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root/OpenMD/branches/development/src/math/Polynomial.hpp

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trunk/src/math/Polynomial.hpp (file contents), Revision 385 by tim, Tue Mar 1 20:10:14 2005 UTC vs.
branches/development/src/math/Polynomial.hpp (file contents), Revision 1874 by gezelter, Wed May 15 15:09:35 2013 UTC

#	Line 1 | Line 1
1	<	/*
1	>	/*
2		* Copyright (c) 2005 The University of Notre Dame. All Rights Reserved.
3		*
4		* The University of Notre Dame grants you ("Licensee") a
#	Line 6 | Line 6
6		* redistribute this software in source and binary code form, provided
7		* that the following conditions are met:
8		*
9	<	* 1. Acknowledgement of the program authors must be made in any
10	<	*    publication of scientific results based in part on use of the
11	<	*    program.  An acceptable form of acknowledgement is citation of
12	<	*    the article in which the program was described (Matthew
13	<	*    A. Meineke, Charles F. Vardeman II, Teng Lin, Christopher
14	<	*    J. Fennell and J. Daniel Gezelter, "OOPSE: An Object-Oriented
15	<	*    Parallel Simulation Engine for Molecular Dynamics,"
16	<	*    J. Comput. Chem. 26, pp. 252-271 (2005))
17	<	*
18	<	* 2. Redistributions of source code must retain the above copyright
9	>	* 1. Redistributions of source code must retain the above copyright
10		*    notice, this list of conditions and the following disclaimer.
11		*
12	<	* 3. Redistributions in binary form must reproduce the above copyright
12	>	* 2. Redistributions in binary form must reproduce the above copyright
13		*    notice, this list of conditions and the following disclaimer in the
14		*    documentation and/or other materials provided with the
15		*    distribution.
#	Line 37 | Line 28
28		* arising out of the use of or inability to use software, even if the
29		* University of Notre Dame has been advised of the possibility of
30		* such damages.
31	+	*
32	+	* SUPPORT OPEN SCIENCE!  If you use OpenMD or its source code in your
33	+	* research, please cite the appropriate papers when you publish your
34	+	* work.  Good starting points are:
35	+	*
36	+	* [1]  Meineke, et al., J. Comp. Chem. 26, 252-271 (2005).
37	+	* [2]  Fennell & Gezelter, J. Chem. Phys. 124, 234104 (2006).
38	+	* [3]  Sun, Lin & Gezelter, J. Chem. Phys. 128, 234107 (2008).
39	+	* [4]  Kuang & Gezelter,  J. Chem. Phys. 133, 164101 (2010).
40	+	* [5]  Vardeman, Stocker & Gezelter, J. Chem. Theory Comput. 7, 834 (2011).
41		*/
42
43		/**
#	Line 53 | Line 54
54		#include <list>
55		#include <map>
56		#include <utility>
57	+	#include <complex>
58	+	#include "config.h"
59	+	#include "math/Eigenvalue.hpp"
60
61	<	namespace oopse {
61	>	namespace OpenMD {
62	>
63	>	template<typename Real> Real fastpow(Real x, int N) {
64	>	Real result(1); //or 1.0?
65
59	–	template<typename ElemType> ElemType pow(ElemType x, int N) {
60	–	ElemType result(1);
61	–
66		for (int i = 0; i < N; ++i) {
67	<	result *= x;
67	>	result *= x;
68		}
69
70		return result;
71	<	}
71	>	}
72
73	<	/**
74	<	* @class Polynomial Polynomial.hpp "math/Polynomial.hpp"
75	<	* A generic Polynomial class
76	<	*/
77	<	template<typename ElemType>
78	<	class Polynomial {
73	>	/**
74	>	* @class Polynomial Polynomial.hpp "math/Polynomial.hpp"
75	>	* A generic Polynomial class
76	>	*/
77	>	template<typename Real>
78	>	class Polynomial {
79
80	<	public:
81	<
82	<	typedef int ExponentType;
83	<	typedef ElemType CoefficientType;
84	<	typedef std::map<ExponentType, CoefficientType> PolynomialPairMap;
85	<	typedef typename PolynomialPairMap::iterator iterator;
86	<	typedef typename PolynomialPairMap::const_iterator const_iterator;
87	<	/**
88	<	* Calculates the value of this Polynomial evaluated at the given x value.
