The R Project SVN R

/*
 *  R : A Computer Language for Statistical Data Analysis
 *  Copyright (C) 1997-1998 Ross Ihaka
 *  Copyright (C) 1999-2001 R Core Team
 *
 *  This program is free software; you can redistribute it and/or modify
 *  it under the terms of the GNU General Public License as published by
 *  the Free Software Foundation; either version 2 of the License, or
 *  (at your option) any later version.
 *
 *  This program is distributed in the hope that it will be useful,
 *  but WITHOUT ANY WARRANTY; without even the implied warranty of
 *  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 *  GNU General Public License for more details.
 *
 *  You should have received a copy of the GNU General Public License
 *  along with this program; if not, a copy is available at
 *  http://www.r-project.org/Licenses/

 *
 *  cpoly finds the zeros of a complex polynomial.
 *
 *  On Entry
 *
 *  opr, opi      -  double precision vectors of real and
 *           imaginary parts of the coefficients in
 *           order of decreasing powers.
 *
 *  degree        -  int degree of polynomial.
 *
 *
 *  On Return
 *
 *  zeror, zeroi  -  output double precision vectors of
 *           real and imaginary parts of the zeros.
 *
 *  fail          -  output int parameter,  true  only if
 *           leading coefficient is zero or if cpoly
 *           has found fewer than degree zeros.
 *
 *  The program has been written to reduce the chance of overflow
 *  occurring. If it does occur, there is still a possibility that
 *  the zerofinder will work provided the overflowed quantity is
 *  replaced by a large number.
 *
 *  This is a C translation of the following.
 *
 *  TOMS Algorithm 419
 *  Jenkins and Traub.
 *  Comm. ACM 15 (1972) 97-99.
 *
 *  Ross Ihaka
 *  February 1997
 */

#ifdef HAVE_CONFIG_H
#include <config.h>
#endif

#include <R_ext/Arith.h> /* for declaration of hypot */
#include <R_ext/Memory.h> /* for declaration of R_alloc */

#include <float.h> /* for FLT_RADIX */

#include <Rmath.h> /* for R_pow_di */
#include <R_ext/Applic.h>

#ifdef HAVE_VISIBILITY_ATTRIBUTE
# define attribute_hidden __attribute__ ((visibility ("hidden")))
#else
# define attribute_hidden
#endif

static void calct(Rboolean *);
static Rboolean fxshft(int, double *, double *);
static Rboolean vrshft(int, double *, double *);
static void nexth(Rboolean);
static void noshft(int);

/* Consider exporting these (via Applic.h): */
static void polyev(int, double, double,
           double *, double *, double *, double *, double *, double *);
static double errev(int, double *, double *, double, double, double, double);
static double cpoly_cauchy(int, double *, double *);
static double cpoly_scale(int, double *, double, double, double, double);
static void cdivid(double, double, double, double, double *, double *);

/* Global Variables (too many!) */

static int nn;
static double *pr, *pi, *hr, *hi, *qpr, *qpi, *qhr, *qhi, *shr, *shi;
static double sr, si;
static double tr, ti;
static double pvr, pvi;

static const double eta =  DBL_EPSILON;
static const double are = /* eta = */DBL_EPSILON;
static const double mre = 2. * M_SQRT2 * /* eta, i.e. */DBL_EPSILON;
static const double infin = DBL_MAX;

attribute_hidden
void R_cpolyroot(double *opr, double *opi, int *degree,
         double *zeror, double *zeroi, Rboolean *fail)
{
    static const double smalno = DBL_MIN;
    static const double base = (double)FLT_RADIX;
    static int d_n, i, i1, i2;
    static double zi, zr, xx, yy;
    static double bnd, xxx;
    Rboolean conv;
    int d1;
    double *tmp;
    static const double cosr =/* cos 94 */ -0.06975647374412529990;
    static const double sinr =/* sin 94 */  0.99756405025982424767;
    xx = M_SQRT1_2;/* 1/sqrt(2) = 0.707.... */

    yy = -xx;
    *fail = FALSE;

    nn = *degree;
    d1 = nn - 1;

