liblinbox-dev 1.4.2-3 / usr / include / linbox / algorithms / rational-solver.inl

/* linbox/algorithms/rational-solver.inl
 * Copyright (C) 2004 Pascal Giorgi
 *
 * Written by Pascal Giorgi  <pascal.giorgi@ens-lyon.fr>
 * Modified by David Pritchard  <daveagp@mit.edu>
 *
 * ========LICENCE========
 * This file is part of the library LinBox.
 *
  * LinBox is free software: you can redistribute it and/or modify
 * it under the terms of the  GNU Lesser General Public
 * License as published by the Free Software Foundation; either
 * version 2.1 of the License, or (at your option) any later version.
 *
 * This library is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.	 See the GNU
 * Lesser General Public License for more details.
 *
 * You should have received a copy of the GNU Lesser General Public
 * License along with this library; if not, write to the Free Software
 * Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301  USA
 * ========LICENCE========
 */

#ifndef __LINBOX_rational_solver_INL
#define __LINBOX_rational_solver_INL

#include "linbox/util/debug.h"
#include "linbox/linbox-config.h"

#include "linbox/matrix/sparse-matrix.h"
#include "linbox/blackbox/lambda-sparse.h"
#include "linbox/blackbox/transpose.h"
#include "linbox/blackbox/diagonal.h"
#include "linbox/blackbox/compose.h"
#include "linbox/algorithms/lifting-container.h"
#include "linbox/algorithms/rational-reconstruction.h"
#include "linbox/algorithms/matrix-inverse.h"
#include "linbox/algorithms/matrix-hom.h"
#include "linbox/algorithms/gauss.h"
#include "linbox/algorithms/blackbox-container.h"
#include "linbox/algorithms/massey-domain.h"
#include "linbox/algorithms/blackbox-block-container.h"
#include "linbox/algorithms/block-massey-domain.h"
#include "linbox/algorithms/vector-fraction.h"
#include <fflas-ffpack/ffpack/ffpack.h>
#include <fflas-ffpack/fflas/fflas.h>
#include "linbox/solutions/methods.h"
#include "linbox/blackbox/block-hankel-inverse.h"

#include "linbox/vector/blas-vector.h"

// #ifdef __LINBOX_BLAS_AVAILABLE
#include "linbox/config-blas.h"
#include "linbox/matrix/dense-matrix.h"
#include "linbox/matrix/matrix-domain.h"
#include "linbox/matrix/factorized-matrix.h"
#include "linbox/util/timer.h"
// #endif

//#define DEBUG_DIXON
//#define DEBUG_INC
//#define SKIP_NONSINGULAR

namespace LinBox
{

	/*! @brief NO DOC !
	 * @bug why is this hard coded ?
	*/
	template <class Prime>
	inline bool checkBlasPrime(const Prime p)
	{
		return p < Prime(67108863);
	}

	template<>
	inline bool checkBlasPrime(const BlasVector<Givaro::ZRing<Integer> > p)
	{
		bool tmp=true;
		for (size_t i=0;i<p.size();++i)
			if  (p[i] >= integer(67108863)) {
				tmp=false;
				break;
			}

		return tmp;
	}


	// SPECIALIZATION FOR WIEDEMANN

	// note: if Vector1 != Vector2 compilation of solve or solveSingluar will fail (via an invalid call to sparseprecondition)!
	// maybe they should not be templated separately, or sparseprecondition should be rewritten

	template <class Ring, class Field, class RandomPrime>
	template <class IMatrix, class Vector1, class Vector2>
	SolverReturnStatus
	RationalSolver<Ring,Field,RandomPrime,WiedemannTraits>::solve (Vector1& num,
								       Integer& den,
								       const IMatrix& A,
								       const Vector2& b,
								       const bool old,
								       int maxPrimes) const
	{
		SolverReturnStatus status=SS_FAILED;

		switch (A.rowdim() == A.coldim() ? solveNonsingular(num, den, A,b) : SS_SINGULAR) {

		case SS_OK:
			status=SS_OK;
			break;

		case SS_SINGULAR:
			commentator().report (Commentator::LEVEL_IMPORTANT, INTERNAL_ERROR) << "switching to singular" << std::endl;
			//std::cerr<<"switching to singular\n";
			status=solveSingular(num, den,A,b);
			break;

		case SS_FAILED:
			break;

		default:
			throw LinboxError ("Bad return value from solveNonsingular");

