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/* |
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* Copyright (c) 2005 The University of Notre Dame. All Rights Reserved. |
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* |
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* The University of Notre Dame grants you ("Licensee") a |
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* non-exclusive, royalty free, license to use, modify and |
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* redistribute this software in source and binary code form, provided |
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* that the following conditions are met: |
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* |
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* 1. Acknowledgement of the program authors must be made in any |
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* publication of scientific results based in part on use of the |
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* program. An acceptable form of acknowledgement is citation of |
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* the article in which the program was described (Matthew |
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* A. Meineke, Charles F. Vardeman II, Teng Lin, Christopher |
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* J. Fennell and J. Daniel Gezelter, "OOPSE: An Object-Oriented |
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* Parallel Simulation Engine for Molecular Dynamics," |
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* J. Comput. Chem. 26, pp. 252-271 (2005)) |
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* |
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* 2. Redistributions of source code must retain the above copyright |
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* notice, this list of conditions and the following disclaimer. |
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* |
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* 3. Redistributions in binary form must reproduce the above copyright |
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* notice, this list of conditions and the following disclaimer in the |
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* documentation and/or other materials provided with the |
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* distribution. |
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* |
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* This software is provided "AS IS," without a warranty of any |
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* kind. All express or implied conditions, representations and |
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* warranties, including any implied warranty of merchantability, |
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* fitness for a particular purpose or non-infringement, are hereby |
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* excluded. The University of Notre Dame and its licensors shall not |
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* be liable for any damages suffered by licensee as a result of |
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* using, modifying or distributing the software or its |
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* derivatives. In no event will the University of Notre Dame or its |
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* licensors be liable for any lost revenue, profit or data, or for |
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* direct, indirect, special, consequential, incidental or punitive |
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* damages, however caused and regardless of the theory of liability, |
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* arising out of the use of or inability to use software, even if the |
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* University of Notre Dame has been advised of the possibility of |
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* such damages. |
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*/ |
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|
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/** |
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* @file Polynomial.hpp |
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* @author teng lin |
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* @date 11/16/2004 |
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* @version 1.0 |
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*/ |
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|
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#ifndef MATH_POLYNOMIAL_HPP |
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#define MATH_POLYNOMIAL_HPP |
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|
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#include <iostream> |
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#include <list> |
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#include <map> |
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#include <utility> |
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#include <complex> |
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#include "config.h" |
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#include "math/Eigenvalue.hpp" |
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|
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namespace oopse { |
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|
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template<typename Real> Real fastpow(Real x, int N) { |
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Real result(1); //or 1.0? |
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|
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for (int i = 0; i < N; ++i) { |
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result *= x; |
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} |
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|