    /* algorithm fails if the leading coefficient is zero. */

    if (opr[0] == 0. && opi[0] == 0.) {
    *fail = TRUE;
    return;
    }

    /* remove the zeros at the origin if any. */

    while (opr[nn] == 0. && opi[nn] == 0.) {
    d_n = d1-nn+1;
    zeror[d_n] = 0.;
    zeroi[d_n] = 0.;
    nn--;
    }
    nn++;
    /*-- Now, global var.  nn := #{coefficients} = (relevant degree)+1 */

    if (nn == 1) return;

    /* Use a single allocation as these as small */
    tmp = (double *) R_alloc((size_t) (10*nn), sizeof(double));
    pr = tmp; pi = tmp + nn; hr = tmp + 2*nn; hi = tmp + 3*nn;
    qpr = tmp + 4*nn; qpi = tmp + 5*nn; qhr = tmp + 6*nn; qhi = tmp + 7*nn;
    shr = tmp + 8*nn; shi = tmp + 9*nn;
    
    /* make a copy of the coefficients and shr[] = | p[] | */
    for (i = 0; i < nn; i++) {
    pr[i] = opr[i];
    pi[i] = opi[i];
    shr[i] = hypot(pr[i], pi[i]);
    }

    /* scale the polynomial with factor 'bnd'. */
    bnd = cpoly_scale(nn, shr, eta, infin, smalno, base);
    if (bnd != 1.) {
    for (i=0; i < nn; i++) {
        pr[i] *= bnd;
        pi[i] *= bnd;
    }
    }

    /* start the algorithm for one zero */

    while (nn > 2) {

    /* calculate bnd, a lower bound on the modulus of the zeros. */

    for (i=0 ; i < nn ; i++)
        shr[i] = hypot(pr[i], pi[i]);
    bnd = cpoly_cauchy(nn, shr, shi);

    /* outer loop to control 2 major passes */
    /* with different sequences of shifts */

    for (i1 = 1; i1 <= 2; i1++) {

        /* first stage calculation, no shift */

        noshft(5);

        /*  inner loop to select a shift */
        for (i2 = 1; i2 <= 9; i2++) {

        /* shift is chosen with modulus bnd */
        /* and amplitude rotated by 94 degrees */
        /* from the previous shift */

        xxx= cosr * xx - sinr * yy;
        yy = sinr * xx + cosr * yy;
        xx = xxx;
        sr = bnd * xx;
        si = bnd * yy;

        /*  second stage calculation, fixed shift */

        conv = fxshft(i2 * 10, &zr, &zi);
        if (conv)
            goto L10;
        }
    }

    /* the zerofinder has failed on two major passes */
    /* return empty handed */

    *fail = TRUE;
    return;

    /* the second stage jumps directly to the third stage iteration.
     * if successful, the zero is stored and the polynomial deflated.
     */
    L10:
    d_n = d1+2 - nn;
    zeror[d_n] = zr;
    zeroi[d_n] = zi;
    --nn;
    for (i=0; i < nn ; i++) {
        pr[i] = qpr[i];
        pi[i] = qpi[i];
    }
    }/*while*/

    /*  calculate the final zero and return */
    cdivid(-pr[1], -pi[1], pr[0], pi[0], &zeror[d1], &zeroi[d1]);

    return;
}


/*  Computes the derivative polynomial as the initial
 *  polynomial and computes l1 no-shift h polynomials.  */

static void noshft(int l1)
{
    int i, j, jj, n = nn - 1, nm1 = n - 1;

    double t1, t2, xni;

    for (i=0; i < n; i++) {
    xni = (double)(nn - i - 1);
    hr[i] = xni * pr[i] / n;
    hi[i] = xni * pi[i] / n;
    }

    for (jj = 1; jj <= l1; jj++) {

    if (hypot(hr[n-1], hi[n-1]) <=
        eta * 10.0 * hypot(pr[n-1], pi[n-1])) {
        /*  If the constant term is essentially zero, */
        /*  shift h coefficients. */

