Polynomial.cpp

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/*
===========================================================================

Doom 3 GPL Source Code
Copyright (C) 1999-2011 id Software LLC, a ZeniMax Media company. 

This file is part of the Doom 3 GPL Source Code (?Doom 3 Source Code?).  

Doom 3 Source Code is free software: you can redistribute it and/or modify
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===========================================================================
*/

#include "../precompiled.h"
#pragma hdrstop

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( void ) {
        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 );
        }
}