//package com.polytechnik.utils;
/*
 * (C) Vladislav Malyshkin 2010
 * This file is under GPL version 3.
 *
 */

/** Polynomial root.
 *  @version $Id: PolynomialRoot.java,v 1.105 2012/08/18 00:00:05 mal Exp $
 *  @author Vladislav Malyshkin mal@gromco.com
 */

public class PolynomialRoot  {


public static int findPolynomialRoots(final int n,
				      final double [] p,
				      final double [] re_root,
				      final double [] im_root)
{
    if(n==4)
    {
	return root4(p,re_root,im_root);
    }
    else if(n==3)
    {
	return root3(p,re_root,im_root);
    }
    else if(n==2)
    {
	return root2(p,re_root,im_root);
    }
    else if(n==1)
    {
	return root1(p,re_root,im_root);
    }
    else
    {
	throw new RuntimeException("n="+n+" is not supported yet");
    }
}



static final double SQRT3=Math.sqrt(3.0),SQRT2=Math.sqrt(2.0);


private static final boolean PRINT_DEBUG=false;

public static int root4(final double [] p,final double [] re_root,final double [] im_root)
{
  if(PRINT_DEBUG) System.err.println("=====================root4:p="+java.util.Arrays.toString(p));
  final double vs=p[4];
  if(PRINT_DEBUG) System.err.println("p[4]="+p[4]);
  if(!(Math.abs(vs)>EPS))
  {
      re_root[0]=re_root[1]=re_root[2]=re_root[3]=
	  im_root[0]=im_root[1]=im_root[2]=im_root[3]=Double.NaN;
      return -1;
  }

/* zsolve_quartic.c - finds the complex roots of
 *  x^4 + a x^3 + b x^2 + c x + d = 0
 */
  final double a=p[3]/vs,b=p[2]/vs,c=p[1]/vs,d=p[0]/vs;
  if(PRINT_DEBUG) System.err.println("input a="+a+" b="+b+" c="+c+" d="+d);


  final double r4 = 1.0 / 4.0;
  final double q2 = 1.0 / 2.0, q4 = 1.0 / 4.0, q8 = 1.0 / 8.0;
  final double q1 = 3.0 / 8.0, q3 = 3.0 / 16.0;
  final int mt;

  /* Deal easily with the cases where the quartic is degenerate. The
   * ordering of solutions is done explicitly. */
  if (0 == b && 0 == c)
  {
      if (0 == d)
      {
	  re_root[0]=-a;
	  im_root[0]=im_root[1]=im_root[2]=im_root[3]=0;
	  re_root[1]=re_root[2]=re_root[3]=0;
	  return 4;
      }
      else if (0 == a)
      {
	  if (d > 0)
	  {
	      final double sq4 = Math.sqrt(Math.sqrt(d));
	      re_root[0]=sq4*SQRT2/2;
	      im_root[0]=re_root[0];
	      re_root[1]=-re_root[0];
	      im_root[1]=re_root[0];
	      re_root[2]=-re_root[0];
	      im_root[2]=-re_root[0];
	      re_root[3]=re_root[0];
	      im_root[3]=-re_root[0];
	      if(PRINT_DEBUG) System.err.println("Path a=0 d>0");
	  }
	  else
	  {
	      final double sq4 = Math.sqrt(Math.sqrt(-d));
	      re_root[0]=sq4;
	      im_root[0]=0;
	      re_root[1]=0;
	      im_root[1]=sq4;
	      re_root[2]=0;
	      im_root[2]=-sq4;
	      re_root[3]=-sq4;
	      im_root[3]=0;
	      if(PRINT_DEBUG) System.err.println("Path a=0 d<0");
	  }
	  return 4;
      }
  }

  if (0.0 == c && 0.0 == d)
  {
      root2(new double []{p[2],p[3],p[4]},re_root,im_root);
      re_root[2]=im_root[2]=re_root[3]=im_root[3]=0;
      return 4;
  }

  if(PRINT_DEBUG) System.err.println("G Path c="+c+" d="+d);
  final double [] u=new double[3];

  if(PRINT_DEBUG) System.err.println("Generic Path");
  /* For non-degenerate solutions, proceed by constructing and
   * solving the resolvent cubic */
  final double aa = a * a;
  final double pp = b - q1 * aa;
  final double qq = c - q2 * a * (b - q4 * aa);
  final double rr = d - q4 * a * (c - q4 * a * (b - q3 * aa));
  final double rc = q2 * pp , rc3 = rc / 3;
  final double sc = q4 * (q4 * pp * pp - rr);
  final double tc = -(q8 * qq * q8 * qq);
  if(PRINT_DEBUG) System.err.println("aa="+aa+" pp="+pp+" qq="+qq+" rr="+rr+" rc="+rc+" sc="+sc+" tc="+tc);
  final boolean flag_realroots;

