605 lines
15 KiB
C
605 lines
15 KiB
C
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// Copyright (C) 2007 Id Software, Inc.
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//
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#ifndef __MATH_POLYNOMIAL_H__
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#define __MATH_POLYNOMIAL_H__
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/*
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===============================================================================
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Polynomial of arbitrary degree with real coefficients.
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===============================================================================
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*/
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class idPolynomial {
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public:
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idPolynomial( void );
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explicit idPolynomial( int d );
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explicit idPolynomial( float a, float b );
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explicit idPolynomial( float a, float b, float c );
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explicit idPolynomial( float a, float b, float c, float d );
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explicit idPolynomial( float a, float b, float c, float d, float e );
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float operator[]( int index ) const;
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float & operator[]( int index );
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idPolynomial operator-() const;
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idPolynomial & operator=( const idPolynomial &p );
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idPolynomial operator+( const idPolynomial &p ) const;
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idPolynomial operator-( const idPolynomial &p ) const;
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idPolynomial operator*( const float s ) const;
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idPolynomial operator/( const float s ) const;
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idPolynomial & operator+=( const idPolynomial &p );
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idPolynomial & operator-=( const idPolynomial &p );
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idPolynomial & operator*=( const float s );
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idPolynomial & operator/=( const float s );
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bool Compare( const idPolynomial &p ) const; // exact compare, no epsilon
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bool Compare( const idPolynomial &p, const float epsilon ) const;// compare with epsilon
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bool operator==( const idPolynomial &p ) const; // exact compare, no epsilon
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bool operator!=( const idPolynomial &p ) const; // exact compare, no epsilon
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void Zero( void );
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void Zero( int d );
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int GetDimension( void ) const; // get the degree of the polynomial
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int GetDegree( void ) const; // get the degree of the polynomial
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float GetValue( const float x ) const; // evaluate the polynomial with the given real value
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idComplex GetValue( const idComplex &x ) const; // evaluate the polynomial with the given complex value
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idPolynomial GetDerivative( void ) const; // get the first derivative of the polynomial
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idPolynomial GetAntiDerivative( void ) const; // get the anti derivative of the polynomial
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int GetRoots( idComplex *roots ) const; // get all roots
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int GetRoots( float *roots ) const; // get the real roots
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static int GetRoots1( float a, float b, float *roots );
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static int GetRoots2( float a, float b, float c, float *roots );
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static int GetRoots3( float a, float b, float c, float d, float *roots );
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static int GetRoots4( float a, float b, float c, float d, float e, float *roots );
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const float * ToFloatPtr( void ) const;
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float * ToFloatPtr( void );
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const char * ToString( int precision = 2 ) const;
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static void Test( void );
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private:
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int degree;
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int allocated;
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float * coefficient;
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void Resize( int d, bool keep );
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int Laguer( const idComplex *coef, const int degree, idComplex &r ) const;
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};
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ID_INLINE idPolynomial::idPolynomial( void ) {
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degree = -1;
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allocated = 0;
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coefficient = NULL;
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}
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ID_INLINE idPolynomial::idPolynomial( int d ) {
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degree = -1;
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allocated = 0;
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coefficient = NULL;
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Resize( d, false );
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}
