vnl_finite_int< N > Class Template Reference

#include <vnl_finite.h>

List of all members.

---

Detailed Description

template<int N>
class vnl_finite_int< N >

finite modulo-N arithmetic.

The templated vnl_finite_int<N> provides arithmetic "modulo N", i.e., arithmetic in the finite (Galois) field GF(N) in case N is a prime or just in the finite ring (or semi-field) of integers modulo N otherwise. In that case division makes no sense (unless no zero divisor is involved), but all other operations remain valid.

Definition at line 47 of file vnl_finite.h.

Public Member Functions
	vnl_finite_int (int x=0)
	Creates a finite int element.
	vnl_finite_int (Base const &x)
	~vnl_finite_int ()
	operator int () const
	operator int ()
	operator long () const
	operator long ()
	operator short () const
	operator short ()
Base &	operator= (Base const &x)
	Assignment.
Base &	operator= (int x)
bool	operator== (Base const &x) const
	Comparison of finite int numbers.
bool	operator!= (Base const &x) const
bool	operator== (int x) const
bool	operator!= (int x) const
Base	operator- () const
	Unary minus - returns the additive inverse.
Base	operator+ () const
	Unary plus - returns the current number.
bool	operator! () const
	Unary not - returns true if finite int number is equal to zero.
Base &	operator+= (Base const &r)
	Plus&assign: replace lhs by lhs + rhs.
Base &	operator+= (int r)
Base &	operator-= (Base const &r)
	Minus&assign: replace lhs by lhs - rhs.
Base &	operator-= (int r)
Base &	operator *= (int r)
	Multiply&assign: replace lhs by lhs * rhs.
Base &	operator *= (Base const &r)
	Multiply&assign: replace lhs by lhs * rhs.
Base	reciproc () const
	Multiplicative inverse.
Base &	operator/= (Base const &r)
	Divide&assign. Uses r.reciproc() for efficient computation.
Base &	operator++ ()
	Pre-increment (++r).
Base &	operator-- ()
	Pre-decrement (--r).
Base	operator++ (int)
	Post-increment (r++).
Base	operator-- (int)
	Post-decrement (r--).
bool	is_unit () const
	Return true only when x is a unit in this ring.
bool	is_zero_divisor () const
	Return true only when x is a zero divisor, i.e., is not a unit.
unsigned int	additive_order () const
	The additive order of x is the smallest nonnegative r such that r*x == 0.
unsigned int	multiplicative_order () const
	The multiplicative order of x is the smallest r (>0) such that x^r == 1.
Base	pow (int r)
	Return the r-th power of this number.
unsigned int	log () const
	Return the smallest nonnegative exponent r for which x=g^r, where g is the smallest generator.
Static Public Member Functions
static unsigned int	cardinality ()
	The number of different finite_int numbers of this type.
static unsigned int	totient ()
	Return the Euler totient function, i.e., the number of units of this ring.
static vcl_vector< unsigned int >	decompose ()
	Write N as the unique product of prime factors.
static bool	is_field ()
	Return true when N is a prime number, i.e., when this ring is a field.
static Base	smallest_generator ()
	Return the smallest multiplicative generator of the units in this ring.
static Base	exp (int r)
	Return the inverse of log(), i.e., return g^r where g is the smallest generator.
static unsigned int	gcd (unsigned int l, unsigned int n)
	Calculate the greatest common divisor of l and N.
static unsigned int	gcd (unsigned int l)
Private Types
typedef vnl_finite_int< N >	Base
Private Member Functions
void	set_log (unsigned int r) const
	private function to set cached value of lp1_ when available.
Private Attributes
int	val_
	value of this number (smallest nonnegative representation)
unsigned int	mo_
	cached value for multiplicative order
unsigned int	lp1_
	cached value for 1+log()
Related Functions
(Note that these are not member functions.)
vcl_ostream &	operator<< (vcl_ostream &s, vnl_finite_int< N > const &r)
	formatted output.
vcl_istream &	operator>> (vcl_istream &s, vnl_finite_int< N > &r)
	simple input.
vnl_finite_int< N >	operator+ (vnl_finite_int< N > const &r1, vnl_finite_int< N > const &r2)
	Returns the sum of two finite int numbers.
vnl_finite_int< N >	operator+ (vnl_finite_int< N > const &r1, int r2)
	Returns the sum of two finite int numbers.
vnl_finite_int< N >	operator+ (int r2, vnl_finite_int< N > const &r1)
	Returns the sum of two finite int numbers.
vnl_finite_int< N >	operator- (vnl_finite_int< N > const &r1, vnl_finite_int< N > const &r2)
	Returns the difference of two finite int numbers.
vnl_finite_int< N >	operator- (vnl_finite_int< N > const &r1, int r2)
	Returns the difference of two finite int numbers.
vnl_finite_int< N >	operator- (int r2, vnl_finite_int< N > const &r1)
	Returns the difference of two finite int numbers.
vnl_finite_int< N >	operator * (vnl_finite_int< N > const &r1, vnl_finite_int< N > const &r2)
	Returns the product of two finite int numbers.
vnl_finite_int< N >	operator * (vnl_finite_int< N > const &r1, int r2)
	Returns the product of two finite int numbers.
vnl_finite_int< N >	operator * (int r2, vnl_finite_int< N > const &r1)
	Returns the product of two finite int numbers.
vnl_finite_int< N >	operator/ (vnl_finite_int< N > const &r1, vnl_finite_int< N > const &r2)
	Returns the quotient of two finite int numbers.
vnl_finite_int< N >	operator/ (vnl_finite_int< N > const &r1, int r2)
	Returns the quotient of two finite int numbers.
vnl_finite_int< N >	operator/ (int r1, vnl_finite_int< N > const &r2)
	Returns the quotient of two finite int numbers.
bool	operator== (int r1, vnl_finite_int< N > const &r2)
bool	operator!= (int r1, vnl_finite_int< N > const &r2)
vnl_finite_int< N >	vnl_math_squared_magnitude (vnl_finite_int< N > const &x)
vnl_finite_int< N >	vnl_math_sqr (vnl_finite_int< N > const &x)
bool	vnl_math_isnan (vnl_finite_int< N > const &)
bool	vnl_math_isfinite (vnl_finite_int< N > const &x)

