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Sprout/sprout/complex.hpp

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#ifndef SPROUT_COMPLEX_HPP
#define SPROUT_COMPLEX_HPP
#include <iosfwd>
#include <ios>
#include <sprout/config.hpp>
#include <sprout/math/constants.hpp>
#include <sprout/math/abs.hpp>
#include <sprout/math/sin.hpp>
#include <sprout/math/cos.hpp>
#include <sprout/math/atan2.hpp>
#include <sprout/math/sinh.hpp>
#include <sprout/math/cosh.hpp>
#include <sprout/math/exp.hpp>
#include <sprout/math/log.hpp>
#include <sprout/math/pow.hpp>
#include <sprout/math/sqrt.hpp>
namespace sprout {
template<typename T>
class complex;
template<typename T>
SPROUT_CONSTEXPR T norm(sprout::complex<T> const& x);
//
// complex
//
template<typename T>
class complex {
public:
typedef T value_type;
private:
T re_;
T im_;
public:
SPROUT_CONSTEXPR complex(T const& re = T(), T const& im = T())
: re_(re)
, im_(im)
{}
SPROUT_CONSTEXPR complex(complex const&) = default;
template<typename X>
SPROUT_CONSTEXPR complex(complex<X> const& other)
: re_(other.real())
, im_(other.imag())
{}
SPROUT_CONSTEXPR T real() const {
return re_;
}
void real(T re) {
re_ = re;
}
SPROUT_CONSTEXPR T imag() const{
return im_;
}
void imag(T im) {
im_ = im;
}
complex<T>& operator=(T const& rhs) {
re_ = rhs;
im_ = T();
return *this;
}
complex<T>& operator+=(T const& rhs) {
re_ += rhs;
return *this;
}
complex<T>& operator-=(T const& rhs) {
re_ -= rhs;
return *this;
}
complex<T>& operator*=(T const& rhs) {
re_ *= rhs;
im_ *= rhs;
return *this;
}
complex<T>& operator/=(T const& rhs) {
re_ /= rhs;
im_ /= rhs;
return *this;
}
complex& operator=(complex const&) = default;
template<typename X>
complex<T>& operator=(complex<X> const& rhs) {
re_ = rhs.real();
im_ = rhs.imag();
return *this;
}
template<typename X>
complex<T>& operator+=(complex<X> const& rhs) {
re_ += rhs.real();
im_ += rhs.imag();
return *this;
}
template<typename X>
complex<T>& operator-=(complex<X> const& rhs) {
re_ -= rhs.real();
im_ -= rhs.imag();
return *this;
}
template<typename X>
complex<T>& operator*=(complex<X> const& rhs) {
return *this = complex<T>(
re_ * rhs.real() - im_ * rhs.imag(),
re_ * rhs.imag() + im_ * rhs.real()
);
}
template<typename X>
complex<T>& operator/=(complex<X> const& rhs) {
T n = sprout::norm(rhs);
return *this = complex<T>(
(re_ * rhs.real() + im_ * rhs.imag()) / n,
(im_ * rhs.real() - re_ * rhs.imag()) / n
);
}
};
// 26.4.6, operators:
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> operator+(sprout::complex<T> const& lhs, sprout::complex<T> const& rhs) {
return sprout::complex<T>(
lhs.real() + rhs.real(),
lhs.imag() + rhs.imag()
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> operator+(sprout::complex<T> const& lhs, T const& rhs) {
return sprout::complex<T>(
lhs.real() + rhs,
lhs.imag()
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> operator+(T const& lhs, sprout::complex<T> const& rhs) {
return sprout::complex<T>(
lhs + rhs.real(),
rhs.imag()
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> operator-(sprout::complex<T> const& lhs, sprout::complex<T> const& rhs) {
return sprout::complex<T>(
lhs.real() - rhs.real(),
lhs.imag() - rhs.imag()
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> operator-(sprout::complex<T> const& lhs, T const& rhs) {
return sprout::complex<T>(
lhs.real() - rhs,
lhs.imag()
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> operator-(T const& lhs, sprout::complex<T> const& rhs) {
return sprout::complex<T>(
lhs - rhs.real(),
-rhs.imag()
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> operator*(sprout::complex<T> const& lhs, sprout::complex<T> const& rhs) {
return sprout::complex<T>(
lhs.real() * rhs.real() - lhs.imag() * rhs.imag(),
lhs.real() * rhs.imag() + lhs.imag() * rhs.real()
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> operator*(sprout::complex<T> const& lhs, T const& rhs) {
return sprout::complex<T>(
lhs.real() * rhs,
lhs.imag() * rhs
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> operator*(T const& lhs, sprout::complex<T> const& rhs) {
return sprout::complex<T>(
lhs * rhs.real(),