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return result; |
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} |
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|
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/** |
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* @class Polynomial Polynomial.hpp "math/Polynomial.hpp" |
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* A generic Polynomial class |
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*/ |
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template<typename Real> |
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class Polynomial { |
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|
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public: |
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typedef Polynomial<Real> PolynomialType; |
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typedef int ExponentType; |
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typedef Real CoefficientType; |
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typedef std::map<ExponentType, CoefficientType> PolynomialPairMap; |
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typedef typename PolynomialPairMap::iterator iterator; |
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typedef typename PolynomialPairMap::const_iterator const_iterator; |
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|
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Polynomial() {} |
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Polynomial(Real v) {setCoefficient(0, v);} |
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/** |
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* Calculates the value of this Polynomial evaluated at the given x value. |
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* @return The value of this Polynomial evaluates at the given x value |
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* @param x the value of the independent variable for this |
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* Polynomial function |
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*/ |
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Real evaluate(const Real& x) { |
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Real result = Real(); |
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ExponentType exponent; |
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CoefficientType coefficient; |
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|
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for (iterator i = polyPairMap_.begin(); i != polyPairMap_.end(); ++i) { |
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exponent = i->first; |
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coefficient = i->second; |
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result += fastpow(x, exponent) * coefficient; |
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} |
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|
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return result; |
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} |
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|
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/** |
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* Returns the first derivative of this polynomial. |
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* @return the first derivative of this polynomial |
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* @param x |
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*/ |
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Real evaluateDerivative(const Real& x) { |
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Real result = Real(); |
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ExponentType exponent; |
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CoefficientType coefficient; |
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|
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for (iterator i = polyPairMap_.begin(); i != polyPairMap_.end(); ++i) { |
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exponent = i->first; |
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coefficient = i->second; |
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result += fastpow(x, exponent - 1) * coefficient * exponent; |
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} |
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|
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return result; |
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} |
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|
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|
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/** |
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* Set the coefficent of the specified exponent, if the |
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* coefficient is already there, it will be overwritten. |
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* @param exponent exponent of a term in this Polynomial |
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* @param coefficient multiplier of a term in this Polynomial |
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*/ |
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void setCoefficient(int exponent, const Real& coefficient) { |
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polyPairMap_[exponent] = coefficient; |
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} |
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|
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/** |
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* Set the coefficent of the specified exponent. If the |
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* coefficient is already there, just add the new coefficient to |
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* the old one, otherwise, just call setCoefficent |
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* @param exponent exponent of a term in this Polynomial |
