text: /*
  ===========================================================================
  
  Doom 3 BFG Edition GPL Source Code
  Copyright (C) 1993-2012 id Software LLC, a ZeniMax Media company. 
  
  This file is part of the Doom 3 BFG Edition GPL Source Code ("Doom 3 BFG Edition Source Code").  
  
  Doom 3 BFG Edition Source Code is free software: you can redistribute it and/or modify
  it under the terms of the GNU General Public License as published by
  the Free Software Foundation, either version 3 of the License, or
  (at your option) any later version.
  
  Doom 3 BFG Edition Source Code is distributed in the hope that it will be useful,
  but WITHOUT ANY WARRANTY; without even the implied warranty of
  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  GNU General Public License for more details.
  
  You should have received a copy of the GNU General Public License
  along with Doom 3 BFG Edition Source Code.  If not, see <http://www.gnu.org/licenses/>.
  
  In addition, the Doom 3 BFG Edition Source Code is also subject to certain additional terms. You should have received a copy of these additional terms immediately following the terms and conditions of the GNU General Public License which accompanied the Doom 3 BFG Edition Source Code.  If not, please request a copy in writing from id Software at the address below.
  
  If you have questions concerning this license or the applicable additional terms, you may contact in writing id Software LLC, c/o ZeniMax Media Inc., Suite 120, Rockville, Maryland 20850 USA.
  
  ===========================================================================
  */
  
  #pragma hdrstop
  #include "../precompiled.h"
  
  const float EPSILON		= 1e-6f;
  
  /*
  =============
  idPolynomial::Laguer
  =============
  */
  int idPolynomial::Laguer( const idComplex *coef, const int degree, idComplex &x ) const {
  	const int MT = 10, MAX_ITERATIONS = MT * 8;
  	static const float frac[] = { 0.0f, 0.5f, 0.25f, 0.75f, 0.13f, 0.38f, 0.62f, 0.88f, 1.0f };
  	int i, j;
  	float abx, abp, abm, err;
  	idComplex dx, cx, b, d, f, g, s, gps, gms, g2;
  
  	for ( i = 1; i <= MAX_ITERATIONS; i++ ) {
  		b = coef[degree];
  		err = b.Abs();
  		d.Zero();
  		f.Zero();
  		abx = x.Abs();
  		for ( j = degree - 1; j >= 0; j-- ) {
  			f = x * f + d;
  			d = x * d + b;
  			b = x * b + coef[j];
  			err = b.Abs() + abx * err;
  		}
  		if ( b.Abs() < err * EPSILON ) {
  			return i;
  		}
  		g = d / b;
  		g2 = g * g;
  		s = ( ( degree - 1 ) * ( degree * ( g2 - 2.0f * f / b ) - g2 ) ).Sqrt();
  		gps = g + s;
  		gms = g - s;
  		abp = gps.Abs();
  		abm = gms.Abs();
  		if ( abp < abm ) {
  			gps = gms;
  		}
  		if ( Max( abp, abm ) > 0.0f ) {
  			dx = degree / gps;
  		} else {
  			dx = idMath::Exp( idMath::Log( 1.0f + abx ) ) * idComplex( idMath::Cos( i ), idMath::Sin( i ) );
  		}
  		cx = x - dx;
  		if ( x == cx ) {
  			return i;
  		}
  		if ( i % MT == 0 ) {
  			x = cx;
  		} else {
  			x -= frac[i/MT] * dx;
  		}
  	}
  	return i;
  }
  
  /*
  =============
  idPolynomial::GetRoots
  =============
  */
  int idPolynomial::GetRoots( idComplex *roots ) const {
  	int i, j;
  	idComplex x, b, c, *coef;
  
  	coef = (idComplex *) _alloca16( ( degree + 1 ) * sizeof( idComplex ) );
  	for ( i = 0; i <= degree; i++ ) {
  		coef[i].Set( coefficient[i], 0.0f );
  	}
  
  	for ( i = degree - 1; i >= 0; i-- ) {
  		x.Zero();
  		Laguer( coef, i + 1, x );
  		if ( idMath::Fabs( x.i ) < 2.0f * EPSILON * idMath::Fabs( x.r ) ) {
  			x.i = 0.0f;
  		}
  		roots[i] = x;
  		b = coef[i+1];
  		for ( j = i; j >= 0; j-- ) {
  			c = coef[j];
  			coef[j] = b;
  			b = x * b + c;
  		}
  	}
  
