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polynomial.inl
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1/*
2 Copyright (c) 2019, Michael Kazhdan
3 All rights reserved.
4
5 Redistribution and use in source and binary forms, with or without modification,
6 are permitted provided that the following conditions are met:
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11 in the documentation and/or other materials provided with the distribution.
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14 may be used to endorse or promote products derived from this software without specific
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27*/
28
29#include <Util/poly34.h>
30#include "Util/exceptions.h"
31
32 namespace Util
33{
35 // Polynomial 1D //
37 template< unsigned int Degree >
38 Polynomial< 1 , Degree >::Polynomial( void ){ memset( _coefficients , 0 , sizeof( _coefficients ) ); }
39
40 template< unsigned int Degree >
41 Polynomial< 1 , Degree >::Polynomial( double c ) : Polynomial() { _coefficients[0] = c; }
42
43 template< unsigned int Degree >
44 template< typename ... Doubles >
45 Polynomial< 1 , Degree >::Polynomial( Doubles ... coefficients )
46 {
47 static_assert( sizeof...(coefficients)<=(Degree+1) , "[ERROR] More coefficients than the degree supports" );
48 memset( _coefficients , 0 , sizeof( _coefficients ) );
49 const double c[] = { coefficients ... };
50 memcpy( _coefficients , c , sizeof(c) );
51 }
52
53 template< unsigned int Degree >
54 template< unsigned int _Degree >
55 Polynomial< 1 , Degree >::Polynomial( const Polynomial< 1 , _Degree > &p )
56 {
57 for( int d=0 ; d<=Degree && d<=_Degree ; d++ ) _coefficients[d] = p._coefficients[d];
58 for( int d=_Degree+1 ; d<=Degree ; d++ ) _coefficients[d] = 0;
59 }
60
61 template< unsigned int Degree >
62 template< unsigned int _Degree >
63 Polynomial< 1 , Degree > &Polynomial< 1 , Degree >::operator= ( const Polynomial< 1 , _Degree > &p )
64 {
65 for( int d=0 ; d<=Degree && d<=_Degree ; d++ ) _coefficients[d] = p._coefficients[d];
66 for( int d=_Degree+1 ; d<=Degree ; d++ ) _coefficients[d] = 0;
67 return *this;
68 }
69
70 template< unsigned int Degree >
71 const double &Polynomial< 1 , Degree >::_coefficient( const unsigned int indices[] , unsigned int maxDegree ) const
72 {
73 if( indices[0]>maxDegree ) ERROR_OUT( "degree out of bounds: %d > %d\n" , indices[0] , maxDegree );
74 return _coefficients[ indices[0] ];
75 }
76
77 template< unsigned int Degree >
78 double &Polynomial< 1 , Degree >::_coefficient( const unsigned int indices[] , unsigned int maxDegree )
79 {
80 if( indices[0]>maxDegree ) ERROR_OUT( "degree out of bounds: %d > %d\n" , indices[0] , maxDegree );
81 return _coefficients[ indices[0] ];
82 }
83
84 template< unsigned int Degree >
85 double Polynomial< 1 , Degree >::_evaluate( const double coordinates[] , unsigned int maxDegree ) const
86 {
87 double value = 0 , tmp = 1;
88 for( unsigned int d=0 ; d<=Degree && d<=maxDegree ; d++ )
89 {
90 value += tmp * _coefficients[d];
91 tmp *= coordinates[0];
92 }
93 return value;
94 }
95
96 template< unsigned int Degree >
97 Polynomial< 1 , Degree > Polynomial< 1 , Degree >::_evaluate( const Ray< 1 > &ray , unsigned int maxDegree ) const
98 {
99 Polynomial< 1 , 1 > _p = Polynomial< 1 , 1 >( ray.position[0] , ray.direction[0] );