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* @param coefficient multiplier of a term in this Polynomial |
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*/ |
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void addCoefficient(int exponent, const Real& coefficient) { |
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iterator i = polyPairMap_.find(exponent); |
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|
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if (i != end()) { |
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i->second += coefficient; |
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} else { |
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setCoefficient(exponent, coefficient); |
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} |
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} |
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|
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/** |
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* Returns the coefficient associated with the given power for |
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* this Polynomial. |
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* @return the coefficient associated with the given power for |
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* this Polynomial |
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* @exponent exponent of any term in this Polynomial |
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*/ |
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Real getCoefficient(ExponentType exponent) { |
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iterator i = polyPairMap_.find(exponent); |
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|
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if (i != end()) { |
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return i->second; |
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} else { |
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return Real(0); |
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} |
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} |
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|
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iterator begin() { |
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return polyPairMap_.begin(); |
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} |
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|
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const_iterator begin() const{ |
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return polyPairMap_.begin(); |
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} |
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|
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iterator end() { |
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return polyPairMap_.end(); |
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} |
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|
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const_iterator end() const{ |
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return polyPairMap_.end(); |
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} |
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|
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iterator find(ExponentType exponent) { |
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return polyPairMap_.find(exponent); |
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} |
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|
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size_t size() { |
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return polyPairMap_.size(); |
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} |
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|
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int degree() { |
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int deg = 0; |
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for (iterator i = polyPairMap_.begin(); i != polyPairMap_.end(); ++i) { |
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if (i->first > deg) |
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deg = i->first; |
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} |
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return deg; |
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} |
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|
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PolynomialType& operator = (const PolynomialType& p) { |
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|
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if (this != &p) // protect against invalid self-assignment |
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{ |
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typename Polynomial<Real>::const_iterator i; |
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|
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polyPairMap_.clear(); // clear out the old map |
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|
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for (i = p.begin(); i != p.end(); ++i) { |
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this->setCoefficient(i->first, i->second); |
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} |
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} |
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// by convention, always return *this |
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return *this; |
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} |
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|
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PolynomialType& operator += (const PolynomialType& p) { |
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typename Polynomial<Real>::const_iterator i; |
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|
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for (i = p.begin(); i != p.end(); ++i) { |
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this->addCoefficient(i->first, i->second); |