  /* This code solves the resolvent cubic in a convenient fashion
   * for this implementation of the quartic. If there are three real
   * roots, then they are placed directly into u[].  If two are
   * complex, then the real root is put into u[0] and the real
   * and imaginary part of the complex roots are placed into
   * u[1] and u[2], respectively. */
  {
      final double qcub = (rc * rc - 3 * sc);
      final double rcub = (rc*(2 * rc * rc - 9 * sc) + 27 * tc);

      final double Q = qcub / 9;
      final double R = rcub / 54;

      final double Q3 = Q * Q * Q;
      final double R2 = R * R;

      final double CR2 = 729 * rcub * rcub;
      final double CQ3 = 2916 * qcub * qcub * qcub;

      if(PRINT_DEBUG) System.err.println("CR2="+CR2+" CQ3="+CQ3+" R="+R+" Q="+Q);

      if (0 == R && 0 == Q)
      {
	  flag_realroots=true;
	  u[0] = -rc3;
	  u[1] = -rc3;
	  u[2] = -rc3;
      }
      else if (CR2 == CQ3)
      {
	  flag_realroots=true;
	  final double sqrtQ = Math.sqrt (Q);
	  if (R > 0)
	  {
	      u[0] = -2 * sqrtQ - rc3;
	      u[1] = sqrtQ - rc3;
	      u[2] = sqrtQ - rc3;
	  }
	  else
	  {
	      u[0] = -sqrtQ - rc3;
	      u[1] = -sqrtQ - rc3;
	      u[2] = 2 * sqrtQ - rc3;
	  }
      }
      else if (R2 < Q3)
      {
	  flag_realroots=true;
	  final double ratio = (R >= 0?1:-1) * Math.sqrt (R2 / Q3);
	  final double theta = Math.acos (ratio);
	  final double norm = -2 * Math.sqrt (Q);

	  u[0] = norm * Math.cos (theta / 3) - rc3;
	  u[1] = norm * Math.cos ((theta + 2.0 * Math.PI) / 3) - rc3;
	  u[2] = norm * Math.cos ((theta - 2.0 * Math.PI) / 3) - rc3;
      }
      else
      {
	  flag_realroots=false;
	  final double A = -(R >= 0?1:-1)*Math.pow(Math.abs(R)+Math.sqrt(R2-Q3),1.0/3.0);
	  final double B = Q / A;

	  u[0] = A + B - rc3;
	  u[1] = -0.5 * (A + B) - rc3;
	  u[2] = -(SQRT3*0.5) * Math.abs (A - B);
      }
      if(PRINT_DEBUG) System.err.println("u[0]="+u[0]+" u[1]="+u[1]+" u[2]="+u[2]+" qq="+qq+" disc="+((CR2 - CQ3) / 2125764.0));
  }
  /* End of solution to resolvent cubic */

  /* Combine the square roots of the roots of the cubic
   * resolvent appropriately. Also, calculate 'mt' which
   * designates the nature of the roots:
   * mt=1 : 4 real roots
   * mt=2 : 0 real roots
   * mt=3 : 2 real roots
   */


