Domain for integer operations. More...

#include <PID-integer.h>

Inherits UnparametricField< K >.

Public Types
Common Object Interface for a LinBox Field.
These methods and member types are required of all LinBox fields. See FieldArchetype for detailed specifications.
typedef UnparametricRandIter< K >	RandIter
	Type of random field element generators.

Public Member Functions
Element &	axpyin (integer &r, const integer &a, const integer &x) const
	axpyin

Element &	axmyin (integer &r, const integer &a, const integer &x) const
	axmyin

Element &	maxpyin (integer &r, const integer &a, const integer &x) const
	maxpyin

Element &	axpy (integer &r, const integer &a, const integer &x, const integer &y) const
	axpy

bool	isUnit (const Element &x) const
	isUnit

Element &	abs (Element &x, const Element &a) const
	abs

Element	abs (const Element &a) const
	abs

long	compare (const Element &a, const Element &b) const
	compare two elements, a and b. More...

Element &	gcd (Element &g, const Element &a, const Element &b) const
	gcd (g, a, b) return g = gcd (a, b)

Element &	gcdin (Element &g, const Element &b) const
	gcdin(g, b) return g = gcd (g, b)

Element &	xgcd (Element &g, Element &s, Element &t, const Element &a, const Element &b) const
	xgcd (g, s, t, a, b) g = gcd(a, b) = as + bt. More...

Element &	lcm (Element &c, const Element &a, const Element &b) const
	lcm (c, a, b) c = lcm (a, b)

Element &	lcmin (Element &l, const Element &b) const
	lcmin (l, b) l = lcm (l, b)

Element &	quo (Element &q, const Element &a, const Element &b) const
	quo (q, x, y) q = floor (x/y);

Element &	rem (Element &r, const Element &a, const Element &b) const
	rem (r, a, b) r = remindar of a / b

Element &	quoin (Element &a, const Element &b) const
	quoin (a, b) a = quotient (a, b)

Element &	remin (Element &a, const Element &b) const
	quoin (a, b) a = quotient (a, b)

void	quoRem (Element &q, Element &r, const Element &a, const Element &b) const
	quoRem (q, r, a, b) q = [a/b], r = a - b*q \|r\| < \|b\|, and if r != 0, sign(r) = sign(b)

bool	isDivisor (const Element &a, const Element &b) const
	isDivisor (a, b) Test if b \| a.

Element &	sqrt (Element &x, const Element &y) const
	sqrt(x,y) x=floor(sqrt(y))

Field Object Basics.
std::istream &	read (std::istream &is)
	Builds this field to have characteristic q and cardinality q^e. More...

std::istream &	read (std::istream &s, Element &a) const
	Builds this field to have characteristic q and cardinality q^e. More...

template<typename Src >
Element &	init (Element &x, const Src &s) const
	Builds this field to have characteristic q and cardinality q^e. More...

Element &	init (Element &x) const
	Builds this field to have characteristic q and cardinality q^e. More...

template<typename T >
T &	convert (T &x, const Element &y) const
	Builds this field to have characteristic q and cardinality q^e. More...

integer &	cardinality (integer &c) const
	Builds this field to have characteristic q and cardinality q^e. More...

integer &	characteristic (integer &c) const
	Builds this field to have characteristic q and cardinality q^e. More...

Implementation-Specific Methods.
These methods are not required of all LinBox fields and are included only for the implementation of this field template.
const K &	operator() (void) const
	Constant access operator. More...

K &	operator() (void)
	Access operator. More...

Protected Member Functions
void	RationalReconstruction (Element &a, Element &b, const Element &f, const Element &m, const Element &k, bool reduce, bool recursive) const
	Rational number reconstruction. More...

Detailed Description

Domain for integer operations.

Examples:: examples/charpoly.C, examples/det.C, examples/doubledet.C, examples/graph-charpoly.C, examples/minpoly.C, examples/smith.C, examples/smithvalence.C, examples/smithvalence.h, examples/solve.C, examples/sparseelimdet.C, and examples/valence.C.

Member Function Documentation

long compare	(	const Element &	a,
		const Element &	b
	)		const

inline

compare two elements, a and b.

return 1, if a > b return 0, if a = b; return -1. if a < b

Element& xgcd	(	Element &	g,
		Element &	s,
		Element &	t,
		const Element &	a,
		const Element &	b
	)		const

inline

xgcd (g, s, t, a, b) g = gcd(a, b) = a*s + b*t.

The coefficients s and t are defined according to the standard Euclidean algorithm applied to |a| and |b|, with the signs then adjusted according to the signs of a and b.

void RationalReconstruction	(	Element &	a,
		Element &	b,
		const Element &	f,
		const Element &	m,
		const Element &	k,
		bool	reduce,
		bool	recursive
	)		const

inlineprotected

Rational number reconstruction.

$\frac{n}{d} \equiv f \mod m$ , with $\vert n \vert <k$ and $0 < \vert d \vert \leq \frac{f}{k}$ .

Bibliography:

von zur Gathen & Gerhard, Modern Computer Algebra, 5.10, Cambridge Univ. Press 1999

std::istream& read ( std::istream & is )

inlineinherited