		}

		return status;
	}

	template <class Ring, class Field, class RandomPrime>
	template <class IMatrix, class Vector1, class Vector2>
	SolverReturnStatus
	RationalSolver<Ring, Field, RandomPrime, WiedemannTraits>::solveNonsingular( Vector1& num,
												      Integer& den,
												      const IMatrix& A,
												      const Vector2& b,
												      int maxPrimes) const
	{
		// checking if matrix is square
		linbox_check(A.rowdim() == A.coldim());

		// checking size of system
		linbox_check(A.rowdim() == b.size());



		SparseMatrix<Field> *Ap;
		FPolynomial MinPoly;
		unsigned long  deg;
		unsigned long issingular = SINGULARITY_THRESHOLD;
		static Field *F=NULL;
		Prime prime = _prime;
		do {
#ifdef RSTIMING
			tNonsingularSetup.clear();
			tNonsingularSetup.start();
#endif
			_prime = prime;
			if (F != NULL) delete F;
			F=new Field(prime);
			Ap = new SparseMatrix<Field>(A, *F);
			typename Field::RandIter random(*F);
			BlackboxContainer<Field,SparseMatrix<Field> > Sequence(Ap,*F,random);
			MasseyDomain<Field,BlackboxContainer<Field,SparseMatrix<Field> > > MD(&Sequence);
#ifdef RSTIMING
			tNonsingularSetup.stop();
			ttNonsingularSetup+=tNonsingularSetup;
			tNonsingularMinPoly.clear();
			tNonsingularMinPoly.start();
#endif
			MD.minpoly(MinPoly,deg);
#ifdef RSTIMING
			tNonsingularMinPoly.stop();
			ttNonsingularMinPoly+=tNonsingularMinPoly;
#endif
			prime = _genprime.randomPrime();
		}
		while(F->isZero(MinPoly.front()) && --issingular );


		if (!issingular){
			std::cerr<<"The Matrix is singular\n";
			delete Ap;
			return SS_SINGULAR;
		}
		else {

			typedef SparseMatrix<Field> FMatrix;

			typedef WiedemannLiftingContainer<Ring, Field, IMatrix, FMatrix, FPolynomial> LiftingContainer;

			LiftingContainer lc(_ring, *F, A, *Ap, MinPoly, b,_prime);

			RationalReconstruction<LiftingContainer> re(lc);

			re.getRational(num, den, 0);
#ifdef RSTIMING
			ttNonsingularSolve.update(re, lc);
#endif
			return SS_OK;
		}
	}

	template <class Ring, class Field, class RandomPrime>
	template <class IMatrix, class Vector1, class Vector2>
	SolverReturnStatus
	RationalSolver<Ring,Field,RandomPrime,WiedemannTraits>:: solveSingular (Vector1& num,
										     Integer& den,
										     const IMatrix& A,
										     const Vector2& b,
										     int maxPrimes) const
	{
		std::cerr<<"in singular solver\n";

		typedef BlasVector<Ring>  IVector;
		typedef SparseMatrix<Field>                  FMatrix;

		// checking size of system
		linbox_check(A.rowdim() == b.size());

		typedef LambdaSparseMatrix<Ring>  IPreconditioner;
		typedef LambdaSparseMatrix<Field> FPreconditioner;

		typedef Compose<IPreconditioner,Compose<IMatrix,IPreconditioner> > IPrecondMatrix;
		typedef Compose<FPreconditioner,Compose<FMatrix,FPreconditioner> > FPrecondMatrix;

		FMatrix               *Ap;
		IPreconditioner *P     =NULL;
		IPreconditioner *Q     =NULL;
		FPreconditioner *Pmodp =NULL;
		FPreconditioner *Qmodp =NULL;
		IPrecondMatrix  *PAQ   =NULL;
		FPrecondMatrix  *PApQ  =NULL;
		IVector Pb;


		FPolynomial MinPoly;
		unsigned long  deg;
		unsigned long badprecondition = BAD_PRECONTITIONER_THRESHOLD;
		Field *F;
		Prime prime = _prime;
		typename Field::Element tmp;
		do {
			if (PApQ != NULL) {
				delete P;
				delete Q;
				delete PApQ;
				delete PAQ;
			}
			_prime = prime;
			F=new Field(prime);//std::cerr<<"here\n";
			Ap = new FMatrix(A, *F);
			sparseprecondition (*F,&A,PAQ,Ap,PApQ,b,Pb,P,Q,Pmodp,Qmodp);
			typename Field::RandIter random(*F);
			BlackboxContainer<Field,FPrecondMatrix> Sequence(PApQ,*F,random);
			MasseyDomain<Field,BlackboxContainer<Field,FPrecondMatrix> > MD(&Sequence);