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ID_INLINE idPolynomial::idPolynomial( float a, float b ) {
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degree = -1;
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allocated = 0;
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coefficient = NULL;
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Resize( 1, false );
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coefficient[0] = b;
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coefficient[1] = a;
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}
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ID_INLINE idPolynomial::idPolynomial( float a, float b, float c ) {
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degree = -1;
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allocated = 0;
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coefficient = NULL;
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Resize( 2, false );
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coefficient[0] = c;
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coefficient[1] = b;
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coefficient[2] = a;
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}
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ID_INLINE idPolynomial::idPolynomial( float a, float b, float c, float d ) {
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degree = -1;
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allocated = 0;
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coefficient = NULL;
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Resize( 3, false );
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coefficient[0] = d;
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coefficient[1] = c;
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coefficient[2] = b;
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coefficient[3] = a;
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}
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ID_INLINE idPolynomial::idPolynomial( float a, float b, float c, float d, float e ) {
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degree = -1;
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allocated = 0;
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coefficient = NULL;
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Resize( 4, false );
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coefficient[0] = e;
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coefficient[1] = d;
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coefficient[2] = c;
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coefficient[3] = b;
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coefficient[4] = a;
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}
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ID_INLINE float idPolynomial::operator[]( int index ) const {
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assert( index >= 0 && index <= degree );
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return coefficient[ index ];
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}
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ID_INLINE float& idPolynomial::operator[]( int index ) {
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assert( index >= 0 && index <= degree );
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return coefficient[ index ];
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}
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ID_INLINE idPolynomial idPolynomial::operator-() const {
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int i;
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idPolynomial n;
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n = *this;
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for ( i = 0; i <= degree; i++ ) {
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n[i] = -n[i];
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}
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return n;
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}
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ID_INLINE idPolynomial &idPolynomial::operator=( const idPolynomial &p ) {
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Resize( p.degree, false );
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for ( int i = 0; i <= degree; i++ ) {
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coefficient[i] = p.coefficient[i];
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}
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return *this;
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}
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ID_INLINE idPolynomial idPolynomial::operator+( const idPolynomial &p ) const {
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int i;
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idPolynomial n;
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if ( degree > p.degree ) {
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n.Resize( degree, false );
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for ( i = 0; i <= p.degree; i++ ) {
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n.coefficient[i] = coefficient[i] + p.coefficient[i];
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}
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for ( ; i <= degree; i++ ) {
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n.coefficient[i] = coefficient[i];
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}
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n.degree = degree;
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} else if ( p.degree > degree ) {
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n.Resize( p.degree, false );
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for ( i = 0; i <= degree; i++ ) {
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n.coefficient[i] = coefficient[i] + p.coefficient[i];
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}
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for ( ; i <= p.degree; i++ ) {
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n.coefficient[i] = p.coefficient[i];
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}
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n.degree = p.degree;
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} else {
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n.Resize( degree, false );
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n.degree = 0;
