DixonSolver< Ring, Field, RandomPrime, Method::DenseElimination > Class Template Reference

partial specialization of p-adic based solver with Dixon algorithm. More...

#include <dixon-solver-dense.h>

Collaboration diagram for DixonSolver< Ring, Field, RandomPrime, Method::DenseElimination >:

Public Member Functions
	DixonSolver (const Ring &r=Ring(), const RandomPrime &rp=RandomPrime())
	Constructor.
	DixonSolver (const Prime &p, const Ring &r=Ring(), const RandomPrime &rp=RandomPrime())
	Constructor, trying the prime p first.
template<class IMatrix , class Vector1 , class Vector2 >
SolverReturnStatus	solve (Vector1 &num, Integer &den, const IMatrix &A, const Vector2 &b, const bool s=false, const int maxPrimes=5, const SolverLevel level=SL_LASVEGAS)
	Solve a linear system Ax=b over quotient field of a ring.
template<class IMatrix , class Vector1 , class Vector2 >
SolverReturnStatus	solve (Vector1 &num, Integer &den, const IMatrix &A, const Vector2 &b, const int maxPrimes, const SolverLevel level=SL_LASVEGAS)
	overload so that the bool 'oldMatrix' argument is not accidentally set to true
template<class IMatrix , class Vector1 , class Vector2 >
SolverReturnStatus	solveNonsingular (Vector1 &num, Integer &den, const IMatrix &A, const Vector2 &b, bool s=false, int maxPrimes=5)
	Solve a nonsingular, square linear system Ax=b over quotient field of a ring.
template<class IMatrix , class Vector1 , class Vector2 >
SolverReturnStatus	solveSingular (Vector1 &num, Integer &den, const IMatrix &A, const Vector2 &b, int maxPrimes=5, const SolverLevel level=SL_LASVEGAS)
	Solve a general rectangular linear system Ax=b over quotient field of a ring.
template<class IMatrix , class Vector1 , class Vector2 >
SolverReturnStatus	findRandomSolution (Vector1 &num, Integer &den, const IMatrix &A, const Vector2 &b, int maxPrimes=5, const SolverLevel level=SL_LASVEGAS)
	Find a random solution of the general linear system Ax=b over quotient field of a ring.
template<class IMatrix , class Vector1 , class Vector2 >
SolverReturnStatus	monolithicSolve (Vector1 &num, Integer &den, const IMatrix &A, const Vector2 &b, Method::Dixon method)
	Big solving routine to perform random solving and certificate generation.

Detailed Description

template<class Ring, class Field, class RandomPrime>
class LinBox::DixonSolver< Ring, Field, RandomPrime, Method::DenseElimination >

partial specialization of p-adic based solver with Dixon algorithm.

See the following reference for details on this algorithm:

Bibliography:

• John D. Dixon Exact Solution of linear equations using p-adic expansions. Numerische Mathematik, volume 40, pages 137-141, 1982.

Constructor & Destructor Documentation

DixonSolver() [1/2]

template<class Ring , class Field , class RandomPrime >

DixonSolver ( const Ring &  r = Ring(), const RandomPrime &  rp = RandomPrime()  )

inline

Constructor.

Parameters

r	a Ring, set by default
rp	a RandomPrime generator, set by default

DixonSolver() [2/2]

template<class Ring , class Field , class RandomPrime >

DixonSolver ( const Prime &  p, const Ring &  r = Ring(), const RandomPrime &  rp = RandomPrime()  )

inline

Constructor, trying the prime p first.

Parameters

p	a Prime
r	a Ring, set by default
rp	a RandomPrime generator, set by default

Member Function Documentation

solve()

template<class Ring , class Field , class RandomPrime >

template<class IMatrix , class Vector1 , class Vector2 >

SolverReturnStatus solve	(	Vector1 &	num,
		Integer &	den,
		const IMatrix &	A,
		const Vector2 &	b,
		const bool	s = false,
		const int	maxPrimes = 5,
		const SolverLevel	level = SL_LASVEGAS
	)

Solve a linear system Ax=b over quotient field of a ring.