			MD.minpoly(MinPoly,deg);
			//MinPoly.resize(3);MinPoly[0]=1;MinPoly[1]=2;MinPoly[2]=1;
			prime = _genprime.randomPrime();
			F->add(tmp,MinPoly.at(1),MinPoly.front());
		}
		while(((F->isZero(tmp) || MinPoly.size() <=2) && --badprecondition ));
		std::cerr<<"minpoly found with size: "<<MinPoly.size()<<std::endl;
		for (size_t i=0;i<MinPoly.size();++i)
			std::cerr<<MinPoly[i]<<"*x^"<<i<<"+";
		std::cerr<<std::endl;

		std::cerr<<"prime is: "<<_prime<<std::endl;
		if (!badprecondition){
			std::cerr<<"Bad Preconditionner\n";

			delete Ap;
			if (PAQ  != NULL) delete PAQ;
			if (PApQ != NULL) delete PApQ;
			if (P    != NULL) delete P;
			if (Q    != NULL) delete Q;

			return SS_BAD_PRECONDITIONER;
		}
		else {

			MinPoly.erase(MinPoly.begin());

			typedef WiedemannLiftingContainer<Ring, Field, IPrecondMatrix, FPrecondMatrix, FPolynomial> LiftingContainer;
			std::cerr<<"before lc\n";
			LiftingContainer lc(_ring, *F, *PAQ, *PApQ, MinPoly, Pb, _prime);
			std::cerr<<"constructing lifting container of length: "<<lc.length()<<std::endl;

			RationalReconstruction<LiftingContainer> re(lc,_ring,2);

			re.getRational(num, den, 0);


			if (Q    != NULL) {

#if 0
				   typename Ring::Element lden;
				   _ring. assign (lden, _ring.one);
				   typename Vector1::iterator p;
				   for (p = answer.begin(); p != answer.end(); ++ p)
				   _ring. lcm (lden, lden, p->second);

#endif

				IVector Qx(num.size());

#if 0
				   typename IVector::iterator p_x;

				   for (p = answer.begin(), p_x = x. begin(); p != answer.end(); ++ p, ++ p_x) {
				   _ring. mul (*p_x, p->first, lden);
				   _ring. divin (*p_x, p->second);
				   }
#endif

				Q->apply(Qx, num);
#if 0
				   for (p=answer.begin(),p_x=Qx.begin(); p != answer.end();++p,++p_x){
				   p->first=*p_x;
				   p->second=lden;
				   }
#endif
				num = Qx;
			}


			delete Ap;
			if (PAQ  != NULL) delete PAQ;
			if (PApQ != NULL) delete PApQ;
			if (P    != NULL) delete P;
			if (Q    != NULL) delete Q;

			return SS_OK;
		}
	}


	template <class Ring, class Field, class RandomPrime>
	template <class IMatrix, class FMatrix, class IVector>
	void
	RationalSolver<Ring,Field,RandomPrime,WiedemannTraits>::sparseprecondition (const Field& F,
										     const IMatrix *A,
										     Compose<LambdaSparseMatrix<Ring>, Compose<IMatrix, LambdaSparseMatrix<Ring> > > *&PAQ,
										     const FMatrix *Ap,
										     Compose<LambdaSparseMatrix<Field>, Compose<FMatrix, LambdaSparseMatrix<Field> > > *&PApQ,
										     const IVector& b,
										     IVector& Pb,
										     LambdaSparseMatrix<Ring> *&P,
										     LambdaSparseMatrix<Ring> *&Q,
										     LambdaSparseMatrix<Field> *&Pmodp,
										     LambdaSparseMatrix<Field> *&Qmodp) const
	{
#if 0
		std::cerr<<"A:\n";
		A->write(std::cerr);
		std::cerr<<"A mod p:\n";
		Ap->write(std::cerr);
#endif
		VectorDomain<Ring> VD(_ring);
#if 0
		std::cerr<<"b:\n";
		VD.write(std::cerr,b)<<std::endl;
#endif


		commentator().start ("Constructing sparse preconditioner");
		typedef LambdaSparseMatrix<Ring>  IPreconditioner;
		typedef LambdaSparseMatrix<Field> FPreconditioner;

		size_t min_dim = A->coldim() < A->rowdim() ? A->coldim() : A->rowdim();

		P = new  IPreconditioner(_ring,min_dim,A->rowdim(),2,3.);
		// 		std::cerr<<"P:\n";
		// 		P->write(std::cerr);

		Q = new  IPreconditioner(_ring,A->coldim(),min_dim,2,3.);
		// 		std::cerr<<"Q:\n";
		// 		Q->write(std::cerr);

