______________________________________________________________________

  26   Numerics library                                   [lib.numerics]

  ______________________________________________________________________

1 This clause describes components that C++ programs may use to  perform
  seminumerical operations.

2 The following subclauses describe components for complex number types,
  numeric ( n-at-a-time) arrays,  generalized  numeric  algorithms,  and
  facilities included from the ISO C library, as summarized in Table 1:

                    Table 1--Numerics library summary

     +--------------------------------------------------------------+
     |                   Subclause                       Header(s)  |
     +--------------------------------------------------------------+
     |_lib.numeric.requirements_ Requirements                       |
     +--------------------------------------------------------------+
     |_lib.complex.numbers_ Complex numbers              <complex>  |
     +--------------------------------------------------------------+
     |_lib.numarray_ Numeric arrays                      <valarray> |
     +--------------------------------------------------------------+
     |_lib.numeric.ops_ Generalized numeric operations   <numeric>  |
     +--------------------------------------------------------------+
     |_lib.c.math_ C library                             <cmath>    |
     |                                                   <cstdlib>  |
     +--------------------------------------------------------------+

  26.1  Numeric type requirements             [lib.numeric.requirements]

1 The  complex  and valarray components are parameterized by the type of
  information they contain and manipulate.  A C++ program shall  instan-
  tiate these components only with a type T that satisfies the following
  requirements:1)

  --T is not an abstract class (it has  no  pure  virtual  member  func-
    tions);

  --T is not a reference type;

  _________________________
  1) In other words, value types.   These  include  built-in  arithmetic
  types,  pointers,  the  library  class  complex, and instantiations of
  valarray for value types.

  --T is not cv-qualified;

  --If T is a class, it has a public default constructor;

  --If T is a class, it has a public copy constructor with the signature
    T::T(const T&)

  --If T is a class, it has a public destructor;

  --If T is a class, it has a public assignment operator whose signature
    is either
    T& T::operator=(const T&) or T& T::operator=(T)

  --If T is a class, its assignment operator, copy and default construc-
    tors, and destructor shall correspond to each other in the following
    sense:  Initialization of raw storage using the default constructor,
    followed by assignment, is semantically equivalent to initialization
    of  raw  storage  using  the  copy  constructor.   Destruction of an
    object, followed by initialization of its raw storage using the copy
    constructor,  is semantically equivalent to assignment to the origi-
    nal object.
    [Note: This rule states that there shall not be any  subtle  differ-
    ences  in  the  semantics of initialization versus assignment.  This
    gives an implementation considerable flexibility in how  arrays  are
    initialized.
    [Example:  An  implementation is allowed to initialize a valarray by
    allocating storage using the new operator (which implies a  call  to
    the  default  constructor  for each element) and then assigning each
    element its value.  Or the implementation can allocate  raw  storage
    and  use  the  copy  constructor to initialize each element.   --end
    example]
    If the distinction between initialization and assignment  is  impor-
    tant  for a class, or if it fails to satisfy any of the other condi-
    tions listed above, the programmer should use vector  (_lib.vector_)
    instead of valarray for that class;  --end note]

  --If T is a class, it does not overload unary operator&.

2 In  addition,  many member and related functions of valarray<T> can be
  successfully instantiated and will exhibit  well-defined  behavior  if
  and  only  if  T  satisfies additional requirements specified for each
  such member or related function.

3 [Example: It is valid to  instantiate  valarray<complex>,  but  opera-
  tor>()  will  not  be  successfully instantiated for valarray<complex>
  operands, since complex does not have any ordering operators.    --end
  example]

  26.2  Complex numbers                            [lib.complex.numbers]

1 The  header <complex> defines a template class, and numerous functions
  for representing and manipulating complex numbers.

2 The  effect  of  instantiating the template complex for any type other
  than float, double or long double is unspecified.

3 If the result of a function is not mathematically defined  or  not  in
  the  range of representable values for its type, the behavior is unde-
  fined.

  26.2.1  Header <complex> synopsis               [lib.complex.synopsis]
  namespace std {
    template<class T> class complex;
    class complex<float>;
    class complex<double>;
    class complex<long double>;
    // _lib.complex.ops_ operators:
    template<class T>
      complex<T> operator+(const complex<T>&, const complex<T>&);
    template<class T> complex<T> operator+(const complex<T>&, const T&);
    template<class T> complex<T> operator+(const T&, const complex<T>&);
    template<class T> complex<T> operator-(const complex<T>&, const complex<T>&);
    template<class T> complex<T> operator-(const complex<T>&, const T&);
    template<class T> complex<T> operator-(const T&, const complex<T>&);
    template<class T> complex<T> operator*(const complex<T>&, const complex<T>&);
    template<class T> complex<T> operator*(const complex<T>&, const T&);
    template<class T> complex<T> operator*(const T&, const complex<T>&);
    template<class T> complex<T> operator/(const complex<T>&, const complex<T>&);
    template<class T> complex<T> operator/(const complex<T>&, const T&);
    template<class T> complex<T> operator/(const T&, const complex<T>&);
    template<class T> complex<T> operator+(const complex<T>&);
    template<class T> complex<T> operator-(const complex<T>&);
    template<class T> bool operator==(const complex<T>&, const complex<T>&);
    template<class T> bool operator==(const complex<T>&, const T&);
    template<class T> bool operator==(const T&, const complex<T>&);
    template<class T> bool operator!=(const complex<T>&, const complex<T>&);
    template<class T> bool operator!=(const complex<T>&, const T&);
    template<class T> bool operator!=(const T&, const complex<T>&);
    template<class T, class charT, class traits>
    basic_istream<charT, traits>&
    operator>>(basic_istream<charT, traits>&, complex<T>&);

    template<class T, class charT, class traits>
    basic_ostream<charT, traits>&
    operator<<(basic_ostream<charT, traits>&, const complex<T>&);
    // _lib.complex.value.ops_ values:
    template<class T> T real(const complex<T>&);
    template<class T> T imag(const complex<T>&);
    template<class T> T abs(const complex<T>&);
    template<class T> T arg(const complex<T>&);
    template<class T> T norm(const complex<T>&);
    template<class T> complex<T> conj(const complex<T>&);
    template<class T> complex<T> polar(const T&, const T&);

    // _lib.complex.transcendentals_ transcendentals:
    template<class T> complex<T> cos  (const complex<T>&);
    template<class T> complex<T> cosh (const complex<T>&);
    template<class T> complex<T> exp  (const complex<T>&);
    template<class T> complex<T> log  (const complex<T>&);
    template<class T> complex<T> log10(const complex<T>&);
    template<class T> complex<T> pow(const complex<T>&, int);
    template<class T> complex<T> pow(const complex<T>&, const T&);
    template<class T> complex<T> pow(const complex<T>&, const complex<T>&);
    template<class T> complex<T> pow(const T&, const complex<T>&);
    template<class T> complex<T> sin  (const complex<T>&);
    template<class T> complex<T> sinh (const complex<T>&);
    template<class T> complex<T> sqrt (const complex<T>&);
    template<class T> complex<T> tan  (const complex<T>&);
    template<class T> complex<T> tanh (const complex<T>&);
  }

  26.2.2  Template class complex                           [lib.complex]
  namespace std {
    template<class T>
    class complex {
    public:
      typedef T value_type;

      complex(const T& re = T(), const T& im = T());
      complex(const complex&);
      template<class X> complex(const complex<X>&);

      T real() const;
      T imag() const;

      complex<T>& operator= (const T&);
      complex<T>& operator+=(const T&);
      complex<T>& operator-=(const T&);
      complex<T>& operator*=(const T&);
      complex<T>& operator/=(const T&);

      complex& operator=(const complex&);
      template<class X> complex<T>& operator= (const complex<X>&);
      template<class X> complex<T>& operator+=(const complex<X>&);
      template<class X> complex<T>& operator-=(const complex<X>&);
      template<class X> complex<T>& operator*=(const complex<X>&);
      template<class X> complex<T>& operator/=(const complex<X>&);
    };

  template<class T> complex<T> operator+(const complex<T>&, const T&);
  template<class T> complex<T> operator+(const T&, const complex<T>&);
  template<class T> complex<T> operator-(const complex<T>&, const T&);
  template<class T> complex<T> operator-(const T&, const complex<T>&);
  template<class T> complex<T> operator*(const complex<T>&, const T&);
  template<class T> complex<T> operator*(const T&, const complex<T>&);
  template<class T> complex<T> operator/(const complex<T>&, const T&);
  template<class T> complex<T> operator/(const T&, const complex<T>&);
  template<class T> complex<T> operator==(const complex<T>&, const T&);
  template<class T> complex<T> operator==(const T&, const complex<T>&);
  template<class T> complex<T> operator!=(const complex<T>&, const T&);
  template<class T> complex<T> operator!=(const T&, const complex<T>&);

1 The class complex describes an object that  can  store  the  Cartesian
  components, real() and imag(), of a complex number.

