Definition of Complex types. More...

#include "plumbing/defs.h"

Include dependency graph for cmplx.h:

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Classes
class	Complex< T >
	Complex definition. More...

struct	hila::is_complex_or_arithmetic< T >
	hila::is_complex_or_arithmetic<T>::value More...

class	Imaginary_t< T >
	Imaginary type, used to represent purely imaginary numbers. More...

Namespaces
namespace	hila
	Implement hila::swap for gauge fields.

Functions
template<typename T >
T	real (const Complex< T > &a)
	Return real value of Complex number.

template<typename T >
T	imag (const Complex< T > &a)
	Retrun imaginary value of Complex number.

template<typename T >
Complex< T >	polar (T r, T arg)
	Return complex number given by polar representation.

template<typename T1 , typename T2 , typename Tr = hila::type_plus<T1, T2>>
Complex< Tr >	operator+ (const Complex< T1 > &a, const Complex< T2 > &b)
	Addition operator Complex + Complex.

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator+ (const Complex< T > &c, const A &a)
	Addition operator Complex + Scalar.

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator+ (const A &a, const Complex< T > &c)
	Addition operator Scalar + Complex.

template<typename T1 , typename T2 , typename Tr = hila::type_plus<T1, T2>>
Complex< Tr >	operator- (const Complex< T1 > &a, const Complex< T2 > &b)
	Subtraction operator Complex - Complex.

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator- (const Complex< T > &c, const A &a)
	Subtraction operator Complex - Scalar.

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator- (const A &a, const Complex< T > &c)
	Subtraction operator Scalar - Complex.

template<typename T1 , typename T2 , typename Tr = hila::type_mul<T1, T2>>
Complex< Tr >	operator* (const Complex< T1 > &a, const Complex< T2 > &b)
	Multiplication operator Complex * Complex.

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator* (const Complex< T > &c, const A &a)
	Multiplication operator Complex * Scalar This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator* (const A &a, const Complex< T > &c)
	Multiplication operator Scalar * Complex This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

template<typename T1 , typename T2 , typename Tr = hila::type_mul<T1, T2>>
Complex< Tr >	operator/ (const Complex< T1 > &a, const Complex< T2 > &b)
	Division operator Complex / Complex.

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator/ (const Complex< T > &c, const A &a)
	Division operator Complex / Scalar.

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
auto	operator/ (const A &a, const Complex< T > &c)
	Division operator Scalar / Complex.

template<typename T >
Complex< T >	mul_add (const Complex< T > &a, const Complex< T > &b, const Complex< T > &c)
	Multiply add with Complex numbers.

template<typename A , typename B >
bool	operator== (const Complex< A > &a, const Complex< B > &b)
	Compare equality of two complex numbers.

template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< B >::value, int > = 0>
bool	operator== (const Complex< A > &a, const B b)
	Compare equality of Complex and scalar.

template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
bool	operator== (const A a, const Complex< B > &b)
	Compare equality of Scalar and Complex.

template<typename A , typename B >
bool	operator!= (const Complex< A > &a, const Complex< B > &b)
	Compare non-equality of two complex numbers.

template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< B >::value, int > = 0>
bool	operator!= (const Complex< A > &a, const B b)
	Compare non-equality of Complex number and Scalar.

template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>
bool	operator!= (const A a, const Complex< B > &b)
	Compare non-equality of Scalar and Complex number.

template<typename T >
T	abs (const Complex< T > &a)
	Return absolute value of Complex number.

template<typename T >
T	arg (const Complex< T > &a)
	Return argument of Complex number.

template<typename T >
Complex< T >	conj (const Complex< T > &val)
	Return conjugate of Complex number.

template<typename T >
Complex< T >	dagger (const Complex< T > &val)
	Return dagger of Complex number.

template<typename T >
auto	squarenorm (const Complex< T > &val)
	Return Squarenorm of Complex number.

template<typename T >
std::ostream &	operator<< (std::ostream &strm, const Complex< T > &A)
	Print a complex value as (re,im)

template<typename T >
std::string	hila::to_string (const Complex< T > &A, int prec=8, char separator=' ')
	Return Complex number as std::string.

template<typename T >
std::string	hila::prettyprint (const Complex< T > &A, int prec=8)
	Return well formatted Complex number as std::string.

constexpr Imaginary_t< double >	I (1.0)
	Imaginary unit I - global variable.

