Logo Search packages:      
Sourcecode: qsapecng version File versions  Download package

rpoly.cpp

/*
 *  http://www.crbond.com/download/misc/rpoly.cpp
 */

/*
 * This software is offered freely and without restriction.
 * (notice added with the permission of the author, see the file
 * README.rpoly)
 */

/*      rpoly.cpp -- Jenkins-Traub real polynomial root finder.
 *
 *      (C) 2000, C. Bond.  All rights reserved.
 *
 *      Translation of TOMS493 from FORTRAN to C. This
 *      implementation of Jenkins-Traub partially adapts
 *      the original code to a C environment by restruction
 *      many of the 'goto' controls to better fit a block
 *      structured form. It also eliminates the global memory
 *      allocation in favor of local, dynamic memory management.
 *
 *      The calling conventions are slightly modified to return
 *      the number of roots found as the function value.
 *
 *      INPUT:
 *      op - double precision vector of coefficients in order of
 *              decreasing powers.
 *      degree - integer degree of polynomial
 *
 *      OUTPUT:
 *      zeror,zeroi - output double precision vectors of the
 *              real and imaginary parts of the zeros.
 *
 *      RETURN:
 *      returnval:   -1 if leading coefficient is zero, otherwise
 *                  number of roots found. 
 */

#include "math.h"

void quad(double a,double b1,double c,double *sr,double *si,
        double *lr,double *li);
void fxshfr(int l2, int *nz);
void quadit(double *uu,double *vv,int *nz);
void realit(double *sss, int *nz, int *iflag);
void calcsc(int *type);
void nextk(int *type);
void newest(int type,double *uu,double *vv);
void quadsd(int n,double *u,double *v,double *p,double *q,
        double *a,double *b);
double *p,*qp,*k,*qk,*svk;
double sr,si,u,v,a,b,c,d,a1,a2;
double a3,a6,a7,e,f,g,h,szr,szi,lzr,lzi;
double eta,are,mre;
int n,nn,nmi,zerok;

int rpoly(double *op, int degree, double *zeror, double *zeroi) 
{
    double t,aa,bb,cc,*temp,factor,rot;
    double *pt;
    double lo,max,min,xx,yy,cosr,sinr,xxx,x,sc,bnd;
    double xm,ff,df,dx,infin,smalno,base;
    int cnt,nz,i,j,jj,l,nm1,zerok;
/*  The following statements set machine constants. */
    base = 2.0;
    eta = 2.22e-16;
    infin = 3.4e38;
    smalno = 1.2e-38;

