[complex.numbers] (original) (raw)

26 Numerics library [numerics]

26.4.1 General [complex.numbers.general]

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

The effect of instantiating the templatecomplexfor any type other than float, double, or long double is unspecified.

The specializationscomplex<float>,complex<double>, andcomplex<long double> are literal types.

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

If z is an lvalue of type cv complex<T> then:

• the expression reinterpret_­cast<cv T(&)[2]>(z) is well-formed,
• reinterpret_­cast<cv T(&)[2]>(z)[0] designates the real part of z, and
• reinterpret_­cast<cv T(&)[2]>(z)[1] designates the imaginary part of z.

Moreover, if a is an expression of type cv complex<T>* and the expression a[i] is well-defined for an integer expression i, then:

• reinterpret_­cast<cv T*>(a)[2*i] designates the real part of a[i], and
• reinterpret_­cast<cv T*>(a)[2*i + 1] designates the imaginary part of a[i].

26.4.3 Class template complex [complex]

namespace std { template<class T> class complex { public: using value_type = T;constexpr complex(const T& re = T(), const T& im = T());constexpr complex(const complex&);template<class X> constexpr complex(const complex<X>&);constexpr T real() const;constexpr void real(T);constexpr T imag() const;constexpr void imag(T);constexpr complex& operator= (const T&);constexpr complex& operator+=(const T&);constexpr complex& operator-=(const T&);constexpr complex& operator*=(const T&);constexpr complex& operator/=(const T&);constexpr complex& operator=(const complex&);template<class X> constexpr complex& operator= (const complex<X>&);template<class X> constexpr complex& operator+=(const complex<X>&);template<class X> constexpr complex& operator-=(const complex<X>&);template<class X> constexpr complex& operator*=(const complex<X>&);template<class X> constexpr complex& operator/=(const complex<X>&);};}

The classcomplexdescribes an object that can store the Cartesian components,real()andimag(), of a complex number.

26.4.4 Specializations [complex.special]

namespace std { template<> class complex<float> { public: using value_type = float;constexpr complex(float re = 0.0f, float im = 0.0f);constexpr complex(const complex<float>&) = default;constexpr explicit complex(const complex<double>&);constexpr explicit complex(const complex<long double>&);constexpr float real() const;constexpr void real(float);constexpr float imag() const;constexpr void imag(float);constexpr complex& operator= (float);constexpr complex& operator+=(float);constexpr complex& operator-=(float);constexpr complex& operator*=(float);constexpr complex& operator/=(float);constexpr complex& operator=(const complex&);template<class X> constexpr complex& operator= (const complex<X>&);template<class X> constexpr complex& operator+=(const complex<X>&);template<class X> constexpr complex& operator-=(const complex<X>&);template<class X> constexpr complex& operator*=(const complex<X>&);template<class X> constexpr complex& operator/=(const complex<X>&);};template<> class complex<double> { public: using value_type = double;constexpr complex(double re = 0.0, double im = 0.0);constexpr complex(const complex<float>&);constexpr complex(const complex<double>&) = default;constexpr explicit complex(const complex<long double>&);constexpr double real() const;constexpr void real(double);constexpr double imag() const;constexpr void imag(double);constexpr complex& operator= (double);constexpr complex& operator+=(double);constexpr complex& operator-=(double);constexpr complex& operator*=(double);constexpr complex& operator/=(double);constexpr complex& operator=(const complex&);template<class X> constexpr complex& operator= (const complex<X>&);template<class X> constexpr complex& operator+=(const complex<X>&);template<class X> constexpr complex& operator-=(const complex<X>&);template<class X> constexpr complex& operator*=(const complex<X>&);template<class X> constexpr complex& operator/=(const complex<X>&);};template<> class complex<long double> { public: using value_type = long double;constexpr complex(long double re = 0.0L, long double im = 0.0L);constexpr complex(const complex<float>&);constexpr complex(const complex<double>&);constexpr complex(const complex<long double>&) = default;constexpr long double real() const;constexpr void real(long double);constexpr long double imag() const;constexpr void imag(long double);constexpr complex& operator= (long double);constexpr complex& operator+=(long double);constexpr complex& operator-=(long double);constexpr complex& operator*=(long double);constexpr complex& operator/=(long double);constexpr complex& operator=(const complex&);template<class X> constexpr complex& operator= (const complex<X>&);template<class X> constexpr complex& operator+=(const complex<X>&);template<class X> constexpr complex& operator-=(const complex<X>&);template<class X> constexpr complex& operator*=(const complex<X>&);template<class X> constexpr complex& operator/=(const complex<X>&);};}

26.4.5 Member functions [complex.members]

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

Postconditions: real() == re && imag() == im is true.

constexpr T real() const;

Returns: The value of the real component.

constexpr void real(T val);

Effects: Assigns val to the real component.

constexpr T imag() const;

Returns: The value of the imaginary component.

constexpr void imag(T val);

Effects: Assigns val to the imaginary component.

26.4.6 Member operators [complex.member.ops]

constexpr complex& operator+=(const T& rhs);

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

constexpr complex& operator-=(const T& rhs);

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

constexpr complex& operator*=(const T& rhs);

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

constexpr complex& operator/=(const T& rhs);

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

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

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

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

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

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

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

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

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

26.4.7 Non-member operations [complex.ops]

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

Returns: complex<T>(lhs).