		Compose<IMatrix,IPreconditioner> *AQ;
		AQ = new Compose<IMatrix,IPreconditioner> (A,Q);

		PAQ = new Compose<IPreconditioner, Compose<IMatrix,IPreconditioner> > (P,AQ);		;
		Pb.resize(min_dim);
		P->apply(Pb,b);
		// 		std::cerr<<"Pb:\n";
		// 		VD.write(std::cerr,Pb)<<std::endl;

		Pmodp = new FPreconditioner(F,*P);
		std::cerr<<"P mod p completed\n";
		Qmodp = new FPreconditioner(F,*Q);
		std::cerr<<"Q mod p completed\n";

		Compose<FMatrix,FPreconditioner> *ApQ;
		ApQ = new Compose<FMatrix,FPreconditioner> (Ap,Qmodp);

		PApQ = new Compose<FPreconditioner, Compose<FMatrix,FPreconditioner> > (Pmodp, ApQ);
		std::cerr<<"Preconditioning done\n";
		commentator().stop ("done");

	}


#if 0
	template <class Ring, class Field, class RandomPrime>
	template <class IMatrix, class FMatrix, class IVector,class FVector>
	void RationalSolver<Ring,Field,RandomPrime,WiedemannTraits>::
	precondition (const Field&                          F,
		      const IMatrix&                        A,
		      BlackboxArchetype<IVector>        *&PAQ,
		      const FMatrix                       *Ap,
		      BlackboxArchetype<FVector>       *&PApQ,
		      const IVector                        &b,
		      IVector                             &Pb,
		      BlackboxArchetype<IVector>          *&P,
		      BlackboxArchetype<IVector>          *&Q) const
	{
		switch (_traits.preconditioner() ) {

		case WiedemannTraits::BUTTERFLY:
			commentator().report (Commentator::LEVEL_IMPORTANT, INTERNAL_ERROR)
			<<"ERROR: Butterfly preconditioner not implemented yet. Sorry." << std::endl;

		case WiedemannTraits::SPARSE:
			{
				commentator().start ("Constructing sparse preconditioner");

				P = new LambdaSparseMatrix<Ring> (_ring,Ap->coldim(),Ap->rowdim(),2);

				PAQ = new Compose<LambdaSparseMatrix<Ring>, IMatrix> (*P,A);

				P->apply(Pb,b);

				LambdaSparseMatrix<Field> Pmodp(F,*P);

				PApQ = new Compose<LambdaSparseMatrix<Field>, FMatrix> (Pmodp, *Ap);

				commentator().stop ("done");
				break;
			}

		case WiedemannTraits::TOEPLITZ:
			commentator().report (Commentator::LEVEL_IMPORTANT, INTERNAL_ERROR)
			<< "ERROR: Toeplitz preconditioner not implemented yet. Sorry." << std::endl;

		case WiedemannTraits::NONE:
			throw PreconditionFailed (__func__, __LINE__, "preconditioner is BUTTERFLY, SPARSE, or TOEPLITZ");

		default:
			throw PreconditionFailed (__func__, __LINE__, "preconditioner is BUTTERFLY, SPARSE, or TOEPLITZ");
		}



	}
#endif


	// SPECIALIZATION FOR BLOCK WIEDEMANN

	// note: if Vector1 != Vector2 compilation of solve or solveSingluar will fail (via an invalid call to sparseprecondition)!
	// maybe they should not be templated separately, or sparseprecondition should be rewritten

	template <class Ring, class Field, class RandomPrime>
	template <class IMatrix, class Vector1, class Vector2>
	SolverReturnStatus
	RationalSolver<Ring,Field,RandomPrime,BlockWiedemannTraits>::solve (Vector1& num, Integer& den,
										     const IMatrix& A,
										     const Vector2& b,
										     const bool old,
										     int maxPrimes) const
	{
		SolverReturnStatus status=SS_FAILED;

		switch (A.rowdim() == A.coldim() ? solveNonsingular(num, den, A,b) : SS_SINGULAR) {

		case SS_OK:
			status=SS_OK;
			break;

		case SS_SINGULAR:
			commentator().report (Commentator::LEVEL_IMPORTANT, INTERNAL_ERROR) << "could switch to singular but not doing it(?)" << std::endl;
			//std::cerr<<"switching to singular\n";
			//status=solveSingular(num, den,A,b);
			break;

		case SS_FAILED:
			break;

		default:
			throw LinboxError ("Bad return value from solveNonsingular");