  final double w1_re,w1_im,w2_re,w2_im,w3_re,w3_im,mod_w1w2,mod_w1w2_squared;
  if (flag_realroots)
  {
      mod_w1w2=-1;
      mt = 2;
      int jmin=0;
      double vmin=Math.abs(u[jmin]);
      for(int j=1;j<3;j++)
      {
	  final double vx=Math.abs(u[j]);
	  if(vx<vmin)
	  {
	      vmin=vx;
	      jmin=j;
	  }
      }
      final double u1=u[(jmin+1)%3],u2=u[(jmin+2)%3];
      mod_w1w2_squared=Math.abs(u1*u2);
      if(u1>=0)
      {
	  w1_re=Math.sqrt(u1);
	  w1_im=0;
      }
      else
      {
	  w1_re=0;
	  w1_im=Math.sqrt(-u1);
      }
      if(u2>=0)
      {
	  w2_re=Math.sqrt(u2);
	  w2_im=0;
      }
      else
      {
	  w2_re=0;
	  w2_im=Math.sqrt(-u2);
      }
      if(PRINT_DEBUG) System.err.println("u1="+u1+" u2="+u2+" jmin="+jmin);
  }
  else
  {
      mt = 3;
      final double w_mod2_sq=u[1]*u[1]+u[2]*u[2],w_mod2=Math.sqrt(w_mod2_sq),w_mod=Math.sqrt(w_mod2);
      if(w_mod2_sq<=0)
      {
	  w1_re=w1_im=0;
      }
      else
      {
	  // calculate square root of a complex number (u[1],u[2])
	  // the result is in the (w1_re,w1_im)
	  final double absu1=Math.abs(u[1]),absu2=Math.abs(u[2]),w;
	  if(absu1>=absu2)
	  {
	      final double t=absu2/absu1;
	      w=Math.sqrt(absu1*0.5 * (1.0 + Math.sqrt(1.0 + t * t)));
	      if(PRINT_DEBUG) System.err.println(" Path1 ");
	  }
	  else
	  {
	      final double t=absu1/absu2;
	      w=Math.sqrt(absu2*0.5 * (t + Math.sqrt(1.0 + t * t)));
	      if(PRINT_DEBUG) System.err.println(" Path1a ");
	  }
	  if(u[1]>=0)
	  {
	      w1_re=w;
	      w1_im=u[2]/(2*w);
	      if(PRINT_DEBUG) System.err.println(" Path2 ");
	  }
	  else
	  {
	      final double vi = (u[2] >= 0) ? w : -w;
	      w1_re=u[2]/(2*vi);
	      w1_im=vi;
	      if(PRINT_DEBUG) System.err.println(" Path2a ");
	  }
      }
      final double absu0=Math.abs(u[0]);
      if(w_mod2>=absu0)
      {
	  mod_w1w2=w_mod2;
	  mod_w1w2_squared=w_mod2_sq;
	  w2_re=w1_re;
	  w2_im=-w1_im;
      }
      else
      {
	  mod_w1w2=-1;
	  mod_w1w2_squared=w_mod2*absu0;
	  if(u[0]>=0)
	  {
	      w2_re=Math.sqrt(absu0);
	      w2_im=0;
	  }
	  else
	  {
	      w2_re=0;
	      w2_im=Math.sqrt(absu0);
	  }
      }
      if(PRINT_DEBUG) System.err.println("u[0]="+u[0]+"u[1]="+u[1]+" u[2]="+u[2]+" absu0="+absu0+" w_mod="+w_mod+" w_mod2="+w_mod2);
  }

  /* Solve the quadratic in order to obtain the roots
   * to the quartic */
  if(mod_w1w2>0)
  {
      // a shorcut to reduce rounding error
      w3_re=qq/(-8)/mod_w1w2;
      w3_im=0;
  }
  else if(mod_w1w2_squared>0)
  {
      // regular path
      final double mqq8n=qq/(-8)/mod_w1w2_squared;
      w3_re=mqq8n*(w1_re*w2_re-w1_im*w2_im);
      w3_im=-mqq8n*(w1_re*w2_im+w2_re*w1_im);
  }
  else
  {
      // typically occur when qq==0
      w3_re=w3_im=0;
  }

  final double h = r4 * a;
  if(PRINT_DEBUG) System.err.println("w1_re="+w1_re+" w1_im="+w1_im+" w2_re="+w2_re+" w2_im="+w2_im+" w3_re="+w3_re+" w3_im="+w3_im+" h="+h);

  re_root[0]=w1_re+w2_re+w3_re-h;
  im_root[0]=w1_im+w2_im+w3_im;
  re_root[1]=-(w1_re+w2_re)+w3_re-h;
  im_root[1]=-(w1_im+w2_im)+w3_im;
  re_root[2]=w2_re-w1_re-w3_re-h;
  im_root[2]=w2_im-w1_im-w3_im;
  re_root[3]=w1_re-w2_re-w3_re-h;
  im_root[3]=w1_im-w2_im-w3_im;