89	<	* @return The value of this Polynomial evaluates at the given x value
90	<	* @param x the value of the independent variable for this Polynomial function
91	<	*/
92	<	ElemType evaluate(const ElemType& x) {
93	<	ElemType result = ElemType();
94	<	ExponentType exponent;
95	<	CoefficientType coefficient;
80	>	public:
81	>	typedef Polynomial<Real> PolynomialType;
82	>	typedef int ExponentType;
83	>	typedef Real CoefficientType;
84	>	typedef std::map<ExponentType, CoefficientType> PolynomialPairMap;
85	>	typedef typename PolynomialPairMap::iterator iterator;
86	>	typedef typename PolynomialPairMap::const_iterator const_iterator;
87	>
88	>	Polynomial() {}
89	>	Polynomial(Real v) {setCoefficient(0, v);}
90	>	/**
91	>	* Calculates the value of this Polynomial evaluated at the given x value.
92	>	* @return The value of this Polynomial evaluates at the given x value
93	>	* @param x the value of the independent variable for this
94	>	* Polynomial function
95	>	*/
96	>	Real evaluate(const Real& x) {
97	>	Real result = Real();
98	>	ExponentType exponent;
99	>	CoefficientType coefficient;
100
101	<	for (iterator i = polyPairMap_.begin(); i != polyPairMap_.end(); ++i) {
102	<	exponent = i->first;
103	<	coefficient = i->second;
104	<	result  += pow(x, exponent) * coefficient;
105	<	}
101	>	for (iterator i = polyPairMap_.begin(); i != polyPairMap_.end(); ++i) {
102	>	exponent = i->first;
103	>	coefficient = i->second;
104	>	result  += fastpow(x, exponent) * coefficient;
105	>	}
106
107	<	return result;
108	<	}
107	>	return result;
108	>	}
109
110	<	/**
111	<	* Returns the first derivative of this polynomial.
112	<	* @return the first derivative of this polynomial
113	<	* @param x
114	<	*/
115	<	ElemType evaluateDerivative(const ElemType& x) {
116	<	ElemType result = ElemType();
117	<	ExponentType exponent;
118	<	CoefficientType coefficient;
110	>	/**
111	>	* Returns the first derivative of this polynomial.
112	>	* @return the first derivative of this polynomial
113	>	* @param x
114	>	*/
115	>	Real evaluateDerivative(const Real& x) {
116	>	Real result = Real();
117	>	ExponentType exponent;
118	>	CoefficientType coefficient;
119
120	<	for (iterator i = polyPairMap_.begin(); i != polyPairMap_.end(); ++i) {
121	<	exponent = i->first;
122	<	coefficient = i->second;
123	<	result  += pow(x, exponent - 1) * coefficient * exponent;
124	<	}
120	>	for (iterator i = polyPairMap_.begin(); i != polyPairMap_.end(); ++i) {
121	>	exponent = i->first;
122	>	coefficient = i->second;
123	>	result  += fastpow(x, exponent - 1) * coefficient * exponent;
124	>	}
125
126	<	return result;
127	<	}
126	>	return result;
127	>	}
128
121	–	/**
122	–	* Set the coefficent of the specified exponent, if the coefficient is already there, it
123	–	* will be overwritten.
124	–	* @param exponent exponent of a term in this Polynomial
125	–	* @param coefficient multiplier of a term in this Polynomial
126	–	*/
127	–
128	–	void setCoefficient(int exponent, const ElemType& coefficient) {
129	–	polyPairMap_.insert(typename PolynomialPairMap::value_type(exponent, coefficient));
130	–	}
129
130	<	/**
131	<	* Set the coefficent of the specified exponent. If the coefficient is already there,  just add the
132	<	* new coefficient to the old one, otherwise,  just call setCoefficent
133	<	* @param exponent exponent of a term in this Polynomial
134	<	* @param coefficient multiplier of a term in this Polynomial
135	<	*/
130	>	/**
131	>	* Set the coefficent of the specified exponent, if the
132	>	* coefficient is already there, it will be overwritten.