  26.2.3  complex specializations                  [lib.complex.special]
    class complex<float> {
    public:
      typedef float value_type;

      complex(float re = 0.0f, float im = 0.0f);
      explicit complex(const complex<double>&);
      explicit complex(const complex<long double>&);

      float real() const;
      float imag() const;

      complex<float>& operator= (float);
      complex<float>& operator+=(float);
      complex<float>& operator-=(float);
      complex<float>& operator*=(float);
      complex<float>& operator/=(float);

      complex<float>& operator=(const complex<float>&);
      template<class X> complex<float>& operator= (const complex<X>&);
      template<class X> complex<float>& operator+=(const complex<X>&);
      template<class X> complex<float>& operator-=(const complex<X>&);
      template<class X> complex<float>& operator*=(const complex<X>&);
      template<class X> complex<float>& operator/=(const complex<X>&);
    };
    class complex<double> {
    public:
      typedef double value_type;

      complex(double re = 0.0, double im = 0.0);
      complex(const complex<float>&);
      explicit complex(const complex<long double>&);
      double real() const;
      double imag() const;

      complex<double>& operator= (double);
      complex<double>& operator+=(double);
      complex<double>& operator-=(double);
      complex<double>& operator*=(double);
      complex<double>& operator/=(double);

      complex<double>& operator=(const complex<double>&);
      template<class X> complex<double>& operator= (const complex<X>&);
      template<class X> complex<double>& operator+=(const complex<X>&);
      template<class X> complex<double>& operator-=(const complex<X>&);
      template<class X> complex<double>& operator*=(const complex<X>&);
      template<class X> complex<double>& operator/=(const complex<X>&);
    };
    class complex<long double> {
    public:
      typedef long double value_type;

      complex(long double re = 0.0L, long double im = 0.0L);
      complex(const complex<float>&);
      complex(const complex<double>&);

      long double real() const;
      long double imag() const;

      complex<long double>& operator=(const complex<long double>&);
      complex<long double>& operator= (long double&);
      complex<long double>& operator+=(long double&);
      complex<long double>& operator-=(long double&);
      complex<long double>& operator*=(long double&);
      complex<long double>& operator/=(long double&);

      template<class X> complex<long double>& operator= (const complex<X>&);
      template<class X> complex<long double>& operator+=(const complex<X>&);
      template<class X> complex<long double>& operator-=(const complex<X>&);
      template<class X> complex<long double>& operator*=(const complex<X>&);
      template<class X> complex<long double>& operator/=(const complex<X>&);
    };

  26.2.4  complex member functions                 [lib.complex.members]

  template<class T> complex(const T& re = T(), const T& im = T());

  Effects:
    Constructs an object of class complex.

1 Postcondition: real() == re  && imag() == im.

  26.2.5  complex member operators              [lib.complex.member.ops]

  complex<T>& operator+=(const T& rhs);

  Effects:
    Adds  the  scalar  value  rhs  to the real part of the complex value
    *this and stores the result in the real part of *this,  leaving  the
    imaginary part unchanged.
  Returns:
    *this.

  complex<T>& operator-=(const T& rhs);

  Effects:
    Subtracts  the  scalar  value  rhs from the real part of the complex
    value *this and stores the result in the real part of *this, leaving
    the imaginary part unchanged.
  Returns:
    *this.

  complex<T>& operator*=(const T& rhs);

  Effects:
    Multiplies  the  scalar  value  rhs  by  the complex value *this and
    stores the result in *this.
  Returns:
    *this.

  complex<T>& operator/=(const T& rhs);

  Effects:
    Divides the scalar value rhs into the complex value *this and stores
    the result in *this.
  Returns:
    *this.

  template<class T> complex<T>& operator+=(const complex<T>& rhs);

  Effects:
    Adds the complex value rhs to the complex value *this and stores the
    sum in *this.
  Returns:
    *this.

  template<class T> complex<T>& operator-=(const complex<T>& rhs);

  Effects:
    Subtracts the complex value rhs from the  complex  value  *this  and
    stores the difference in *this.
  Returns:
    *this.

  template<class T> complex<T>& operator*=(const complex<T>& rhs);

  Effects:
    Multiplies  the  complex  value  rhs  by the complex value *this and
    stores the product in *this.
  Returns:
    *this.

  template<class T> complex<T>& operator/=(const complex<T>& rhs);

  Effects:
    Divides the complex value rhs  into  the  complex  value  *this  and
    stores the quotient in *this.
  Returns:
    *this.

  26.2.6  complex non-member operations                [lib.complex.ops]

  template<class T> complex<T> operator+(const complex<T>& lhs);

  Notes:
    unary operator.
  Returns:
    complex<T>(lhs).

  template<class T>
    complex<T> operator+(const complex<T>& lhs, const complex<T>& rhs);
  template<class T> complex<T> operator+(const complex<T>& lhs, const T& rhs);
  template<class T> complex<T> operator+(const T& lhs, const complex<T>& rhs);

  Returns:
    complex<T>(lhs) += rhs.

  template<class T> complex<T> operator-(const complex<T>& lhs);

  Notes:
    unary operator.
  Returns:
    complex<T>(-lhs.real(),-lhs.imag()).

  template<class T>
    complex<T> operator-(const complex<T>& lhs, const complex<T>& rhs);
  template<class T> complex<T> operator-(const complex<T>& lhs, const T& rhs);
  template<class T> complex<T> operator-(const T& lhs, const complex<T>& rhs);

  Returns:
    complex<T>(lhs) -= rhs.

  template<class T>
    complex<T> operator*(const complex<T>& lhs, const complex<T>& rhs);
  template<class T> complex<T> operator*(const complex<T>& lhs, const T& rhs);
  template<class T> complex<T> operator*(const T& lhs, const complex<T>& rhs);

  Returns:
    complex<T>(lhs) *= rhs.

  template<class T>
    complex<T> operator/(const complex<T>& lhs, const complex<T>& rhs);
  template<class T> complex<T> operator/(const complex<T>& lhs, const T& rhs);
  template<class T> complex<T> operator/(const T& lhs, const complex<T>& rhs);

  Returns:
    complex<T>(lhs) /= rhs.

  template<class T>
    bool operator==(const complex<T>& lhs, const complex<T>& >rhs);
  template<class T> bool operator==(const complex<T>& lhs, const T& rhs);
  template<class T> bool operator==(const T& lhs, const complex<T>& rhs);

  Returns:
    lhs.real() == rhs.real() && lhs.imag() == rhs.imag().
  Notes:
    The  imaginary  part  is  assumed to be T(), or 0.0, for the T argu-
    ments.

  template<class T>
    bool operator!=(complex<T>& lhs, complex<T>& rhs);
  template<class T> bool operator!=(complex<T>& lhs, const T& rhs);
  template<class T> bool operator!=(const T& lhs, complex<T>& rhs);

  Returns:
    rhs.real() != lhs.real() || rhs.imag() != lhs.imag().

  template<class T, class charT, class traits>
  basic_istream<charT, traits>&
  operator>>(basic_istream<charT, traits>& is, complex<T>& x);

  Effects:
    Extracts a complex number x of the form: u, (u), or (u,v),  where  u
    is  the  real part and v is the imaginary part (_lib.istream.format-
    ted_).
  Requires:
    The input values be convertible to T.
    If bad input is encountered, calls is.setstate(ios::failbit)  (which
    may throw ios::failure (_lib.iostate.flags_).
  Returns:
    is.

  template<class T, class charT, class traits>
  basic_ostream<charT, traits>&
  operator<<(basic_ostream<charT, traits>& o, const complex<T>& x);

  Effects:
    inserts  the complex number x onto the stream o as if it were imple-
    mented as follows:

      template<class T, class charT, class traits>
      basic_ostream<charT, traits>&
      operator<<(basic_ostream<charT, traits>& o, const complex<T>& x)
      {
              basic_ostringstream<charT, traits> s;
              s.flags(o.flags());
              s.imbue(o.getloc());
              s.precision(o.precision());
              s << '(' << x.real() << "," << x.imag() << ')' << ends;
              return o << s.str();
      }

  26.2.7  complex value operations               [lib.complex.value.ops]

  template<class T> T real(const complex<T>& x);

  Returns:
    x.real().

  template<class T> T imag(const complex<T>& x);

  Returns:
    x.imag().

  template<class T> T abs(const complex<T>& x);

  Returns:
    the magnitude of x.

  template<class T> T arg(const complex<T>& x);

  Returns:
    the phase angle of x, or atan2(imag(x), real(x)).

  template<class T> T norm(const complex<T>& x);

  Returns:
    the squared magnitude of x.

  template<class T> complex<T> conj(const complex<T>& x);

  Returns:
    the complex conjugate of x.

  template<class T> complex<T> polar(const T& rho, const T& theta = 0);

  Returns:
    the complex value corresponding to a complex number whose  magnitude
    is rho and whose phase angle is theta.

  26.2.8  complex transcendentals          [lib.complex.transcendentals]

  template<class T> complex<T> cos(const complex<T>& x);

  Returns:
    the complex cosine of x.

  template<class T> complex<T> cosh(const complex<T>& x);

  Returns:
    the complex hyperbolic cosine of x.

  template<class T> complex<T> exp(const complex<T>& x);

  Returns:
    the complex base e exponential of x.

  template<class T> complex<T> log(const complex<T>& x);

  Notes:
    the branch cuts are along the negative real axis.
  Returns:
    the complex natural (base e) logarithm of x, in the range of a strip
    mathematically unbounded along the real axis and in the interval [-i
    times  pi,  i times pi ] along the imaginary axis. When x is a nega-
    tive real number, imag(log(x)) is pi.

  template<class T> complex<T> log10(const complex<T>& x);

  Notes:
    the branch cuts are along the negative real axis.
  Returns:
    the  complex  common   (base   10)logarithm   of   x,   defined   as
    log(x)/log(10).

  template<class T> complex<T> pow(const complex<T>& x, int y);
  template<class T>
    complex<T> pow(const complex<T>& x, const complex<T>& y);
  template<class T> complex<T> pow  (const complex<T>& x, const T& y);
  template<class T> complex<T> pow  (const T& x, const complex<T>& y);

  Notes:
    the branch cuts are along the negative real axis.