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} |
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|
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return *this; |
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} |
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|
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PolynomialType& operator -= (const PolynomialType& p) { |
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typename Polynomial<Real>::const_iterator i; |
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for (i = p.begin(); i != p.end(); ++i) { |
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this->addCoefficient(i->first, -i->second); |
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} |
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return *this; |
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} |
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|
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PolynomialType& operator *= (const PolynomialType& p) { |
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typename Polynomial<Real>::const_iterator i; |
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typename Polynomial<Real>::const_iterator j; |
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Polynomial<Real> p2(*this); |
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|
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polyPairMap_.clear(); // clear out old map |
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for (i = p2.begin(); i !=p2.end(); ++i) { |
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for (j = p.begin(); j !=p.end(); ++j) { |
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this->addCoefficient( i->first + j->first, i->second * j->second); |
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} |
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} |
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return *this; |
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} |
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|
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//PolynomialType& operator *= (const Real v) |
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PolynomialType& operator *= (const Real v) { |
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typename Polynomial<Real>::const_iterator i; |
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//Polynomial<Real> result; |
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|
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for (i = this->begin(); i != this->end(); ++i) { |
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this->setCoefficient( i->first, i->second*v); |
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} |
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|
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return *this; |
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} |
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|
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PolynomialType& operator += (const Real v) { |
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this->addCoefficient( 0, v); |
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return *this; |
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} |
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|
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/** |
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* Returns the first derivative of this polynomial. |
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* @return the first derivative of this polynomial |
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*/ |
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PolynomialType & getDerivative() { |
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Polynomial<Real> p(); |
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|
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typename Polynomial<Real>::const_iterator i; |
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ExponentType exponent; |
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CoefficientType coefficient; |
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|
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for (i = this->begin(); i != this->end(); ++i) { |
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exponent = i->first; |
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coefficient = i->second; |
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p.setCoefficient(exponent-1, coefficient * exponent); |
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} |
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|
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return p; |
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} |
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|
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// Creates the Companion matrix for a given polynomial |
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DynamicRectMatrix<Real> CreateCompanion() { |
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int rank = degree(); |
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DynamicRectMatrix<Real> mat(rank, rank); |
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Real majorCoeff = getCoefficient(rank); |
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for(int i = 0; i < rank; ++i) { |
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for(int j = 0; j < rank; ++j) { |
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if(i - j == 1) { |
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mat(i, j) = 1; |
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} else if(j == rank-1) { |
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mat(i, j) = -1 * getCoefficient(i) / majorCoeff; |
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} |
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} |
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} |
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return mat; |