100 Polynomial< 1 , Degree > p( 0. ) , __p( 1. );
101 for( unsigned int d=0 ; d<=maxDegree ; d++ )
102 {
103 p += __p * _coefficients[d];
104 __p = __p * _p;
105 }
106 return p;
107 }
108
109 template< unsigned int Degree >
110 bool Polynomial< 1 , Degree >::_isZero( unsigned int maxDegree ) const
111 {
112 for( unsigned int d=0 ; d<=maxDegree ; d++ ) if( _coefficients[d]!=0 ) return false;
113 return true;
114 }
115
116 template< unsigned int Degree >
117 bool Polynomial< 1 , Degree >::_isConstant( unsigned int maxDegree ) const
118 {
119 for( unsigned int d=1 ; d<=maxDegree ; d++ ) if( _coefficients[d]!=0 ) return false;
120 return true;
121 }
122
123 template< unsigned int Degree >
124 const double &Polynomial< 1 , Degree >::coefficient( unsigned int d ) const { return _coefficient( &d , Degree ); }
125
126 template< unsigned int Degree >
127 double &Polynomial< 1 , Degree >::coefficient( unsigned int d ) { return _coefficient( &d , Degree ); }
128
129 template< unsigned int Degree >
130 double Polynomial< 1 , Degree >::operator()( double x ) const { return _evaluate( &x , Degree ); }
131
132 template< unsigned int Degree >
133 Polynomial< 1 , Degree-1 > Polynomial< 1 , Degree >::d( unsigned int ) const
134 {
135 Polynomial< 1 , Degree-1 > derivative;
136 for( int i=0 ; i<Degree ; i++ ) derivative._coefficients[i] = _coefficients[i+1]*(i+1);
137 return derivative;
138 }
139
140 template< unsigned int Degree >
141 Polynomial< 1 , Degree > Polynomial< 1 , Degree >::operator()( const Ray< 1 > &ray ) const { return _evaluate( ray , Degree ); }
142
143 template< unsigned int Degree >
144 Polynomial< 1 , Degree > Polynomial< 1 , Degree >::operator * ( double s ) const
145 {
146 Polynomial q;
147 for( int d=0 ; d<=Degree ; d++ ) q._coefficients[d] = _coefficients[d]*s;
148 return q;
149 }
150
151 template< unsigned int Degree >
152 Polynomial< 1 , Degree > Polynomial< 1 , Degree >::operator + ( const Polynomial< 1 , Degree > &p ) const
153 {
154 Polynomial< 1 , Degree > q;
155 for( int d=0 ; d<=Degree ; d++ ) q._coefficients[d] = _coefficients[d] + p._coefficients[d];
156 return q;
157 }
158
159 template< unsigned int Degree >
160 unsigned int Polynomial< 1 , Degree >::roots( double *r ) const
161 {
162 ERROR_OUT( "Root functionality not supported for polynomial of degree = %d" , Degree );
163 return 0;
164 }
165
166 template<>
167 inline unsigned int Polynomial< 1 , 1 >::roots( double *r ) const
168 {
169 if( _coefficients[1]==0 ) return 0;
170 else
171 {
172 r[0] = -_coefficients[0] / _coefficients[1];
173 return 1;
174 }
175 }
176
177 template<>
178 inline unsigned int Polynomial< 1 , 2 >::roots( double *r ) const
179 {
180 double disc = _coefficients[1]*_coefficients[1] - 4. * _coefficients[0] * _coefficients[2];
181 if( disc<0 ) return 0;
182 else if( disc==0 )
183 {
184 r[0] = - _coefficients[1] / ( 2 * _coefficients[2] );
185 return 1;
186 }
187 else
188 {
189 disc = sqrt(disc);
190 r[0] = ( -_coefficients[1] - disc ) / (2 * _coefficients[2] );
191 r[1] = ( -_coefficients[1] + disc ) / (2 * _coefficients[2] );
192 return 2;
193 }
194 }
195
196 template<>
197 inline unsigned int Polynomial< 1 , 3 >::roots( double *r ) const
198 {
199 return poly34::SolveP3( r , _coefficients[2]/_coefficients[3] , _coefficients[1]/_coefficients[3] , _coefficients[0]/_coefficients[3] );
200 }
201
202 template<>