133	>	* @param exponent exponent of a term in this Polynomial
134	>	* @param coefficient multiplier of a term in this Polynomial
135	>	*/
136	>	void setCoefficient(int exponent, const Real& coefficient) {
137	>	polyPairMap_[exponent] = coefficient;
138	>	}
139	>
140	>	/**
141	>	* Set the coefficent of the specified exponent. If the
142	>	* coefficient is already there, just add the new coefficient to
143	>	* the old one, otherwise, just call setCoefficent
144	>	* @param exponent exponent of a term in this Polynomial
145	>	* @param coefficient multiplier of a term in this Polynomial
146	>	*/
147	>	void addCoefficient(int exponent, const Real& coefficient) {
148	>	iterator i = polyPairMap_.find(exponent);
149	>
150	>	if (i != end()) {
151	>	i->second += coefficient;
152	>	} else {
153	>	setCoefficient(exponent, coefficient);
154	>	}
155	>	}
156	>
157	>	/**
158	>	* Returns the coefficient associated with the given power for
159	>	* this Polynomial.
160	>	* @return the coefficient associated with the given power for
161	>	* this Polynomial
162	>	* @param exponent exponent of any term in this Polynomial
163	>	*/
164	>	Real getCoefficient(ExponentType exponent) {
165	>	iterator i = polyPairMap_.find(exponent);
166	>
167	>	if (i != end()) {
168	>	return i->second;
169	>	} else {
170	>	return Real(0);
171	>	}
172	>	}
173	>
174	>	iterator begin() {
175	>	return polyPairMap_.begin();
176	>	}
177	>
178	>	const_iterator begin() const{
179	>	return polyPairMap_.begin();
180	>	}
181
182	<	void addCoefficient(int exponent, const ElemType& coefficient) {
183	<	iterator i = polyPairMap_.find(exponent);
182	>	iterator end() {
183	>	return polyPairMap_.end();
184	>	}
185
186	<	if (i != end()) {
187	<	i->second += coefficient;
188	<	} else {
145	<	setCoefficient(exponent, coefficient);
146	<	}
147	<	}
186	>	const_iterator end() const{
187	>	return polyPairMap_.end();
188	>	}
189
190	+	iterator find(ExponentType exponent) {
191	+	return polyPairMap_.find(exponent);
192	+	}
193
194	<	/**
195	<	* Returns the coefficient associated with the given power for this Polynomial.
196	<	* @return the coefficient associated with the given power for this Polynomial
153	<	* @exponent exponent of any term in this Polynomial
154	<	*/
155	<	ElemType getCoefficient(ExponentType exponent) {
156	<	iterator i = polyPairMap_.find(exponent);
194	>	size_t size() {
195	>	return polyPairMap_.size();
196	>	}
197
198	<	if (i != end()) {
199	<	return i->second;
200	<	} else {
201	<	return ElemType(0);
202	<	}
203	<	}
198	>	int degree() {
199	>	int deg = 0;
200	>	for (iterator i = polyPairMap_.begin(); i != polyPairMap_.end(); ++i) {
201	>	if (i->first > deg)
202	>	deg = i->first;
203	>	}
204	>	return deg;
205	>	}
206
207	<	iterator begin() {
166	<	return polyPairMap_.begin();
167	<	}
207	>	PolynomialType& operator = (const PolynomialType& p) {
208
209	<	const_iterator begin() const{
210	<	return polyPairMap_.begin();
211	<	}
172	<
173	<	iterator end() {
174	<	return polyPairMap_.end();
175	<	}
209	>	if (this != &p)  // protect against invalid self-assignment
210	>	{
211	>	typename Polynomial<Real>::const_iterator i;
212
213	<	const_iterator end() const{
214	<	return polyPairMap_.end();
213	>	polyPairMap_.clear();  // clear out the old map
214	>
215	>	for (i =  p.begin(); i != p.end(); ++i) {
216	>	this->setCoefficient(i->first, i->second);
217	>	}
218		}
219	+	// by convention, always return *this
220	+	return *this;
221	+	}
222
223	<	iterator find(ExponentType exponent) {
224	<	return polyPairMap_.find(exponent);
183	<	}
223	>	PolynomialType& operator += (const PolynomialType& p) {
224	>	typename Polynomial<Real>::const_iterator i;
225
226	<	size_t size() {
227	<	return polyPairMap_.size();
228	<	}
188	<
189	<	private:
190	<
191	<	PolynomialPairMap polyPairMap_;
192	<	};
226	>	for (i =  p.begin(); i  != p.end(); ++i) {
227	>	this->addCoefficient(i->first, i->second);
228	>	}
229
230	+	return *this;
231	+	}
232
233	<	/**
234	<	* Generates and returns the product of two given Polynomials.