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

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

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

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

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

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

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

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

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

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

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

Returns: lhs.real() == rhs.real() && lhs.imag() == rhs.imag().

Remarks: The imaginary part is assumed to beT(), or 0.0, for theTarguments.

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

Preconditions: The input values are convertible toT.

Effects: Extracts a complex number x of the form:u,(u), or(u,v), whereuis the real part andvis the imaginary part ([istream.formatted]).

If bad input is encountered, callsis.setstate(ios_­base​::​failbit)(which may throwios_­base​::​​failure ([iostate.flags])).

Remarks: This extraction is performed as a series of simpler extractions.

Therefore, the skipping of whitespace is specified to be the same for each of the simpler extractions.

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 xonto the stream o as if it were implemented as follows:basic_ostringstream<charT, traits> s; s.flags(o.flags()); s.imbue(o.getloc()); s.precision(o.precision()); s << '(' << x.real() << "," << x.imag() << ')';return o << s.str();

[Note 1:

In a locale in which comma is used as a decimal point character, the use of comma as a field separator can be ambiguous.

Insertingshowpoint into the output stream forces all outputs to show an explicit decimal point character; as a result, all inserted sequences of complex numbers can be extracted unambiguously.

— _end note_]

26.4.8 Value operations [complex.value.ops]

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

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

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> constexpr T norm(const complex<T>& x);

Returns: The squared magnitude of x.

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

Returns: The complex conjugate of x.

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

Returns: The projection of x onto the Riemann sphere.

Remarks: Behaves the same as the C function cproj.

See also: ISO C 7.3.9.5

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

Preconditions: rho is non-negative and non-NaN.

theta is finite.

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

26.4.9 Transcendentals [complex.transcendentals]

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

Returns: The complex arc cosine of x.

Remarks: Behaves the same as the C function cacos.

See also: ISO C 7.3.5.1

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

Returns: The complex arc sine of x.

Remarks: Behaves the same as the C function casin.

See also: ISO C 7.3.5.2

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

Returns: The complex arc tangent of x.

Remarks: Behaves the same as the C function catan.

See also: ISO C 7.3.5.3

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

Returns: The complex arc hyperbolic cosine of x.

Remarks: Behaves the same as the C function cacosh.

See also: ISO C 7.3.6.1

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

Returns: The complex arc hyperbolic sine of x.

Remarks: Behaves the same as the C function casinh.

See also: ISO C 7.3.6.2

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

Returns: The complex arc hyperbolic tangent of x.

Remarks: Behaves the same as the C function catanh.

See also: ISO C 7.3.6.3

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);

Returns: The complex natural (base-e) logarithm of x.

For all x,imag(log(x)) lies in the interval [, π].

[Note 1:

The semantics of this function are intended to be the same in C++ as they are for clog in C.

— _end note_]

Remarks: The branch cuts are along the negative real axis.

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

Returns: The complex common (base-10) logarithm of x, defined aslog(x) / log(10).

Remarks: The branch cuts are along the negative real axis.

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);

Returns: The complex power of base x raised to the power, defined asexp(y * log(x)).

The value returned forpow(0, 0)is implementation-defined.

Remarks: The branch cuts are along the negative real axis.

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);

Returns: The complex square root of x, in the range of the right half-plane.

[Note 2:

The semantics of this function are intended to be the same in C++ as they are for csqrt in C.

— _end note_]

Remarks: The branch cuts are along the negative real 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.

26.4.10 Additional overloads [cmplx.over]

The following function templates shall have additional overloads:arg norm conj proj imag realwhere norm, conj, imag, and real are constexpr overloads.

The additional overloads shall be sufficient to ensure:

• If the argument has type long double, then it is effectively cast to complex<long double>.
• Otherwise, if the argument has type double or an integer type, then it is effectively cast to complex<​double>.
• Otherwise, if the argument has type float, then it is effectively cast to complex<float>.

Function template pow shall have additional overloads sufficient to ensure, for a call with at least one argument of type complex<T>:

• If either argument has type complex<long double> or type long double, then both arguments are effectively cast tocomplex<long double>.
• Otherwise, if either argument has type complex<double>, double, or an integer type, then both arguments are effectively cast tocomplex<double>.
• Otherwise, if either argument has type complex<float> or float, then both arguments are effectively cast to complex<float>.

26.4.11 Suffixes for complex number literals [complex.literals]

This subclause describes literal suffixes for constructing complex number literals.

The suffixes i, il, and if create complex numbers of the types complex<double>, complex<long double>, andcomplex<float> respectively, with their imaginary part denoted by the given literal number and the real part being zero.

constexpr complex<long double> operator""il(long double d);constexpr complex<long double> operator""il(unsigned long long d);

Returns: complex<long double>{0.0L, static_­cast<long double>(d)}.

constexpr complex<double> operator""i(long double d);constexpr complex<double> operator""i(unsigned long long d);

Returns: complex<double>{0.0, static_­cast<double>(d)}.

constexpr complex<float> operator""if(long double d);constexpr complex<float> operator""if(unsigned long long d);

Returns: complex<float>{0.0f, static_­cast<float>(d)}.