		}

		return status;
	}

	template <class Ring, class Field, class RandomPrime>
	template <class IMatrix, class Vector1, class Vector2>
	SolverReturnStatus
	RationalSolver<Ring,Field,RandomPrime,BlockWiedemannTraits>::solveNonsingular (Vector1& num,
										       Integer& den,
										       const IMatrix& A,
										       const Vector2& b,
										       int maxPrimes) const
	{
		// checking if matrix is square
		linbox_check(A.rowdim() == A.coldim());

		// checking size of system
		linbox_check(A.rowdim() == b.size());

		size_t m,n;
		integer tmp,tmproot;
		tmp=A.coldim();
#if 0
		m = n = tmp.bitsize();
		m = n = sqrt(tmp);
		m = n = root(tmp,3); // wrong # args to root. -bds
#endif
		// m = n =
		root(tmproot, tmp,3);
		m = n = uint32_t(tmproot);
		// 		std::cout<<"block factor= "<<m<<"\n";;
		typedef SparseMatrix<Field> FMatrix;

		Field F(_prime);
		FMatrix Ap(A, F);
		Transpose<FMatrix > Bp(Ap);
		// 		std::cout<<"Ap:\n";
		// 		Ap.write(std::cout);
		typedef BlockWiedemannLiftingContainer<Ring, Field, Transpose<IMatrix >, Transpose<FMatrix > > LiftingContainer;

		Transpose<IMatrix> B(A);

		LiftingContainer lc(_ring, F, B, Bp, b,_prime, m, n);

		RationalReconstruction<LiftingContainer> re(lc);

		re.getRational(num, den, 0);
#ifdef RSTIMING
		ttNonsingularSolve.update(re, lc);
#endif

		return SS_OK;
	}

	// END OF SPECIALIZATION FOR BLOCK WIEDEMANN



	// SPECIALIZATION FOR DIXON

	template <class Ring, class Field, class RandomPrime>
	template <class IMatrix, class Vector1, class Vector2>
	SolverReturnStatus
	RationalSolver<Ring,Field,RandomPrime,DixonTraits>::solve (Vector1& num,
								   Integer& den,
								   const IMatrix& A,
								   const Vector2& b,
								   const bool old,
								   int maxP,
								   const SolverLevel level ) const
	{

		SolverReturnStatus status;
		int maxPrimes=maxP;
		while (maxPrimes > 0)
		{
#ifdef SKIP_NONSINGULAR
			switch (SS_SINGULAR)
#else
				switch (A.rowdim() == A.coldim() ? solveNonsingular(num, den,A,b,old,maxPrimes) : SS_SINGULAR)
#endif
				{

				case SS_OK:
#ifdef DEBUG_DIXON
					std::cout <<"nonsingular worked\n";
#endif
					return SS_OK;
					break;

				case SS_SINGULAR:
#ifdef DEBUG_DIXON
					std::cout<<"switching to singular\n";
#endif
					status = solveSingular(num, den,A,b,maxPrimes,level);
					if (status != SS_FAILED)
						return status;
					break;

				case SS_FAILED:
					//std::cout <<"nonsingular failed\n";  // BDS: in what sense is this part of the spec of solve?
					break;

				default:
					throw LinboxError ("Bad return value from solveNonsingular");

				}
			maxPrimes--;
			if (maxPrimes > 0) chooseNewPrime();
		}
		return SS_FAILED;
	}


	template <class Ring, class Field, class RandomPrime>
	template <class IMatrix, class Vector1, class Vector2>
	SolverReturnStatus
	RationalSolver<Ring,Field,RandomPrime,DixonTraits>::solveNonsingular(Vector1& num,
									     Integer& den,
									     const IMatrix& A,
									     const Vector2& b,
									     bool oldMatrix,
									     int maxPrimes) const
	{

		// std::cout<<"DIXON\n\n\n\n";
#ifdef DEBUG_DIXON
		std::cout << "entering nonsingular solver\n";
#endif
		int trials = 0, notfr;

		// history sensitive data for optimal reason
		// static const IMatrix* IMP;

		BlasMatrix<Field>* FMP = NULL;
		Field *F=NULL;

		do
		{
#if 0
			if (trials == maxPrimes) return SS_SINGULAR;
			if (trials != 0) chooseNewPrime();
			++trials;
#endif
#ifdef DEBUG_DIXON
			//std::cout << "_prime: "<<_prime<<"\n";
			std::cout<<"A:=\n";
			A.write(std::cout);
			std::cout<<"b:=\n";
			for (size_t i=0;i<b.size();++i) std::cout<<b[i]<<" , ";
			std::cout<<std::endl;
#endif
#ifdef RSTIMING
			tNonsingularSetup.start();
#endif
			// typedef typename Field::Element Element;
			// typedef typename Ring::Element Integer;

