[i__ - (*n << 1)] - avg;
        }
        dlassq_(n, &work[(*n << 1) + 1], &c__1, &scale, &sumsq);
        std = scale * sqrt(sumsq / *n);
        if (std < tol * avg) {
            goto L999;
        }
        i__1 = *n;
        for (i__ = 1; i__ <= i__1; ++i__) {
            t = (d__1 = a[i__ + i__ * a_dim1], abs(d__1));
            si = s[i__];
            c2 = (*n - 1) * t;
            c1 = (*n - 2) * (work[i__] - t * si);
            c0 = -(t * si) * si + work[i__] * 2 * si - *n * avg;
            d__ = c1 * c1 - c0 * 4 * c2;
            if (d__ <= 0.) {
                *info = -1;
                return 0;
            }
            si = c0 * -2 / (c1 + sqrt(d__));
            d__ = si - s[i__];
            u = 0.;
            if (up) {
                i__2 = i__;
                for (j = 1; j <= i__2; ++j) {
                    t = (d__1 = a[j + i__ * a_dim1], abs(d__1));
                    u += s[j] * t;
                    work[j] += d__ * t;
                }
                i__2 = *n;
                for (j = i__ + 1; j <= i__2; ++j) {
                    t = (d__1 = a[i__ + j * a_dim1], abs(d__1));
                    u += s[j] * t;
                    work[j] += d__ * t;
                }
            } else {
                i__2 = i__;
                for (j = 1; j <= i__2; ++j) {
                    t = (d__1 = a[i__ + j * a_dim1], abs(d__1));
                    u += s[j] * t;
                    work[j] += d__ * t;
                }
                i__2 = *n;
                for (j = i__ + 1; j <= i__2; ++j) {
                    t = (d__1 = a[j + i__ * a_dim1], abs(d__1));
                    u += s[j] * t;
                    work[j] += d__ * t;
                }
            }
            avg += (u + work[i__]) * d__ / *n;
            s[i__] = si;
        }
    }
L999:
    smlnum = dlamch_("SAFEMIN");
    bignum = 1. / smlnum;
    smin = bignum;
    smax = 0.;
    t = 1. / sqrt(avg);
    base = dlamch_("B");
    u = 1. / log(base);
    i__1 = *n;
    for (i__ = 1; i__ <= i__1; ++i__) {
        i__2 = (integer) (u * log(s[i__] * t));
        s[i__] = pow_di(&base, &i__2);
/* Computing MIN */
        d__1 = smin, d__2 = s[i__];
        smin = min(d__1,d__2);
/* Computing MAX */
        d__1 = smax, d__2 = s[i__];
        smax = max(d__1,d__2);
    }
    *scond = max(smin,smlnum) / min(smax,bignum);

    return 0;
} /* dsyequb_ */