  Returns:
    the  complex  power  of  base x raised to the y-th power, defined as
    exp(y*log(x)).  The value returned for pow(0,0)  is  implementation-
    defined.

  template<class T> complex<T> sin  (const complex<T>& x);

  Returns:
    the complex sine of x.

  template<class T> complex<T> sinh (const complex<T>& x);

  Returns:
    the complex hyperbolic sine of x.

  template<class T> complex<T> sqrt (const complex<T>& x);

  Notes:
    the branch cuts are along the negative real axis.
  Returns:
    the  complex square root of x, in the range of the right half-plane.
    If the argument is a negative real number, the value  returned  lies
    on the positive imaginary axis.

  template<class T> complex<T> tan  (const complex<T>& x);

  Returns:
    the complex tangent of x.

  template<class T> complex<T> tanh (const complex<T>& x);

  Returns:
    the complex hyperbolic tangent of x.

  26.3  Numeric arrays                                    [lib.numarray]

  26.3.1  Header <valarray> synopsis             [lib.valarray.synopsis]
  namespace std {
    template<class T> class valarray;       // An array of type T
    class slice;                            // a BLAS-like slice out of an array
    template<class T> class slice_array;
    class gslice;                           // a generalized slice out of an array
    template<class T> class gslice_array;
    template<class T> class mask_array;     // a masked array
    template<class T> class indirect_array; // an indirected array

    template<class T> valarray<T> operator*
      (const valarray<T>&, const valarray<T>&);
    template<class T> valarray<T> operator* (const valarray<T>&, const T&);
    template<class T> valarray<T> operator* (const T&, const valarray<T>&);
    template<class T> valarray<T> operator/
      (const valarray<T>&, const valarray<T>&);
    template<class T> valarray<T> operator/ (const valarray<T>&, const T&);
    template<class T> valarray<T> operator/ (const T&, const valarray<T>&);
    template<class T> valarray<T> operator%
      (const valarray<T>&, const valarray<T>&);
    template<class T> valarray<T> operator% (const valarray<T>&, const T&);
    template<class T> valarray<T> operator% (const T&, const valarray<T>&);
    template<class T> valarray<T> operator+
      (const valarray<T>&, const valarray<T>&);
    template<class T> valarray<T> operator+ (const valarray<T>&, const T&);
    template<class T> valarray<T> operator+ (const T&, const valarray<T>&);
    template<class T> valarray<T> operator-
      (const valarray<T>&, const valarray<T>&);
    template<class T> valarray<T> operator- (const valarray<T>&, const T&);
    template<class T> valarray<T> operator- (const T&, const valarray<T>&);
    template<class T> valarray<T> operator^
      (const valarray<T>&, const valarray<T>&);
    template<class T> valarray<T> operator^ (const valarray<T>&, const T&);
    template<class T> valarray<T> operator^ (const T&, const valarray<T>&);
    template<class T> valarray<T> operator&
      (const valarray<T>&, const valarray<T>&);
    template<class T> valarray<T> operator& (const valarray<T>&, const T&);
    template<class T> valarray<T> operator& (const T&, const valarray<T>&);
    template<class T> valarray<T> operator|
      (const valarray<T>&, const valarray<T>&);
    template<class T> valarray<T> operator| (const valarray<T>&, const T&);
    template<class T> valarray<T> operator| (const T&, const valarray<T>&);
    template<class T> valarray<T> operator<<
      (const valarray<T>&, const valarray<T>&);
    template<class T> valarray<T> operator<<(const valarray<T>&, const T&);
    template<class T> valarray<T> operator<<(const T&, const valarray<T>&);
    template<class T> valarray<T> operator>>
      (const valarray<T>&, const valarray<T>&);
    template<class T> valarray<T> operator>>(const valarray<T>&, const T&);
    template<class T> valarray<T> operator>>(const T&, const valarray<T>&);
    template<class T> valarray<bool> operator&&
      (const valarray<T>&, const valarray<T>&);
    template<class T> valarray<bool> operator&&(const valarray<T>&, const T&);
    template<class T> valarray<bool> operator&&(const T&, const valarray<T>&);
    template<class T> valarray<bool> operator||
      (const valarray<T>&, const valarray<T>&);
    template<class T> valarray<bool> operator||(const valarray<T>&, const T&);
    template<class T> valarray<bool> operator||(const T&, const valarray<T>&);

    template<class T>
      valarray<bool> operator==(const valarray<T>&, const valarray<T>&);
    template<class T> valarray<bool> operator==(const valarray<T>&, const T&);
    template<class T> valarray<bool> operator==(const T&, const valarray<T>&);
    template<class T>
      valarray<bool> operator!=(const valarray<T>&, const valarray<T>&);
    template<class T> valarray<bool> operator!=(const valarray<T>&, const T&);
    template<class T> valarray<bool> operator!=(const T&, const valarray<T>&);
    template<class T>
      valarray<bool> operator< (const valarray<T>&, const valarray<T>&);
    template<class T> valarray<bool> operator< (const valarray<T>&, const T&);
    template<class T> valarray<bool> operator< (const T&, const valarray<T>&);
    template<class T>
      valarray<bool> operator> (const valarray<T>&, const valarray<T>&);
    template<class T> valarray<bool> operator> (const valarray<T>&, const T&);
    template<class T> valarray<bool> operator> (const T&, const valarray<T>&);
    template<class T>
      valarray<bool> operator<=(const valarray<T>&, const valarray<T>&);
    template<class T> valarray<bool> operator<=(const valarray<T>&, const T&);
    template<class T> valarray<bool> operator<=(const T&, const valarray<T>&);
    template<class T>
      valarray<bool> operator>=(const valarray<T>&, const valarray<T>&);
    template<class T> valarray<bool> operator>=(const valarray<T>&, const T&);
    template<class T> valarray<bool> operator>=(const T&, const valarray<T>&);
    template<class T> T min(const valarray<T>&);
    template<class T> T max(const valarray<T>&);
    template<class T> valarray<T> abs  (const valarray<T>&);
    template<class T> valarray<T> acos (const valarray<T>&);
    template<class T> valarray<T> asin (const valarray<T>&);
    template<class T> valarray<T> atan (const valarray<T>&);
    template<class T> valarray<T> atan2(const valarray<T>&, const valarray<T>&);
    template<class T> valarray<T> atan2(const valarray<T>&, const T&);
    template<class T> valarray<T> atan2(const T&, const valarray<T>&);
    template<class T> valarray<T> cos  (const valarray<T>&);
    template<class T> valarray<T> cosh (const valarray<T>&);
    template<class T> valarray<T> exp  (const valarray<T>&);
    template<class T> valarray<T> log  (const valarray<T>&);
    template<class T> valarray<T> log10(const valarray<T>&);
    template<class T> valarray<T> pow  (const valarray<T>&, const valarray<T>&);
    template<class T> valarray<T> pow  (const valarray<T>&, const T&);
    template<class T> valarray<T> pow  (const T&, const valarray<T>&);
    template<class T> valarray<T> sin  (const valarray<T>&);
    template<class T> valarray<T> sinh (const valarray<T>&);
    template<class T> valarray<T> sqrt (const valarray<T>&);
    template<class T> valarray<T> tan  (const valarray<T>&);
    template<class T> valarray<T> tanh (const valarray<T>&);
  }

1 The  header  <valarray>  defines  five  template  classes  ( valarray,
  slice_array,  gslice_array,  mask_array,  and   indirect_array),   two
  classes  (  slice and gslice), and a series of related function signa-
  tures for representing and manipulating arrays of values.

2 The valarray array classes are defined to be free of certain forms  of
  aliasing, thus allowing operations on these classes to be optimized.

3 Any  function returning a valarray<T> is permitted to return an object
  of another type, provided all the const  member  functions  of  valar-
  ray<T>  are  also  applicable to this type. This return type shall not
  add more than two levels of template  nesting  over  the  most  deeply
  nested argument type.2)

4 Implementations introducing such replacement types shall provide addi-
  tional functions and operators as follows:

  --for every functions taking a const valarray<T>&, identical functions
    taking the replacement types shall be added;

  --for  every function taking two const valarray<T>& arguments, identi-
    cal functions taking every combination  of  const  valarray<T>&  and
    replacement types shall be added.

5 In  particular, an implementation shall allow a valarray<T> to be con-
  structed from such replacement types and shall allow  assignments  and
  computed  assignments  of  such  types to valarray<T>, slice_array<T>,
  gslice_array<T>, mask_array<T> and indirect_array<T> objects.

6 These  library  functions  are  permitted   to   throw   a   bad_alloc
  (_lib.bad.alloc_)  exception  if  there  are  not sufficient resources
  available to carry out the operation.  Note that the exception is  not
  mandated.

