text: /*******************************************************************************
  
  "A Collection of Useful C++ Classes for Digital Signal Processing"
   By Vincent Falco
  
  Official project location:
  http://code.google.com/p/dspfilterscpp/
  
  See Documentation.cpp for contact information, notes, and bibliography.
  
  --------------------------------------------------------------------------------
  
  License: MIT License (http://www.opensource.org/licenses/mit-license.php)
  Copyright (c) 2009 by Vincent Falco
  
  Permission is hereby granted, free of charge, to any person obtaining a copy
  of this software and associated documentation files (the "Software"), to deal
  in the Software without restriction, including without limitation the rights
  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
  copies of the Software, and to permit persons to whom the Software is
  furnished to do so, subject to the following conditions:
  
  The above copyright notice and this permission notice shall be included in
  all copies or substantial portions of the Software.
  
  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
  IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
  FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
  AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
  LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
  OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
  THE SOFTWARE.
  
  *******************************************************************************/
  
  #include "DspFilters/Common.h"
  #include "DspFilters/RootFinder.h"
  #include <stdexcept>
  
  namespace Dsp {
  
  void RootFinderBase::solve (int degree,
                              bool polish,
                              bool doSort)
  {
    assert (degree <= m_maxdegree);
  
    const double EPS = 1.0e-30;
  
    int its;
    complex_t x, b, c;
  
    int m = degree;
  
    // copy coefficients
    for (int j = 0; j <= m; ++j)
      m_ad[j] = m_a[j];
  
    // for each root
    for (int j = m - 1; j >= 0; --j)
    {
      // initial guess at 0
      x = 0.0;
      laguerre (j + 1, m_ad, x, its);
      
      if (fabs (std::imag(x)) <= 2.0 * EPS * fabs (std::real(x)))
        x = complex_t (std::real(x), 0.0);
  
      m_root[j] = x;
  
      // deflate
      b = m_ad[j+1];
      for (int jj = j; jj >= 0; --jj)
      {
        c = m_ad[jj];
        m_ad[jj] = b;
        b = x * b + c;
      }
    }
  
    if (polish)
      for (int j = 0; j < m; ++j)
        laguerre (degree, m_a, m_root[j], its);
  
    if (doSort)
      sort (degree);
  }
  
  void RootFinderBase::sort (int degree)
  {
    for (int j = 1; j < degree; ++j)
    {
      complex_t x = m_root[j];
  
      int i;
      for (i = j - 1; i >= 0; --i )
      {
        if (m_root[i].imag() >= x.imag())
          break;
  
        m_root[i+1] = m_root[i];
      }
  
      m_root[i+1] = x;
    }
  }
  
  //------------------------------------------------------------------------------
  
  void RootFinderBase::laguerre (int degree,
                             complex_t a[],
                             complex_t& x,
                             int& its)
  {
    const int MR = 8, MT = 10, MAXIT = MT * MR;
    const double EPS = std::numeric_limits<double>::epsilon();
  
    static const double frac[MR + 1] =
      {0.0, 0.5, 0.25, 0.75, 0.13, 0.38, 0.62, 0.88, 1.0};
  
    complex_t dx, x1, b, d, f, g, h, sq, gp, gm, g2;
  
    int m = degree;
    for (int iter = 1; iter <= MAXIT; ++iter)
    {
      its = iter;
      b = a[m];
      double err = std::abs(b);
      d = f = 0.0;
      double abx = std::abs(x);
      for (int j = m - 1; j >= 0; --j)
      {
        f = x * f + d;
        d = x * d + b;
        b = x * b + a[j];
        err = std::abs(b) + abx * err;
      }
      err *= EPS;
      if (std::abs(b) <= err)
        return;
      g  = d / b;
      g2 = g * g;
      h  = g2 - 2.0 * f / b;
      
      sq = sqrt (double(m - 1) * (double(m) * h - g2));
      gp = g + sq;
      gm = g - sq;
      
      double abp = std::abs (gp);
      double abm = std::abs (gm);
      if (abp < abm)
        gp = gm;
      dx = std::max(abp, abm) > 0.0 ? double(m) / gp : std::polar (1 + abx, double(iter));
      x1 = x - dx;
      if (x == x1)
        return;
      if (iter % MT != 0)
        x = x1;
      else
        x -= frac[iter / MT] * dx;
    }
  
    throw std::logic_error ("laguerre failed");
  }
  
  //------------------------------------------------------------------------------
  
  complex_t RootFinderBase::eval (int degree,
                              const complex_t& x )
  {
    complex_t y;
  
    if (x != 0.)
    {
      for (int i = 0; i <= degree; ++i)
        y += m_a[i] * pow (x, double(i));
    }
    else
    {
      y = m_a[0];
    }
  
    return y;
  }
  
  
  }
  