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  class Polynomial { | 
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  public: | 
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    typedef Polynomial<ElemType> PolynomialType;     | 
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    typedef int ExponentType; | 
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    typedef ElemType 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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    Polynomial() {} | 
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    Polynomial(ElemType 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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    size_t size() { | 
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      return polyPairMap_.size(); | 
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    } | 
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    PolynomialType& operator += (const PolynomialType& p) { | 
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        typename Polynomial<ElemType>::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<ElemType>::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<ElemType>::const_iterator i; | 
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    typename Polynomial<ElemType>::const_iterator j; | 
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     | 
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    for (i = this->begin(); i !=this->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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  private: | 
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    PolynomialPairMap polyPairMap_; | 
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    return p; | 
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  } | 
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 | 
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  template<typename ElemType> | 
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  Polynomial<ElemType> operator *(const Polynomial<ElemType>& p, const ElemType v) { | 
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    typename Polynomial<ElemType>::const_iterator i; | 
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    Polynomial<ElemType> result; | 
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     | 
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    for (i = p.begin(); i !=p.end(); ++i) { | 
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        result.addCoefficient( i->first , i->second * v); | 
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    } | 
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    return result; | 
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  } | 
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 | 
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  template<typename ElemType> | 
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  Polynomial<ElemType> operator *( const ElemType v, const Polynomial<ElemType>& p) { | 
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    typename Polynomial<ElemType>::const_iterator i; | 
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    Polynomial<ElemType> result; | 
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     | 
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    for (i = p.begin(); i !=p.end(); ++i) { | 
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        result.addCoefficient( i->first , i->second * v); | 
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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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   * Generates and returns the sum of two given Polynomials. | 
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   * @param p1 the first polynomial | 
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 | 
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  /** | 
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   * Tests if two polynomial have the same exponents | 
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   * @return true if these all of the exponents in these Polynomial are identical | 
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   * @return true if all of the exponents in these Polynomial are identical | 
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   * @param p1 the first polynomial | 
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   * @param p2 the second polynomial | 
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   * @note this function does not compare the coefficient |