  26.3.2  Template class valarray                [lib.template.valarray]
  namespace std {
    template<class T> class valarray {
    public:
      typedef T value_type;

      // _lib.valarray.cons_ construct/destroy:
      valarray();
      explicit valarray(size_t);
      valarray(const T&, size_t);
      valarray(const T*, size_t);
      valarray(const valarray&);
      valarray(const slice_array<T>&);
      valarray(const gslice_array<T>&);
      valarray(const mask_array<T>&);
      valarray(const indirect_array<T>&);
     ~valarray();

  _________________________
  2) _limits_ recommends a minimum number of recursively nested template
  instantiations. This requirement thus indirectly  suggests  a  minimum
  allowable complexity for valarray expressions.

    // _lib.valarray.assign_ assignment:
      valarray<T>& operator=(const valarray<T>&);
      valarray<T>& operator=(const T&);
      valarray<T>& operator=(const slice_array<T>&);
      valarray<T>& operator=(const gslice_array<T>&);
      valarray<T>& operator=(const mask_array<T>&);
      valarray<T>& operator=(const indirect_array<T>&);
    // _lib.valarray.access_ element access:
      T                 operator[](size_t) const;
      T&                operator[](size_t);
    // _lib.valarray.sub_ subset operations:
      valarray<T>       operator[](slice) const;
      slice_array<T>    operator[](slice);
      valarray<T>       operator[](const gslice&) const;
      gslice_array<T>   operator[](const gslice&);
      valarray<T>       operator[](const valarray<bool>&) const;
      mask_array<T>     operator[](const valarray<bool>&);
      valarray<T>       operator[](const valarray<size_t>&) const;
      indirect_array<T> operator[](const valarray<size_t>&);
    // _lib.valarray.unary_ unary operators:
      valarray<T> operator+() const;
      valarray<T> operator-() const;
      valarray<T> operator~() const;
      valarray<T> operator!() const;
    // _lib.valarray.cassign_ computed assignment:
      valarray<T>& operator*= (const T&);
      valarray<T>& operator/= (const T&);
      valarray<T>& operator%= (const T&);
      valarray<T>& operator+= (const T&);
      valarray<T>& operator-= (const T&);
      valarray<T>& operator^= (const T&);
      valarray<T>& operator&= (const T&);
      valarray<T>& operator|= (const T&);
      valarray<T>& operator<<=(const T&);
      valarray<T>& operator>>=(const T&);
      valarray<T>& operator*= (const valarray<T>&);
      valarray<T>& operator/= (const valarray<T>&);
      valarray<T>& operator%= (const valarray<T>&);
      valarray<T>& operator+= (const valarray<T>&);
      valarray<T>& operator-= (const valarray<T>&);
      valarray<T>& operator^= (const valarray<T>&);
      valarray<T>& operator|= (const valarray<T>&);
      valarray<T>& operator&= (const valarray<T>&);
      valarray<T>& operator<<=(const valarray<T>&);
      valarray<T>& operator>>=(const valarray<T>&);
    // _lib.valarray.members_ member functions:
      size_t size() const;
      T    sum() const;

      valarray<T> shift (int) const;
      valarray<T> cshift(int) const;
      valarray<T> apply(T func(T)) const;
      valarray<T> apply(T func(const T&)) const;
      void resize(size_t sz, T c = T());
    };
  }

1 The  template class valarray<T> is a one-dimensional smart array, with
  elements numbered sequentially from zero.  It is a  representation  of
  the mathematical concept of an ordered set of values.  The illusion of
  higher dimensionality may be produced by the familiar  idiom  of  com-
  puted indices, together with the powerful subsetting capabilities pro-
  vided by the generalized subscript operators.3)

2 An implementation  is  permitted  to  qualify  any  of  the  functions
  declared in <valarray> as inline.

  26.3.2.1  valarray constructors                    [lib.valarray.cons]

  valarray();

  Effects:
    Constructs an object of class valarray<T>,4) which has  zero  length
    until it is passed into a library function as a modifiable lvalue or
    through a non-constant this pointer.5)

  explicit valarray(size_t);

1 The  array created by this constructor has a length equal to the value
  of the argument.  The elements of the array are constructed using  the
  default constructor for the instantiating type T.

  valarray(const T&, size_t);

  _________________________
  3) The intent is to specify an array template  that  has  the  minimum
  functionality  necessary  to address aliasing ambiguities and the pro-
  liferation of temporaries.  Thus, the valarray template is  neither  a
  matrix class nor a field class.  However, it is a very useful building
  block for designing such classes.
  4)  For  convenience,  such  objects  are  referred  to  as ``arrays''
  throughout the remainder of subclause _lib.numarray_.
  5) This default constructor is essential, since arrays of valarray are
  likely  to prove useful.  There shall also be a way to change the size
  of an array after initialization; this is supplied by the semantics of
  the resize member function.

2 The array created by this constructor has a length equal to the second
  argument.  The elements of the array are initialized with the value of
  the first argument.

  valarray(const T*, size_t);

3 The array created by this constructor has a length equal to the second
  argument n.  The values of the elements of the array  are  initialized
  with  the  first  n  values pointed to by the first argument.6) If the
  value of the second argument is greater  than  the  number  of  values
  pointed to by the first argument, the behavior is undefined.

  valarray(const valarray<T>&);

4 The array created by this constructor has the same length as the argu-
  ment array. The elements are initialized with the values of the corre-
  sponding elements of the argument array.7)

  valarray(const slice_array<T>&);
  valarray(const gslice_array<T>&);
  valarray(const mask_array<T>&);
  valarray(const indirect_array<T>&);

5 These  conversion  constructors convert one of the four reference tem-
  plates to a valarray.

  ~valarray();

6 The destructor is applied to every element of *this; an implementation
  may return all allocated memory.

  26.3.2.2  valarray assignment                    [lib.valarray.assign]

  valarray<T>& operator=(const valarray<T>&);

  _________________________
  6)  This  constructor is the preferred method for converting a C array
  to a valarray object.
  7) This copy constructor creates  a  distinct  array  rather  than  an
  alias.   Implementations  in which arrays share storage are permitted,
  but they shall implement a copy-on-reference mechanism to ensure  that
  arrays are conceptually distinct.

1 Each  element  of  the *this array is assigned the value of the corre-
  sponding element of the argument array.   The  resulting  behavior  is
  undefined  if  the  length  of  the argument array is not equal to the
  length of the *this array.

  valarray<T>& operator=(const T&);

2 The scalar assignment operator causes each element of the *this  array
  to be assigned the value of the argument.

  valarray<T>& operator=(const slice_array<T>&);
  valarray<T>& operator=(const gslice_array<T>&);
  valarray<T>& operator=(const mask_array<T>&);
  valarray<T>& operator=(const indirect_array<T>&);

3 These operators allow the results of a generalized subscripting opera-
  tion to be assigned directly to a valarray.

4 If the value of an element in the left hand side of a valarray assign-
  ment  operator  depends  on  the value of another element in that left
  hand side, the resulting behavior is undefined.

  26.3.2.3  valarray element access                [lib.valarray.access]

  T  operator[](size_t) const;
  T& operator[](size_t);

1 When applied to a constant array, the subscript operator  returns  the
  value  of  the  corresponding element of the array.  When applied to a
  non-constant array, the subscript operator returns a reference to  the
  corresponding element of the array.

2 Thus,  the  expression (a[i] = q, a[i]) == q evaluates as true for any
  non-constant valarray<T> a, any T q, and for any size_t  i  such  that
  the value of i is less than the length of a.

3 The expression &a[i+j] == &a[i] + j evaluates as true for all size_t i
  and size_t j such that i+j is less than the length of the non-constant
  array a.

4 Likewise,  the expression &a[i] != &b[j] evaluates as true for any two
  non-constant arrays a and b and for any size_t i  and  size_t  j  such
  that  i  is less than the length of a and j is less than the length of
  b.  This property indicates an absence of aliasing and may be used  to
  advantage by optimizing compilers.8)
  _________________________
  8)  Compilers  may  take  advantage of inlining, constant propagation,

5 The  reference  returned  by the subscript operator for a non-constant
  array  is  guaranteed  to  be  valid   until   the   member   function
  resize(size_t, T) (_lib.valarray.members_) is called for that array or
  until the lifetime of that array ends, whichever happens first.

6 If the subscript operator is invoked  with  a  size_t  argument  whose
  value  is not less than the length of the array, the behavior is unde-
  fined.

  26.3.2.4  valarray subset operations                [lib.valarray.sub]

  valarray<T>       operator[](slice) const;
  slice_array<T>    operator[](slice);
  valarray<T>       operator[](const gslice&) const;
  gslice_array<T>   operator[](const gslice&);
  valarray<T>       operator[](const valarray<bool>&) const;
  mask_array<T>     operator[](const valarray<bool>&);
  valarray<T>       operator[](const valarray<size_t>&) const;
  indirect_array<T> operator[](const valarray<size_t>&);

1 Each of these operations returns a subset of the  array.   The  const-
  qualified  versions  return  this  subset as a new valarray.  The non-
  const versions return a class  template  object  which  has  reference
  semantics to the original array.

  26.3.2.5  valarray unary operators                [lib.valarray.unary]

  valarray<T> operator+() const;
  valarray<T> operator-() const;
  valarray<T> operator~() const;
  valarray<bool> operator!() const;

1 Each of these operators may only be instantiated for a type T to which
  the indicated operator can be applied  and  for  which  the  indicated
  operator  returns  a  value which is of type T (bool for operator!) or
  which may be unambiguously converted to type T (bool for operator!).

