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std::norm(std::complex)

Defined in header <complex>
(1) 
template< class T > 
T norm( const std::complex<T>& z );
 (until C++20) 
template< class T > 
constexpr T norm( const std::complex<T>& z );
 (since C++20)
Additional overloads (since C++11)
Defined in header <complex>
(A) 
float       norm( float f );
double      norm( double f );
long double norm( long double f );
 (until C++20) 
constexpr float       norm( float f );
constexpr double      norm( double f );
constexpr long double norm( long double f );
 (since C++20)
(until C++23)		 
template< class FloatingPoint >
constexpr FloatingPoint norm( FloatingPoint f );
 (since C++23)
(B) 
template< class Integer > 
double norm( Integer i );
 (until C++20) 
template< class Integer > 
constexpr double norm( Integer i );
 (since C++20)
1) Returns the squared magnitude of the complex number z.
A,B) Additional overloads are provided for all integer and floating-point types, which are treated as complex numbers with zero imaginary component. (since C++11)

Parameters

z - complex value
f - floating-point value
i - integer value

Return value

1) The squared magnitude of z.
A) The square of f.
B) The square of i.

Notes

The norm calculated by this function is also known as field norm or absolute square.

The Euclidean norm of a complex number is provided by std::abs, which is more costly to compute. In some situations, it may be replaced by std::norm, for example, if abs(z1) > abs(z2) then norm(z1) > norm(z2).

The additional overloads are not required to be provided exactly as (A,B). They only need to be sufficient to ensure that for their argument num:

  • If num has a standard(until C++23) floating-point type T, then std::norm(num) has the same effect as std::norm(std::complex<T>(num)).
  • Otherwise, if num has an integer type, then std::norm(num) has the same effect as std::norm(std::complex<double>(num)).

Example

#include <cassert>
#include <complex>
#include <iostream>
 
int main()
{
    constexpr std::complex<double> z {3.0, 4.0};
    static_assert(std::norm(z) == (z.real() * z.real() + z.imag() * z.imag()));
    static_assert(std::norm(z) == (z * std::conj(z)));
           assert(std::norm(z) == (std::abs(z) * std::abs(z)));
    std::cout << "std::norm(" << z << ") = " << std::norm(z) << '\n';
}

Output:

std::norm((3,4)) = 25

See also

returns the magnitude of a complex number
(function template)
returns the complex conjugate
(function template)
constructs a complex number from magnitude and phase angle
(function template)

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