  	for ( i = 0; i <= degree; i++ ) {
  		coef[i].Set( coefficient[i], 0.0f );
  	}
  	for ( i = 0; i < degree; i++ ) {
  		Laguer( coef, degree, roots[i] );
  	}
  
  	for ( i = 1; i < degree; i++ ) {
  		x = roots[i];
  		for ( j = i - 1; j >= 0; j-- ) {
  			if ( roots[j].r <= x.r ) {
  				break;
  			}
  			roots[j+1] = roots[j];
  		}
  		roots[j+1] = x;
  	}
  
  	return degree;
  }
  
  /*
  =============
  idPolynomial::GetRoots
  =============
  */
  int idPolynomial::GetRoots( float *roots ) const {
  	int i, num;
  	idComplex *complexRoots;
  
  	switch( degree ) {
  		case 0: return 0;
  		case 1: return GetRoots1( coefficient[1], coefficient[0], roots );
  		case 2: return GetRoots2( coefficient[2], coefficient[1], coefficient[0], roots );
  		case 3: return GetRoots3( coefficient[3], coefficient[2], coefficient[1], coefficient[0], roots );
  		case 4: return GetRoots4( coefficient[4], coefficient[3], coefficient[2], coefficient[1], coefficient[0], roots );
  	}
  
  	// The Abel-Ruffini theorem states that there is no general solution
  	// in radicals to polynomial equations of degree five or higher.
  	// A polynomial equation can be solved by radicals if and only if
  	// its Galois group is a solvable group.
  
  	complexRoots = (idComplex *) _alloca16( degree * sizeof( idComplex ) );
  
  	GetRoots( complexRoots );
  
  	for ( num = i = 0; i < degree; i++ ) {
  		if ( complexRoots[i].i == 0.0f ) {
  			roots[i] = complexRoots[i].r;
  			num++;
  		}
  	}
  	return num;
  }
  
  /*
  =============
  idPolynomial::ToString
  =============
  */
  const char *idPolynomial::ToString( int precision ) const {
  	return idStr::FloatArrayToString( ToFloatPtr(), GetDimension(), precision );
  }
  
  /*
  =============
  idPolynomial::Test
  =============
  */
  void idPolynomial::Test() {
  	int i, num;
  	float roots[4], value;
  	idComplex complexRoots[4], complexValue;
  	idPolynomial p;
  
  	p = idPolynomial( -5.0f, 4.0f );
  	num = p.GetRoots( roots );
  	for ( i = 0; i < num; i++ ) {
  		value = p.GetValue( roots[i] );
  		assert( idMath::Fabs( value ) < 1e-4f );
  	}
  
  	p = idPolynomial( -5.0f, 4.0f, 3.0f );
  	num = p.GetRoots( roots );
  	for ( i = 0; i < num; i++ ) {
  		value = p.GetValue( roots[i] );
  		assert( idMath::Fabs( value ) < 1e-4f );
  	}
  
  	p = idPolynomial( 1.0f, 4.0f, 3.0f, -2.0f );
  	num = p.GetRoots( roots );
  	for ( i = 0; i < num; i++ ) {
  		value = p.GetValue( roots[i] );
  		assert( idMath::Fabs( value ) < 1e-4f );
  	}
  
  	p = idPolynomial( 5.0f, 4.0f, 3.0f, -2.0f );
  	num = p.GetRoots( roots );
  	for ( i = 0; i < num; i++ ) {
  		value = p.GetValue( roots[i] );
  		assert( idMath::Fabs( value ) < 1e-4f );
  	}
  
  	p = idPolynomial( -5.0f, 4.0f, 3.0f, 2.0f, 1.0f );
  	num = p.GetRoots( roots );
  	for ( i = 0; i < num; i++ ) {
  		value = p.GetValue( roots[i] );
  		assert( idMath::Fabs( value ) < 1e-4f );
  	}
  
  	p = idPolynomial( 1.0f, 4.0f, 3.0f, -2.0f );
  	num = p.GetRoots( complexRoots );
  	for ( i = 0; i < num; i++ ) {
  		complexValue = p.GetValue( complexRoots[i] );
  		assert( idMath::Fabs( complexValue.r ) < 1e-4f && idMath::Fabs( complexValue.i ) < 1e-4f );
  	}
  
  	p = idPolynomial( 5.0f, 4.0f, 3.0f, -2.0f );
  	num = p.GetRoots( complexRoots );
  	for ( i = 0; i < num; i++ ) {
  		complexValue = p.GetValue( complexRoots[i] );
  		assert( idMath::Fabs( complexValue.r ) < 1e-4f && idMath::Fabs( complexValue.i ) < 1e-4f );
  	}
  }
  