			// checking size of system
			linbox_check(A.rowdim() == A.coldim());
			linbox_check(A.rowdim() == b.size());

			LinBox::integer tmp;

			// if input matrix A is different one.
			if (!oldMatrix) {
				if (trials == maxPrimes) return SS_SINGULAR;
				if (trials != 0) chooseNewPrime();
				++trials;

				// Could delete a non allocated matrix -> segfault
				if (FMP != NULL) delete FMP;

				// IMP = &A;

				if (F != NULL) delete F;

				F= new Field (_prime);

				FMP = new BlasMatrix<Field>(*F, A.rowdim(),A.coldim());

				MatrixHom::map (*FMP, A ); // use MatrixHom to reduce matrix PG 2005-06-16
#if 0
				typename BlasMatrix<Field>::Iterator iter_p  = FMP->Begin();
				typename IMatrix::ConstIterator iter  = A.Begin();
				for (;iter != A.End();++iter,++iter_p)
					F->init(*iter_p, _ring.convert(tmp,*iter));
#endif

#ifdef DEBUG_DIXON
				std::cout<< "p = ";
				F->write(std::cout);
				std::cout<<" A mod p :=\n";
				FMP->write(std::cout);
#endif

				if (!checkBlasPrime(_prime)){
					if (FMP != NULL) delete FMP;
					FMP = new BlasMatrix<Field>(*F, A.rowdim(),A.coldim());
					notfr = (int)MatrixInverse::matrixInverseIn(*F,*FMP);
				}
				else {
					BlasMatrix<Field> *invA = new BlasMatrix<Field>(*F, A.rowdim(),A.coldim());
					BlasMatrixDomain<Field> BMDF(*F);
#ifdef RSTIMING
					tNonsingularSetup.stop();
					ttNonsingularSetup += tNonsingularSetup;
					tNonsingularInv.start();
#endif
					assert(FMP != NULL);
					BMDF.invin(*invA, *FMP, notfr); //notfr <- nullity
					// if (FMP != NULL)  // useless
					delete FMP;
					FMP = invA;
#if 0
					std::cout << "notfr = " << notfr << std::endl;
					std::cout << "inverse mod p: " << std::endl;
					FMP->write(std::cout, *F);
#endif
#ifdef RSTIMING
					tNonsingularInv.stop();
					ttNonsingularInv += tNonsingularInv;
#endif
				}
			}
			else {
#ifdef RSTIMING
				tNonsingularSetup.stop();
				ttNonsingularSetup += tNonsingularSetup;
#endif
				notfr = 0;
			}
		} while (notfr);

#ifdef DEBUG_DIXON
		std::cout<<"A^-1 mod p :=\n";
		FMP->write(std::cout);
#endif

		typedef DixonLiftingContainer<Ring,Field,IMatrix,BlasMatrix<Field> > LiftingContainer;
		LiftingContainer lc(_ring, *F, A, *FMP, b, _prime);
		RationalReconstruction<LiftingContainer > re(lc);
		if (!re.getRational(num, den,0)){
			delete FMP;
			return SS_FAILED;
		}
#ifdef RSTIMING
		ttNonsingularSolve.update(re, lc);
#endif
		if (F!=NULL)
			delete F;
		if (FMP != NULL)
			delete FMP;
		return SS_OK;
	}

	template <class Ring, class Field, class RandomPrime>
	template <class IMatrix, class Vector1, class Vector2>
	SolverReturnStatus
	RationalSolver<Ring,Field,RandomPrime,DixonTraits>::solveSingular (Vector1& num,
									   Integer& den,
									   const IMatrix& A,
									   const Vector2& b,
									   int maxPrimes,
									   const SolverLevel level) const
	{
		return monolithicSolve (num, den, A, b, false, false, maxPrimes, level);
	}

	template <class Ring, class Field, class RandomPrime>
	template <class IMatrix, class Vector1, class Vector2>
	SolverReturnStatus
	RationalSolver<Ring,Field,RandomPrime,DixonTraits>::findRandomSolution (Vector1& num,
										     Integer& den,
										     const IMatrix& A,
										     const Vector2& b,
										     int maxPrimes,
										     const SolverLevel level ) const
	{

		return monolithicSolve (num, den, A, b, false, true, maxPrimes, level);
	}


	// Most solving is done by the routine below.
	// There used to be one for random and one for deterministic, but they have been merged to ease with
	//  repeated code (certifying inconsistency, optimization are 2 examples)