        for (i = 1; i <= nm1; i++) {
        j = nn - i;
        hr[j-1] = hr[j-2];
        hi[j-1] = hi[j-2];
        }
        hr[0] = 0.;
        hi[0] = 0.;
    }
    else {
        cdivid(-pr[nn-1], -pi[nn-1], hr[n-1], hi[n-1], &tr, &ti);
        for (i = 1; i <= nm1; i++) {
        j = nn - i;
        t1 = hr[j-2];
        t2 = hi[j-2];
        hr[j-1] = tr * t1 - ti * t2 + pr[j-1];
        hi[j-1] = tr * t2 + ti * t1 + pi[j-1];
        }
        hr[0] = pr[0];
        hi[0] = pi[0];
    }
    }
}


/*  Computes l2 fixed-shift h polynomials and tests for convergence.
 *  initiates a variable-shift iteration and returns with the
 *  approximate zero if successful.
 */
static Rboolean fxshft(int l2, double *zr, double *zi)
{
/*  l2    - limit of fixed shift steps
 *  zr,zi - approximate zero if convergence (result TRUE)
 *
 * Return value indicates convergence of stage 3 iteration
 *
 * Uses global (sr,si), nn, pr[], pi[], .. (all args of polyev() !)
*/

    Rboolean pasd, bool, test;
    static double svsi, svsr;
    static int i, j, n;
    static double oti, otr;

    n = nn - 1;

    /* evaluate p at s. */

    polyev(nn, sr, si, pr, pi, qpr, qpi, &pvr, &pvi);

    test = TRUE;
    pasd = FALSE;

    /* calculate first t = -p(s)/h(s). */

    calct(&bool);

    /* main loop for one second stage step. */

    for (j=1; j<=l2; j++) {

    otr = tr;
    oti = ti;

    /* compute next h polynomial and new t. */

    nexth(bool);
    calct(&bool);
    *zr = sr + tr;
    *zi = si + ti;

    /* test for convergence unless stage 3 has */
    /* failed once or this is the last h polynomial. */

    if (!bool && test && j != l2) {
        if (hypot(tr - otr, ti - oti) >= hypot(*zr, *zi) * 0.5) {
        pasd = FALSE;
        }
        else if (! pasd) {
        pasd = TRUE;
        }
        else {

        /* the weak convergence test has been */
        /* passed twice, start the third stage */
        /* iteration, after saving the current */
        /* h polynomial and shift. */

        for (i = 0; i < n; i++) {
            shr[i] = hr[i];
            shi[i] = hi[i];
        }
        svsr = sr;
        svsi = si;
        if (vrshft(10, zr, zi)) {
            return TRUE;
        }

        /* the iteration failed to converge. */
        /* turn off testing and restore */
        /* h, s, pv and t. */

        test = FALSE;
        for (i=1 ; i<=n ; i++) {
            hr[i-1] = shr[i-1];
            hi[i-1] = shi[i-1];
        }
        sr = svsr;
        si = svsi;
        polyev(nn, sr, si, pr, pi, qpr, qpi, &pvr, &pvi);
        calct(&bool);
        }
    }
    }

    /* attempt an iteration with final h polynomial */
    /* from second stage. */

    return(vrshft(10, zr, zi));
}


/* carries out the third stage iteration.
 */
static Rboolean vrshft(int l3, double *zr, double *zi)
{
/*  l3      - limit of steps in stage 3.
 *  zr,zi   - on entry contains the initial iterate;
 *        if the iteration converges it contains
 *        the final iterate on exit.
 * Returns TRUE if iteration converges
 *
 * Assign and uses  GLOBAL sr, si
*/
    Rboolean bool, b;
    static int i, j;
    static double r1, r2, mp, ms, tp, relstp;
    static double omp;

    b = FALSE;
    sr = *zr;
    si = *zi;