  return 4;
}



    static void setRandomP(final double [] p,final int n,java.util.Random r)
    {
	if(r.nextDouble()<0.1)
	{
	    // integer coefficiens
	    for(int j=0;j<p.length;j++)
	    {
		if(j<=n)
		{
		    p[j]=(r.nextInt(2)<=0?-1:1)*r.nextInt(10);
		}
		else
		{
		    p[j]=0;
		}
	    }
	}
	else
	{
	    // real coefficiens
	    for(int j=0;j<p.length;j++)
	    {
		if(j<=n)
		{
		    p[j]=-1+2*r.nextDouble();
		}
		else
		{
		    p[j]=0;
		}
	    }
	}
	if(Math.abs(p[n])<1e-2)
	{
	    p[n]=(r.nextInt(2)<=0?-1:1)*(0.1+r.nextDouble());
	}
    }


    static void checkValues(final double [] p,
			    final int n,
			    final double rex,
			    final double imx,
			    final double eps,
			    final String txt)
    {
	double res=0,ims=0,sabs=0;
	final double xabs=Math.abs(rex)+Math.abs(imx);
	for(int k=n;k>=0;k--)
	{
	    final double res1=(res*rex-ims*imx)+p[k];
	    final double ims1=(ims*rex+res*imx);
	    res=res1;
	    ims=ims1;
	    sabs+=xabs*sabs+p[k];
	}
	sabs=Math.abs(sabs);
	if(false && sabs>1/eps?
	   (!(Math.abs(res/sabs)<=eps)||!(Math.abs(ims/sabs)<=eps))
	   :
	   (!(Math.abs(res)<=eps)||!(Math.abs(ims)<=eps)))
	{
	    throw new RuntimeException(
		getPolinomTXT(p)+"\n"+
		"\t x.r="+rex+" x.i="+imx+"\n"+
		"res/sabs="+(res/sabs)+" ims/sabs="+(ims/sabs)+
		" sabs="+sabs+
		"\nres="+res+" ims="+ims+" n="+n+" eps="+eps+" "+
		" sabs>1/eps="+(sabs>1/eps)+
		" f1="+(!(Math.abs(res/sabs)<=eps)||!(Math.abs(ims/sabs)<=eps))+
		" f2="+(!(Math.abs(res)<=eps)||!(Math.abs(ims)<=eps))+
		" "+txt);
	}
    }

    static String getPolinomTXT(final double [] p)
    {
	final StringBuilder buf=new StringBuilder();
	buf.append("order="+(p.length-1)+"\t");
	for(int k=0;k<p.length;k++)
	{
	    buf.append("p["+k+"]="+p[k]+";");
	}
	return buf.toString();
    }

    static String getRootsTXT(int nr,final double [] re,final double [] im)
    {
	final StringBuilder buf=new StringBuilder();
	for(int k=0;k<nr;k++)
	{
	    buf.append("x."+k+"("+re[k]+","+im[k]+")\n");
	}
	return buf.toString();
    }

    static void testRoots(final int n,
			  final int n_tests,
			  final java.util.Random rn,
			  final double eps)
    {
	final double [] p=new double [n+1];
	final double [] rex=new double [n],imx=new double [n];
	for(int i=0;i<n_tests;i++)
	{
	  for(int dg=n;dg-->-1;)
	  {
	    for(int dr=3;dr-->0;)
	    {
	      setRandomP(p,n,rn);
	      for(int j=0;j<=dg;j++)
	      {
		  p[j]=0;
	      }
	      if(dr==0)
	      {
		  p[0]=-1+2.0*rn.nextDouble();
	      }
	      else if(dr==1)
	      {
		  p[0]=p[1]=0;
	      }

	      findPolynomialRoots(n,p,rex,imx);

	      for(int j=0;j<n;j++)
	      {
		  //System.err.println("j="+j);
		  checkValues(p,n,rex[j],imx[j],eps," t="+i);
	      }
	    }
	  }
	}
	System.err.println("testRoots(): n_tests="+n_tests+" OK, dim="+n);
    }




    static final double EPS=0;