2 Each of these operators returns an array whose length is equal to  the
  length  of  the array.  Each element of the returned array is initial-
  ized with the result of applying the indicated operator to the  corre-
  sponding element of the array.

  _________________________
  loop fusion, tracking of pointers obtained from operator new, and oth-
  er techniques to generate efficient valarrays.

  26.3.2.6  valarray computed assignment          [lib.valarray.cassign]

  valarray<T>& operator*= (const valarray<T>&);
  valarray<T>& operator/= (const valarray<T>&);
  valarray<T>& operator%= (const valarray<T>&);
  valarray<T>& operator+= (const valarray<T>&);
  valarray<T>& operator-= (const valarray<T>&);
  valarray<T>& operator^= (const valarray<T>&);
  valarray<T>& operator&= (const valarray<T>&);
  valarray<T>& operator|= (const valarray<T>&);
  valarray<T>& operator<<=(const valarray<T>&);
  valarray<T>& operator>>=(const valarray<T>&);

1 Each of these operators may only be instantiated for a type T to which
  the indicated operator can be applied.  Each of these  operators  per-
  forms  the  indicated operation on each of its elements and the corre-
  sponding element of the argument array.

2 The array is then returned by reference.

3 If the array and the argument array do not have the same  length,  the
  behavior  is  undefined.   The appearance of an array on the left hand
  side of a computed assignment does not invalidate references or point-
  ers.

4 If  the  value  of an element in the left hand side of a valarray com-
  puted assignment operator depends on the value of another  element  in
  that left hand side, the resulting behavior is undefined.

  valarray<T>& operator*= (const T&);
  valarray<T>& operator/= (const T&);
  valarray<T>& operator%= (const T&);
  valarray<T>& operator+= (const T&);
  valarray<T>& operator-= (const T&);
  valarray<T>& operator^= (const T&);
  valarray<T>& operator&= (const T&);
  valarray<T>& operator|= (const T&);
  valarray<T>& operator<<=(const T&);
  valarray<T>& operator>>=(const T&);

5 Each of these operators may only be instantiated for a type T to which
  the indicated operator can be applied.

6 Each of these operators applies the indicated operation to  each  ele-
  ment of the array and the non-array argument.

7 The array is then returned by reference.

8 The appearance of an array on the left hand side of a computed assign-
  ment does not invalidate references or pointers to the elements of the
  array.

  26.3.2.7  valarray member functions             [lib.valarray.members]

  size_t size() const;

1 This function returns the number of elements in the array.

  T sum() const;

  This  function  may  only be instantiated for a type T to which opera-
  tor+= can be applied.  This function returns the sum of all  the  ele-
  ments of the array.

2 If  the  array  has length 0, the behavior is undefined.  If the array
  has length 1, sum() returns the value of element  0.   Otherwise,  the
  returned  value  is  calculated by applying operator+= to a copy of an
  element of the array and all other elements of the array in an unspec-
  ified order.

  T min() const;

3 This function returns the minimum value contained in *this.  The value
  returned for an array of length 0 is undefined. For an array of length
  1,  the  value  of element 0 is returned. For all other array lengths,
  the determination is made using operator<.

  T max() const;

4 This function returns the maximum value contained in *this.  The value
  returned for an array of length 0 is undefined. For an array of length
  1, the value of element 0 is returned. For all  other  array  lengths,
  the determination is made using operator<.

  valarray<T> shift(int n) const;

5 This function returns an object of class valarray<T> of length size(),
  each of whose elements I is (*this)[I+n] if I+n  is  non-negative  and
  less than size(), otherwise T().  Thus if element zero is taken as the
  leftmost element, a positive value of n shifts  the  elements  left  n
  places, with zero fill.

6 [Example:  If the argument has the value -2, the first two elements of
  the result will be constructed  using  the  default  constructor;  the
  third  element  of  the result will be assigned the value of the first
  element of the argument; etc.   --end example]

  valarray<T> cshift(int n) const;

7 This function returns  an  object  of  class  valarray<T>,  of  length
  size(),  each  of whose elements I is (*this)[(I+n)%size()].  Thus, if
  element zero is taken as the leftmost element, a positive value  of  n
  shifts the elements circularly left n places.

  valarray<T> apply(T func(T)) const;
  valarray<T> apply(T func(const T&)) const;

8 These  functions  return  an array whose length is equal to the array.
  Each element of the returned array is assigned the value  returned  by
  applying  the  argument  function  to the corresponding element of the
  array.

  void resize(size_t sz, T c = T());

9 This member function changes the length of the *this array to  sz  and
  then assigns to each element the value of the second argument.  Resiz-
  ing invalidates all pointers and references to elements in the  array.

  26.3.3  valarray non-member operations       [lib.valarray.nonmembers]

  26.3.3.1  valarray binary operators              [lib.valarray.binary]

  template<class T> valarray<T> operator*
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator/
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator%
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator+
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator-
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator^
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator&
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator|
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator<<
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> operator>>
      (const valarray<T>&, const valarray<T>&);

1 Each of these operators may only be instantiated for a type T to which
  the indicated operator can be applied  and  for  which  the  indicated
  operator  returns a value which is of type T or which can be unambigu-
  ously converted to type T.

2 Each of these operators returns an array whose length is equal to  the
  lengths of the argument arrays.  Each element of the returned array is
  initialized with the result of applying the indicated operator to  the
  corresponding elements of the argument arrays.

3 If  the  argument  arrays do not have the same length, the behavior is
  undefined.

  template<class T> valarray<T> operator* (const valarray<T>&, const T&);
  template<class T> valarray<T> operator* (const T&, const valarray<T>&);
  template<class T> valarray<T> operator/ (const valarray<T>&, const T&);
  template<class T> valarray<T> operator/ (const T&, const valarray<T>&);
  template<class T> valarray<T> operator% (const valarray<T>&, const T&);
  template<class T> valarray<T> operator% (const T&, const valarray<T>&);
  template<class T> valarray<T> operator+ (const valarray<T>&, const T&);
  template<class T> valarray<T> operator+ (const T&, const valarray<T>&);
  template<class T> valarray<T> operator- (const valarray<T>&, const T&);
  template<class T> valarray<T> operator- (const T&, const valarray<T>&);
  template<class T> valarray<T> operator^ (const valarray<T>&, const T&);
  template<class T> valarray<T> operator^ (const T&, const valarray<T>&);
  template<class T> valarray<T> operator& (const valarray<T>&, const T&);
  template<class T> valarray<T> operator& (const T&, const valarray<T>&);
  template<class T> valarray<T> operator| (const valarray<T>&, const T&);
  template<class T> valarray<T> operator| (const T&, const valarray<T>&);
  template<class T> valarray<T> operator<<(const valarray<T>&, const T&);
  template<class T> valarray<T> operator<<(const T&, const valarray<T>&);
  template<class T> valarray<T> operator>>(const valarray<T>&, const T&);
  template<class T> valarray<T> operator>>(const T&, const valarray<T>&);

4 Each of these operators may only be instantiated for a type T to which
  the  indicated  operator  can  be  applied and for which the indicated
  operator returns a value which is of type T or which can be  unambigu-
  ously converted to type T.

5 Each  of these operators returns an array whose length is equal to the
  length of the array argument.  Each element of the returned  array  is
  initialized  with the result of applying the indicated operator to the
  corresponding element of the array argument and  the  non-array  argu-
  ment.

  26.3.3.2  valarray logical operators         [lib.valarray.comparison]

  template<class T> valarray<bool> operator==
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<bool> operator!=
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<bool> operator<
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<bool> operator>
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<bool> operator<=
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<bool> operator>=
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<bool> operator&&
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<bool> operator||
      (const valarray<T>&, const valarray<T>&);

1 Each of these operators may only be instantiated for a type T to which
  the indicated operator can be applied  and  for  which  the  indicated
  operator  returns  a value which is of type bool or which can be unam-
  biguously converted to type bool.

2 Each of these operators returns a bool array whose length is equal  to
  the length of the array arguments.  Each element of the returned array
  is initialized with the result of applying the indicated  operator  to
  the corresponding elements of the argument arrays.

3 If  the  two array arguments do not have the same length, the behavior
  is undefined.

  template<class T> valarray<bool> operator==(const valarray&, const T&);
  template<class T> valarray<bool> operator==(const T&, const valarray&);
  template<class T> valarray<bool> operator!=(const valarray&, const T&);
  template<class T> valarray<bool> operator!=(const T&, const valarray&);
  template<class T> valarray<bool> operator< (const valarray&, const T&);
  template<class T> valarray<bool> operator< (const T&, const valarray&);
  template<class T> valarray<bool> operator> (const valarray&, const T&);
  template<class T> valarray<bool> operator> (const T&, const valarray&);
  template<class T> valarray<bool> operator<=(const valarray&, const T&);
  template<class T> valarray<bool> operator<=(const T&, const valarray&);
  template<class T> valarray<bool> operator>=(const valarray&, const T&);
  template<class T> valarray<bool> operator>=(const T&, const valarray&);
  template<class T> valarray<bool> operator&&(const valarray<T>&, const T&);
  template<class T> valarray<bool> operator&&(const T&, const valarray<T>&);
  template<class T> valarray<bool> operator||(const valarray<T>&, const T&);
  template<class T> valarray<bool> operator||(const T&, const valarray<T>&);

4 Each of these operators may only be instantiated for a type T to which
  the  indicated  operator  can  be  applied and for which the indicated
  operator returns a value which is of type bool or which can  be  unam-
  biguously converted to type bool.