235	<	* @return A Polynomial containing the product of the two given Polynomial parameters
236	<	*/
237	<	template<typename ElemType>
238	<	Polynomial<ElemType> operator *(const Polynomial<ElemType>& p1, const Polynomial<ElemType>& p2) {
239	<	typename Polynomial<ElemType>::const_iterator i;
202	<	typename Polynomial<ElemType>::const_iterator j;
203	<	Polynomial<ElemType> p;
233	>	PolynomialType& operator -= (const PolynomialType& p) {
234	>	typename Polynomial<Real>::const_iterator i;
235	>	for (i =  p.begin(); i  != p.end(); ++i) {
236	>	this->addCoefficient(i->first, -i->second);
237	>	}
238	>	return *this;
239	>	}
240
241	<	for (i = p1.begin(); i !=p1.end(); ++i) {
242	<	for (j = p2.begin(); j !=p2.end(); ++j) {
243	<	p.addCoefficient( i->first + j->first, i->second * j->second);
241	>	PolynomialType& operator *= (const PolynomialType& p) {
242	>	typename Polynomial<Real>::const_iterator i;
243	>	typename Polynomial<Real>::const_iterator j;
244	>	Polynomial<Real> p2(*this);
245	>
246	>	polyPairMap_.clear();  // clear out old map
247	>	for (i = p2.begin(); i !=p2.end(); ++i) {
248	>	for (j = p.begin(); j !=p.end(); ++j) {
249	>	this->addCoefficient( i->first + j->first, i->second * j->second);
250		}
251	+	}
252	+	return *this;
253		}
254
255	<	return p;
256	<	}
255	>	//PolynomialType& operator *= (const Real v)
256	>	PolynomialType& operator *= (const Real v) {
257	>	typename Polynomial<Real>::const_iterator i;
258	>	//Polynomial<Real> result;
259	>
260	>	for (i = this->begin(); i != this->end(); ++i) {
261	>	this->setCoefficient( i->first, i->second*v);
262	>	}
263	>
264	>	return *this;
265	>	}
266
267	<	/**
268	<	* Generates and returns the sum of two given Polynomials.
269	<	* @param p1 the first polynomial
270	<	* @param p2 the second polynomial
218	<	*/
219	<	template<typename ElemType>
220	<	Polynomial<ElemType> operator +(const Polynomial<ElemType>& p1, const Polynomial<ElemType>& p2) {
221	<	Polynomial<ElemType> p(p1);
267	>	PolynomialType& operator += (const Real v) {
268	>	this->addCoefficient( 0, v);
269	>	return *this;
270	>	}
271
272	<	typename Polynomial<ElemType>::const_iterator i;
272	>	/**
273	>	* Returns the first derivative of this polynomial.
274	>	* @return the first derivative of this polynomial
275	>	*/
276	>	PolynomialType* getDerivative() {
277	>	Polynomial<Real>* p = new Polynomial<Real>();
278	>
279	>	typename Polynomial<Real>::const_iterator i;
280	>	ExponentType exponent;
281	>	CoefficientType coefficient;
282	>
283	>	for (i =  this->begin(); i  != this->end(); ++i) {
284	>	exponent = i->first;
285	>	coefficient = i->second;
286	>	p->setCoefficient(exponent-1, coefficient * exponent);
287	>	}
288	>
289	>	return p;
290	>	}
291	>
292	>	// Creates the Companion matrix for a given polynomial
293	>	DynamicRectMatrix<Real> CreateCompanion() {
294	>	int rank = degree();
295	>	DynamicRectMatrix<Real> mat(rank, rank);
296	>	Real majorCoeff = getCoefficient(rank);
297	>	for(int i = 0; i < rank; ++i) {