    are = eta;
    mre = eta;
    lo = smalno/eta;
/*  Initialization of constants for shift rotation. */        
    xx = sqrt(0.5);
    yy = -xx;
    rot = 94.0;
    rot *= 0.017453293;
    cosr = cos(rot);
    sinr = sin(rot);
    n = degree;
/*  Algorithm fails of the leading coefficient is zero. */
    if (op[0] == 0.0) return -1;
/*  Remove the zeros at the origin, if any. */
    while (op[n] == 0.0) {
        j = degree - n;
        zeror[j] = 0.0;
        zeroi[j] = 0.0;
        n--;
    }
    if (n < 1) return -1;
/*
 *  Allocate memory here
 */
    temp = new double [degree+1];
    pt = new double [degree+1];
    p = new double [degree+1];
    qp = new double [degree+1];
    k = new double [degree+1];
    qk = new double [degree+1];
    svk = new double [degree+1];
/*  Make a copy of the coefficients. */
    for (i=0;i<=n;i++)
        p[i] = op[i];
/*  Start the algorithm for one zero. */
_40:        
    if (n == 1) {
        zeror[degree-1] = -p[1]/p[0];
        zeroi[degree-1] = 0.0;
        n -= 1;
        goto _99;
    }
/*  Calculate the final zero or pair of zeros. */
    if (n == 2) {
        quad(p[0],p[1],p[2],&zeror[degree-2],&zeroi[degree-2],
            &zeror[degree-1],&zeroi[degree-1]);
        n -= 2;
        goto _99;
    }
/*  Find largest and smallest moduli of coefficients. */
    max = 0.0;
    min = infin;
    for (i=0;i<=n;i++) {
        x = fabs(p[i]);
        if (x > max) max = x;
        if (x != 0.0 && x < min) min = x;
    }
/*  Scale if there are large or very small coefficients.
 *  Computes a scale factor to multiply the coefficients of the
 *  polynomial. The scaling si done to avoid overflow and to
 *  avoid undetected underflow interfering with the convergence
 *  criterion. The factor is a power of the base.
 */
    sc = lo/min;
    if (sc > 1.0 && infin/sc < max) goto _110;
    if (sc <= 1.0) {
        if (max < 10.0) goto _110;
        if (sc == 0.0)
            sc = smalno;
    }
    l = (int)(log(sc)/log(base) + 0.5);
    factor = pow(base*1.0,l);
    if (factor != 1.0) {
        for (i=0;i<=n;i++) 
            p[i] = factor*p[i];     /* Scale polynomial. */
    }
_110:
/*  Compute lower bound on moduli of roots. */
    for (i=0;i<=n;i++) {
        pt[i] = (fabs(p[i]));
    }
    pt[n] = - pt[n];
/*  Compute upper estimate of bound. */
    x = exp((log(-pt[n])-log(pt[0])) / (double)n);
/*  If Newton step at the origin is better, use it. */        
    if (pt[n-1] != 0.0) {
        xm = -pt[n]/pt[n-1];
        if (xm < x)  x = xm;
    }
/*  Chop the interval (0,x) until ff <= 0 */
    while (1) {
        xm = x*0.1;
        ff = pt[0];
        for (i=1;i<=n;i++) 
            ff = ff*xm + pt[i];
        if (ff <= 0.0) break;
        x = xm;
    }
    dx = x;
/*  Do Newton interation until x converges to two 
 *  decimal places. 
 */
    while (fabs(dx/x) > 0.005) {
        ff = pt[0];
        df = ff;
        for (i=1;i<n;i++) { 
            ff = ff*x + pt[i];
            df = df*x + ff;
        }
        ff = ff*x + pt[n];
        dx = ff/df;
        x -= dx;
    }
    bnd = x;
/*  Compute the derivative as the initial k polynomial
 *  and do 5 steps with no shift.
 */
    nm1 = n - 1;
    for (i=1;i<n;i++)
        k[i] = (double)(n-i)*p[i]/(double)n;
    k[0] = p[0];
    aa = p[n];
    bb = p[n-1];
    zerok = (k[n-1] == 0);
    for(jj=0;jj<5;jj++) {
        cc = k[n-1];
        if (!zerok) {
/*  Use a scaled form of recurrence if value of k at 0 is nonzero. */             
            t = -aa/cc;
            for (i=0;i<nm1;i++) {
                j = n-i-1;
                k[j] = t*k[j-1]+p[j];
            }
            k[0] = p[0];
            zerok = (fabs(k[n-1]) <= fabs(bb)*eta*10.0);
        }
        else {
/*  Use unscaled form of recurrence. */
            for (i=0;i<nm1;i++) {
                j = n-i-1;
                k[j] = k[j-1];
            }
            k[0] = 0.0;
            zerok = (k[n-1] == 0.0);
        }
    }
/*  Save k for restarts with new shifts. */
    for (i=0;i<n;i++) 
        temp[i] = k[i];
/*  Loop to select the quadratic corresponding to each new shift. */
    for (cnt = 0;cnt < 20;cnt++) {
/*  Quadratic corresponds to a double shift to a            
 *  non-real point and its complex conjugate. The point
 *  has 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;
        u = -2.0 * sr;
        v = bnd;
        fxshfr(20*(cnt+1),&nz);
        if (nz != 0) {
/*  The second stage jumps directly to one of the third
 *  stage iterations and returns here if successful.
 *  Deflate the polynomial, store the zero or zeros and
 *  return to the main algorithm.
 */
            j = degree - n;
            zeror[j] = szr;
            zeroi[j] = szi;
            n -= nz;
            for (i=0;i<=n;i++)
                p[i] = qp[i];
            if (nz != 1) {
                zeror[j+1] = lzr;
                zeroi[j+1] = lzi;
            }
            goto _40;
        }
/*  If the iteration is unsuccessful another quadratic
 *  is chosen after restoring k.
 */
        for (i=0;i<n;i++) {
            k[i] = temp[i];
        }
    } 
/*  Return with failure if no convergence with 20 shifts. */
_99:
    delete [] svk;
    delete [] qk;
    delete [] k;
    delete [] qp;
    delete [] p;
    delete [] pt;
    delete [] temp;