5 Each  of these operators returns a bool array whose length is equal to
  the length of the array argument.  Each element of the returned  array
  is  initialized  with the result of applying the indicated operator to
  the corresponding element of the array and the non-array argument.

  26.3.3.3  valarray transcendentals            [lib.valarray.transcend]

  template<class T> valarray<T> abs  (const valarray<T>&);
  template<class T> valarray<T> acos (const valarray<T>&);
  template<class T> valarray<T> asin (const valarray<T>&);
  template<class T> valarray<T> atan (const valarray<T>&);
  template<class T> valarray<T> atan2
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> atan2(const valarray<T>&, const T&);
  template<class T> valarray<T> atan2(const T&, const valarray<T>&);
  template<class T> valarray<T> cos  (const valarray<T>&);
  template<class T> valarray<T> cosh (const valarray<T>&);
  template<class T> valarray<T> exp  (const valarray<T>&);
  template<class T> valarray<T> log  (const valarray<T>&);
  template<class T> valarray<T> log10(const valarray<T>&);
  template<class T> valarray<T> pow
      (const valarray<T>&, const valarray<T>&);
  template<class T> valarray<T> pow  (const valarray<T>&, const T&);
  template<class T> valarray<T> pow  (const T&, const valarray<T>&);
  template<class T> valarray<T> sin  (const valarray<T>&);
  template<class T> valarray<T> sinh (const valarray<T>&);
  template<class T> valarray<T> sqrt (const valarray<T>&);
  template<class T> valarray<T> tan  (const valarray<T>&);
  template<class T> valarray<T> tanh (const valarray<T>&);

1 Each of these functions may only be instantiated for a type T to which
  a  unique function with the indicated name can be applied.  This func-
  tion shall return a value which is of type T or which can be unambigu-
  ously converted to type T.

  26.3.4  Class slice                                  [lib.class.slice]
  namespace std {
    class slice {
    public:
      slice();
      slice(size_t, size_t, size_t);

      size_t start() const;
      size_t size() const;
      size_t stride() const;
    };
  }

1 The  slice  class  represents a BLAS-like slice from an array.  Such a
  slice is specified by a starting index, a length, and a stride.9)

  _________________________
  9) BLAS stands for Basic Linear Algebra Subprograms.  C++ programs may
  instantiate  this  class.   See, for example, Dongarra, Du Croz, Duff,
  and Hammerling: A set of Level 3  Basic  Linear  Algebra  Subprograms;
  Technical Report MCS-P1-0888, Argonne National Laboratory (USA), Math-
  ematics and Computer Science Division, August, 1988.

  26.3.4.1  slice constructors                          [lib.cons.slice]

  slice();
  slice(size_t start, size_t length, size_t stride);
  slice(const slice&);

1 The default constructor for slice creates a slice which  specifies  no
  elements.  A default constructor is provided only to permit the decla-
  ration of arrays of slices.  The  constructor  with  arguments  for  a
  slice takes a start, length, and stride parameter.

2 [Example:  slice(3, 8, 2) constructs a slice which selects elements 3,
  5, 7, ... 17 from an array.   --end example]

  26.3.4.2  slice access functions                    [lib.slice.access]

  size_t start() const;
  size_t size() const;
  size_t stride() const;

1 These functions return the start, length, or  stride  specified  by  a
  slice object.

  26.3.5  Template class slice_array          [lib.template.slice.array]
  namespace std {
    template <class T> class slice_array {
    public:
      typedef T value_type;

      void operator=  (const valarray<T>&) const;
      void operator*= (const valarray<T>&) const;
      void operator/= (const valarray<T>&) const;
      void operator%= (const valarray<T>&) const;
      void operator+= (const valarray<T>&) const;
      void operator-= (const valarray<T>&) const;
      void operator^= (const valarray<T>&) const;
      void operator&= (const valarray<T>&) const;
      void operator|= (const valarray<T>&) const;
      void operator<<=(const valarray<T>&) const;
      void operator>>=(const valarray<T>&) const;
      void fill(const T&);
     ~slice_array();
    private:
      slice_array();
      slice_array(const slice_array&);
      slice_array& operator=(const slice_array&);
      //   remainder implementation defined
    };
  }

1 The  slice_array  template is a helper template used by the slice sub-
  script operator
  slice_array<T> valarray<T>::operator[](slice);
  It has reference semantics to a subset of  an  array  specified  by  a
  slice object.

2 [Example:  The  expression  a[slice(1,  5,  3)] = b; has the effect of
  assigning the elements of b to a slice of the elements in a.  For  the
  slice  shown,  the elements selected from a are 1, 4, ..., 13.   --end
  example]

3 [Note: C++ programs may not instantiate  slice_array,  since  all  its
  constructors are private.  It is intended purely as a helper class and
  should be transparent to the user.   --end note]

  26.3.5.1  slice_array constructors                [lib.cons.slice.arr]

  slice_array();
  slice_array(const slice_array&);

1 The slice_array template has no public constructors.  These  construc-
  tors  are  declared  to  be  private.   These constructors need not be
  defined.

  26.3.5.2  slice_array assignment                [lib.slice.arr.assign]

  void         operator=(const valarray<T>&) const;
  slice_array& operator=(const slice_array&);

1 The second of these two assignment operators is declared  private  and
  need not be defined.  The first has reference semantics, assigning the
  values of the argument array elements  to  selected  elements  of  the
  valarray<T> object to which the slice_array object refers.

  26.3.5.3  slice_array computed             [lib.slice.arr.comp.assign]
       assignment

  void operator*= (const valarray<T>&) const;
  void operator/= (const valarray<T>&) const;
  void operator%= (const valarray<T>&) const;
  void operator+= (const valarray<T>&) const;
  void operator-= (const valarray<T>&) const;
  void operator^= (const valarray<T>&) const;
  void operator&= (const valarray<T>&) const;
  void operator|= (const valarray<T>&) const;
  void operator<<=(const valarray<T>&) const;
  void operator>>=(const valarray<T>&) const;

1 These computed assignments  have  reference  semantics,  applying  the
  indicated operation to the elements of the argument array and selected
  elements of the valarray<T> object to  which  the  slice_array  object
  refers.

  26.3.5.4  slice_array fill function               [lib.slice.arr.fill]

  void fill(const T&);

1 This  function  has  reference  semantics,  assigning the value of its
  argument to the elements  of  the  valarray<T>  object  to  which  the
  slice_array object refers.

  26.3.6  The gslice class                            [lib.class.gslice]
  namespace std {
    class gslice {
    public:
      gslice();
      gslice(size_t s, const valarray<size_t>& l, const valarray<size_t>& d);

      size_t           start() const;
      valarray<size_t> size() const;
      valarray<size_t> stride() const;
    };
  }

1 This  class  represents a generalized slice out of an array.  A gslice
  is defined by a starting offset (s), a set of lengths (lj), and a  set
  of  strides  (dj).   The  number  of lengths shall equal the number of
  strides.

2 A gslice represents a mapping from a set of  indices  (ij),  equal  in
  number  to  the  number of strides, to a single index k.  It is useful
  for building multidimensional array classes using  the  valarray  tem-
  plate,  which  is  one-dimensional.   The set of one-dimensional index
  values specified by a gslice are k=s+>ijdj where the  multidimensional
  indices ij range in value from 0 to ljij-1.

3 [Example: The gslice specification
  start  = 3
  length = {2, 4, 3}
  stride = {19, 4, 1}
  yields the sequence of one-dimensional indices

                    k=3+(0,1)19+(0,1,2,3)4+(0,1,2)1
  which are ordered as shown in the following table:

      (i0, i1, i2, k) =
              (0, 0, 0, 3),
              (0, 0, 1, 4),
              (0, 0, 2, 5),
              (0, 1, 0, 7),
              (0, 1, 1, 8),
              (0, 1, 2, 9),
              (0, 2, 0, 11),
              (0, 2, 1, 12),
              (0, 2, 2, 13),
              (0, 3, 0, 15),
              (0, 3, 1, 16),
              (0, 3, 2, 17),
              (1, 0, 0, 22),
              (1, 0, 1, 23),
              ...
              (1, 3, 2, 36)
  That is, the highest-ordered index turns fastest.   --end example]

4 It  is  possible  to  have  degenerate  generalized slices in which an
  address is repeated.

5 [Example: If the stride parameters in the previous example are changed
  to  {1,  1,  1},  the  first few elements of the resulting sequence of
  indices will be
              (0, 0, 0, 3),
              (0, 0, 1, 4),
              (0, 0, 2, 5),
              (0, 1, 0, 4),
              (0, 1, 1, 5),
              (0, 1, 2, 6),
              ...
   --end example]

6 If a degenerate slice is used as the argument to the non-const version
  of operator[](const gslice&), the resulting behavior is undefined.

  26.3.6.1  gslice constructors                        [lib.gslice.cons]

  gslice();
  gslice(size_t start, const valarray<size_t>& lengths,
                             const valarray<size_t>& strides);
  gslice(const gslice&);

1 The  default constructor creates a gslice which specifies no elements.
  The constructor with arguments builds a gslice based on  a  specifica-
  tion of start, lengths, and strides, as explained in the previous sec-
  tion.