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} |
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|
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// Find the Roots of a given polynomial |
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std::vector<complex<Real> > FindRoots() { |
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int rank = degree(); |
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DynamicRectMatrix<Real> companion = CreateCompanion(); |
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JAMA::Eigenvalue<Real> eig(companion); |
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DynamicVector<Real> reals, imags; |
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eig.getRealEigenvalues(reals); |
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eig.getImagEigenvalues(imags); |
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|
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std::vector<complex<Real> > roots; |
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for (int i = 0; i < rank; i++) { |
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roots.push_back(complex(reals(i), imags(i))); |
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} |
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|
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return roots; |
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} |
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|
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std::vector<Real> FindRealRoots() { |
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|
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const Real fEpsilon = 1.0e-8; |
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std::vector<Real> roots; |
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roots.clear(); |
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|
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const int deg = degree(); |
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|
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switch (deg) { |
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case 1: { |
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Real fC1 = getCoefficient(1); |
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Real fC0 = getCoefficient(0); |
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roots.push_back( -fC0 / fC1); |
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return roots; |
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} |
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break; |
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case 2: { |
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Real fC2 = getCoefficient(2); |
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Real fC1 = getCoefficient(1); |
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Real fC0 = getCoefficient(0); |
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Real fDiscr = fC1*fC1 - 4.0*fC0*fC2; |
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if (abs(fDiscr) <= fEpsilon) { |
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fDiscr = (Real)0.0; |
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} |
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|
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if (fDiscr < (Real)0.0) { // complex roots only |
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return roots; |
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} |
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|
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Real fTmp = ((Real)0.5)/fC2; |
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|
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if (fDiscr > (Real)0.0) { // 2 real roots |
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fDiscr = sqrt(fDiscr); |
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roots.push_back(fTmp*(-fC1 - fDiscr)); |
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roots.push_back(fTmp*(-fC1 + fDiscr)); |
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} else { |
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roots.push_back(-fTmp * fC1); // 1 real root |
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} |
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} |
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return roots; |
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break; |
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|
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case 3: { |
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Real fC3 = getCoefficient(3); |
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Real fC2 = getCoefficient(2); |
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Real fC1 = getCoefficient(1); |
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Real fC0 = getCoefficient(0); |
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|
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// make polynomial monic, x^3+c2*x^2+c1*x+c0 |
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Real fInvC3 = ((Real)1.0)/fC3; |
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fC0 *= fInvC3; |
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fC1 *= fInvC3; |
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fC2 *= fInvC3; |
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|
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// convert to y^3+a*y+b = 0 by x = y-c2/3 |
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const Real fThird = (Real)1.0/(Real)3.0; |
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const Real fTwentySeventh = (Real)1.0/(Real)27.0; |
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Real fOffset = fThird*fC2; |
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Real fA = fC1 - fC2*fOffset; |
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Real fB = fC0+fC2*(((Real)2.0)*fC2*fC2-((Real)9.0)*fC1)*fTwentySeventh; |
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Real fHalfB = ((Real)0.5)*fB; |