    public static int root1(final double [] p,final double [] re_root,final double [] im_root)
    {
	if(!(Math.abs(p[1])>EPS))
	{
	    re_root[0]=im_root[0]=Double.NaN;
	    return -1;
	}
	re_root[0]=-p[0]/p[1];
	im_root[0]=0;
	return 1;
    }

    public static int root2(final double [] p,final double [] re_root,final double [] im_root)
    {
	if(!(Math.abs(p[2])>EPS))
	{
	    re_root[0]=re_root[1]=im_root[0]=im_root[1]=Double.NaN;
	    return -1;
	}
	final double b2=0.5*(p[1]/p[2]),c=p[0]/p[2],d=b2*b2-c;
	if(d>=0)
	{
	    final double sq=Math.sqrt(d);
	    if(b2<0)
	    {
		re_root[1]=-b2+sq;
		re_root[0]=c/re_root[1];
	    }
	    else if(b2>0)
	    {
		re_root[0]=-b2-sq;
		re_root[1]=c/re_root[0];
	    }
	    else
	    {
		re_root[0]=-b2-sq;
		re_root[1]=-b2+sq;
	    }
	    im_root[0]=im_root[1]=0;
	}
	else
	{
	    final double sq=Math.sqrt(-d);
	    re_root[0]=re_root[1]=-b2;
	    im_root[0]=sq;
	    im_root[1]=-sq;
	}
	return 2;
    }

    public static int root3(final double [] p,final double [] re_root,final double [] im_root)
    {
	final double vs=p[3];
	if(!(Math.abs(vs)>EPS))
	{
	    re_root[0]=re_root[1]=re_root[2]=
		im_root[0]=im_root[1]=im_root[2]=Double.NaN;
	    return -1;
	}
	final double a=p[2]/vs,b=p[1]/vs,c=p[0]/vs;
	/* zsolve_cubic.c - finds the complex roots of x^3 + a x^2 + b x + c = 0
	 */
	final double q = (a * a - 3 * b);
	final double r = (a*(2 * a * a - 9 * b) + 27 * c);

	final double Q = q / 9;
	final double R = r / 54;

	final double Q3 = Q * Q * Q;
	final double R2 = R * R;

	final double CR2 = 729 * r * r;
	final double CQ3 = 2916 * q * q * q;
	final double a3=a/3;

	if (R == 0 && Q == 0)
	{
	    re_root[0]=re_root[1]=re_root[2]=-a3;
	    im_root[0]=im_root[1]=im_root[2]=0;
	    return 3;
	}
	else if (CR2 == CQ3)
	{
	    /* this test is actually R2 == Q3, written in a form suitable
	       for exact computation with integers */

	    /* Due to finite precision some double roots may be missed, and
	       will be considered to be a pair of complex roots z = x +/-
	       epsilon i close to the real axis. */

	    final double sqrtQ = Math.sqrt (Q);

	    if (R > 0)
	    {
		re_root[0] = -2 * sqrtQ - a3;
		re_root[1]=re_root[2]=sqrtQ - a3;
		im_root[0]=im_root[1]=im_root[2]=0;
	    }
	    else
	    {
		re_root[0]=re_root[1] = -sqrtQ - a3;
		re_root[2]=2 * sqrtQ - a3;
		im_root[0]=im_root[1]=im_root[2]=0;
	    }
	    return 3;
	}
	else if (R2 < Q3)
	{
	    final double sgnR = (R >= 0 ? 1 : -1);
	    final double ratio = sgnR * Math.sqrt (R2 / Q3);
	    final double theta = Math.acos (ratio);
	    final double norm = -2 * Math.sqrt (Q);
	    final double r0 = norm * Math.cos (theta/3) - a3;
	    final double r1 = norm * Math.cos ((theta + 2.0 * Math.PI) / 3) - a3;
	    final double r2 = norm * Math.cos ((theta - 2.0 * Math.PI) / 3) - a3;

	    re_root[0]=r0;
	    re_root[1]=r1;
	    re_root[2]=r2;
	    im_root[0]=im_root[1]=im_root[2]=0;
	    return 3;
	}
	else
	{
	    final double sgnR = (R >= 0 ? 1 : -1);
	    final double A = -sgnR * Math.pow (Math.abs (R) + Math.sqrt (R2 - Q3), 1.0 / 3.0);
	    final double B = Q / A;

	    re_root[0]=A + B - a3;
	    im_root[0]=0;
	    re_root[1]=-0.5 * (A + B) - a3;
	    im_root[1]=-(SQRT3*0.5) * Math.abs(A - B);
	    re_root[2]=re_root[1];
	    im_root[2]=-im_root[1];
	    return 3;
	}