lhs * rhs.imag()
);
}
namespace detail {
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> divides_impl(
sprout::complex<T> const& lhs,
sprout::complex<T> const& rhs,
T const& n
)
{
return sprout::complex<T>(
(lhs.real() * rhs.real() + lhs.imag() * rhs.imag()) / n,
(lhs.imag() * rhs.real() - lhs.real() * rhs.imag()) / n
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> divides_impl(
T const& lhs,
sprout::complex<T> const& rhs,
T const& n
)
{
return sprout::complex<T>(
lhs * rhs.real() / n,
-lhs * rhs.imag() / n
);
}
} // namespace detail
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> operator/(sprout::complex<T> const& lhs, sprout::complex<T> const& rhs) {
return sprout::detail::divides_impl(lhs, rhs, sprout::norm(rhs));
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> operator/(sprout::complex<T> const& lhs, T const& rhs) {
return sprout::complex<T>(
lhs.real() / rhs,
lhs.imag() / rhs
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> operator/(T const& lhs, sprout::complex<T> const& rhs) {
return sprout::detail::divides_impl(lhs, rhs, sprout::norm(rhs));
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> operator+(sprout::complex<T> const& x) {
return x;
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> operator-(sprout::complex<T> const& x) {
return sprout::complex<T>(-x.real(), -x.imag());
}
template<typename T>
SPROUT_CONSTEXPR bool operator==(sprout::complex<T> const& lhs, sprout::complex<T> const& rhs) {
return lhs.real() == rhs.real() && lhs.imag() == rhs.imag();
}
template<typename T>
SPROUT_CONSTEXPR bool operator==(sprout::complex<T> const& lhs, T const& rhs) {
return lhs.real() == rhs && lhs.imag() == T();
}
template<typename T>
SPROUT_CONSTEXPR bool operator==(T const& lhs, sprout::complex<T> const& rhs) {
return lhs == rhs.real() && T() == rhs.imag();
}
template<typename T>
SPROUT_CONSTEXPR bool operator!=(sprout::complex<T> const& lhs, sprout::complex<T> const& rhs) {
return !(lhs == rhs);
}
template<typename T>
SPROUT_CONSTEXPR bool operator!=(sprout::complex<T> const& lhs, T const& rhs) {
return !(lhs == rhs);
}
template<typename T>
SPROUT_CONSTEXPR bool operator!=(T const& lhs, sprout::complex<T> const& rhs) {
return !(lhs == rhs);
}
template<typename T, typename Char, typename Traits>
std::basic_istream<Char, Traits>& operator>>(std::basic_istream<Char, Traits>& lhs, sprout::complex<T>& rhs) {
T re, im;
Char ch;
lhs >> ch;
if (ch == '(') {
lhs >> re >> ch;
if (ch == ',') {
lhs >> im >> ch;
if (ch == ')') {
rhs = sprout::complex<T>(re, im);
} else {
lhs.setstate(std::ios_base::failbit);
}
} else if (ch == ')') {
rhs = re;
} else {
lhs.setstate(std::ios_base::failbit);
}
} else {
lhs.putback(ch);
lhs >> re;
rhs = re;
}
return lhs;
}
template<typename T, typename Char, typename Traits>
std::basic_ostream<Char, Traits>& operator<<(std::basic_ostream<Char, Traits>& lhs, sprout::complex<T> const& rhs) {
return lhs << '(' << rhs.real() << ',' << rhs.imag() << ')';
}
// 26.4.7, values:
template<typename T>
SPROUT_CONSTEXPR T real(sprout::complex<T> const& x) {
return x.real();
}
template<typename T>
SPROUT_CONSTEXPR T imag(sprout::complex<T> const& x) {
return x.imag();
}
template<typename T>
SPROUT_CONSTEXPR T abs(sprout::complex<T> const& x) {
using sprout::sqrt;
return sqrt(sprout::norm(x));
}
template<typename T>
SPROUT_CONSTEXPR T arg(sprout::complex<T> const& x) {
using sprout::atan2;
return atan2(x.imag(), x.real());
}
template<typename T>
SPROUT_CONSTEXPR T norm(sprout::complex<T> const& x) {
return x.real() * x.real() + x.imag() * x.imag();
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> conj(sprout::complex<T> const& x) {
return sprout::complex<T>(x.real(), -x.imag());
}
namespace detail {
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> proj_impl(sprout::complex<T> const& x, T const& den) {
return sprout::complex<T>(
T(2) * x.real() / den,
T(2) * x.imag() / den
);
}
} // detail
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> proj(sprout::complex<T> const& x) {
return sprout::detail::proj_impl(
x,
sprout::norm(x) + T(1)