---

Member Typedef Documentation

template<int N>

typedef vnl_finite_int<N> vnl_finite_int< N >::Base [private]

Definition at line 51 of file vnl_finite.h.

---

Constructor & Destructor Documentation

template<int N>

vnl_finite_int< N >::vnl_finite_int

(

int

x = 0

)

[inline]

Creates a finite int element.

Default constructor gives 0. Also serves as automatic cast from int to vnl_finite_int.

Definition at line 59 of file vnl_finite.h.

template<int N>

vnl_finite_int< N >::vnl_finite_int

(

Base const &

x

)

[inline]

Definition at line 61 of file vnl_finite.h.

template<int N>

vnl_finite_int< N >::~vnl_finite_int

(

)

[inline]

Definition at line 63 of file vnl_finite.h.

---

Member Function Documentation

template<int N>

static unsigned int vnl_finite_int< N >::cardinality

(

)

[inline, static]

The number of different finite_int numbers of this type.

Definition at line 54 of file vnl_finite.h.

template<int N>

vnl_finite_int< N >::operator int

(

)

const [inline]

Definition at line 65 of file vnl_finite.h.

template<int N>

vnl_finite_int< N >::operator int

(

)

[inline]

Definition at line 66 of file vnl_finite.h.

template<int N>

vnl_finite_int< N >::operator long

(

)

const [inline]

Definition at line 67 of file vnl_finite.h.

template<int N>

vnl_finite_int< N >::operator long

(

)

[inline]

Definition at line 68 of file vnl_finite.h.

template<int N>

vnl_finite_int< N >::operator short

(

)

const [inline]

Definition at line 69 of file vnl_finite.h.

template<int N>

vnl_finite_int< N >::operator short

(

)

[inline]

Definition at line 70 of file vnl_finite.h.

template<int N>

Base& vnl_finite_int< N >::operator=

(

Base const &

x

)

[inline]

Assignment.