    return degree - n;
}
/*  Computes up to L2 fixed shift k-polynomials,
 *  testing for convergence in the linear or quadratic
 *  case. Initiates one of the variable shift
 *  iterations and returns with the number of zeros
 *  found.
 */
void fxshfr(int l2,int *nz)
{
    double svu,svv,ui,vi,s;
    double betas,betav,oss,ovv,ss,vv,ts,tv;
    double ots,otv,tvv,tss;
  int type, i,j,iflag,vpass,spass,vtry,stry;

  *nz = 0;
  betav = 0.25;
  betas = 0.25;
  oss = sr;
  ovv = v;
/*  Evaluate polynomial by synthetic division. */
    quadsd(n,&u,&v,p,qp,&a,&b);
  calcsc(&type);
  for (j=0;j<l2;j++) {
/*  Calculate next k polynomial and estimate v. */
    nextk(&type);
    calcsc(&type);
    newest(type,&ui,&vi);
    vv = vi;
/*  Estimate s. */
        ss = 0.0;
        if (k[n-1] != 0.0) ss = -p[n]/k[n-1];
    tv = 1.0;
    ts = 1.0;
    if (j == 0 || type == 3) goto _70;
/*  Compute relative measures of convergence of s and v sequences. */
        if (vv != 0.0) tv = fabs((vv-ovv)/vv);
        if (ss != 0.0) ts = fabs((ss-oss)/ss);
/*  If decreasing, multiply two most recent convergence measures. */
    tvv = 1.0;
    if (tv < otv) tvv = tv*otv;
    tss = 1.0;
    if (ts < ots) tss = ts*ots;
/*  Compare with convergence criteria. */
    vpass = (tvv < betav);
    spass = (tss < betas);
    if (!(spass || vpass)) goto _70;
/*  At least one sequence has passed the convergence test.
 *  Store variables before iterating.
 */
    svu = u;
    svv = v;
    for (i=0;i<n;i++) {
      svk[i] = k[i];
    }
    s = ss;
/*  Choose iteration according to the fastest converging
 *  sequence.
 */
    vtry = 0;
    stry = 0;
    if (spass && (!vpass) || tss < tvv) goto _40;
_20:        
    quadit(&ui,&vi,nz);
        if (*nz > 0) return;
/*  Quadratic iteration has failed. Flag that it has
 *  been tried and decrease the convergence criterion.
 */
    vtry = 1;
    betav *= 0.25;
/*  Try linear iteration if it has not been tried and
 *  the S sequence is converging.
 */
    if (stry || !spass) goto _50;
    for (i=0;i<n;i++) {
      k[i] = svk[i];
    }
_40:
        realit(&s,nz,&iflag);
    if (*nz > 0) return;
/*  Linear iteration has failed. Flag that it has been
 *  tried and decrease the convergence criterion.
 */
    stry = 1;
    betas *=0.25;
    if (iflag == 0) goto _50;
/*  If linear iteration signals an almost double real
 *  zero attempt quadratic iteration.
 */
    ui = -(s+s);
    vi = s*s;
    goto _20;
/*  Restore variables. */
_50:
    u = svu;
    v = svv;
    for (i=0;i<n;i++) {
      k[i] = svk[i];
    }
/*  Try quadratic iteration if it has not been tried
 *  and the V sequence is convergin.
 */
    if (vpass && !vtry) goto _20;
/*  Recompute QP and scalar values to continue the
 *  second stage.
 */
        quadsd(n,&u,&v,p,qp,&a,&b);
    calcsc(&type);
_70:
    ovv = vv;
    oss = ss;
    otv = tv;
    ots = ts;
  }
}
/*  Variable-shift k-polynomial iteration for a
 *  quadratic factor converges only if the zeros are
 *  equimodular or nearly so.
 *  uu, vv - coefficients of starting quadratic.
 *  nz - number of zeros found.
 */
void quadit(double *uu,double *vv,int *nz)
{
    double ui,vi;
    double mp,omp,ee,relstp,t,zm;
  int type,i,j,tried;