203 inline unsigned int Polynomial< 1 , 4 >::roots( double *r ) const
204 {
205 return poly34::SolveP4( r , _coefficients[3]/_coefficients[4] , _coefficients[2]/_coefficients[4] , _coefficients[1]/_coefficients[4] , _coefficients[0]/_coefficients[4] );
206 }
207
208 template<>
209 inline unsigned int Polynomial< 1 , 5 >::roots( double *r ) const
210 {
211 return poly34::SolveP5( r , _coefficients[4]/_coefficients[5] , _coefficients[3]/_coefficients[5] , _coefficients[2]/_coefficients[5] , _coefficients[1]/_coefficients[5] , _coefficients[0]/_coefficients[5] );
212 }
213
214 template< unsigned int Degree1 , unsigned int Degree2 >
215 Polynomial< 1 , Degree1 + Degree2 > operator * ( const Polynomial< 1 , Degree1 > &p1 , const Polynomial< 1 , Degree2 > &p2 )
216 {
217 Polynomial< 1 , Degree1 + Degree2 > p;
218 for( int d1=0 ; d1<=Degree1 ; d1++ ) for( int d2=0 ; d2<=Degree2 ; d2++ ) p._coefficients[ d1+d2 ] += p1._coefficients[d1] * p2._coefficients[d2];
219 return p;
220 }
221
222 template< unsigned int Degree1 , unsigned int Degree2 >
223 Polynomial< 1 , Max< Degree1 , Degree2 >::Value > operator + ( const Polynomial< 1 , Degree1 > &p1 , const Polynomial< 1 , Degree2 > &p2 )
224 {
225 Polynomial< 1 , Max< Degree1 , Degree2 >::Value > p;
226 for( int d=0 ; d<=Degree1 ; d++ ) p._coefficients[d] += p1._coefficients[d];
227 for( int d=0 ; d<=Degree2 ; d++ ) p._coefficients[d] += p2._coefficients[d];
228 return p;
229 }
230
231 template< unsigned int Degree1 , unsigned int Degree2 >
232 Polynomial< 1 , Max< Degree1 , Degree2 >::Value > operator - ( const Polynomial< 1 , Degree1 > &p1 , const Polynomial< 1 , Degree2 > &p2 ){ return p1 + (-p2); }
233
234 template< unsigned int Degree >
235 std::ostream &operator << ( std::ostream &stream , const Polynomial< 1 , Degree > &poly )
236 {
237 static const unsigned int Dim = 1;
238 auto PrintCoefficient = [&]( double c , bool first )
239 {
240 if( c<0 ) stream << " - ";
241 else if( c>0 && !first ) stream << " + ";
242 stream << fabs( c );
243 };
244
245 auto PrintMonomial = [&]( unsigned int d )
246 {
247 if ( d==0 ) ;
248 else if( d==1 ) stream << " * x_" << Dim;
249 else stream << " * x_" << Dim << "^" << d;
250 };
251
252 bool first = true;
253 if( poly._isZero( Degree ) ) stream << "0";
254 for( int d=0 ; d<=Degree ; d++ )
255 {
256 if( poly._coefficients[d] )
257 {
258 PrintCoefficient( poly._coefficients[d] , first );
259 PrintMonomial( d );
260 first = false;
261 }
262 }
263 return stream;
264 }
265
267 // Polynomial Dim-dimensions //
269 template< unsigned int Dim , unsigned int Degree > Polynomial< Dim , Degree >::Polynomial( void ){}
270
271 template< unsigned int Dim , unsigned int Degree > Polynomial< Dim , Degree >::Polynomial( double c ){ _polynomials[0] = Polynomial< Dim-1 , Degree >( c ); }
272
273 template< unsigned int Dim , unsigned int Degree >
274 template< unsigned int _Degree >
275 Polynomial< Dim , Degree >::Polynomial( const Polynomial< Dim , _Degree > &p )
276 {
277 for( int d=0 ; d<=Degree && d<=_Degree ; d++ ) _polynomials[d] = p._polynomials[d];
278 for( int d=_Degree+1 ; d<=Degree ; d++ ) _polynomials[d] = Polynomial< Dim-1 , Degree >();
279 }
280
281 template< unsigned int Dim , unsigned int Degree >
282 template< unsigned int _Degree >
283 Polynomial< Dim , Degree > &Polynomial< Dim , Degree >::operator= ( const Polynomial< Dim , _Degree > &p )