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for ( i = 0; i <= degree; i++ ) {
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n.coefficient[i] = coefficient[i] + p.coefficient[i];
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if ( n.coefficient[i] != 0.0f ) {
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n.degree = i;
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}
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}
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}
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return n;
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}
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ID_INLINE idPolynomial idPolynomial::operator-( const idPolynomial &p ) const {
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int i;
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idPolynomial n;
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if ( degree > p.degree ) {
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n.Resize( degree, false );
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for ( i = 0; i <= p.degree; i++ ) {
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n.coefficient[i] = coefficient[i] - p.coefficient[i];
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}
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for ( ; i <= degree; i++ ) {
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n.coefficient[i] = coefficient[i];
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}
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n.degree = degree;
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} else if ( p.degree >= degree ) {
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n.Resize( p.degree, false );
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for ( i = 0; i <= degree; i++ ) {
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n.coefficient[i] = coefficient[i] - p.coefficient[i];
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}
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for ( ; i <= p.degree; i++ ) {
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n.coefficient[i] = - p.coefficient[i];
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}
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n.degree = p.degree;
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} else {
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n.Resize( degree, false );
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n.degree = 0;
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for ( i = 0; i <= degree; i++ ) {
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n.coefficient[i] = coefficient[i] - p.coefficient[i];
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if ( n.coefficient[i] != 0.0f ) {
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n.degree = i;
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}
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}
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}
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return n;
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}
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ID_INLINE idPolynomial idPolynomial::operator*( const float s ) const {
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idPolynomial n;
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if ( s == 0.0f ) {
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n.degree = 0;
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} else {
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n.Resize( degree, false );
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for ( int i = 0; i <= degree; i++ ) {
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n.coefficient[i] = coefficient[i] * s;
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}
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}
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return n;
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}
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ID_INLINE idPolynomial idPolynomial::operator/( const float s ) const {
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float invs;
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idPolynomial n;
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assert( s != 0.0f );
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n.Resize( degree, false );
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invs = 1.0f / s;
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for ( int i = 0; i <= degree; i++ ) {
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n.coefficient[i] = coefficient[i] * invs;
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}
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return n;
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}
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ID_INLINE idPolynomial &idPolynomial::operator+=( const idPolynomial &p ) {
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int i;
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if ( degree > p.degree ) {
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for ( i = 0; i <= p.degree; i++ ) {
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coefficient[i] += p.coefficient[i];
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}
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} else if ( p.degree > degree ) {
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Resize( p.degree, true );
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for ( i = 0; i <= degree; i++ ) {
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coefficient[i] += p.coefficient[i];
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}
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for ( ; i <= p.degree; i++ ) {
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coefficient[i] = p.coefficient[i];
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}
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} else {
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for ( i = 0; i <= degree; i++ ) {
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coefficient[i] += p.coefficient[i];
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if ( coefficient[i] != 0.0f ) {
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degree = i;
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}
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}
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}
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return *this;
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}