    /* main loop for stage three */

    for (i = 1; i <= l3; i++) {

    /* evaluate p at s and test for convergence. */
    polyev(nn, sr, si, pr, pi, qpr, qpi, &pvr, &pvi);

    mp = hypot(pvr, pvi);
    ms = hypot(sr, si);
    if (mp <=  20. * errev(nn, qpr, qpi, ms, mp, /*are=*/eta, mre)) {
        goto L_conv;
    }

    /* polynomial value is smaller in value than */
    /* a bound on the error in evaluating p, */
    /* terminate the iteration. */

    if (i != 1) {

        if (!b && mp >= omp && relstp < .05) {

        /* iteration has stalled. probably a */
        /* cluster of zeros. do 5 fixed shift */
        /* steps into the cluster to force */
        /* one zero to dominate. */

        tp = relstp;
        b = TRUE;
        if (relstp < eta)
            tp = eta;
        r1 = sqrt(tp);
        r2 = sr * (r1 + 1.) - si * r1;
        si = sr * r1 + si * (r1 + 1.);
        sr = r2;
        polyev(nn, sr, si, pr, pi, qpr, qpi, &pvr, &pvi);
        for (j = 1; j <= 5; ++j) {
            calct(&bool);
            nexth(bool);
        }
        omp = infin;
        goto L10;
        }
        else {

        /* exit if polynomial value */
        /* increases significantly. */

        if (mp * .1 > omp)
            return FALSE;
        }
    }
    omp = mp;

    /* calculate next iterate. */

    L10:
    calct(&bool);
    nexth(bool);
    calct(&bool);
    if (!bool) {
        relstp = hypot(tr, ti) / hypot(sr, si);
        sr += tr;
        si += ti;
    }
    }
    return FALSE;

L_conv:
    *zr = sr;
    *zi = si;
    return TRUE;
}

static void calct(Rboolean *bool)
{
    /* computes  t = -p(s)/h(s).
     * bool   - logical, set true if h(s) is essentially zero.  */

    int n = nn - 1;
    double hvi, hvr;

    /* evaluate h(s). */
    polyev(n, sr, si, hr, hi,
       qhr, qhi, &hvr, &hvi);

    *bool = hypot(hvr, hvi) <= are * 10. * hypot(hr[n-1], hi[n-1]);
    if (!*bool) {
    cdivid(-pvr, -pvi, hvr, hvi, &tr, &ti);
    }
    else {
    tr = 0.;
    ti = 0.;
    }
}

static void nexth(Rboolean bool)
{
    /* calculates the next shifted h polynomial.
     * bool :   if TRUE  h(s) is essentially zero
     */
    int j, n = nn - 1;
    double t1, t2;

    if (!bool) {
    for (j=1; j < n; j++) {
        t1 = qhr[j - 1];
        t2 = qhi[j - 1];
        hr[j] = tr * t1 - ti * t2 + qpr[j];
        hi[j] = tr * t2 + ti * t1 + qpi[j];
    }
    hr[0] = qpr[0];
    hi[0] = qpi[0];
    }
    else {
    /* if h(s) is zero replace h with qh. */

    for (j=1; j < n; j++) {
        hr[j] = qhr[j-1];
        hi[j] = qhi[j-1];
    }
    hr[0] = 0.;
    hi[0] = 0.;
    }
}

/*--------------------- Independent Complex Polynomial Utilities ----------*/

static
void polyev(int n,
        double s_r, double s_i,
        double *p_r, double *p_i,
        double *q_r, double *q_i,
        double *v_r, double *v_i)
{
    /* evaluates a polynomial  p  at  s  by the horner recurrence
     * placing the partial sums in q and the computed value in v_.
     */
    int i;
    double t;

    q_r[0] = p_r[0];
    q_i[0] = p_i[0];
    *v_r = q_r[0];
    *v_i = q_i[0];
    for (i = 1; i < n; i++) {
    t = *v_r * s_r - *v_i * s_i + p_r[i];
    q_i[i] = *v_i = *v_r * s_i + *v_i * s_r + p_i[i];
    q_r[i] = *v_r = t;
    }
}

