    /**************************************************************************\

    MODULE: vec_RR

    SUMMARY:

    Defines the class vec_RR.

    \**************************************************************************/

    typedef Vec<RR> vec_RR; // backward compatibility

    void mul(vec_RR& x, const vec_RR& a, const RR& b);
    void mul(vec_RR& x, const vec_RR& a, double b);

    void mul(vec_RR& x, const RR& a, const vec_RR& b);
    void mul(vec_RR& x, double a, const vec_RR& b);
    // x = a * b

    void add(vec_RR& x, const vec_RR& a, const vec_RR& b);
    // x = a + b

    void sub(vec_RR& x, const vec_RR& a, const vec_RR& b);
    // x = a - b

    void clear(vec_RR& x);
    // x = 0 (length unchanged)

    void negate(vec_RR& x, const vec_RR& a);
    // x = -a

    long IsZero(const vec_RR& a);
    // test if a is the zero vector

    void InnerProduct(RR& x, const vec_RR& a, const vec_RR& b);
    // x = inner product of a and b, padded with zeros to make the lengths
    // even.

    void VectorCopy(vec_RR& x, const vec_RR& a, long n);
    vec_RR VectorCopy(const vec_RR& a, long n);
    // x = a copy of a of length exactly n.
    // The input is truncated or padded with zeroes, as necessary.

    // operator notation:

    vec_RR operator+(const vec_RR& a, const vec_RR& b);
    vec_RR operator-(const vec_RR& a, const vec_RR& b);

    vec_RR operator-(const vec_RR& a);

    // vector/scalar multiplication:

    vec_RR operator*(const vec_RR& a, const RR& b);
    vec_RR operator*(const vec_RR& a, double b);

    vec_RR operator*(const RR& a, const vec_RR& b);
    vec_RR operator*(double a, const vec_RR& b);

    // inner product:

    RR operator*(const vec_RR& a, const vec_RR& b);

    // assignment operator notation:

    vec_RR& operator+=(vec_RR& x, const vec_RR& a);
    vec_RR& operator-=(vec_RR& x, const vec_RR& a);

    vec_RR& operator*=(vec_RR& x, const RR& a);
    vec_RR& operator*=(vec_RR& x, double a);