Definition at line 73 of file vnl_finite.h.

template<int N>

Base& vnl_finite_int< N >::operator=

(

int

x

)

[inline]

Definition at line 74 of file vnl_finite.h.

template<int N>

bool vnl_finite_int< N >::operator==

(

Base const &

x

)

const [inline]

Comparison of finite int numbers.

Note that finite ints have no order, so < and > make no sense.

Definition at line 78 of file vnl_finite.h.

template<int N>

bool vnl_finite_int< N >::operator!=

(

Base const &

x

)

const [inline]

Definition at line 79 of file vnl_finite.h.

template<int N>

bool vnl_finite_int< N >::operator==

(

int

x

)

const [inline]

Definition at line 80 of file vnl_finite.h.

template<int N>

bool vnl_finite_int< N >::operator!=

(

int

x

)

const [inline]

Definition at line 81 of file vnl_finite.h.

template<int N>

Base vnl_finite_int< N >::operator-

(

)

const [inline]

Unary minus - returns the additive inverse.

Definition at line 84 of file vnl_finite.h.

template<int N>

Base vnl_finite_int< N >::operator+

(

)

const [inline]

Unary plus - returns the current number.

Definition at line 86 of file vnl_finite.h.

template<int N>

bool vnl_finite_int< N >::operator!

(

)

const [inline]

Unary not - returns true if finite int number is equal to zero.

Definition at line 88 of file vnl_finite.h.

template<int N>

Base& vnl_finite_int< N >::operator+=

(

Base const &

r

)

[inline]

Plus&assign: replace lhs by lhs + rhs.

Definition at line 91 of file vnl_finite.h.

template<int N>

Base& vnl_finite_int< N >::operator+=

(

int

r

)

[inline]

Definition at line 92 of file vnl_finite.h.

template<int N>

Base& vnl_finite_int< N >::operator-=

(

Base const &

r

)

[inline]

Minus&assign: replace lhs by lhs - rhs.

Definition at line 94 of file vnl_finite.h.

template<int N>

Base& vnl_finite_int< N >::operator-=

(

int

r

)

[inline]

Definition at line 95 of file vnl_finite.h.

template<int N>

Base& vnl_finite_int< N >::operator *=

(

int

r

)

[inline]

Multiply&assign: replace lhs by lhs * rhs.

Definition at line 97 of file vnl_finite.h.

template<int N>

Base& vnl_finite_int< N >::operator *=

(

Base const &

r

)

[inline]

Multiply&assign: replace lhs by lhs * rhs.

Definition at line 104 of file vnl_finite.h.

template<int N>

static unsigned int vnl_finite_int< N >::totient

(

)

[inline, static]

Return the Euler totient function, i.e., the number of units of this ring.

This number also equals the periodicity of the exponent: every unit, when raised to this power, yields 1.

Definition at line 116 of file vnl_finite.h.

template<int N>

Base vnl_finite_int< N >::reciproc

(

)

const [inline]

Multiplicative inverse.

Uses exp() and log() for efficient computation, unless log() is not defined.

Definition at line 132 of file vnl_finite.h.

template<int N>

Base& vnl_finite_int< N >::operator/=

(

Base const &

r

)

[inline]

Divide&assign. Uses r.reciproc() for efficient computation.

Definition at line 145 of file vnl_finite.h.

template<int N>

Base& vnl_finite_int< N >::operator++

(

)

[inline]

Pre-increment (++r).

Definition at line 151 of file vnl_finite.h.

template<int N>

Base& vnl_finite_int< N >::operator--

(

)

[inline]

Pre-decrement (--r).

Definition at line 153 of file vnl_finite.h.

template<int N>

Base vnl_finite_int< N >::operator++

(

int

)

[inline]

Post-increment (r++).

