rootQ.cpp

Go to the documentation of this file.

/* $Id: rootQ.cpp 513 2011-03-03 01:43:48Z pbelotti $
 *
 * Name:    rootQ.cpp
 * Author:  Pietro Belotti
 * Purpose: find roots of polynomial Q^k(x) (see Liberti and Pantelides, 2003)
 *
 * (C) Carnegie-Mellon University, 2006-11.
 * This file is licensed under the Eclipse Public License (EPL)
 */

#include <math.h>
#include <stdio.h>

#include "CouenneTypes.hpp"
#include "CouennePrecisions.hpp"

namespace Couenne {

/* compute Q(x) */

CouNumber Q (register int k, CouNumber x) {

  register CouNumber xp = x, Q = 1.;

  k *= 2;

  for (register int i=2; i<=k; i++) {

    Q += (CouNumber) i * xp;
    xp *= x;
  }

  return Q;
}


/*
 * Find roots of polynomial $Q^k(x) = \sum_{i=1}^{2k} i x^{i-1}$. Used
 *  in convexification of powers with odd exponent
 */

CouNumber rootQ (int k) {

  if (k==1) return - 0.5; // for x^3, solution is -1/2
  else {

    register CouNumber 
      l  = - 1.0 + 0.5 / k, 
      u  = - 0.5,
      /* Ql = Q (k, l), Qu = Q (k, u), */
      Qm,
      midpoint;
    do {

      midpoint = 0.5 * (l+u); /* (- Ql * u + Qu * l) / (Qu - Ql); */
      Qm = Q (k, midpoint);

      /*      printf ("[%.4f, %.4f] --> %.4f: %.24f\n", l, u, midpoint, Qm); */

      if (Qm<0) {l = midpoint; /* Ql = Qm; */ Qm = - Qm;} // invert sign to avoid fabs() in termination check
      else      {u = midpoint; /* Qu = Qm; */}

    } while (Qm > 1e-15);

    return midpoint;
  }
}

}

#ifdef DEBUG_ROOTQ
int main () {

  register int k;
  CouNumber x, q;

  for (k=6; --k;) {

    printf ("root, %3d -> %.15f\n", 2*k+1, rootQ (k));
    /*
    printf ("k=%3d: ", 2*k+1);
    for (x = -1.0; x < 0.4; x += 0.1) {
      //      Q (k, x, &q, NULL, NULL); 
      printf ("[%.2f, %.3f] ", x, rootQ);
    }
    printf ("\n");
    */
  }
}

#endif