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> polar(T const& rho, T const& theta = 0) {
using sprout::cos;
using sprout::sin;
return sprout::complex<T>(rho * cos(theta), rho * sin(theta));
}
// 26.4.8, transcendentals:
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> acos(sprout::complex<T> const& x);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> asin(sprout::complex<T> const& x);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> atan(sprout::complex<T> const& x);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> acosh(sprout::complex<T> const& x);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> asinh(sprout::complex<T> const& x);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> atanh(sprout::complex<T> const& x);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> cos(sprout::complex<T> const& x);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> cosh(sprout::complex<T> const& x);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> exp(sprout::complex<T> const& x);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> log(sprout::complex<T> const& x);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> log10(sprout::complex<T> const& x);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> pow(sprout::complex<T> const& x, T const& y);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> pow(sprout::complex<T> const& x, sprout::complex<T> const& y);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> pow(T const& x, sprout::complex<T> const& y);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> sin(sprout::complex<T> const& x);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> sinh(sprout::complex<T> const& x);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> sqrt(sprout::complex<T> const& x);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> tan(sprout::complex<T> const& x);
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> tanh(sprout::complex<T> const& x);
namespace detail {
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> acos_impl(sprout::complex<T> const& t) {
return sprout::complex<T>(sprout::math::half_pi<T>() - t.real(), -t.imag());
}
} // namespace detail
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> acos(sprout::complex<T> const& x) {
return sprout::detail::acos_impl(sprout::asin(x));
}
namespace detail {
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> asin_impl(sprout::complex<T> const& t) {
return sprout::complex<T>(t.imag(), -t.real());
}
} // namespace detail
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> asin(sprout::complex<T> const& x) {
return sprout::detail::asin_impl(sprout::asinh(sprout::complex<T>(-x.imag(), x.real())));
}
namespace detail {
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> atan_impl_1(
sprout::complex<T> const& x,
T const& r2,
T const& z,
T const& num,
T const& den
)
{
using sprout::atan2;
using sprout::log;
return sprout::complex<T>(
T(0.5) * atan2(T(2) * x.real(), z),
T(0.25) * log((r2 + num * num) / (r2 + den * den))
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> atan_impl(
sprout::complex<T> const& x,
T const& r2
)
{
return sprout::detail::atan_impl_1(
x,
r2,
T(1) - r2 - x.imag() * x.imag(),
x.imag() + T(1),
x.imag() - T(1)
);
}
} // namespace detail
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> atan(sprout::complex<T> const& x) {
return sprout::detail::atan_impl(x, x.real() * x.real());
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> acosh(sprout::complex<T> const& x) {
return T(2) * sprout::log(sprout::sqrt(T(0.5) * (x + T(1))) + sprout::sqrt(T(0.5) * (x - T(1))));
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> asinh(sprout::complex<T> const& x) {
return sprout::log(
sprout::sqrt(
sprout::complex<T>(
(x.real() - x.imag()) * (x.real() + x.imag()) + T(1),
T(2) * x.real() * x.imag()
)
)
+ x
);
}
namespace detail {
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> atanh_impl_1(
sprout::complex<T> const& x,
T const& i2,
T const& z,
T const& num,
T const& den
)
{
using sprout::atan2;
using sprout::log;
return sprout::complex<T>(
T(0.25) * (log(i2 + num * num) - log(i2 + den * den)),