template<typename T >
Complex< T >	exp (const Complex< T > z)
	\(\exp(z)\)

template<typename T , std::enable_if_t< hila::is_arithmetic< T >::value, int > = 0>
Complex< T >	expi (T a)
	\(\exp(i\cdot x)\)

template<typename T >
Complex< T >	log (Complex< T > z)
	\(\log{z}\)

template<typename T >
Complex< T >	sqrt (Complex< T > z)
	\(\sqrt{z}\)

template<typename T >
Complex< T >	cbrt (Complex< T > z)
	\(\sqrt[3]{z}\)

template<typename A , typename B >
auto	pow (Complex< A > z, Complex< B > p)
	pow(z.p) = \(z^p\) = \(exp(p*log(z))\)

template<typename T >
Complex< T >	sin (Complex< T > z)

template<typename T >
Complex< T >	cos (Complex< T > z)

template<typename T >
Complex< T >	tan (Complex< T > z)
	\(\tan(z) = \frac{\sin(z)}{\cos(z)}\) - rely on optimizer to simplify

template<typename T >
Complex< T >	sinh (Complex< T > z)

template<typename T >
Complex< T >	cosh (Complex< T > z)

template<typename T >
Complex< T >	tanh (Complex< T > z)
	\(\tanh(z)\)

template<typename T >
Complex< T >	atan (Complex< T > z)
	\(\arctan(z)\)

template<typename T >
Complex< T >	asin (Complex< T > z)
	\(\arcsin(z)\)

template<typename T >
Complex< T >	acos (Complex< T > z)
	\(\arccos(z)\)

template<typename T >
Complex< T >	atanh (Complex< T > z)
	\(\text{artanh}(z)\)

template<typename T >
Complex< T >	asinh (Complex< T > z)
	\(\text{arsinh}(z)\)

template<typename T >
Complex< T >	acosh (Complex< T > z)
	\(\text{arcosh}(z)\)

constexpr Imaginary_t< double >	operator""_i (long double a)

Detailed Description

Definition of Complex types.

This file contains definitions and methods for Complex numbers and Imaginary type.

‍NOTE: All overloads for operators +,-,/,* are not documented separately since there exists a function for each combinations of scalar,imaginary and complex number representations. All versions are documented in the Complex – Complex definitions.

Definition in file cmplx.h.

Function Documentation

◆ abs()

template<typename T >

T abs ( const Complex< T > & a )

inline

Return absolute value of Complex number.

Wrapper around Complex::abs

Template Parameters

T	Arithmetic type of a

Parameters

a	Complex number to get abs from

Returns: T

Definition at line 1187 of file cmplx.h.

◆ arg()

template<typename T >

T arg ( const Complex< T > & a )

inline

Return argument of Complex number.

Wrapper around Complex::arg

Template Parameters

T	Arithmetic type of a

Parameters

a	Complex number to get arg from

Returns: T

Definition at line 1199 of file cmplx.h.

◆ conj()

template<typename T >

Complex< T > conj ( const Complex< T > & val )

inline

Return conjugate of Complex number.

Wrapper around Complex::conj

Template Parameters

T	Arithmetic type of a

Parameters

a	Complex number to get conjugate from

Returns: Complex<T>

Definition at line 1211 of file cmplx.h.

◆ cos()

template<typename T >

Complex< T > cos ( Complex< T > z )

inline

\begin{align}\cos(z) &= \cos(\Re{z})\cos(i \Im(z)) - \sin(\Re{z})\sin(i \Im{z}) \\ &= \cos(\Re{z})\cosh(\Im(z)) - i \sin(\Re{z})\sinh(\Im(z))\end{align}

Definition at line 1476 of file cmplx.h.

◆ cosh()

template<typename T >

Complex< T > cosh ( Complex< T > z )

inline

cosh(z) = cosh(re)cosh(i im) - sinh(re)sinh(i im) = cosh(re)cos(im) - i sinh(re)sin(im)

Definition at line 1496 of file cmplx.h.

◆ dagger()

template<typename T >

Complex< T > dagger ( const Complex< T > & val )

inline

Return dagger of Complex number.

Wrapper around Complex::conj

Template Parameters

T	Arithmetic type of a

Parameters

a	Complex number to get conjugate from

Returns: Complex<T>

Definition at line 1223 of file cmplx.h.