284 {
285 for( int d=0 ; d<=Degree && d<=_Degree ; d++ ) _polynomials[d] = p._polynomials[d];
286 for( int d=_Degree+1 ; d<=Degree ; d++ ) _polynomials[d] = Polynomial< Dim-1 , Degree >();
287 return *this;
288 }
289
290 template< unsigned int Dim , unsigned int Degree >
291 const double &Polynomial< Dim , Degree >::_coefficient( const unsigned int indices[] , unsigned int maxDegree ) const
292 {
293 if( indices[0]>maxDegree ) ERROR_OUT( "degree out of bounds: %d > %d\n" , indices[0] , maxDegree );
294 return _polynomials[ indices[0] ]._coefficient( indices+1 , maxDegree-indices[0] );
295 }
296
297 template< unsigned int Dim , unsigned int Degree >
298 double& Polynomial< Dim , Degree >::_coefficient( const unsigned int indices[] , unsigned int maxDegree )
299 {
300 if( indices[0]>maxDegree ) ERROR_OUT( "degree out of bounds: %d > %d\n" , indices[0] , maxDegree );
301 return _polynomials[ indices[0] ]._coefficient( indices+1 , maxDegree-indices[0] );
302 }
303
304 template< unsigned int Dim , unsigned int Degree >
305 double Polynomial< Dim , Degree >::_evaluate( const double coordinates[] , unsigned int maxDegree ) const
306 {
307 double sum = 0 , tmp = 1;
308 for( unsigned int d=0 ; d<=maxDegree ; d++ )
309 {
310 sum += _polynomials[d]._evaluate( coordinates+1 , maxDegree-d ) * tmp;
311 tmp *= coordinates[0];
312 }
313 return sum;
314 }
315
316 template< unsigned int Dim , unsigned int Degree >
317 Polynomial< 1 , Degree > Polynomial< Dim , Degree >::_evaluate( const Ray< Dim > &ray , unsigned int maxDegree ) const
318 {
319 Polynomial< 1 , 1 > _p = Polynomial< 1 , 1 >( ray.position[0] , ray.direction[0] );
320 Ray< Dim-1 > _ray;
321 for( int d=1 ; d<Dim ; d++ ) _ray.position[d-1] = ray.position[d] , _ray.direction[d-1] = ray.direction[d];
322
323 Polynomial< 1 , Degree > p( 0. ) , __p( 1. );
324 for( unsigned int d=0 ; d<=maxDegree ; d++ )
325 {
326 p += _polynomials[d]._evaluate( _ray , maxDegree-d ) * __p;
327 __p = __p * _p;
328 }
329 return p;
330 }
331
332 template< unsigned int Dim , unsigned int Degree >
333 bool Polynomial< Dim , Degree >::_isZero( unsigned int maxDegree ) const
334 {
335 for( unsigned int d=0 ; d<=maxDegree ; d++ ) if( !_polynomials[d]._isZero( maxDegree-d ) ) return false;
336 return true;
337 }
338
339 template< unsigned int Dim , unsigned int Degree >
340 bool Polynomial< Dim , Degree >::_isConstant( unsigned int maxDegree ) const
341 {
342 if( !_polynomials[0]._isConstant( Degree ) ) return false;
343 for( unsigned int d=1 ; d<=maxDegree ; d++ ) if( !_polynomials[d]._isZero( maxDegree-d ) ) return false;
344 return true;
345 }
346
347 template< unsigned int Dim , unsigned int Degree >
348 template< typename ... UnsignedInts >
349 const double &Polynomial< Dim , Degree >::coefficient( UnsignedInts ... indices ) const
350 {
351 static_assert( sizeof...(indices)==Dim , "[ERROR] Polynomial< Dim , Degree >::coefficient: Invalid number of indices" );
352 unsigned int _indices[] = { indices ... };
353 return _coefficient( _indices , Degree );
354 }
355
356 template< unsigned int Dim , unsigned int Degree >
357 template< typename ... UnsignedInts >
358 double &Polynomial< Dim , Degree >::coefficient( UnsignedInts ... indices )
359 {
360 static_assert( sizeof...(indices)==Dim , "[ERROR] Polynomial< Dim , Degree >::coefficient: Invalid number of indices" );