	template <class Ring, class Field, class RandomPrime>
	template <class IMatrix, class Vector1, class Vector2>
	SolverReturnStatus
	RationalSolver<Ring,Field,RandomPrime,DixonTraits>::monolithicSolve (Vector1& num,
										     Integer& den,
										     const IMatrix& A,
										     const Vector2& b,
										     bool makeMinDenomCert,
										     bool randomSolution,
										     int maxPrimes,
										     const SolverLevel level) const
	{

		if (level == SL_MONTECARLO && maxPrimes > 1)
			std::cout << "WARNING: Even if maxPrimes > 1, SL_MONTECARLO uses just one prime." << std::endl;
#if 0
		if (makeMinDenomCert && !randomSolution)
			std::cout << "WARNING: Will not compute a certificate of minimal denominator deterministically." << std::endl;
#endif
		if (makeMinDenomCert && level == SL_MONTECARLO)
			std::cout << "WARNING: No certificate of min-denominality generated due to  level=SL_MONTECARLO" << std::endl;
		int trials = 0;
		while (trials < maxPrimes){
			if (trials != 0) chooseNewPrime();
			++trials;
#ifdef DEBUG_DIXON
			std::cout << "_prime: "<<_prime<<"\n";
#endif
#ifdef RSTIMING
			tSetup.start();
#endif

			// typedef typename Field::Element Element;
			// typedef typename Ring::Element  Integer_t;
			typedef DixonLiftingContainer<Ring, Field,
				BlasMatrix<Ring>, BlasMatrix<Field> > LiftingContainer;

			// checking size of system
			linbox_check(A.rowdim() == b.size());

			LinBox::integer tmp;
			Field F (_prime);
			BlasMatrixDomain<Ring>  BMDI(_ring);
			BlasMatrixDomain<Field> BMDF(F);
			BlasApply<Ring> BAR(_ring);
			MatrixDomain<Ring> MD(_ring);
			VectorDomain<Ring> VDR(_ring);

			BlasMatrix<Ring> A_check(A); // used to check answer later

			// TAS_xxx stands for Transpose Augmented System (A|b)t
			// this provides a factorization (A|b) = TAS_Pt . TAS_Ut . TAS_Qt . TAS_Lt
			// such that
			// - TAS_P . (A|b) . TAS_Q   has nonzero principal minors up to TAS_rank
			// - TAS_Q permutes b to the (TAS_rank)th column of A iff the system is inconsistent mod p
			BlasMatrix<Field>* TAS_factors = new BlasMatrix<Field>(F, A.coldim()+1, A.rowdim());
			Hom<Ring, Field> Hmap(_ring, F);

			BlasMatrix<Field> Ap(F, A.rowdim(), A.coldim());
			MatrixHom::map(Ap, A);
			for (size_t i=0;i<A.rowdim();++i)
				for (size_t j=0;j<A.coldim();++j)
					TAS_factors->setEntry(j,i, Ap.getEntry(i,j));

			for (size_t i=0;i<A.rowdim();++i){
				typename Field::Element tmpe;
				F.init(tmpe);
				F.init(tmpe,_ring.convert(tmp,b[i]));
				TAS_factors->setEntry(A.coldim(),i, tmpe);
			}
#ifdef RSTIMING
			tSetup.stop();
			ttSetup += tSetup;
			tLQUP.start();
#endif
			BlasPermutation<size_t>  TAS_P(TAS_factors->coldim()) ;
			BlasPermutation<size_t>  TAS_Qt(TAS_factors->rowdim()) ;

			LQUPMatrix<Field>* TAS_LQUP = new LQUPMatrix<Field>(*TAS_factors,TAS_P,TAS_Qt);
			size_t TAS_rank = TAS_LQUP->getRank();

			// check consistency. note, getQ returns Qt.
			// BlasPermutation<size_t>  TAS_P = TAS_LQUP->getP();
			// BlasPermutation<size_t>  TAS_Qt = TAS_LQUP->getQ();
			std::vector<size_t> srcRow(A.rowdim()), srcCol(A.coldim()+1);
			std::vector<size_t>::iterator sri = srcRow.begin(), sci = srcCol.begin();
			for (size_t i=0; i<A.rowdim(); ++i, ++sri) *sri = i;
			for (size_t i=0; i<A.coldim()+1; ++i, ++sci) *sci = i;
			indexDomain iDom;
			BlasMatrixDomain<indexDomain> BMDs(iDom);
			BMDs.mulin_right(TAS_Qt, srcCol);
			BMDs.mulin_right(TAS_P, srcRow);