◆ I()

constexpr Imaginary_t< double > I ( 1. 0 )

constexpr

Imaginary unit I - global variable.

◆ imag()

template<typename T >

T imag ( const Complex< T > & a )

inline

Retrun imaginary value of Complex number.

Wrapper around Complex::imag

Template Parameters

T	Arithmetic type of a

Parameters

a	Complex number to get imaginary value from

Returns: T

Definition at line 803 of file cmplx.h.

◆ mul_add()

template<typename T >

Complex< T > mul_add	(	const Complex< T > &	a,
		const Complex< T > &	b,
		const Complex< T > &	c
	)

inline

Multiply add with Complex numbers.

Defined as

mul_add(a,b,c) = a*b + c;

Template Parameters

T	Arithmetic type of a,b and c

Parameters

a	Complex number to multiply
b	Complex number to multiply
c	Complex number to add to result of multiplication

Returns: Complex<T>

Definition at line 1049 of file cmplx.h.

◆ operator!=() [1/3]

template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

bool operator!=	(	const A	a,
		const Complex< B > &	b
	)

inline

Compare non-equality of Scalar and Complex number.

Negation of operator==()

Template Parameters

A	Arithmetic type of a
B	Arithmetic type of b

Parameters

a	Scalar to compare
b	Complex number to compare

Returns: true if values are not arithmetically equal

Definition at line 1158 of file cmplx.h.

◆ operator!=() [2/3]

template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< B >::value, int > = 0>

bool operator!=	(	const Complex< A > &	a,
		const B	b
	)

inline

Compare non-equality of Complex number and Scalar.

Negation of operator==()

Template Parameters

A	Arithmetic type of a
B	Arithmetic type of b

Parameters

a	Complex number to compare
b	Scalar to compare

Returns: true if values are not arithmetically equal

Definition at line 1142 of file cmplx.h.

◆ operator!=() [3/3]

template<typename A , typename B >

bool operator!=	(	const Complex< A > &	a,
		const Complex< B > &	b
	)

inline

Compare non-equality of two complex numbers.

Negation of operator==()

Template Parameters

A	Arithmetic type of a
B	Arithmetic type of b

Parameters

a	Complex number to compare
b	Complex number to compare

Returns: true if values are not arithmetically equal

Definition at line 1126 of file cmplx.h.

◆ operator""_i()

constexpr Imaginary_t< double > operator""_i ( long double a )

constexpr

Operators to implement imaginary unit 1_i, enablig expressions 3 + 2_i etc. This is defined as an user-defined literal, which requires an underscore.

Definition at line 1548 of file cmplx.h.

◆ operator*() [1/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator*	(	const A &	a,
		const Complex< T > &	c
	)

inline

Multiplication operator Scalar * Complex This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Multiplication between Complex and scalar is defined in the usual way

\begin{align}z &= x + iy \in \mathbb{C}, a \in \mathbb{R} \\ z * a &= (x\cdot a + iy\cdot a)\end{align}

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

c	Complex number to multiply
a	Scalar to multiply

Returns: auto

Definition at line 967 of file cmplx.h.

◆ operator*() [2/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator*	(	const Complex< T > &	c,
		const A &	a
	)

inline

Multiplication operator Complex * Scalar This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Multiplication between Complex and scalar is defined in the usual way

\begin{align}z &= x + iy \in \mathbb{C}, a \in \mathbb{R} \\ z * a &= (x\cdot a + iy\cdot a)\end{align}

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

c	Complex number to multiply
a	Scalar to multiply

Returns: auto

Definition at line 950 of file cmplx.h.

◆ operator*() [3/3]

template<typename T1 , typename T2 , typename Tr = hila::type_mul<T1, T2>>

Complex< Tr > operator*	(	const Complex< T1 > &	a,
		const Complex< T2 > &	b
	)

inline

Multiplication operator Complex * Complex.

Defined in the usual way

\begin{align}z &= x + iy, w = x' + iy' \in \mathbb{C} \\ z w &= (x + iy)(x' + iy') = (xx'-yy') + i(xy' + yx')\end{align}

Template Parameters

T1	Arithmetic type of a
T2	Arithmetic type of b
Tr	Resulting type after multiplication

Parameters

a
b

Returns: Complex<Tr>

Definition at line 933 of file cmplx.h.