361 unsigned int _indices[] = { indices ... };
362 return _coefficient( _indices , Degree );
363 }
364
365 template< unsigned int Dim , unsigned int Degree >
366 template< typename ... Doubles >
367 double Polynomial< Dim , Degree >::operator()( Doubles ... coordinates ) const
368 {
369 static_assert( sizeof...(coordinates)==Dim , "[ERROR] Polynomial< Dim , Degree >::operator(): Invalid number of coordinates" );
370 double _coordinates[] = { coordinates... };
371 return _evaluate( _coordinates , Degree );
372 }
373
374 template< unsigned int Dim , unsigned int Degree >
375 double Polynomial< Dim , Degree >::operator()( Point< Dim > p ) const { return _evaluate( &p[0] , Degree ); }
376
378 template< unsigned int Dim , unsigned int Degree >
379 Polynomial< Dim , Degree-1 > Polynomial< Dim , Degree >::d( int dim ) const
380 {
381 Polynomial< Dim , Degree-1 > derivative;
382 if( dim==0 ) for( int d=0 ; d<Degree ; d++ ) derivative._polynomials[d] = _polynomials[d+1] * (d+1);
383 else for( int d=0 ; d<Degree ; d++ ) derivative._polynomials[d] = _polynomials[d].d( dim-1 );
384 return derivative;
385 }
386
387 template< unsigned int Dim , unsigned int Degree >
388 Polynomial< 1 , Degree > Polynomial< Dim , Degree >::operator()( const Ray< Dim > &ray ) const { return _evaluate( ray , Degree ); }
389
390 template< unsigned int Dim , unsigned int Degree >
391 std::ostream &operator << ( std::ostream &stream , const Polynomial< Dim , Degree > &poly )
392 {
393 auto PrintSign = [&]( bool first )
394 {
395 if( !first ) stream << " + ";
396 };
397
398 auto PrintMonomial = [&]( unsigned int d )
399 {
400 if ( d==0 ) ;
401 else if( d==1 ) stream << " * x_" << Dim;
402 else stream << " * x_" << Dim << "^" << d;
403 };
404
405 bool first = true;
406 if( poly._isZero( Degree ) ) stream << "0";
407 for( int d=0 ; d<=Degree ; d++ )
408 {
409 if( !poly._polynomials[d]._isZero( Degree-d ) )
410 {
411 PrintSign( first );
412 if( poly._polynomials[d]._isConstant( Degree-d ) ) stream << poly._polynomials[d] ;
413 else stream << "( " << poly._polynomials[d] << " )";
414 PrintMonomial( d );
415 first = false;
416 }
417 }
418 return stream;
419 }
420
421 template< unsigned int Dim , unsigned int Degree >
422 Polynomial< Dim , Degree > Polynomial< Dim , Degree >::operator * ( double s ) const
423 {
424 Polynomial p;
425 for( int d=0 ; d<=Degree ; d++ ) p._polynomials[d] = _polynomials[d] * s;
426 return p;
427 }
428
429 template< unsigned int Dim , unsigned int Degree >
430 Polynomial< Dim , Degree > Polynomial< Dim , Degree >::operator + ( const Polynomial< Dim , Degree > &p ) const
431 {
432 Polynomial< Dim , Degree > q;
433 for( int d=0 ; d<=Degree ; d++ ) q._polynomials[d] = _polynomials[d] + p._polynomials[d];
434 return q;
435 }
436
437 template< unsigned int Dim , unsigned int Degree1 , unsigned int Degree2 >
438 Polynomial< Dim , Degree1 + Degree2 > operator * ( const Polynomial< Dim , Degree1 > &p1 , const Polynomial< Dim , Degree2 > &p2 )
439 {
440 Polynomial< Dim , Degree1 + Degree2 > p;
441 for( int d1=0 ; d1<=Degree1 ; d1++ ) for( int d2=0 ; d2<=Degree2 ; d2++ ) p._polynomials[ d1+d2 ] += p1._polynomials[d1] * p2._polynomials[d2];
442 return p;
443 }
444
445 template< unsigned int Dim , unsigned int Degree1 , unsigned int Degree2 >