298	>	for(int j = 0; j < rank; ++j) {
299	>	if(i - j == 1) {
300	>	mat(i, j) = 1;
301	>	} else if(j == rank-1) {
302	>	mat(i, j) = -1 * getCoefficient(i) / majorCoeff;
303	>	}
304	>	}
305	>	}
306	>	return mat;
307	>	}
308	>
309	>	// Find the Roots of a given polynomial
310	>	std::vector<complex<Real> > FindRoots() {
311	>	int rank = degree();
312	>	DynamicRectMatrix<Real> companion = CreateCompanion();
313	>	JAMA::Eigenvalue<Real> eig(companion);
314	>	DynamicVector<Real> reals, imags;
315	>	eig.getRealEigenvalues(reals);
316	>	eig.getImagEigenvalues(imags);
317	>
318	>	std::vector<complex<Real> > roots;
319	>	for (int i = 0; i < rank; i++) {
320	>	roots.push_back(complex<Real>(reals(i), imags(i)));
321	>	}
322
323	+	return roots;
324	+	}
325	+
326	+	std::vector<Real> FindRealRoots() {
327	+
328	+	const Real fEpsilon = 1.0e-8;
329	+	std::vector<Real> roots;
330	+	roots.clear();
331	+
332	+	const int deg = degree();
333	+
334	+	switch (deg) {
335	+	case 1: {
336	+	Real fC1 = getCoefficient(1);
337	+	Real fC0 = getCoefficient(0);
338	+	roots.push_back( -fC0 / fC1);
339	+	return roots;
340	+	}
341	+	case 2: {
342	+	Real fC2 = getCoefficient(2);
343	+	Real fC1 = getCoefficient(1);
344	+	Real fC0 = getCoefficient(0);
345	+	Real fDiscr = fC1*fC1 - 4.0*fC0*fC2;
346	+	if (abs(fDiscr) <= fEpsilon) {
347	+	fDiscr = (Real)0.0;
348	+	}
349	+
350	+	if (fDiscr < (Real)0.0) {  // complex roots only
351	+	return roots;
352	+	}
353	+
354	+	Real fTmp = ((Real)0.5)/fC2;
355	+
356	+	if (fDiscr > (Real)0.0) { // 2 real roots
357	+	fDiscr = sqrt(fDiscr);
358	+	roots.push_back(fTmp*(-fC1 - fDiscr));
359	+	roots.push_back(fTmp*(-fC1 + fDiscr));
360	+	} else {
361	+	roots.push_back(-fTmp * fC1);  // 1 real root
362	+	}
363	+	}
364	+	return roots;
365	+	case 3: {
366	+	Real fC3 = getCoefficient(3);
367	+	Real fC2 = getCoefficient(2);
368	+	Real fC1 = getCoefficient(1);
369	+	Real fC0 = getCoefficient(0);
370	+
371	+	// make polynomial monic, x^3+c2*x^2+c1*x+c0
372	+	Real fInvC3 = ((Real)1.0)/fC3;
373	+	fC0 *= fInvC3;
374	+	fC1 *= fInvC3;
375	+	fC2 *= fInvC3;
376	+
377	+	// convert to y^3+a*y+b = 0 by x = y-c2/3
378	+	const Real fThird = (Real)1.0/(Real)3.0;
379	+	const Real fTwentySeventh = (Real)1.0/(Real)27.0;
380	+	Real fOffset = fThird*fC2;
381	+	Real fA = fC1 - fC2*fOffset;
382	+	Real fB = fC0+fC2*(((Real)2.0)*fC2*fC2-((Real)9.0)*fC1)*fTwentySeventh;
383	+	Real fHalfB = ((Real)0.5)*fB;
384	+
385	+	Real fDiscr = fHalfB*fHalfB + fA*fA*fA*fTwentySeventh;
386	+	if (fabs(fDiscr) <= fEpsilon) {
387	+	fDiscr = (Real)0.0;
388	+	}
389	+
390	+	if (fDiscr > (Real)0.0) {  // 1 real, 2 complex roots
391	+
392	+	fDiscr = sqrt(fDiscr);
393	+	Real fTemp = -fHalfB + fDiscr;
394	+	Real root;
395	+	if (fTemp >= (Real)0.0) {
396	+	root = pow(fTemp,fThird);
397	+	} else {
398	+	root = -pow(-fTemp,fThird);
399	+	}
400	+	fTemp = -fHalfB - fDiscr;
401	+	if ( fTemp >= (Real)0.0 ) {
402	+	root += pow(fTemp,fThird);
403	+	} else {
404	+	root -= pow(-fTemp,fThird);