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ID_INLINE idPolynomial &idPolynomial::operator-=( const idPolynomial &p ) {
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int i;
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if ( degree > p.degree ) {
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for ( i = 0; i <= p.degree; i++ ) {
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coefficient[i] -= p.coefficient[i];
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}
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} else if ( p.degree > degree ) {
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Resize( p.degree, true );
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for ( i = 0; i <= degree; i++ ) {
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coefficient[i] -= p.coefficient[i];
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}
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for ( ; i <= p.degree; i++ ) {
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coefficient[i] = - p.coefficient[i];
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}
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} else {
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for ( i = 0; i <= degree; i++ ) {
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coefficient[i] -= p.coefficient[i];
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if ( coefficient[i] != 0.0f ) {
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degree = i;
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}
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}
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}
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return *this;
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}
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ID_INLINE idPolynomial &idPolynomial::operator*=( const float s ) {
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if ( s == 0.0f ) {
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degree = 0;
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} else {
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for ( int i = 0; i <= degree; i++ ) {
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coefficient[i] *= s;
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}
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}
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return *this;
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}
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ID_INLINE idPolynomial &idPolynomial::operator/=( const float s ) {
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float invs;
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assert( s != 0.0f );
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invs = 1.0f / s;
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for ( int i = 0; i <= degree; i++ ) {
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coefficient[i] = invs;
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}
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return *this;;
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}
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ID_INLINE bool idPolynomial::Compare( const idPolynomial &p ) const {
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if ( degree != p.degree ) {
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return false;
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}
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for ( int i = 0; i <= degree; i++ ) {
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if ( coefficient[i] != p.coefficient[i] ) {
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return false;
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}
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}
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return true;
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}
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ID_INLINE bool idPolynomial::Compare( const idPolynomial &p, const float epsilon ) const {
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if ( degree != p.degree ) {
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return false;
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}
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for ( int i = 0; i <= degree; i++ ) {
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if ( idMath::Fabs( coefficient[i] - p.coefficient[i] ) > epsilon ) {
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return false;
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}
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}
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return true;
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}
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ID_INLINE bool idPolynomial::operator==( const idPolynomial &p ) const {
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return Compare( p );
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}
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ID_INLINE bool idPolynomial::operator!=( const idPolynomial &p ) const {
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return !Compare( p );
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}
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ID_INLINE void idPolynomial::Zero( void ) {
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degree = 0;
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}
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ID_INLINE void idPolynomial::Zero( int d ) {
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Resize( d, false );
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for ( int i = 0; i <= degree; i++ ) {
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coefficient[i] = 0.0f;
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}
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}
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ID_INLINE int idPolynomial::GetDimension( void ) const {
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return degree;
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}
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ID_INLINE int idPolynomial::GetDegree( void ) const {
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return degree;
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}
|
||
|
|
|
||
|
|
ID_INLINE float idPolynomial::GetValue( const float x ) const {
|
||
|
|
float y, z;
|
||
|
|
y = coefficient[0];
|
||
|
|
z = x;
|
||
|
|