◆ operator+() [1/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator+	(	const A &	a,
		const Complex< T > &	c
	)

inline

Addition operator Scalar + Complex.

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

a	Scalar to sum
c	Complex number to sum

Returns: auto

Definition at line 870 of file cmplx.h.

◆ operator+() [2/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator+	(	const Complex< T > &	c,
		const A &	a
	)

inline

Addition operator Complex + Scalar.

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Defined in the usual way

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

c	Complex number to sum
a	Scalar to sum

Returns: auto

Definition at line 856 of file cmplx.h.

◆ operator+() [3/3]

template<typename T1 , typename T2 , typename Tr = hila::type_plus<T1, T2>>

Complex< Tr > operator+	(	const Complex< T1 > &	a,
		const Complex< T2 > &	b
	)

inline

Addition operator Complex + Complex.

Addition between Complex numbers is defined in the usual way

Template Parameters

T1	Arithmetic type of a
T2	Arithmetic type of b
Tr	Resulting type after addition

Parameters

a
b

Returns: Complex<Tr>

Definition at line 839 of file cmplx.h.

◆ operator-() [1/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator-	(	const A &	a,
		const Complex< T > &	c
	)

inline

Subtraction operator Scalar - Complex.

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

a	Scalar to subtract
c	Complex number to subtract

Returns: auto

Definition at line 914 of file cmplx.h.

◆ operator-() [2/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator-	(	const Complex< T > &	c,
		const A &	a
	)

inline

Subtraction operator Complex - Scalar.

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Defined in the usual way

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

c	Complex number to subtract
a	Scalar to subtract

Returns: auto

Definition at line 900 of file cmplx.h.

◆ operator-() [3/3]

template<typename T1 , typename T2 , typename Tr = hila::type_plus<T1, T2>>

Complex< Tr > operator-	(	const Complex< T1 > &	a,
		const Complex< T2 > &	b
	)

inline

Subtraction operator Complex - Complex.

Subtraction between Complex numbers is defined in the usual way

Template Parameters

T1	Arithmetic type of a
T2	Arithmetic type of b
Tr	Resulting type after subtraction

Parameters

a
b

Returns: Complex<Tr>

Definition at line 885 of file cmplx.h.

◆ operator/() [1/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator/	(	const A &	a,
		const Complex< T > &	c
	)

inline

Division operator Scalar / Complex.

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Defined in the usual way

\begin{align} z=x + iy &\in \mathbb{C}, a \in \mathbb{R} \\ \frac{a}{z} &= \frac{az^*}{|z|^2} \end{align}

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

c	Complex number to divide
a	Scalar to divide with

Returns: auto

Definition at line 1030 of file cmplx.h.

◆ operator/() [2/3]

template<typename T , typename A , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

auto operator/	(	const Complex< T > &	c,
		const A &	a
	)

inline

Division operator Complex / Scalar.

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Defined in the usual way

\begin{align} z=x + iy &\in \mathbb{C}, a \in \mathbb{R} \\ \frac{z}{a} &= \frac{x}{a} + i\cdot \frac{y}{a} \end{align}

Template Parameters

T	Arithmetic type of c
A	Arithmetic type of a

Parameters

c	Complex number to divide
a	Scalar to divide with

Returns: auto

Definition at line 1011 of file cmplx.h.

◆ operator/() [3/3]

template<typename T1 , typename T2 , typename Tr = hila::type_mul<T1, T2>>

Complex< Tr > operator/	(	const Complex< T1 > &	a,
		const Complex< T2 > &	b
	)

inline

Division operator Complex / Complex.

Defined in the usual way

\begin{align} a,b &\in \mathbb{C} \\ \frac{a}{b} &= \frac{a\cdot b^{*}}{|b|^2} \end{align}

Template Parameters

T1	Arithmetic type of a
T2	Arithmetic type of b
Tr	Resulting type after division

Parameters

a	Complex number to divide
b	Complex number to divide with

Returns: Complex<Tr>

Definition at line 992 of file cmplx.h.

◆ operator<<()

template<typename T >

std::ostream & operator<<	(	std::ostream &	strm,
		const Complex< T > &	A
	)

Print a complex value as (re,im)

Stream operator

Print Complex number as "A.re A.im"

Template Parameters

T	Arithmetic type of A

Parameters

strm
A	Complex number to print

Returns: std::ostream&

Definition at line 1253 of file cmplx.h.