    }


    static void root3a(final double [] p,final double [] re_root,final double [] im_root)
    {
	if(Math.abs(p[3])>EPS)
	{
	    final double v=p[3],
		a=p[2]/v,b=p[1]/v,c=p[0]/v,
		a3=a/3,a3a=a3*a,
		pd3=(b-a3a)/3,
		qd2=a3*(a3a/3-0.5*b)+0.5*c,
		Q=pd3*pd3*pd3+qd2*qd2;
	    if(Q<0)
	    {
		// three real roots
		final double SQ=Math.sqrt(-Q);
		final double th=Math.atan2(SQ,-qd2);
		im_root[0]=im_root[1]=im_root[2]=0;
		final double f=2*Math.sqrt(-pd3);
		re_root[0]=f*Math.cos(th/3)-a3;
		re_root[1]=f*Math.cos((th+2*Math.PI)/3)-a3;
		re_root[2]=f*Math.cos((th+4*Math.PI)/3)-a3;
		//System.err.println("3r");
	    }
	    else
	    {
		// one real & two complex roots
		final double SQ=Math.sqrt(Q);
		final double r1=-qd2+SQ,r2=-qd2-SQ;
		final double v1=Math.signum(r1)*Math.pow(Math.abs(r1),1.0/3),
		    v2=Math.signum(r2)*Math.pow(Math.abs(r2),1.0/3),
		    sv=v1+v2;
		// real root
		re_root[0]=sv-a3;
		im_root[0]=0;
		// complex roots
		re_root[1]=re_root[2]=-0.5*sv-a3;
		im_root[1]=(v1-v2)*(SQRT3*0.5);
		im_root[2]=-im_root[1];
		//System.err.println("1r2c");
	    }
	}
	else
	{
	    re_root[0]=re_root[1]=re_root[2]=im_root[0]=im_root[1]=im_root[2]=Double.NaN;
	}
    }


    static void printSpecialValues()
    {
	for(int st=0;st<6;st++)
	{
	    //final double [] p=new double []{8,1,3,3.6,1};
	    final double [] re_root=new double [4],im_root=new double [4];
	    final double [] p;
	    final int n;
	    if(st<=3)
	    {
		if(st<=0)
		{
		    p=new double []{2,-4,6,-4,1};
		    //p=new double []{-6,6,-6,8,-2};
		}
		else if(st==1)
		{
		    p=new double []{0,-4,8,3,-9};
		}
		else if(st==2)
		{
		    p=new double []{-1,0,2,0,-1};
		}
		else
		{
		    p=new double []{-5,2,8,-2,-3};
		}
		root4(p,re_root,im_root);
		n=4;
	    }
	    else
	    {
		p=new double []{0,2,0,1};
		if(st==4)
		{
		    p[1]=-p[1];
		}
		root3(p,re_root,im_root);
		n=3;
	    }
	    System.err.println("======== n="+n);
	    for(int i=0;i<=n;i++)
	    {
		if(i<n)
		{
		    System.err.println(String.valueOf(i)+"\t"+
				       p[i]+"\t"+
				       re_root[i]+"\t"+
				       im_root[i]);
		}
		else
		{
		    System.err.println(String.valueOf(i)+"\t"+p[i]+"\t");
		}
	    }
	}
    }



    public static void main(final String [] args)
    {
	final long t0=System.currentTimeMillis();
	final double eps=1e-6;
	//checkRoots();
	final java.util.Random r=new java.util.Random(-1381923);
	printSpecialValues();

	final int n_tests=10000000;
	//testRoots(2,n_tests,r,eps);
	//testRoots(3,n_tests,r,eps);
	testRoots(4,n_tests,r,eps);
	final long t1=System.currentTimeMillis();
	System.err.println("PolynomialRoot.main: "+n_tests+" tests OK done in "+(t1-t0)+" milliseconds. ver=$Id: PolynomialRoot.java,v 1.105 2012/08/18 00:00:05 mal Exp $");
    }



}