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|
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Real fDiscr = fHalfB*fHalfB + fA*fA*fA*fTwentySeventh; |
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if (fabs(fDiscr) <= fEpsilon) { |
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fDiscr = (Real)0.0; |
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} |
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|
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if (fDiscr > (Real)0.0) { // 1 real, 2 complex roots |
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|
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fDiscr = sqrt(fDiscr); |
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Real fTemp = -fHalfB + fDiscr; |
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Real root; |
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if (fTemp >= (Real)0.0) { |
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root = pow(fTemp,fThird); |
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} else { |
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root = -pow(-fTemp,fThird); |
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} |
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fTemp = -fHalfB - fDiscr; |
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if ( fTemp >= (Real)0.0 ) { |
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root += pow(fTemp,fThird); |
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} else { |
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root -= pow(-fTemp,fThird); |
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} |
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root -= fOffset; |
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|
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roots.push_back(root); |
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} else if (fDiscr < (Real)0.0) { |
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const Real fSqrt3 = sqrt((Real)3.0); |
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Real fDist = sqrt(-fThird*fA); |
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Real fAngle = fThird*atan2(sqrt(-fDiscr), -fHalfB); |
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Real fCos = cos(fAngle); |
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Real fSin = sin(fAngle); |
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roots.push_back(((Real)2.0)*fDist*fCos-fOffset); |
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roots.push_back(-fDist*(fCos+fSqrt3*fSin)-fOffset); |
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roots.push_back(-fDist*(fCos-fSqrt3*fSin)-fOffset); |
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} else { |
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Real fTemp; |
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if (fHalfB >= (Real)0.0) { |
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fTemp = -pow(fHalfB,fThird); |
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} else { |
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fTemp = pow(-fHalfB,fThird); |
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} |
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roots.push_back(((Real)2.0)*fTemp-fOffset); |
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roots.push_back(-fTemp-fOffset); |
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roots.push_back(-fTemp-fOffset); |
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} |
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} |
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return roots; |
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break; |
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case 4: { |
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Real fC4 = getCoefficient(4); |
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Real fC3 = getCoefficient(3); |
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Real fC2 = getCoefficient(2); |
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Real fC1 = getCoefficient(1); |
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Real fC0 = getCoefficient(0); |
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|
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// make polynomial monic, x^4+c3*x^3+c2*x^2+c1*x+c0 |
| 442 |
Real fInvC4 = ((Real)1.0)/fC4; |
| 443 |
fC0 *= fInvC4; |
| 444 |
fC1 *= fInvC4; |
| 445 |
fC2 *= fInvC4; |
| 446 |
fC3 *= fInvC4; |
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|
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// reduction to resolvent cubic polynomial y^3+r2*y^2+r1*y+r0 = 0 |
| 449 |
Real fR0 = -fC3*fC3*fC0 + ((Real)4.0)*fC2*fC0 - fC1*fC1; |
| 450 |
Real fR1 = fC3*fC1 - ((Real)4.0)*fC0; |
| 451 |
Real fR2 = -fC2; |
| 452 |
Polynomial<Real> tempCubic; |
| 453 |
tempCubic.setCoefficient(0, fR0); |
| 454 |
tempCubic.setCoefficient(1, fR1); |
| 455 |
tempCubic.setCoefficient(2, fR2); |
| 456 |
tempCubic.setCoefficient(3, 1.0); |
| 457 |
std::vector<Real> cubeRoots = tempCubic.FindRealRoots(); // always |
| 458 |
// produces |
| 459 |
// at |
| 460 |
// least |
| 461 |
// one |
| 462 |
// root |
| 463 |
Real fY = cubeRoots[0]; |
| 464 |
|
| 465 |
Real fDiscr = ((Real)0.25)*fC3*fC3 - fC2 + fY; |
| 466 |
if (fabs(fDiscr) <= fEpsilon) { |
| 467 |
fDiscr = (Real)0.0; |
| 468 |
} |
| 469 |
|
| 470 |
if (fDiscr > (Real)0.0) { |
| 471 |
Real fR = sqrt(fDiscr); |
| 472 |
Real fT1 = ((Real)0.75)*fC3*fC3 - fR*fR - ((Real)2.0)*fC2; |
| 473 |
Real fT2 = (((Real)4.0)*fC3*fC2 - ((Real)8.0)*fC1 - fC3*fC3*fC3) / |
| 474 |
(((Real)4.0)*fR); |
| 475 |
|
| 476 |
Real fTplus = fT1+fT2; |
| 477 |
Real fTminus = fT1-fT2; |
| 478 |
if (fabs(fTplus) <= fEpsilon) { |
| 479 |
fTplus = (Real)0.0; |
| 480 |
} |
| 481 |
if (fabs(fTminus) <= fEpsilon) { |
| 482 |
fTminus = (Real)0.0; |
| 483 |
} |
| 484 |
|
| 485 |
if (fTplus >= (Real)0.0) { |
| 486 |
Real fD = sqrt(fTplus); |
| 487 |
roots.push_back(-((Real)0.25)*fC3+((Real)0.5)*(fR+fD)); |
| 488 |
roots.push_back(-((Real)0.25)*fC3+((Real)0.5)*(fR-fD)); |
| 489 |
} |
| 490 |
if (fTminus >= (Real)0.0) { |