  *nz = 0;
  tried = 0;
  u = *uu;
  v = *vv;
  j = 0;
/*  Main loop. */
_10:    
  quad(1.0,u,v,&szr,&szi,&lzr,&lzi);
/*  Return if roots of the quadratic are real and not
 *  close to multiple or nearly equal and of opposite
 *  sign.
 */
    if (fabs(fabs(szr)-fabs(lzr)) > 0.01 * fabs(lzr)) return;
/*  Evaluate polynomial by quadratic synthetic division. */
    quadsd(n,&u,&v,p,qp,&a,&b);
    mp = fabs(a-szr*b) + fabs(szi*b);
/*  Compute a rigorous bound on the rounding error in
 *  evaluating p.
 */
    zm = sqrt(fabs(v));
    ee = 2.0*fabs(qp[0]);
  t = -szr*b;
  for (i=1;i<n;i++) {
        ee = ee*zm + fabs(qp[i]);
  }
    ee = ee*zm + fabs(a+t);
    ee *= (5.0 *mre + 4.0*are);
       ee = ee - (5.0*mre+2.0*are)*(fabs(a+t)+fabs(b)*zm)+2.0*are*fabs(t);
/*  Iteration has converged sufficiently if the
 *  polynomial value is less than 20 times this bound.
 */
    if (mp <= 20.0*ee) {
        *nz = 2;
        return;
    }
  j++;
/*  Stop iteration after 20 steps. */
  if (j > 20) return;
  if (j < 2) goto _50;
    if (relstp > 0.01 || mp < omp || tried) goto _50;
/*  A cluster appears to be stalling the convergence.
 *  Five fixed shift steps are taken with a u,v close
 *  to the cluster.
 */
  if (relstp < eta) relstp = eta;
  relstp = sqrt(relstp);
  u = u - u*relstp;
  v = v + v*relstp;
    quadsd(n,&u,&v,p,qp,&a,&b);
  for (i=0;i<5;i++) {
    calcsc(&type);
    nextk(&type);
  }
  tried = 1;
  j = 0;
_50:
  omp = mp;
/*  Calculate next k polynomial and new u and v. */
  calcsc(&type);
  nextk(&type);
  calcsc(&type);
  newest(type,&ui,&vi);
/*  If vi is zero the iteration is not converging. */
    if (vi == 0.0) return;
    relstp = fabs((vi-v)/vi);
  u = ui;
  v = vi;
  goto _10;
}
/*  Variable-shift H polynomial iteration for a real zero.
 *  sss - starting iterate
 *  nz  - number of zeros found
 *  iflag - flag to indicate a pair of zeros near real axis.
 */
void realit(double *sss, int *nz, int *iflag)
{
    double pv,kv,t,s;
    double ms,mp,omp,ee;
    int i,j;