◆ operator==() [1/3]

template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< A >::value, int > = 0>

bool operator==	(	const A	a,
		const Complex< B > &	b
	)

inline

Compare equality of Scalar and Complex.

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Two numbers are equal, if the arithmetic values are equal: thus, complex and real comparison (a + i b) == a is true if b == 0.

Template Parameters

A	Arithmetic type of a
B	Arithmetic type of b

Parameters

b	Complex number to compare
a	Scalar to compare

Returns: true if values compare to equal

Definition at line 1110 of file cmplx.h.

◆ operator==() [2/3]

template<typename A , typename B , std::enable_if_t< hila::is_arithmetic< B >::value, int > = 0>

bool operator==	(	const Complex< A > &	a,
		const B	b
	)

inline

Compare equality of Complex and scalar.

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Two numbers are equal, if the arithmetic values are equal: thus, complex and real comparison (a + i b) == a is true if b == 0.

Template Parameters

A	Arithmetic type of a
B	Arithmetic type of b

Parameters

a	Complex number to compare
b	Scalar to compare

Returns: true if values compare to equal

Definition at line 1091 of file cmplx.h.

◆ operator==() [3/3]

template<typename A , typename B >

bool operator==	(	const Complex< A > &	a,
		const Complex< B > &	b
	)

inline

Compare equality of two complex numbers.

Two numbers are equal, if the real and imaginary components are respectively equal

Template Parameters

A	Arithmetic type of a
B	Arithmetic type of b

Parameters

a	Complex number to compare
b	Complex number to compare

Returns: true if values compare to equal

Definition at line 1072 of file cmplx.h.

◆ polar()

template<typename T >

Complex< T > polar	(	T	r,
		T	arg
	)

Return complex number given by polar representation.

Same as Complex::polar

Template Parameters

T	Arithmetic type r and arg

Parameters

r	Radial component of Complex number
arg	Argument of Complex number

Returns: Complex<T>

Definition at line 816 of file cmplx.h.

◆ real()

template<typename T >

T real ( const Complex< T > & a )

inline

Return real value of Complex number.

Wrapper around Complex::real

Template Parameters

T	Arithmetic type of a

Parameters

a	Complex number to get real value from

Returns: T

Definition at line 791 of file cmplx.h.

◆ sin()

template<typename T >

Complex< T > sin ( Complex< T > z )

inline

\begin{align}sin(z) &= \sin(\Re(z) + i \Im(z)) \\ &= \sin(\Re(z))\cos(i \Im(z)) + \cos(\Re(z))\sin(i \Im(z)) \\ &= \sin(\Re(z)) \cosh(\Im(z)) + i \cos(\Re(z)) \sinh(\Im(z)) \end{align}

Definition at line 1469 of file cmplx.h.

◆ sinh()

template<typename T >

Complex< T > sinh ( Complex< T > z )

inline

sinh(z) = sinh(re)cosh(i im) + cosh(re)sinh(i im) = sinh(re)cos(im) + i cosh(re)sin(im)

Definition at line 1489 of file cmplx.h.

◆ squarenorm()

template<typename T >

auto squarenorm ( const Complex< T > & val )

inline

Return Squarenorm of Complex number.

Wrapper around Complex::squarenorm

Template Parameters

T	Arithmetic type of val

Parameters

val	Complex number to compute squarenorm of

Returns: T

Definition at line 1235 of file cmplx.h.

Classes

Namespaces

Functions

Detailed Description

Function Documentation

◆ abs()

◆ arg()

◆ conj()

◆ cos()

◆ cosh()

◆ dagger()

◆ I()

◆ imag()

◆ mul_add()

◆ operator!=() [1/3]

◆ operator!=() [2/3]

◆ operator!=() [3/3]

◆ operator""_i()

◆ operator*() [1/3]

◆ operator*() [2/3]

◆ operator*() [3/3]

◆ operator+() [1/3]

◆ operator+() [2/3]

◆ operator+() [3/3]

◆ operator-() [1/3]

◆ operator-() [2/3]

◆ operator-() [3/3]

◆ operator/() [1/3]

◆ operator/() [2/3]

◆ operator/() [3/3]

◆ operator<<()

◆ operator==() [1/3]

◆ operator==() [2/3]

◆ operator==() [3/3]

◆ polar()

◆ real()

◆ sin()

◆ sinh()

◆ squarenorm()