405	+	}
406	+	root -= fOffset;
407	+
408	+	roots.push_back(root);
409	+	} else if (fDiscr < (Real)0.0) {
410	+	const Real fSqrt3 = sqrt((Real)3.0);
411	+	Real fDist = sqrt(-fThird*fA);
412	+	Real fAngle = fThird*atan2(sqrt(-fDiscr), -fHalfB);
413	+	Real fCos = cos(fAngle);
414	+	Real fSin = sin(fAngle);
415	+	roots.push_back(((Real)2.0)*fDist*fCos-fOffset);
416	+	roots.push_back(-fDist*(fCos+fSqrt3*fSin)-fOffset);
417	+	roots.push_back(-fDist*(fCos-fSqrt3*fSin)-fOffset);
418	+	} else {
419	+	Real fTemp;
420	+	if (fHalfB >= (Real)0.0) {
421	+	fTemp = -pow(fHalfB,fThird);
422	+	} else {
423	+	fTemp = pow(-fHalfB,fThird);
424	+	}
425	+	roots.push_back(((Real)2.0)*fTemp-fOffset);
426	+	roots.push_back(-fTemp-fOffset);
427	+	roots.push_back(-fTemp-fOffset);
428	+	}
429	+	}
430	+	return roots;
431	+	case 4: {
432	+	Real fC4 = getCoefficient(4);
433	+	Real fC3 = getCoefficient(3);
434	+	Real fC2 = getCoefficient(2);
435	+	Real fC1 = getCoefficient(1);
436	+	Real fC0 = getCoefficient(0);
437	+
438	+	// make polynomial monic, x^4+c3*x^3+c2*x^2+c1*x+c0
439	+	Real fInvC4 = ((Real)1.0)/fC4;
440	+	fC0 *= fInvC4;
441	+	fC1 *= fInvC4;
442	+	fC2 *= fInvC4;
443	+	fC3 *= fInvC4;
444	+
445	+	// reduction to resolvent cubic polynomial y^3+r2*y^2+r1*y+r0 = 0
446	+	Real fR0 = -fC3*fC3*fC0 + ((Real)4.0)*fC2*fC0 - fC1*fC1;
447	+	Real fR1 = fC3*fC1 - ((Real)4.0)*fC0;
448	+	Real fR2 = -fC2;
449	+	Polynomial<Real> tempCubic;
450	+	tempCubic.setCoefficient(0, fR0);
451	+	tempCubic.setCoefficient(1, fR1);
452	+	tempCubic.setCoefficient(2, fR2);
453	+	tempCubic.setCoefficient(3, 1.0);
454	+	std::vector<Real> cubeRoots = tempCubic.FindRealRoots(); // always
455	+	// produces
456	+	// at
457	+	// least
458	+	// one
459	+	// root
460	+	Real fY = cubeRoots[0];
461	+
462	+	Real fDiscr = ((Real)0.25)*fC3*fC3 - fC2 + fY;
463	+	if (fabs(fDiscr) <= fEpsilon) {
464	+	fDiscr = (Real)0.0;
465	+	}
466	+
467	+	if (fDiscr > (Real)0.0) {
468	+	Real fR = sqrt(fDiscr);
469	+	Real fT1 = ((Real)0.75)*fC3*fC3 - fR*fR - ((Real)2.0)*fC2;
470	+	Real fT2 = (((Real)4.0)*fC3*fC2 - ((Real)8.0)*fC1 - fC3*fC3*fC3) /
471	+	(((Real)4.0)*fR);
472	+
473	+	Real fTplus = fT1+fT2;
474	+	Real fTminus = fT1-fT2;
475	+	if (fabs(fTplus) <= fEpsilon) {
476	+	fTplus = (Real)0.0;
477	+	}
478	+	if (fabs(fTminus) <= fEpsilon) {
479	+	fTminus = (Real)0.0;
480	+	}
481	+
482	+	if (fTplus >= (Real)0.0) {
483	+	Real fD = sqrt(fTplus);
484	+	roots.push_back(-((Real)0.25)*fC3+((Real)0.5)*(fR+fD));
485	+	roots.push_back(-((Real)0.25)*fC3+((Real)0.5)*(fR-fD));
486	+	}
487	+	if (fTminus >= (Real)0.0) {
488	+	Real fE = sqrt(fTminus);
489	+	roots.push_back(-((Real)0.25)*fC3+((Real)0.5)*(fE-fR));
490	+	roots.push_back(-((Real)0.25)*fC3-((Real)0.5)*(fE+fR));
491	+	}
492	+	} else if (fDiscr < (Real)0.0) {
493	+	//roots.clear();
494	+	} else {
495	+	Real fT2 = fY*fY-((Real)4.0)*fC0;
496	+	if (fT2 >= -fEpsilon) {
497	+	if (fT2 < (Real)0.0) { // round to zero
498	+	fT2 = (Real)0.0;
499	+	}
500	+	fT2 = ((Real)2.0)*sqrt(fT2);