    *nz = 0;
    s = *sss;
    *iflag = 0;
  j = 0;
/*  Main loop */
    while (1) {
        pv = p[0];
/*  Evaluate p at s. */
        qp[0] = pv;
        for (i=1;i<=n;i++) {
            pv = pv*s + p[i];
            qp[i] = pv;
        }
        mp = fabs(pv);
/*  Compute a rigorous bound on the error in evaluating p. */
        ms = fabs(s);
        ee = (mre/(are+mre))*fabs(qp[0]);
        for (i=1;i<=n;i++) {
            ee = ee*ms + fabs(qp[i]);
        }
/*  Iteration has converged sufficiently if the polynomial
 *  value is less than 20 times this bound.
 */
        if (mp <= 20.0*((are+mre)*ee-mre*mp)) {
            *nz = 1;
            szr = s;
            szi = 0.0;
            return;
        }
        j++;
/*  Stop iteration after 10 steps. */
        if (j > 10) return;
        if (j < 2) goto _50;
        if (fabs(t) > 0.001*fabs(s-t) || mp < omp) goto _50;
/*  A cluster of zeros near the real axis has been
 *  encountered. Return with iflag set to initiate a
 *  quadratic iteration.
 */
        *iflag = 1;
        *sss = s;
        return;
/*  Return if the polynomial value has increased significantly. */
_50:
        omp = mp;
/*  Compute t, the next polynomial, and the new iterate. */
        kv = k[0];
        qk[0] = kv;
        for (i=1;i<n;i++) {
            kv = kv*s + k[i];
            qk[i] = kv;
        }
        if (fabs(kv) <= fabs(k[n-1])*10.0*eta) {
/*  Use unscaled form. */
            k[0] = 0.0;
            for (i=1;i<n;i++) {
                k[i] = qk[i-1];
            }
        }
        else {
/*  Use the scaled form of the recurrence if the value
 *  of k at s is nonzero.
 */
            t = -pv/kv;
            k[0] = qp[0];
            for (i=1;i<n;i++) {
                k[i] = t*qk[i-1] + qp[i];
            }
        }
        kv = k[0];
        for (i=1;i<n;i++) {
            kv = kv*s + k[i];
        }
        t = 0.0;
        if (fabs(kv) > fabs(k[n-1]*10.0*eta)) t = -pv/kv;
        s += t;
    }
}

/*  This routine calculates scalar quantities used to
 *  compute the next k polynomial and new estimates of
 *  the quadratic coefficients.
 *  type - integer variable set here indicating how the
 *  calculations are normalized to avoid overflow.
 */
void calcsc(int *type)
{
/*  Synthetic division of k by the quadratic 1,u,v */    
    quadsd(n-1,&u,&v,k,qk,&c,&d);
    if (fabs(c) > fabs(k[n-1]*100.0*eta)) goto _10;
    if (fabs(d) > fabs(k[n-2]*100.0*eta)) goto _10;
  *type = 3;
/*  Type=3 indicates the quadratic is almost a factor of k. */
  return;
_10:
    if (fabs(d) < fabs(c)) {
        *type = 1;
/*  Type=1 indicates that all formulas are divided by c. */   
        e = a/c;
        f = d/c;
        g = u*e;
        h = v*b;
        a3 = a*e + (h/c+g)*b;
        a1 = b - a*(d/c);
        a7 = a + g*d + h*f;
        return;
    }
    *type = 2;
/*  Type=2 indicates that all formulas are divided by d. */
  e = a/d;
  f = c/d;
  g = u*b;
  h = v*b;
  a3 = (a+g)*e + h*(b/d);
  a1 = b*f-a;
  a7 = (f+u)*a + h;
}
/*  Computes the next k polynomials using scalars 
 *  computed in calcsc.
 */
void nextk(int *type)
{
    double temp;
  int i;

    if (*type == 3) {
/*  Use unscaled form of the recurrence if type is 3. */
        k[0] = 0.0;
        k[1] = 0.0;
        for (i=2;i<n;i++) {
            k[i] = qk[i-2];
        }
        return;
    }
  temp = a;
  if (*type == 1) temp = b;
    if (fabs(a1) <= fabs(temp)*eta*10.0) {
/*  If a1 is nearly zero then use a special form of the
 *  recurrence.
 */
        k[0] = 0.0;
        k[1] = -a7*qp[0];
        for(i=2;i<n;i++) {
            k[i] = a3*qk[i-2] - a7*qp[i-1];
        }
        return;
    }
/*  Use scaled form of the recurrence. */
  a7 /= a1;
  a3 /= a1;
  k[0] = qp[0];
  k[1] = qp[1] - a7*qp[0];
  for (i=2;i<n;i++) {
    k[i] = a3*qk[i-2] - a7*qp[i-1] + qp[i];
  }
}
/*  Compute new estimates of the quadratic coefficients
 *  using the scalars computed in calcsc.
 */
void newest(int type,double *uu,double *vv)
{
    double a4,a5,b1,b2,c1,c2,c3,c4,temp;