| 491 |
Real fE = sqrt(fTminus); |
| 492 |
roots.push_back(-((Real)0.25)*fC3+((Real)0.5)*(fE-fR)); |
| 493 |
roots.push_back(-((Real)0.25)*fC3-((Real)0.5)*(fE+fR)); |
| 494 |
} |
| 495 |
} else if (fDiscr < (Real)0.0) { |
| 496 |
//roots.clear(); |
| 497 |
} else { |
| 498 |
Real fT2 = fY*fY-((Real)4.0)*fC0; |
| 499 |
if (fT2 >= -fEpsilon) { |
| 500 |
if (fT2 < (Real)0.0) { // round to zero |
| 501 |
fT2 = (Real)0.0; |
| 502 |
} |
| 503 |
fT2 = ((Real)2.0)*sqrt(fT2); |
| 504 |
Real fT1 = ((Real)0.75)*fC3*fC3 - ((Real)2.0)*fC2; |
| 505 |
if (fT1+fT2 >= fEpsilon) { |
| 506 |
Real fD = sqrt(fT1+fT2); |
| 507 |
roots.push_back( -((Real)0.25)*fC3+((Real)0.5)*fD); |
| 508 |
roots.push_back( -((Real)0.25)*fC3-((Real)0.5)*fD); |
| 509 |
} |
| 510 |
if (fT1-fT2 >= fEpsilon) { |
| 511 |
Real fE = sqrt(fT1-fT2); |
| 512 |
roots.push_back( -((Real)0.25)*fC3+((Real)0.5)*fE); |
| 513 |
roots.push_back( -((Real)0.25)*fC3-((Real)0.5)*fE); |
| 514 |
} |
| 515 |
} |
| 516 |
} |
| 517 |
} |
| 518 |
return roots; |
| 519 |
break; |
| 520 |
default: { |
| 521 |
DynamicRectMatrix<Real> companion = CreateCompanion(); |
| 522 |
JAMA::Eigenvalue<Real> eig(companion); |
| 523 |
DynamicVector<Real> reals, imags; |
| 524 |
eig.getRealEigenvalues(reals); |
| 525 |
eig.getImagEigenvalues(imags); |
| 526 |
|
| 527 |
for (int i = 0; i < deg; i++) { |
| 528 |
if (fabs(imags(i)) < fEpsilon) |
| 529 |
roots.push_back(reals(i)); |
| 530 |
} |
| 531 |
} |
| 532 |
return roots; |
| 533 |
break; |
| 534 |
} |
| 535 |
|
| 536 |
return roots; // should be empty if you got here |
| 537 |
} |
| 538 |
|
| 539 |
private: |
| 540 |
|
| 541 |
PolynomialPairMap polyPairMap_; |
| 542 |
}; |
| 543 |
|
| 544 |
|
| 545 |
/** |
| 546 |
* Generates and returns the product of two given Polynomials. |
| 547 |
* @return A Polynomial containing the product of the two given Polynomial parameters |
| 548 |
*/ |
| 549 |
template<typename Real> |
| 550 |
Polynomial<Real> operator *(const Polynomial<Real>& p1, const Polynomial<Real>& p2) { |
| 551 |
typename Polynomial<Real>::const_iterator i; |
| 552 |
typename Polynomial<Real>::const_iterator j; |
| 553 |
Polynomial<Real> p; |
| 554 |
|
| 555 |
for (i = p1.begin(); i !=p1.end(); ++i) { |
| 556 |
for (j = p2.begin(); j !=p2.end(); ++j) { |
| 557 |
p.addCoefficient( i->first + j->first, i->second * j->second); |
| 558 |
} |
| 559 |
} |
| 560 |
|
| 561 |
return p; |
| 562 |
} |
| 563 |
|
| 564 |
template<typename Real> |
| 565 |
Polynomial<Real> operator *(const Polynomial<Real>& p, const Real v) { |
| 566 |
typename Polynomial<Real>::const_iterator i; |
| 567 |
Polynomial<Real> result; |
| 568 |
|
| 569 |
for (i = p.begin(); i !=p.end(); ++i) { |
| 570 |
result.setCoefficient( i->first , i->second * v); |
| 571 |
} |
| 572 |
|
| 573 |
return result; |
| 574 |
} |
| 575 |
|
| 576 |
template<typename Real> |
| 577 |
Polynomial<Real> operator *( const Real v, const Polynomial<Real>& p) { |
| 578 |
typename Polynomial<Real>::const_iterator i; |
| 579 |
Polynomial<Real> result; |
| 580 |
|
| 581 |
for (i = p.begin(); i !=p.end(); ++i) { |
| 582 |
result.setCoefficient( i->first , i->second * v); |
| 583 |
} |
| 584 |
|
| 585 |
return result; |
| 586 |
} |
| 587 |
|
| 588 |
/** |
| 589 |
* Generates and returns the sum of two given Polynomials. |
| 590 |
* @param p1 the first polynomial |
| 591 |
* @param p2 the second polynomial |
| 592 |
*/ |
| 593 |
template<typename Real> |
| 594 |
Polynomial<Real> operator +(const Polynomial<Real>& p1, const Polynomial<Real>& p2) { |
| 595 |
Polynomial<Real> p(p1); |
| 596 |
|
| 597 |
typename Polynomial<Real>::const_iterator i; |
| 598 |
|
| 599 |
for (i = p2.begin(); i != p2.end(); ++i) { |
| 600 |
p.addCoefficient(i->first, i->second); |
| 601 |
} |
| 602 |
|
| 603 |
return p; |
| 604 |
|
| 605 |
} |
| 606 |
|
| 607 |
/** |
| 608 |
* Generates and returns the difference of two given Polynomials. |
| 609 |
* @return |
| 610 |
* @param p1 the first polynomial |
| 611 |
* @param p2 the second polynomial |
| 612 |
*/ |
| 613 |
template<typename Real> |
| 614 |
Polynomial<Real> operator -(const Polynomial<Real>& p1, const Polynomial<Real>& p2) { |
| 615 |
Polynomial<Real> p(p1); |
| 616 |
|
| 617 |
typename Polynomial<Real>::const_iterator i; |
| 618 |
|
| 619 |
for (i = p2.begin(); i != p2.end(); ++i) { |
| 620 |
p.addCoefficient(i->first, -i->second); |
| 621 |
} |
| 622 |
|
| 623 |
return p; |
| 624 |
|
| 625 |
} |
| 626 |
|
| 627 |
/** |
| 628 |
* Returns the first derivative of this polynomial. |
| 629 |
* @return the first derivative of this polynomial |
| 630 |
*/ |
| 631 |
template<typename Real> |
| 632 |
Polynomial<Real> getDerivative(const Polynomial<Real>& p1) { |
| 633 |
Polynomial<Real> p(); |
| 634 |
|
| 635 |
typename Polynomial<Real>::const_iterator i; |
| 636 |
ExponentType exponent; |
| 637 |
CoefficientType coefficient; |
| 638 |
|
| 639 |
for (i = p1.begin(); i != p1.end(); ++i) { |
| 640 |
exponent = i->first; |
| 641 |
coefficient = i->second; |
| 642 |
p.setCoefficient(exponent-1, coefficient * exponent); |
| 643 |
} |
| 644 |
|
| 645 |
return p; |
| 646 |
} |
| 647 |
|
| 648 |
/** |
| 649 |
* Tests if two polynomial have the same exponents |
| 650 |
* @return true if all of the exponents in these Polynomial are identical |
| 651 |
* @param p1 the first polynomial |
| 652 |
* @param p2 the second polynomial |
| 653 |
* @note this function does not compare the coefficient |
| 654 |
*/ |
| 655 |
template<typename Real> |
| 656 |
bool equal(const Polynomial<Real>& p1, const Polynomial<Real>& p2) { |
| 657 |
|
| 658 |
typename Polynomial<Real>::const_iterator i; |
| 659 |
typename Polynomial<Real>::const_iterator j; |
| 660 |
|
| 661 |
if (p1.size() != p2.size() ) { |
| 662 |
return false; |
| 663 |
} |
| 664 |
|
| 665 |
for (i = p1.begin(), j = p2.begin(); i != p1.end() && j != p2.end(); ++i, ++j) { |
| 666 |
if (i->first != j->first) { |
| 667 |
return false; |
| 668 |
} |
| 669 |
} |
| 670 |
|
| 671 |
return true; |
| 672 |
} |
| 673 |
|
| 674 |
|
| 675 |
|
| 676 |
typedef Polynomial<RealType> DoublePolynomial; |
| 677 |
|
| 678 |
} //end namespace oopse |
| 679 |
#endif //MATH_POLYNOMIAL_HPP |