  26.3.6.2  gslice access functions                  [lib.gslice.access]

  size_t           start()  const;
  valarray<size_t> size() const;
  valarray<size_t> stride() const;

  These  access  functions  return  the  representation  of  the  start,
  lengths, or strides specified for the gslice.

  26.3.7  Template class gslice_array        [lib.template.gslice.array]
  namespace std {
    template <class T> class gslice_array {
    public:
      typedef T value_type;

      void operator=  (const valarray<T>&) const;
      void operator*= (const valarray<T>&) const;
      void operator/= (const valarray<T>&) const;
      void operator%= (const valarray<T>&) const;
      void operator+= (const valarray<T>&) const;
      void operator-= (const valarray<T>&) const;
      void operator^= (const valarray<T>&) const;
      void operator&= (const valarray<T>&) const;
      void operator|= (const valarray<T>&) const;
      void operator<<=(const valarray<T>&) const;
      void operator>>=(const valarray<T>&) const;
      void fill(const T&);
     ~gslice_array();
    private:
      gslice_array();
      gslice_array(const gslice_array&);
      gslice_array& operator=(const gslice_array&);
      //  remainder implementation defined
    };
  }

1 This  template is a helper template used by the slice subscript opera-
  tor
  gslice_array<T> valarray<T>::operator[](const gslice&);
  It has reference semantics to a subset of  an  array  specified  by  a
  gslice object.

2 Thus,  the  expression a[gslice(1, length, stride)] = b has the effect
  of assigning the elements of b to a generalized slice of the  elements
  in a.

3 [Note:  C++  programs  may not instantiate gslice_array, since all its
  constructors are private.  It is intended purely as a helper class and
  should be transparent to the user.   --end note]

  26.3.7.1  gslice_array constructors            [lib.gslice.array.cons]

  gslice_array();
  gslice_array(const gslice_array&);

1 The gslice_array template has no public constructors.  It declares the
  above constructors to be private.   These  constructors  need  not  be
  defined.

  26.3.7.2  gslice_array assignment            [lib.gslice.array.assign]

  void operator=(const valarray<T>&) const;
  gslice_array& operator=(const gslice_array&);

1 The  second  of these two assignment operators is declared private and
  need not be defined.  The first has reference semantics, assigning the
  values  of  the  argument  array  elements to selected elements of the
  valarray<T> object to which the gslice_array refers.

  26.3.7.3  gslice_array computed         [lib.gslice.array.comp.assign]
       assignment

  void operator*= (const valarray<T>&) const;
  void operator/= (const valarray<T>&) const;
  void operator%= (const valarray<T>&) const;
  void operator+= (const valarray<T>&) const;
  void operator-= (const valarray<T>&) const;
  void operator^= (const valarray<T>&) const;
  void operator&= (const valarray<T>&) const;
  void operator|= (const valarray<T>&) const;
  void operator<<=(const valarray<T>&) const;
  void operator>>=(const valarray<T>&) const;

1 These  computed  assignments  have  reference  semantics, applying the
  indicated operation to the elements of the argument array and selected
  elements  of  the  valarray<T> object to which the gslice_array object
  refers.

  26.3.7.4  gslice_array fill function           [lib.gslice.array.fill]

  void fill(const T&);

1 This function has reference semantics,  assigning  the  value  of  its
  argument  to  the  elements  of  the  valarray<T>  object to which the
  gslice_array object refers.

  26.3.8  Template class mask_array            [lib.template.mask.array]
  namespace std {
    template <class T> class mask_array {
    public:
      typedef T value_type;

      void operator=  (const valarray<T>&) const;
      void operator*= (const valarray<T>&) const;
      void operator/= (const valarray<T>&) const;
      void operator%= (const valarray<T>&) const;
      void operator+= (const valarray<T>&) const;
      void operator-= (const valarray<T>&) const;
      void operator^= (const valarray<T>&) const;
      void operator&= (const valarray<T>&) const;
      void operator|= (const valarray<T>&) const;
      void operator<<=(const valarray<T>&) const;
      void operator>>=(const valarray<T>&) const;
      void fill(const T&);
     ~mask_array();
    private:
      mask_array();
      mask_array(const mask_array&);
      mask_array& operator=(const mask_array&);
      //  remainder implementation defined
    };
  }

1 This template is a helper template used by the mask  subscript  opera-
  tor:
    mask_array<T> valarray<T>::operator[](const valarray<bool>&).
  It  has  reference  semantics  to  a subset of an array specified by a
  boolean mask.  Thus, the expression a[mask] = b;  has  the  effect  of
  assigning  the  elements  of  b to the masked elements in a (those for
  which the corresponding element in mask is true.)

2 [Note: C++ programs may not declare instances of mask_array, since all
  its  constructors  are  private.   It  is  intended purely as a helper
  class, and should be transparent to the user.   --end note]

  26.3.8.1  mask_array constructors                [lib.mask.array.cons]

  mask_array();
  mask_array(const mask_array&);

1 The mask_array template has no public constructors.  It  declares  the
  above  constructors  to  be  private.   These constructors need not be
  defined.

  26.3.8.2  mask_array assignment                [lib.mask.array.assign]

  void operator=(const valarray<T>&) const;
  mask_array& operator=(const mask_array&);

1 The second of these two assignment operators is declared  private  and
  need not be defined.  The first has reference semantics, assigning the
  values of the argument array elements  to  selected  elements  of  the
  valarray<T> object to which it refers.

  26.3.8.3  mask_array computed             [lib.mask.array.comp.assign]
       assignment

  void operator*= (const valarray<T>&) const;
  void operator/= (const valarray<T>&) const;
  void operator%= (const valarray<T>&) const;
  void operator+= (const valarray<T>&) const;
  void operator-= (const valarray<T>&) const;
  void operator^= (const valarray<T>&) const;
  void operator&= (const valarray<T>&) const;
  void operator|= (const valarray<T>&) const;
  void operator<<=(const valarray<T>&) const;
  void operator>>=(const valarray<T>&) const;

1 These computed assignments  have  reference  semantics,  applying  the
  indicated operation to the elements of the argument array and selected
  elements of the valarray<T> object to which the mask object refers.

  26.3.8.4  mask_array fill function               [lib.mask.array.fill]

  void fill(const T&);

  This function has reference semantics,  assigning  the  value  of  its
  argument  to  the  elements  of  the  valarray<T>  object to which the
  mask_array object refers.

  26.3.9  Template class                   [lib.template.indirect.array]
       indirect_array

  namespace std {
    template <class T> class indirect_array {
    public:
      typedef T value_type;

      void operator=  (const valarray<T>&) const;
      void operator*= (const valarray<T>&) const;
      void operator/= (const valarray<T>&) const;
      void operator%= (const valarray<T>&) const;
      void operator+= (const valarray<T>&) const;
      void operator-= (const valarray<T>&) const;
      void operator^= (const valarray<T>&) const;
      void operator&= (const valarray<T>&) const;
      void operator|= (const valarray<T>&) const;
      void operator<<=(const valarray<T>&) const;
      void operator>>=(const valarray<T>&) const;
      void fill(const T&);
     ~indirect_array();
    private:
      indirect_array();
      indirect_array(const indirect_array&);
      indirect_array& operator=(const indirect_array&);
      //  remainder implementation defined
    };
  }

1 This  template  is  a  helper  template used by the indirect subscript
  operator
    indirect_array<T> valarray<T>::operator[](const  valarray<size_t>&).
  It  has  reference  semantics  to a subset of an array specified by an
  indirect_array.  Thus the expression a[indirect] = b; has  the  effect
  of  assigning  the  elements  of  b to the elements in a whose indices
  appear in indirect.

2 [Note: C++ programs may not declare instances of indirect_array, since
  all  its  constructors are private.  It is intended purely as a helper
  class, and should be transparent to the user.   --end note]

  26.3.9.1  indirect_array constructors        [lib.indirect.array.cons]

  indirect_array();
  indirect_array(const indirect_array&);

  The indirect_array template has no public constructors.  The construc-
  tors  listed  above  are  private.   These  constructors  need  not be
  defined.

  26.3.9.2  indirect_array assignment        [lib.indirect.array.assign]

  void operator=(const valarray<T>&) const;
  indirect_array& operator=(const indirect_array&);

1 The second of these two assignment operators is declared  private  and
  need not be defined.  The first has reference semantics, assigning the
  values of the argument array elements  to  selected  elements  of  the
  valarray<T> object to which it refers.

2 If  the  indirect_array specifies an element in the valarray<T> object
  to which it refers more than once, the behavior is undefined.

3 [Example:
  int addr[] = {2, 3, 1, 4, 4};
  valarray<size_t> indirect(addr, 5);
  valarray<double> a(0., 10), b(1., 5);
  array[indirect] = b;
  results in undefined behavior since element 4 is  specified  twice  in
  the indirection.   --end example]

  26.3.9.3  indirect_array              [lib.indirect.array.comp.assign]
       computed assignment

  void operator*= (const valarray<T>&) const;
  void operator/= (const valarray<T>&) const;
  void operator%= (const valarray<T>&) const;
  void operator+= (const valarray<T>&) const;
  void operator-= (const valarray<T>&) const;
  void operator^= (const valarray<T>&) const;
  void operator&= (const valarray<T>&) const;
  void operator|= (const valarray<T>&) const;
  void operator<<=(const valarray<T>&) const;
  void operator>>=(const valarray<T>&) const;

1 These computed assignments  have  reference  semantics,  applying  the
  indicated operation to the elements of the argument array and selected
  elements of the valarray<T> object to which the indirect_array  object
  refers.