/* Use formulas appropriate to setting of type. */
    if (type == 3) {
/*  If type=3 the quadratic is zeroed. */
        *uu = 0.0;
        *vv = 0.0;
        return;
    }
    if (type == 2) {
        a4 = (a+g)*f + h;
        a5 = (f+u)*c + v*d;
    }
    else {
        a4 = a + u*b +h*f;
        a5 = c + (u+v*f)*d;
    }
/*  Evaluate new quadratic coefficients. */
    b1 = -k[n-1]/p[n];
    b2 = -(k[n-2]+b1*p[n-1])/p[n];
  c1 = v*b2*a1;
  c2 = b1*a7;
  c3 = b1*b1*a3;
  c4 = c1 - c2 - c3;
  temp = a5 + b1*a4 - c4;
    if (temp == 0.0) {
        *uu = 0.0;
        *vv = 0.0;
        return;
    }
  *uu = u - (u*(c3+c2)+v*(b1*a1+b2*a7))/temp;
  *vv = v*(1.0+c4/temp);
  return;
}

/*  Divides p by the quadratic 1,u,v placing the quotient
 *  in q and the remainder in a,b.
 */
void quadsd(int nn,double *u,double *v,double *p,double *q,
    double *a,double *b)
{
    double c;
  int i;
  *b = p[0];
  q[0] = *b;
    *a = p[1] - (*b)*(*u);
  q[1] = *a;
    for (i=2;i<=nn;i++) {
        c = p[i] - (*a)*(*u) - (*b)*(*v);
    q[i] = c;
    *b = *a;
    *a = c;
  }
}
/*  Calculate the zeros of the quadratic a*z^2 + b1*z + c.
 *  The quadratic formula, modified to avoid overflow, is used 
 *  to find the larger zero if the zeros are real and both
 *  are complex. The smaller real zero is found directly from 
 *  the product of the zeros c/a.
 */
void quad(double a,double b1,double c,double *sr,double *si,
        double *lr,double *li)
{
        double b,d,e;

        if (a == 0.0) {         /* less than two roots */
            if (b1 != 0.0)     
        *sr = -c/b1;
      else 
                *sr = 0.0;
            *lr = 0.0;
            *si = 0.0;
            *li = 0.0;
      return;
    }
        if (c == 0.0) {         /* one real root, one zero root */
            *sr = 0.0;
      *lr = -b1/a;
            *si = 0.0;
            *li = 0.0;
      return;
    }
/* Compute discriminant avoiding overflow. */
    b = b1/2.0;
        if (fabs(b) < fabs(c)) { 
            if (c < 0.0) 
        e = -a;
      else
        e = a;
            e = b*(b/fabs(c)) - e;
            d = sqrt(fabs(e))*sqrt(fabs(c));
    }
    else {
      e = 1.0 - (a/b)*(c/b);
            d = sqrt(fabs(e))*fabs(b);
    }
        if (e < 0.0) {      /* complex conjugate zeros */
      *sr = -b/a;
      *lr = *sr;
            *si = fabs(d/a);
      *li = -(*si);
    }
    else {
            if (b >= 0.0)   /* real zeros. */
        d = -d;
        *lr = (-b+d)/a;
                *sr = 0.0;
                if (*lr != 0.0) 
          *sr = (c/ *lr)/a;
                *si = 0.0;
                *li = 0.0;
    }
}

Generated by  Doxygen 1.6.0   Back to index