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polrt.c

/*                                        polrt.c
 *
 *    Find roots of a polynomial
 *
 *
 *
 * SYNOPSIS:
 *
 * typedef struct
 *    {
 *    double r;
 *    double i;
 *    }cmplx;
 *
 * double xcof[], cof[];
 * int m;
 * cmplx root[];
 *
 * polrt( xcof, cof, m, root )
 *
 *
 *
 * DESCRIPTION:
 *
 * Iterative determination of the roots of a polynomial of
 * degree m whose coefficient vector is xcof[].  The
 * coefficients are arranged in ascending order; i.e., the
 * coefficient of x**m is xcof[m].
 *
 * The array cof[] is working storage the same size as xcof[].
 * root[] is the output array containing the complex roots.
 *
 *
 * ACCURACY:
 *
 * Termination depends on evaluation of the polynomial at
 * the trial values of the roots.  The values of multiple roots
 * or of roots that are nearly equal may have poor relative
 * accuracy after the first root in the neighborhood has been
 * found.
 *
 */

/*                                        polrt */
/* Complex roots of real polynomial */
/* number of coefficients is m + 1 ( i.e., m is degree of polynomial) */

#include "mconf.h"

int polrt( double *xcof, double *cof, int m, cmplx *root )
{
register double *p, *q;
int i, j, nsav, n, n1, n2, nroot, iter, retry;
int final;
double mag, cofj;
cmplx x0, x, xsav, dx, t, t1, u, ud;

final = 0;
n = m;
if( n <= 0 )
      return(1);
if( n > 36 )
      return(2);
if( xcof[m] == 0.0 )
      return(4);

n1 = n;
n2 = n;
nroot = 0;
nsav = n;
q = &xcof[0];
p = &cof[n];
for( j=0; j<=nsav; j++ )
      *p-- = *q++;      /*    cof[ n-j ] = xcof[j];*/
xsav.r = 0.0;
xsav.i = 0.0;

nxtrut:
x0.r = 0.00500101;
x0.i = 0.01000101;
retry = 0;

tryagn:
retry += 1;
x.r = x0.r;

x0.r = -10.0 * x0.i;
x0.i = -10.0 * x.r;

x.r = x0.r;
x.i = x0.i;

finitr:
iter = 0;

while( iter < 500 )
{
u.r = cof[n];
if( u.r == 0.0 )
      {           /* this root is zero */
      x.r = 0;
      n1 -= 1;
      n2 -= 1;
      goto zerrut;
      }
u.i = 0;
ud.r = 0;
ud.i = 0;
t.r = 1.0;
t.i = 0;
p = &cof[n-1];
for( i=0; i<n; i++ )
      {
      t1.r = x.r * t.r  -  x.i * t.i;
      t1.i = x.r * t.i  +  x.i * t.r;
      cofj = *p--;            /* evaluate polynomial */
      u.r += cofj * t1.r;
      u.i += cofj * t1.i;
      cofj = cofj * (i+1);    /* derivative */
      ud.r += cofj * t.r;
      ud.i -= cofj * t.i;
      t.r = t1.r;
      t.i = t1.i;
      }

mag = ud.r * ud.r  +  ud.i * ud.i;
if( mag == 0.0 )
      {
      if( !final )
            goto tryagn;
      x.r = xsav.r;
      x.i = xsav.i;
      goto findon;
      }
dx.r = (u.i * ud.i  -  u.r * ud.r)/mag;
x.r += dx.r;
dx.i = -(u.r * ud.i  +  u.i * ud.r)/mag;
x.i += dx.i;
if( (fabs(dx.i) + fabs(dx.r)) < 1.0e-6 )
      goto lupdon;
iter += 1;
}     /* while iter < 500 */

if( final )
      goto lupdon;
if( retry < 5 )
      goto tryagn;
return(3);

lupdon:
/* Swap original and reduced polynomials */
q = &xcof[nsav];
p = &cof[0];
for( j=0; j<=n2; j++ )
      {
      cofj = *q;
      *q-- = *p;
      *p++ = cofj;
      }
i = n;
n = n1;
n1 = i;

if( !final )
      {
      final = 1;
      if( fabs(x.i/x.r) < 1.0e-4 )
            x.i = 0.0;
      xsav.r = x.r;
      xsav.i = x.i;
      goto finitr;      /* do final iteration on original polynomial */
      }

findon:
final = 0;
if( fabs(x.i/x.r) >= 1.0e-5 )
      {
      cofj = x.r + x.r;
      mag = x.r * x.r  +  x.i * x.i;
      n -= 2;
      }
else
      {           /* root is real */
zerrut:
      x.i = 0;
      cofj = x.r;
      mag = 0;
      n -= 1;
      }
/* divide working polynomial cof(z) by z - x */
p = &cof[1];
*p += cofj * *(p-1);
for( j=1; j<n; j++ )
      {
      *(p+1) += cofj * *p  -  mag * *(p-1);
      p++;
      }

setrut:
root[nroot].r = x.r;
root[nroot].i = x.i;
nroot += 1;
if( mag != 0.0 )
      {
      x.i = -x.i;
      mag = 0;
      goto setrut;      /* fill in the complex conjugate root */
      }
if( n > 0 )
      goto nxtrut;
return(0);
}

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