Parameters

num	Vector of numerators of the solution
den	The common denominator. 1/den * num is the rational solution of Ax=b.
A	Matrix of linear system
b	Right-hand side of system
s
maxPrimes	maximum number of moduli to try
level	level of certification to be used

Returns

status of solution. if (return != SS_FAILED), and (level >= SL_LASVEGAS), solution is guaranteed correct. SS_FAILED - all primes used were bad SS_OK - solution found. SS_INCONSISTENT - system appreared inconsistent. certificate is in lastCertificate if (level >= SL_CERTIFIED)

solveNonsingular()

template<class Ring , class Field , class RandomPrime >

template<class IMatrix , class Vector1 , class Vector2 >

SolverReturnStatus solveNonsingular	(	Vector1 &	num,
		Integer &	den,
		const IMatrix &	A,
		const Vector2 &	b,
		bool	s = false,
		int	maxPrimes = 5
	)

Solve a nonsingular, square linear system Ax=b over quotient field of a ring.

Parameters

num	Vector of numerators of the solution
den	The common denominator. 1/den * num is the rational solution of Ax = b
A	Matrix of linear system (it must be square)
b	Right-hand side of system
s	unused
maxPrimes	maximum number of moduli to try

Returns

status of solution :

• SS_FAILED all primes used were bad;
• SS_OK solution found, guaranteed correct;
• SS_SINGULAR system appreared singular mod all primes.

solveSingular()

template<class Ring , class Field , class RandomPrime >

template<class IMatrix , class Vector1 , class Vector2 >

SolverReturnStatus solveSingular	(	Vector1 &	num,
		Integer &	den,
		const IMatrix &	A,
		const Vector2 &	b,
		int	maxPrimes = 5,
		const SolverLevel	level = SL_LASVEGAS
	)

Solve a general rectangular linear system Ax=b over quotient field of a ring.

If A is known to be square and nonsingular, calling solveNonsingular is more efficient.

Parameters

num	Vector of numerators of the solution
den	The common denominator. 1/den * num is the rational solution of Ax = b
A	Matrix of linear system
b	Right-hand side of system
maxPrimes	maximum number of moduli to try
level	level of certification to be used

Returns

status of solution. if (return != SS_FAILED), and (level >= SL_LASVEGAS), solution is guaranteed correct.

• SS_FAILED all primes used were bad
• SS_OK solution found.
• SS_INCONSISTENT system appeared inconsistent. certificate is in lastCertificate if (level >= SL_CERTIFIED)

findRandomSolution()

template<class Ring , class Field , class RandomPrime >

template<class IMatrix , class Vector1 , class Vector2 >

SolverReturnStatus findRandomSolution	(	Vector1 &	num,
		Integer &	den,
		const IMatrix &	A,
		const Vector2 &	b,
		int	maxPrimes = 5,
		const SolverLevel	level = SL_LASVEGAS
	)

Find a random solution of the general linear system Ax=b over quotient field of a ring.

Parameters

num	Vector of numerators of the solution
den	The common denominator. 1/den * num is the rational solution of Ax = b.
A	Matrix of linear system
b	Right-hand side of system
maxPrimes	maximum number of moduli to try
level	level of certification to be used

Returns

status of solution. if (return != SS_FAILED), and (level >= SL_LASVEGAS), solution is guaranteed correct.

• SS_FAILED all primes used were bad
• SS_OK solution found.
• SS_INCONSISTENT system appreared inconsistent. certificate is in lastCertificate if (level >= SL_CERTIFIED)

monolithicSolve()

template<class Ring , class Field , class RandomPrime >

template<class IMatrix , class Vector1 , class Vector2 >

SolverReturnStatus monolithicSolve	(	Vector1 &	num,
		Integer &	den,
		const IMatrix &	A,
		const Vector2 &	b,
		Method::Dixon	method
	)

Big solving routine to perform random solving and certificate generation.

Same arguments and return as findRandomSolution, except

Parameters

num	Vector of numerators of the solution
den	The common denominator. 1/den * num is the rational solution of Ax = b
A
b
randomSolution	parameter to determine whether to randomize or not (since solveSingular calls this function as well)
makeMinDenomCert	determines whether a partial certificate for the minimal denominator of a rational solution is made
maxPrimes
level

When (randomSolution == true && makeMinDenomCert == true),

• If (level == SL_MONTECARLO) this function has the same effect as calling findRandomSolution.
• If (level >= SL_LASVEGAS && return == SS_OK), lastCertifiedDenFactor contains a certified factor of the min-solution's denominator.
• If (level >= SL_CERTIFIED && return == SS_OK), lastZBNumer and lastCertificate are updated as well.

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

• dixon-solver-dense.h
• dixon-solver-dense.inl