Definition at line 155 of file vnl_finite.h.

template<int N>

Base vnl_finite_int< N >::operator--

(

int

)

[inline]

Post-decrement (r--).

Definition at line 157 of file vnl_finite.h.

template<int N>

static vcl_vector<unsigned int> vnl_finite_int< N >::decompose

(

)

[inline, static]

Write N as the unique product of prime factors.

Definition at line 160 of file vnl_finite.h.

template<int N>

static bool vnl_finite_int< N >::is_field

(

)

[inline, static]

Return true when N is a prime number, i.e., when this ring is a field.

Definition at line 171 of file vnl_finite.h.

template<int N>

bool vnl_finite_int< N >::is_unit

(

)

const [inline]

Return true only when x is a unit in this ring.

In a field, all numbers except 0 are units. The total number of units is given by the Euler totient function.

Definition at line 179 of file vnl_finite.h.

template<int N>

bool vnl_finite_int< N >::is_zero_divisor

(

)

const [inline]

Return true only when x is a zero divisor, i.e., is not a unit.

Definition at line 182 of file vnl_finite.h.

template<int N>

unsigned int vnl_finite_int< N >::additive_order

(

)

const [inline]

The additive order of x is the smallest nonnegative r such that r*x == 0.

Definition at line 185 of file vnl_finite.h.

template<int N>

unsigned int vnl_finite_int< N >::multiplicative_order

(

)

const [inline]

The multiplicative order of x is the smallest r (>0) such that x^r == 1.

Definition at line 188 of file vnl_finite.h.

template<int N>

static Base vnl_finite_int< N >::smallest_generator

(

)

[inline, static]

Return the smallest multiplicative generator of the units in this ring.

This is only possible if the units form a cyclic group for multiplication. If not, smallest_generator() returns 1 to indicate this fact. Note that the multiplication group of a finite *field* is always cyclic.

Definition at line 200 of file vnl_finite.h.

template<int N>

Base vnl_finite_int< N >::pow

(

int

r

)

[inline]

Return the r-th power of this number.

Definition at line 218 of file vnl_finite.h.

template<int N>

unsigned int vnl_finite_int< N >::log

(

)

const [inline]

Return the smallest nonnegative exponent r for which x=g^r, where g is the smallest generator.

Definition at line 227 of file vnl_finite.h.

template<int N>

static Base vnl_finite_int< N >::exp

(

int

r

)

[inline, static]

Return the inverse of log(), i.e., return g^r where g is the smallest generator.

Definition at line 241 of file vnl_finite.h.

template<int N>

static unsigned int vnl_finite_int< N >::gcd	(	unsigned int	l,
		unsigned int	n
	)			[inline, static]

Calculate the greatest common divisor of l and N.

Definition at line 248 of file vnl_finite.h.

template<int N>

static unsigned int vnl_finite_int< N >::gcd

(

unsigned int

l

)

[inline, static]

Definition at line 253 of file vnl_finite.h.

template<int N>

void vnl_finite_int< N >::set_log

(

unsigned int

r

)

const [inline, private]

private function to set cached value of lp1_ when available.

Definition at line 257 of file vnl_finite.h.

---

Friends And Related Function Documentation

template<int N>

vcl_ostream & operator<<	(	vcl_ostream &	s,
		vnl_finite_int< N > const &	r
	)			[related]

formatted output.

Definition at line 266 of file vnl_finite.h.

template<int N>

vcl_istream & operator>>	(	vcl_istream &	s,
		vnl_finite_int< N > &	r
	)			[related]

simple input.

Definition at line 274 of file vnl_finite.h.

template<int N>

vnl_finite_int< N > operator+	(	vnl_finite_int< N > const &	r1,
		vnl_finite_int< N > const &	r2
	)			[related]

Returns the sum of two finite int numbers.

Definition at line 282 of file vnl_finite.h.

template<int N>

vnl_finite_int< N > operator+	(	vnl_finite_int< N > const &	r1,
		int	r2
	)			[related]

Returns the sum of two finite int numbers.