501	+	Real fT1 = ((Real)0.75)*fC3*fC3 - ((Real)2.0)*fC2;
502	+	if (fT1+fT2 >= fEpsilon) {
503	+	Real fD = sqrt(fT1+fT2);
504	+	roots.push_back( -((Real)0.25)*fC3+((Real)0.5)*fD);
505	+	roots.push_back( -((Real)0.25)*fC3-((Real)0.5)*fD);
506	+	}
507	+	if (fT1-fT2 >= fEpsilon) {
508	+	Real fE = sqrt(fT1-fT2);
509	+	roots.push_back( -((Real)0.25)*fC3+((Real)0.5)*fE);
510	+	roots.push_back( -((Real)0.25)*fC3-((Real)0.5)*fE);
511	+	}
512	+	}
513	+	}
514	+	}
515	+	return roots;
516	+	default: {
517	+	DynamicRectMatrix<Real> companion = CreateCompanion();
518	+	JAMA::Eigenvalue<Real> eig(companion);
519	+	DynamicVector<Real> reals, imags;
520	+	eig.getRealEigenvalues(reals);
521	+	eig.getImagEigenvalues(imags);
522	+
523	+	for (int i = 0; i < deg; i++) {
524	+	if (fabs(imags(i)) < fEpsilon)
525	+	roots.push_back(reals(i));
526	+	}
527	+	}
528	+	return roots;
529	+	}
530	+	}
531	+
532	+	private:
533	+
534	+	PolynomialPairMap polyPairMap_;
535	+	};
536	+
537	+
538	+	/**
539	+	* Generates and returns the product of two given Polynomials.
540	+	* @return A Polynomial containing the product of the two given Polynomial parameters
541	+	*/
542	+	template<typename Real>
543	+	Polynomial<Real> operator *(const Polynomial<Real>& p1, const Polynomial<Real>& p2) {
544	+	typename Polynomial<Real>::const_iterator i;
545	+	typename Polynomial<Real>::const_iterator j;
546	+	Polynomial<Real> p;
547	+
548	+	for (i = p1.begin(); i !=p1.end(); ++i) {
549	+	for (j = p2.begin(); j !=p2.end(); ++j) {
550	+	p.addCoefficient( i->first + j->first, i->second * j->second);
551	+	}
552	+	}
553	+
554	+	return p;
555	+	}
556	+
557	+	template<typename Real>
558	+	Polynomial<Real> operator *(const Polynomial<Real>& p, const Real v) {
559	+	typename Polynomial<Real>::const_iterator i;
560	+	Polynomial<Real> result;
561	+
562	+	for (i = p.begin(); i !=p.end(); ++i) {
563	+	result.setCoefficient( i->first , i->second * v);
564	+	}
565	+
566	+	return result;
567	+	}
568	+
569	+	template<typename Real>
570	+	Polynomial<Real> operator *( const Real v, const Polynomial<Real>& p) {
571	+	typename Polynomial<Real>::const_iterator i;
572	+	Polynomial<Real> result;
573	+
574	+	for (i = p.begin(); i !=p.end(); ++i) {
575	+	result.setCoefficient( i->first , i->second * v);
576	+	}
577	+
578	+	return result;
579	+	}
580	+
581	+	/**
582	+	* Generates and returns the sum of two given Polynomials.
583	+	* @param p1 the first polynomial
584	+	* @param p2 the second polynomial
585	+	*/
586	+	template<typename Real>
587	+	Polynomial<Real> operator +(const Polynomial<Real>& p1, const Polynomial<Real>& p2) {
588	+	Polynomial<Real> p(p1);
589	+
590	+	typename Polynomial<Real>::const_iterator i;
591	+
592		for (i =  p2.begin(); i  != p2.end(); ++i) {
593	<	p.addCoefficient(i->first, i->second);
593	>	p.addCoefficient(i->first, i->second);
594		}
595
596		return p;
597
598	<	}
598	>	}
599
600	<	/**
601	<	* Generates and returns the difference of two given Polynomials.