static
double errev(int n, double *qr, double *qi,
         double ms, double mp, double a_re, double m_re)
{
    /*  bounds the error in evaluating the polynomial by the horner
     *  recurrence.
     *
     *  qr,qi    - the partial sum vectors
     *  ms   - modulus of the point
     *  mp   - modulus of polynomial value
     * a_re,m_re - error bounds on complex addition and multiplication
     */
    double e;
    int i;

    e = hypot(qr[0], qi[0]) * m_re / (a_re + m_re);
    for (i=0; i < n; i++)
    e = e*ms + hypot(qr[i], qi[i]);

    return e * (a_re + m_re) - mp * m_re;
}


static
double cpoly_cauchy(int n, double *pot, double *q)
{
    /* Computes a lower bound on the moduli of the zeros of a polynomial
     * pot[1:nn] is the modulus of the coefficients.
     */
    double f, x, delf, dx, xm;
    int i, n1 = n - 1;

    pot[n1] = -pot[n1];

    /* compute upper estimate of bound. */

    x = exp((log(-pot[n1]) - log(pot[0])) / (double) n1);

    /* if newton step at the origin is better, use it. */

    if (pot[n1-1] != 0.) {
    xm = -pot[n1] / pot[n1-1];
    if (xm < x)
        x = xm;
    }

    /* chop the interval (0,x) unitl f le 0. */

    for(;;) {
    xm = x * 0.1;
    f = pot[0];
    for (i = 1; i < n; i++)
        f = f * xm + pot[i];
    if (f <= 0.0) {
        break;
    }
    x = xm;
    }

    dx = x;

    /* do Newton iteration until x converges to two decimal places. */

    while (fabs(dx / x) > 0.005) {
    q[0] = pot[0];
    for(i = 1; i < n; i++)
        q[i] = q[i-1] * x + pot[i];
    f = q[n1];
    delf = q[0];
    for(i = 1; i < n1; i++)
        delf = delf * x + q[i];
    dx = f / delf;
    x -= dx;
    }
    return x;
}

static
double cpoly_scale(int n, double *pot,
           double eps, double BIG, double small, double base)
{
    /* Returns a scale factor to multiply the coefficients of the polynomial.
     * The scaling is done to avoid overflow and to avoid
     *  undetected underflow interfering with the convergence criterion.
     * The factor is a power of the base.

     * pot[1:n] : modulus of coefficients of p
     * eps,BIG,
     * small,base - constants describing the floating point arithmetic.
     */

    int i, ell;
    double x, high, sc, lo, min_, max_;

    /* find largest and smallest moduli of coefficients. */
    high = sqrt(BIG);
    lo = small / eps;
    max_ = 0.;
    min_ = BIG;
    for (i = 0; i < n; i++) {
    x = pot[i];
    if (x > max_) max_ = x;
    if (x != 0. && x < min_)
        min_ = x;
    }

    /* scale only if there are very large or very small components. */

    if (min_ < lo || max_ > high) {
    x = lo / min_;
    if (x <= 1.)
        sc = 1. / (sqrt(max_) * sqrt(min_));
    else {
        sc = x;
        if (BIG / sc > max_)
        sc = 1.0;
    }
    ell = (int) (log(sc) / log(base) + 0.5);
    return R_pow_di(base, ell);
    }
    else return 1.0;
}


static
void cdivid(double ar, double ai, double br, double bi,
        double *cr, double *ci)
{
/* complex division c = a/b, i.e., (cr +i*ci) = (ar +i*ai) / (br +i*bi),
   avoiding overflow. */

    double d, r;

    if (br == 0. && bi == 0.) {
    /* division by zero, c = infinity. */
    *cr = *ci = R_PosInf;
    }
    else if (fabs(br) >= fabs(bi)) {
    r = bi / br;
    d = br + r * bi;
    *cr = (ar + ai * r) / d;
    *ci = (ai - ar * r) / d;
    }
    else {
    r = br / bi;
    d = bi + r * br;
    *cr = (ar * r + ai) / d;
    *ci = (ai * r - ar) / d;
    }
}

/* static double cpoly_cmod(double *r, double *i)
 * --> replaced by hypot() everywhere
*/