2 If  the  indirect_array specifies an element in the valarray<T> object
  to which it refers more than once, the behavior is undefined.

  26.3.9.4  indirect_array fill function       [lib.indirect.array.fill]

  void fill(const T&);

1 This function has reference semantics,  assigning  the  value  of  its
  argument  to the elements of the valarray<T> object to which the indi-
  rect_array object refers.

  26.4  Generalized numeric operations                 [lib.numeric.ops]

  Header <numeric> synopsis

  namespace std {
    template <class InputIterator, class T>
      T accumulate(InputIterator first, InputIterator last, T init);
    template <class InputIterator, class T, class BinaryOperation>
      T accumulate(InputIterator first, InputIterator last, T init,
                   BinaryOperation binary_op);
    template <class InputIterator1, class InputIterator2, class T>
      T inner_product(InputIterator1 first1, InputIterator1 last1,
                      InputIterator2 first2, T init);
    template <class InputIterator1, class InputIterator2, class T,
              class BinaryOperation1, class BinaryOperation2>
      T inner_product(InputIterator1 first1, InputIterator1 last1,
                      InputIterator2 first2, T init,
                      BinaryOperation1 binary_op1, BinaryOperation2 binary_op2);
    template <class InputIterator, class OutputIterator>
      OutputIterator partial_sum(InputIterator first, InputIterator last,
                                 OutputIterator result);
    template <class InputIterator, class OutputIterator, class BinaryOperation>
      OutputIterator partial_sum(InputIterator first, InputIterator last,
                                 OutputIterator result, BinaryOperation binary_op);
    template <class InputIterator, class OutputIterator>
      OutputIterator adjacent_difference(InputIterator first, InputIterator last,
                                         OutputIterator result);
    template <class InputIterator, class OutputIterator, class BinaryOperation>
      OutputIterator adjacent_difference(InputIterator first, InputIterator last,
                                         OutputIterator result,
                                         BinaryOperation binary_op);
  }

1 The requirements on  the  types  of  algorithms'  arguments  that  are
  described in the introduction to clause _lib.algorithms_ also apply to
  the following algorithms.

  26.4.1  Accumulate                                    [lib.accumulate]

  template <class InputIterator, class T>
    T accumulate(InputIterator first, InputIterator last, T init);
  template <class InputIterator, class T, class BinaryOperation>
    T accumulate(InputIterator first, InputIterator last, T init,
                 BinaryOperation binary_op);

  Effects:
    Initializes the accumulator acc with the initial value init and then
    modifies  it  with  acc  =  acc + *i or acc = binary_op(acc, *i) for
    every iterator i in the range [first, last) in order.10)
  _________________________
  10) accumulate is similar to the APL  reduction  operator  and  Common
  Lisp reduce function, but it avoids the difficulty of defining the re-

  Requires:
    binary_op shall not cause side effects.

  26.4.2  Inner product                              [lib.inner.product]

  template <class InputIterator1, class InputIterator2, class T>
    T inner_product(InputIterator1 first1, InputIterator1 last1,
                    InputIterator2 first2, T init);
  template <class InputIterator1, class InputIterator2, class T,
            class BinaryOperation1, class BinaryOperation2>
    T inner_product(InputIterator1 first1, InputIterator1 last1,
                    InputIterator2 first2, T init,
                    BinaryOperation1 binary_op1,
                    BinaryOperation2 binary_op2);

  Effects:
    Computes  its  result  by  initializing the accumulator acc with the
    initial value init and then modifying it with acc = acc  +  (*i1)  *
    (*i2) or acc = binary_op1(acc, binary_op2(*i1, *i2)) for every iter-
    ator i1 in the range [first, last) and  iterator  i2  in  the  range
    [first2, first2 + (last - first)) in order.
  Requires:
    binary_op1 and binary_op2 shall not cause side effects.

  26.4.3  Partial sum                                  [lib.partial.sum]

  template <class InputIterator, class OutputIterator>
    OutputIterator
      partial_sum(InputIterator first, InputIterator last,
                  OutputIterator result);
  template
    <class InputIterator, class OutputIterator, class BinaryOperation>
      OutputIterator
        partial_sum(InputIterator first, InputIterator last,
                    OutputIterator result, BinaryOperation binary_op);

  Effects:
    Assigns  to  every  element  referred  to by iterator i in the range
    [result, result + (last - first)) a value correspondingly equal to
    ((...(*first + *(first + 1)) + ...) + *(first + (i - result)))
    or
    binary_op(binary_op(...,  binary_op(*first,  *(first   +   1)),...),
    *(first + (i - result)))
  Returns:
    result + (last - first).
  Complexity:
    Exactly (last - first) - 1 applications of binary_op.

  _________________________
  sult of reduction on an empty sequence by always requiring an  initial
  value.

  Requires:
    binary_op is expected not to have any side effects.
  Notes:
    result may be equal to first.

  26.4.4  Adjacent difference                  [lib.adjacent.difference]

  template <class InputIterator, class OutputIterator>
    OutputIterator
      adjacent_difference(InputIterator first, InputIterator last,
                          OutputIterator result);
  template
    <class InputIterator, class OutputIterator, class BinaryOperation>
      OutputIterator
        adjacent_difference(InputIterator first, InputIterator last,
                            OutputIterator result,
                            BinaryOperation binary_op);

  Effects:
    Assigns  to  every  element  referred  to by iterator i in the range
    [result + 1, result + (last - first)) a value correspondingly  equal
    to
    *(first + (i - result)) - *(first + (i - result) - 1)
    or
    binary_op(*(first + (i - result)), *(first + (i - result) - 1)).
    result gets the value of *first.
  Requires:
    binary_op shall not have any side effects.
  Notes:
    result may be equal to first.
  Returns:
    result + (last - first).
  Complexity:
    Exactly (last - first) - 1 applications of binary_op.

  26.5  C Library                                           [lib.c.math]

1 Tables  2 and 3 describe headers <cmath> and <cstdlib> ( abs(), div(),
  rand(), srand()), respectively.

                     Table 2--Header <cmath> synopsis

          +-----------------------------------------------------+
          |          Type    Name(s)                            |
          +-----------------------------------------------------+
          |Macro:  HUGE_VAL                                     |
          +-----------------------------------------------------+
          |Functions:                                           |
          |acos    cos     fmod    modf    tan                  |
          |asin    cosh    frexp   pow     tanh                 |
          |atan    exp     ldexp   sin                          |
          |atan2   fabs    log     sinh                         |
          |ceil    floor   log10   sqrt                         |
          +-----------------------------------------------------+

                    Table 2--Header <cstdlib> synopsis

                     +-------------------------------+
                     |   Type    Name(s)             |
                     +-------------------------------+
                     |Macros: RAND_MAX               |
                     +-------------------------------+
                     |Types:  div_t   ldiv_t         |
                     +-------------------------------+
                     |Functions:                     |
                     |abs     labs    srand          |
                     |div     ldiv    rand           |
                     +-------------------------------+

2 The contents of these headers are the same as the Standard  C  library
  headers <math.h> and <stdlib.h> respectively, with the following addi-
  tions:

3 In addition to the int versions of certain  math  functions  in  <cst-
  dlib>,  C++ adds long overloaded versions of these functions, with the
  same semantics.

4 The added signatures are:

  long   abs(long);        // labs()
  ldiv_t div(long, long);  // ldiv()

5 In addition to the double versions of the math functions  in  <cmath>,
  C++ adds float and long double overloaded versions of these functions,
  with the same semantics.

6 The added signatures are:

  float abs  (float);
  float acos (float);
  float asin (float);
  float atan (float);
  float atan2(float, float);
  float ceil (float);
  float cos  (float);
  float cosh (float);
  float exp  (float);
  float fabs (float);
  float floor(float);
  float fmod (float, float);
  float frexp(float, int*);
  float ldexp(float, int);
  float log  (float);
  float log10(float);
  float modf (float, float*);
  float pow  (float, float);
  float pow  (float, int);
  float sin  (float);
  float sinh (float);
  float sqrt (float);
  float tan  (float);
  float tanh (float);

  double abs(double);         // fabs()
  double pow(double, int);

  long double abs  (long double);
  long double acos (long double);
  long double asin (long double);
  long double atan (long double);
  long double atan2(long double, long double);
  long double ceil (long double);
  long double cos  (long double);
  long double cosh (long double);
  long double exp  (long double);
  long double fabs (long double);
  long double floor(long double);
  long double frexp(long double, int*);
  long double fmod (long double, long double);
  long double frexp(long double, int*);
  long double ldexp(long double, int);
  long double log  (long double);
  long double log10(long double);
  long double modf (long double, long double*);
  long double pow  (long double, long double);
  long double pow  (long double, int);
  long double sin  (long double);
  long double sinh (long double);
  long double sqrt (long double);
  long double tan  (long double);
  long double tanh (long double);

  SEE ALSO: ISO C subclauses 7.5, 7.10.2, 7.10.6.