602	<	* @return
603	<	* @param p1 the first polynomial
604	<	* @param p2 the second polynomial
605	<	*/
606	<	template<typename ElemType>
607	<	Polynomial<ElemType> operator -(const Polynomial<ElemType>& p1, const Polynomial<ElemType>& p2) {
608	<	Polynomial<ElemType> p(p1);
600	>	/**
601	>	* Generates and returns the difference of two given Polynomials.
602	>	* @return
603	>	* @param p1 the first polynomial
604	>	* @param p2 the second polynomial
605	>	*/
606	>	template<typename Real>
607	>	Polynomial<Real> operator -(const Polynomial<Real>& p1, const Polynomial<Real>& p2) {
608	>	Polynomial<Real> p(p1);
609
610	<	typename Polynomial<ElemType>::const_iterator i;
610	>	typename Polynomial<Real>::const_iterator i;
611
612		for (i =  p2.begin(); i  != p2.end(); ++i) {
613	<	p.addCoefficient(i->first, -i->second);
613	>	p.addCoefficient(i->first, -i->second);
614		}
615
616		return p;
617
618	<	}
618	>	}
619
620	<	/**
621	<	* Tests if two polynomial have the same exponents
622	<	* @return true if these all of the exponents in these Polynomial are identical
623	<	* @param p1 the first polynomial
624	<	* @param p2 the second polynomial
625	<	* @note this function does not compare the coefficient
626	<	*/
627	<	template<typename ElemType>
628	<	bool equal(const Polynomial<ElemType>& p1, const Polynomial<ElemType>& p2) {
620	>	/**
621	>	* Returns the first derivative of this polynomial.
622	>	* @return the first derivative of this polynomial
623	>	*/
624	>	template<typename Real>
625	>	Polynomial<Real> * getDerivative(const Polynomial<Real>& p1) {
626	>	Polynomial<Real> * p = new Polynomial<Real>();
627	>
628	>	typename Polynomial<Real>::const_iterator i;
629	>	int exponent;
630	>	Real coefficient;
631	>
632	>	for (i =  p1.begin(); i  != p1.end(); ++i) {
633	>	exponent = i->first;
634	>	coefficient = i->second;
635	>	p->setCoefficient(exponent-1, coefficient * exponent);
636	>	}
637	>
638	>	return p;
639	>	}
640
641	<	typename Polynomial<ElemType>::const_iterator i;
642	<	typename Polynomial<ElemType>::const_iterator j;
641	>	/**
642	>	* Tests if two polynomial have the same exponents
643	>	* @return true if all of the exponents in these Polynomial are identical
644	>	* @param p1 the first polynomial
645	>	* @param p2 the second polynomial
646	>	* @note this function does not compare the coefficient
647	>	*/
648	>	template<typename Real>
649	>	bool equal(const Polynomial<Real>& p1, const Polynomial<Real>& p2) {
650
651	+	typename Polynomial<Real>::const_iterator i;
652	+	typename Polynomial<Real>::const_iterator j;
653	+
654		if (p1.size() != p2.size() ) {
655	<	return false;
655	>	return false;
656		}
657
658		for (i =  p1.begin(), j = p2.begin(); i  != p1.end() && j != p2.end(); ++i, ++j) {
659	<	if (i->first != j->first) {
660	<	return false;
661	<	}
659	>	if (i->first != j->first) {
660	>	return false;
661	>	}
662		}
663
664		return true;
665	<	}
665	>	}
666
279	–	typedef Polynomial<double> DoublePolynomial;
667
668	<	} //end namespace oopse
668	>
669	>	typedef Polynomial<RealType> DoublePolynomial;
670	>
671	>	} //end namespace OpenMD
672		#endif //MATH_POLYNOMIAL_HPP

Comparing:
trunk/src/math/Polynomial.hpp (property svn:keywords), Revision 385 by tim, Tue Mar 1 20:10:14 2005 UTC vs.
branches/development/src/math/Polynomial.hpp (property svn:keywords), Revision 1874 by gezelter, Wed May 15 15:09:35 2013 UTC

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