446 Polynomial< Dim , Max< Degree1 , Degree2 >::Value > operator + ( const Polynomial< Dim , Degree1 > &p1 , const Polynomial< Dim , Degree2 > &p2 )
447 {
448 Polynomial< Dim , Max< Degree1 , Degree2 >::Value > p;
449 for( int d=0 ; d<=Degree1 ; d++ ) p._polynomials[d] += p1._polynomials[d];
450 for( int d=0 ; d<=Degree2 ; d++ ) p._polynomials[d] += p2._polynomials[d];
451 return p;
452 }
453
454 template< unsigned int Dim , unsigned int Degree1 , unsigned int Degree2 >
455 Polynomial< Dim , Max< Degree1 , Degree2 >::Value > operator - ( const Polynomial< Dim , Degree1 > &p1 , const Polynomial< Dim , Degree2 > &p2 ){ return p1 + (-p2); }
456}
Util::Point
Definition geometry.h:52
Util::Polynomial
Definition polynomial.h:45
Util::Polynomial::_isConstant
bool _isConstant(unsigned int maxDegree) const
Definition polynomial.inl:117
Util::Polynomial::operator*
friend Polynomial< _Dim, Degree1+Degree2 > operator*(const Polynomial< _Dim, Degree1 > &, const Polynomial< _Dim, Degree2 > &)
Util::Polynomial::_evaluate
double _evaluate(const double coordinates[], unsigned int maxDegree) const
Definition polynomial.inl:85
Util::Polynomial::coefficient
const double & coefficient(UnsignedInts ... indices) const
Definition polynomial.inl:349
Util::Polynomial< 2, Degree >::d
Polynomial< Dim, Degree-1 > d(int dim) const
Util::Polynomial::_polynomials
Polynomial< Dim-1, Degree > _polynomials[Degree+1]
Definition polynomial.h:54
Util::Polynomial::Polynomial
friend class Polynomial
Definition polynomial.h:46
Util::Polynomial::operator()
double operator()(Doubles ... coordinates) const
Definition polynomial.inl:367
Util::Polynomial::roots
unsigned int roots(double *r) const
Definition polynomial.inl:167
Util::Polynomial::_isZero
bool _isZero(unsigned int maxDegree) const
Definition polynomial.inl:110
Util::Polynomial::operator+
friend Polynomial< _Dim, Max< Degree1, Degree2 >::Value > operator+(const Polynomial< _Dim, Degree1 > &, const Polynomial< _Dim, Degree2 > &)
Util::Polynomial::operator=
Polynomial & operator=(const Polynomial< Dim, _Degree > &p)
Util::Polynomial::_coefficient
const double & _coefficient(const unsigned int indices[], unsigned int maxDegree) const
Definition polynomial.inl:71
Util::Ray< 1 >
Util::Ray::direction
Point< Dim > direction
Definition geometry.h:305
Util::Ray::position
Point< Dim > position
Definition geometry.h:302
exceptions.h
ERROR_OUT
#define ERROR_OUT(...)
Definition exceptions.h:154
Ray
Definition box.h:7
Util
Definition algebra.h:7
Util::operator<<
std::ostream & operator<<(std::ostream &stream, const Point< Dim > &p)
Definition geometry.inl:127
Util::operator-
Polynomial< Dim, Max< Degree1, Degree2 >::Value > operator-(const Polynomial< Dim, Degree1 > &p1, const Polynomial< Dim, Degree2 > &p2)
Definition polynomial.inl:455
Util::operator*
Ray< Dim > operator*(const Matrix< Dim+1, Dim+1 > &m, const Ray< Dim > &ray)
Definition geometry.inl:493
Util::operator+
Polynomial< Dim, Max< Degree1, Degree2 >::Value > operator+(const Polynomial< Dim, Degree1 > &p1, const Polynomial< Dim, Degree2 > &p2)
Definition polynomial.inl:446
poly34::SolveP4
int SolveP4(double *x, double a, double b, double c, double d)
Definition poly34.cpp:275
poly34::SolveP3
int SolveP3(double *x, double a, double b, double c)
Definition poly34.cpp:73
poly34::SolveP5
int SolveP5(double *x, double a, double b, double c, double d, double e)
Definition poly34.cpp:359
poly34.h