Definition at line 290 of file vnl_finite.h.

template<int N>

vnl_finite_int< N > operator+	(	int	r2,
		vnl_finite_int< N > const &	r1
	)			[related]

Returns the sum of two finite int numbers.

Definition at line 298 of file vnl_finite.h.

template<int N>

vnl_finite_int< N > operator-	(	vnl_finite_int< N > const &	r1,
		vnl_finite_int< N > const &	r2
	)			[related]

Returns the difference of two finite int numbers.

Definition at line 306 of file vnl_finite.h.

template<int N>

vnl_finite_int< N > operator-	(	vnl_finite_int< N > const &	r1,
		int	r2
	)			[related]

Returns the difference of two finite int numbers.

Definition at line 314 of file vnl_finite.h.

template<int N>

vnl_finite_int< N > operator-	(	int	r2,
		vnl_finite_int< N > const &	r1
	)			[related]

Returns the difference of two finite int numbers.

Definition at line 322 of file vnl_finite.h.

template<int N>

vnl_finite_int< N > operator *	(	vnl_finite_int< N > const &	r1,
		vnl_finite_int< N > const &	r2
	)			[related]

Returns the product of two finite int numbers.

Definition at line 330 of file vnl_finite.h.

template<int N>

vnl_finite_int< N > operator *	(	vnl_finite_int< N > const &	r1,
		int	r2
	)			[related]

Returns the product of two finite int numbers.

Definition at line 338 of file vnl_finite.h.

template<int N>

vnl_finite_int< N > operator *	(	int	r2,
		vnl_finite_int< N > const &	r1
	)			[related]

Returns the product of two finite int numbers.

Definition at line 346 of file vnl_finite.h.

template<int N>

vnl_finite_int< N > operator/	(	vnl_finite_int< N > const &	r1,
		vnl_finite_int< N > const &	r2
	)			[related]

Returns the quotient of two finite int numbers.

Uses r2.reciproc() for efficient computation.

Definition at line 355 of file vnl_finite.h.

template<int N>

vnl_finite_int< N > operator/	(	vnl_finite_int< N > const &	r1,
		int	r2
	)			[related]

Returns the quotient of two finite int numbers.

Definition at line 363 of file vnl_finite.h.

template<int N>

vnl_finite_int< N > operator/	(	int	r1,
		vnl_finite_int< N > const &	r2
	)			[related]

Returns the quotient of two finite int numbers.

Definition at line 371 of file vnl_finite.h.

template<int N>

bool operator==	(	int	r1,
		vnl_finite_int< N > const &	r2
	)			[related]

Definition at line 379 of file vnl_finite.h.

template<int N>

bool operator!=	(	int	r1,
		vnl_finite_int< N > const &	r2
	)			[related]

Definition at line 384 of file vnl_finite.h.

template<int N>

vnl_finite_int< N > vnl_math_squared_magnitude

(

vnl_finite_int< N > const &

x

)

[related]

Definition at line 389 of file vnl_finite.h.

template<int N>

vnl_finite_int< N > vnl_math_sqr

(

vnl_finite_int< N > const &

x

)

[related]

Definition at line 394 of file vnl_finite.h.

template<int N>

bool vnl_math_isnan

(

vnl_finite_int< N > const &

)

[related]

Definition at line 399 of file vnl_finite.h.

template<int N>

bool vnl_math_isfinite

(

vnl_finite_int< N > const &

x

)

[related]

Definition at line 404 of file vnl_finite.h.

---

Member Data Documentation

template<int N>

int vnl_finite_int< N >::val_ [private]

value of this number (smallest nonnegative representation)

Definition at line 49 of file vnl_finite.h.

template<int N>

unsigned int vnl_finite_int< N >::mo_ [mutable, private]

cached value for multiplicative order

Definition at line 259 of file vnl_finite.h.

template<int N>

unsigned int vnl_finite_int< N >::lp1_ [mutable, private]

cached value for 1+log()

Definition at line 260 of file vnl_finite.h.

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The documentation for this class was generated from the following file:

• core/vnl/vnl_finite.h

---