for ( int i = 1; i <= degree; i++ ) {
|
||
|
|
y += coefficient[i] * z;
|
||
|
|
z *= x;
|
||
|
|
}
|
||
|
|
return y;
|
||
|
|
}
|
||
|
|
|
||
|
|
ID_INLINE idComplex idPolynomial::GetValue( const idComplex &x ) const {
|
||
|
|
idComplex y, z;
|
||
|
|
y.Set( coefficient[0], 0.0f );
|
||
|
|
z = x;
|
||
|
|
for ( int i = 1; i <= degree; i++ ) {
|
||
|
|
y += coefficient[i] * z;
|
||
|
|
z *= x;
|
||
|
|
}
|
||
|
|
return y;
|
||
|
|
}
|
||
|
|
|
||
|
|
ID_INLINE idPolynomial idPolynomial::GetDerivative( void ) const {
|
||
|
|
idPolynomial n;
|
||
|
|
|
||
|
|
if ( degree == 0 ) {
|
||
|
|
return n;
|
||
|
|
}
|
||
|
|
n.Resize( degree - 1, false );
|
||
|
|
for ( int i = 1; i <= degree; i++ ) {
|
||
|
|
n.coefficient[i-1] = i * coefficient[i];
|
||
|
|
}
|
||
|
|
return n;
|
||
|
|
}
|
||
|
|
|
||
|
|
ID_INLINE idPolynomial idPolynomial::GetAntiDerivative( void ) const {
|
||
|
|
idPolynomial n;
|
||
|
|
|
||
|
|
if ( degree == 0 ) {
|
||
|
|
return n;
|
||
|
|
}
|
||
|
|
n.Resize( degree + 1, false );
|
||
|
|
n.coefficient[0] = 0.0f;
|
||
|
|
for ( int i = 0; i <= degree; i++ ) {
|
||
|
|
n.coefficient[i+1] = coefficient[i] / ( i + 1 );
|
||
|
|
}
|
||
|
|
return n;
|
||
|
|
}
|
||
|
|
|
||
|
|
ID_INLINE int idPolynomial::GetRoots1( float a, float b, float *roots ) {
|
||
|
|
assert( a != 0.0f );
|
||
|
|
roots[0] = - b / a;
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
|
||
|
|
ID_INLINE int idPolynomial::GetRoots2( float a, float b, float c, float *roots ) {
|
||
|
|
float inva, ds;
|
||
|
|
|
||
|
|
if ( a != 1.0f ) {
|
||
|
|
assert( a != 0.0f );
|
||
|
|
inva = 1.0f / a;
|
||
|
|
c *= inva;
|
||
|
|
b *= inva;
|
||
|
|
}
|
||
|
|
ds = b * b - 4.0f * c;
|
||
|
|
if ( ds < 0.0f ) {
|
||
|
|
return 0;
|
||
|
|
} else if ( ds > 0.0f ) {
|
||
|
|
ds = idMath::Sqrt( ds );
|
||
|
|
roots[0] = 0.5f * ( -b - ds );
|
||
|
|
roots[1] = 0.5f * ( -b + ds );
|
||
|
|
return 2;
|
||
|
|
} else {
|
||
|
|
roots[0] = 0.5f * -b;
|
||
|
|
return 1;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
ID_INLINE int idPolynomial::GetRoots3( float a, float b, float c, float d, float *roots ) {
|
||
|
|
float inva, f, g, halfg, ofs, ds, dist, angle, cs, ss, t;
|
||
|
|
|
||
|
|
if ( a != 1.0f ) {
|
||
|
|
assert( a != 0.0f );
|
||
|
|
inva = 1.0f / a;
|
||
|
|
d *= inva;
|
||
|
|
c *= inva;
|
||
|
|
b *= inva;
|
||
|
|
}
|
||
|
|
|
||
|
|
f = ( 1.0f / 3.0f ) * ( 3.0f * c - b * b );
|
||
|
|
g = ( 1.0f / 27.0f ) * ( 2.0f * b * b * b - 9.0f * c * b + 27.0f * d );
|
||
|
|
halfg = 0.5f * g;
|
||
|
|
ofs = ( 1.0f / 3.0f ) * b;
|
||
|
|
ds = 0.25f * g * g + ( 1.0f / 27.0f ) * f * f * f;
|
||
|
|
|
||
|
|
if ( ds < 0.0f ) {
|
||
|
|
dist = idMath::Sqrt( ( -1.0f / 3.0f ) * f );
|
||
|
|
angle = ( 1.0f / 3.0f ) * idMath::ATan( idMath::Sqrt( -ds ), -halfg );
|
||
|
|
cs = idMath::Cos( angle );
|
||
|
|
ss = idMath::Sin( angle );
|
||
|
|
roots[0] = 2.0f * dist * cs - ofs;
|
||
|
|
roots[1] = -dist * ( cs + idMath::SQRT_THREE * ss ) - ofs;
|
||
|
|
roots[2] = -dist * ( cs - idMath::SQRT_THREE * ss ) - ofs;
|
||
|
|
return 3;
|
||
|
|
} else if ( ds > 0.0f ) {
|
||
|
|
ds = idMath::Sqrt( ds );
|
||
|
|
t = -halfg + ds;
|
||
|
|
if ( t >= 0.0f ) {
|
||
|
|
roots[0] = idMath::Pow( t, ( 1.0f / 3.0f ) );
|
||
|
|
} else {
|
||
|
|
roots[0] = -idMath::Pow( -t, ( 1.0f / 3.0f ) );
|
||
|
|
}
|
||
|
|
t = -halfg - ds;
|
||
|
|
if ( t >= 0.0f ) {
|
||
|
|
roots[0] += idMath::Pow( t, ( 1.0f / 3.0f ) );
|
||
|
|
} else {
|
||
|
|
roots[0] -= idMath::Pow( -t, ( 1.0f / 3.0f ) );
|
||
|
|
}
|
||
|
|
roots[0] -= ofs;
|
||
|
|
return 1;
|
||
|
|
} else {
|
||
|
|
if ( halfg >= 0.0f ) {
|
||
|
|
t = -idMath::Pow( halfg, ( 1.0f / 3.0f ) );
|
||
|
|
} else {
|
||
|
|
t = idMath::Pow( -halfg, ( 1.0f / 3.0f ) );
|
||
|
|
}
|
||
|
|
roots[0] = 2.0f * t - ofs;
|
||
|
|
roots[1] = -t - ofs;
|
||
|
|
roots[2] = roots[1];
|
||
|
|
return 3;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
ID_INLINE int idPolynomial::GetRoots4( float a, float b, float c, float d, float e, float *roots ) {
|
||
|
|
int count;
|
||
|
|
float inva, y, ds, r, s1, s2, t1, t2, tp, tm;
|
||
|
|
float roots3[3];
|
||
|
|
|
||
|
|
if ( a != 1.0f ) {
|
||
|
|
assert( a != 0.0f );
|
||
|
|
inva = 1.0f / a;
|
||
|
|
e *= inva;
|
||
|
|
d *= inva;
|
||
|
|
c *= inva;
|
||
|
|
b *= inva;
|
||
|
|
}
|
||
|
|
|
||
|
|
count = 0;
|
||
|
|
|
||
|
|
GetRoots3( 1.0f, -c, b * d - 4.0f * e, -b * b * e + 4.0f * c * e - d * d, roots3 );
|
||
|
|
y = roots3[0];
|
||
|
|
ds = 0.25f * b * b - c + y;
|
||
|
|
|
||
|
|
if ( ds < 0.0f ) {
|
||
|
|
return 0;
|
||
|
|
} else if ( ds > 0.0f ) {
|
||
|
|
r = idMath::Sqrt( ds );
|
||
|
|
t1 = 0.75f * b * b - r * r - 2.0f * c;
|
||
|
|
t2 = ( 4.0f * b * c - 8.0f * d - b * b * b ) / ( 4.0f * r );
|
||
|
|
tp = t1 + t2;
|
||
|
|
tm = t1 - t2;
|
||
|
|
|
||
|
|
if ( tp >= 0.0f ) {
|
||
|
|
s1 = idMath::Sqrt( tp );
|
||
|
|
roots[count++] = -0.25f * b + 0.5f * ( r + s1 );
|
||
|
|
roots[count++] = -0.25f * b + 0.5f * ( r - s1 );
|
||
|
|
}
|
||
|
|
if ( tm >= 0.0f ) {
|
||
|
|
s2 = idMath::Sqrt( tm );
|
||
|
|
roots[count++] = -0.25f * b + 0.5f * ( s2 - r );
|
||
|
|
roots[count++] = -0.25f * b - 0.5f * ( s2 + r );
|
||
|
|
}
|
||
|
|
return count;
|
||
|
|
} else {
|
||
|
|
t2 = y * y - 4.0f * e;
|
||
|
|
if ( t2 >= 0.0f ) {
|
||
|
|
t2 = 2.0f * idMath::Sqrt( t2 );
|
||
|
|
t1 = 0.75f * b * b - 2.0f * c;
|
||
|
|
if ( t1 + t2 >= 0.0f ) {
|
||
|
|
s1 = idMath::Sqrt( t1 + t2 );
|
||
|
|
roots[count++] = -0.25f * b + 0.5f * s1;
|
||
|
|
roots[count++] = -0.25f * b - 0.5f * s1;
|
||
|
|
}
|
||
|
|
if ( t1 - t2 >= 0.0f ) {
|
||
|
|
s2 = idMath::Sqrt( t1 - t2 );
|
||
|
|
roots[count++] = -0.25f * b + 0.5f * s2;
|
||
|
|
roots[count++] = -0.25f * b - 0.5f * s2;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return count;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
ID_INLINE const float *idPolynomial::ToFloatPtr( void ) const {
|
||
|
|
return coefficient;
|
||
|
|
}
|
||
|
|
|
||
|
|
ID_INLINE float *idPolynomial::ToFloatPtr( void ) {
|
||
|
|
return coefficient;
|
||
|
|
}
|
||
|
|
|
||
|
|
ID_INLINE void idPolynomial::Resize( int d, bool keep ) {
|
||
|
|
int alloc = ( d + 1 + 3 ) & ~3;
|
||
|
|
if ( alloc > allocated ) {
|
||
|
|
float *ptr = (float *) Mem_AllocAligned( alloc * sizeof( float ), ALIGN_16 );
|
||
|
|
if ( coefficient != NULL ) {
|
||
|
|
if ( keep ) {
|
||
|
|
for ( int i = 0; i <= degree; i++ ) {
|
||
|
|
ptr[i] = coefficient[i];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
Mem_FreeAligned( coefficient );
|
||
|
|
}
|
||
|
|
allocated = alloc;
|
||
|
|
coefficient = ptr;
|
||
|
|
}
|
||
|
|
degree = d;
|
||
|
|
}
|
||
|
|
|
||
|
|
#endif /* !__MATH_POLYNOMIAL_H__ */
|