T(0.5) * atan2(T(2) * x.imag(), z)
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> atanh_impl(
sprout::complex<T> const& x,
T const& i2
)
{
return sprout::detail::atanh_impl_1(
x,
i2,
T(1) - i2 - x.real() * x.real(),
T(1) + x.imag(),
T(1) - x.imag()
);
}
} // namespace detail
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> atanh(sprout::complex<T> const& x) {
return sprout::detail::atanh_impl(x, x.imag() * x.imag());
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> cos(sprout::complex<T> const& x) {
using sprout::cos;
using sprout::sin;
using sprout::cosh;
using sprout::sinh;
return sprout::complex<T>(
cos(x.real()) * cosh(x.imag()),
-(sin(x.real()) * sinh(x.imag()))
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> cosh(sprout::complex<T> const& x) {
using sprout::cos;
using sprout::sin;
using sprout::cosh;
using sprout::sinh;
return sprout::complex<T>(
cosh(x.real()) * cos(x.imag()),
sinh(x.real()) * sin(x.imag())
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> exp(sprout::complex<T> const& x) {
using sprout::exp;
return sprout::polar(exp(x.real()), x.imag());
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> log(sprout::complex<T> const& x) {
return sprout::complex<T>(sprout::log(sprout::abs(x)), sprout::arg(x));
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> log10(sprout::complex<T> const& x) {
using sprout::log;
return sprout::log(x) / log(T(10));
}
namespace detail {
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> pow_impl(sprout::complex<T> const& t, T const& y) {
using sprout::exp;
return sprout::polar(exp(y * t.real()), y * t.imag());
}
} // namespace detail
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> pow(sprout::complex<T> const& x, T const& y) {
using sprout::pow;
return x == T() ? T()
: x.imag() == T() && x.real() > T() ? pow(x.real(), y)
: sprout::detail::pow_impl(sprout::log(x), y)
;
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> pow(sprout::complex<T> const& x, sprout::complex<T> const& y) {
return x == T() ? T()
: sprout::exp(y * sprout::log(x))
;
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> pow(T const& x, sprout::complex<T> const& y) {
using sprout::log;
return x > T() ? sprout::polar(sprout::pow(x, y.real()), y.imag() * log(x))
: sprout::pow(sprout::complex<T>(x), y)
;
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> sin(sprout::complex<T> const& x) {
using sprout::cos;
using sprout::sin;
using sprout::cosh;
using sprout::sinh;
return sprout::complex<T>(
sin(x.real()) * cosh(x.imag()),
cos(x.real()) * sinh(x.imag())
);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> sinh(sprout::complex<T> const& x) {
using sprout::cos;
using sprout::sin;
using sprout::cosh;
using sprout::sinh;
return sprout::complex<T>(
sinh(x.real()) * cos(x.imag()),
cosh(x.real()) * sin(x.imag())
);
}
namespace detail {
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> sqrt_impl_1(sprout::complex<T> const& x, T const& t) {
return sprout::complex<T>(t, x.imag() < T() ? -t : t);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> sqrt_impl_2_1(sprout::complex<T> const& x, T const& t, T const& u) {
using sprout::abs;
return x.real() > T() ? sprout::complex<T>(u, x.imag() / t)
: sprout::complex<T>(abs(x.imag()) / t, x.imag() < T() ? -u : u)
;
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> sqrt_impl_2(sprout::complex<T> const& x, T const& t) {
return sqrt_impl_2(x, t, t / 2);
}
} // namespace detail
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> sqrt(sprout::complex<T> const& x) {
using sprout::sqrt;
using sprout::abs;
return x.real() == T() ? sprout::detail::sqrt_impl_1(x, sqrt(abs(x.imag()) / 2))
: sprout::detail::sqrt_impl_2(x, sqrt(2 * (sprout::abs(x) + abs(x.real()))))
;
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> tan(sprout::complex<T> const& x) {
return sprout::sin(x) / sprout::cos(x);
}
template<typename T>
SPROUT_CONSTEXPR sprout::complex<T> tanh(sprout::complex<T> const& x) {
return sprout::sinh(x) / sprout::cosh(x);
}
} // namespace sprout
#endif // #ifndef SPROUT_COMPLEX_HPP
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