#ifdef DEBUG_INC
			std::cout << "P takes (0 1 ...) to (";
			for (size_t i=0; i<A.rowdim(); ++i) std::cout << srcRow[i] << ' '; std::cout << ')' << std::endl;
			std::cout << "Q takes (0 1 ...) to (";
			for (size_t i=0; i<A.coldim()+1; ++i) std::cout << srcCol[i] << ' '; std::cout << ')' << std::endl;
#endif

			bool appearsInconsistent = (srcCol[TAS_rank-1] == A.coldim());
			size_t rank = TAS_rank - (appearsInconsistent ? 1 : 0);
#ifdef DIXON_DEBUG
			std::cout << "TAS_rank, rank: " << TAS_rank << ' ' << rank << std::endl;
#endif
#ifdef RSTIMING
			tLQUP.stop();
			ttLQUP += tLQUP;
#endif
			if (rank == 0) {
				delete TAS_LQUP;
				delete TAS_factors;
				//special case when A = 0, mod p. dealt with to avoid crash later
				bool aEmpty = true;
				if (level >= SL_LASVEGAS) { // in monte carlo, we assume A is actually empty
					typename BlasMatrix<Ring>::Iterator iter = A_check.Begin();
					for (; aEmpty && iter != A_check.End(); ++iter)
						aEmpty &= _ring.isZero(*iter);
				}
				if (aEmpty) {
					for (size_t i=0; i<b.size(); ++i)
						if (!_ring.areEqual(b[i], _ring.zero)) {
							if (level >= SL_CERTIFIED) {
								lastCertificate.clearAndResize(b.size());
								_ring.assign(lastCertificate.numer[i], _ring.one);
							}
							return SS_INCONSISTENT;
						}
#if 0
					// both A and b are all zero.
					for (size_t i=0; i<answer.size(); ++i) {
						answer[i].first = _ring.zero;
						answer[i].second = _ring.one;
					}
#endif
					_ring. assign (den, _ring.one);
					for (typename Vector1::iterator p = num. begin(); p != num. end(); ++ p)
						_ring. assign (*p, _ring.zero);

					if (level >= SL_LASVEGAS)
						_ring.assign(lastCertifiedDenFactor, _ring.one);
					if (level == SL_CERTIFIED) {
						_ring.assign(lastZBNumer, _ring.zero);
						lastCertificate.clearAndResize(b.size());
					}
					return SS_OK;
				}
				// so a was empty mod p but not over Z.
				continue; //try new prime
			}

			BlasMatrix<Field>* Atp_minor_inv = NULL;

			if ((appearsInconsistent && level > SL_MONTECARLO) || randomSolution == false) {
				// take advantage of the (LQUP)t factorization to compute
				// an inverse to the leading minor of (TAS_P . (A|b) . TAS_Q)
#ifdef RSTIMING
				tFastInvert.start();
#endif
				Atp_minor_inv = new BlasMatrix<Field>(F, rank, rank);


				FFPACK::LQUPtoInverseOfFullRankMinor(F, rank, TAS_factors->getPointer(), A.rowdim(),
								     TAS_Qt.getPointer(),
								     Atp_minor_inv->getPointer(), rank);
#ifdef RSTIMING
				tFastInvert.stop();
				ttFastInvert += tFastInvert;
#endif
			}

			delete TAS_LQUP;
			delete TAS_factors;

			if (appearsInconsistent && level <= SL_MONTECARLO)
				return SS_INCONSISTENT;

			if (appearsInconsistent) {
#ifdef RSTIMING
				tCheckConsistency.start();
#endif
				Givaro::ZRing<Integer> Z;
				BlasVector<Givaro::ZRing<Integer> > zt(Z,rank);
				for (size_t i=0; i<rank; ++i)
					_ring.assign(zt[i], A.getEntry(srcRow[rank], srcCol[i]));

				BlasMatrix<Ring> At_minor(_ring, rank, rank);
				for (size_t i=0; i<rank; ++i)
					for (size_t j=0; j<rank; ++j)
						_ring.assign(At_minor.refEntry(j, i), A.getEntry(srcRow[i], srcCol[j]));
#ifdef DEBUG_INC
				At_minor.write(std::cout << "At_minor:" << std::endl);//, _ring);
				Atp_minor_inv->write(std::cout << "Atp_minor_inv:" << std::endl);//, F);
				std::cout << "zt: "; for (size_t i=0; i<rank; ++i) std::cout << zt[i] <<' '; std::cout << std::endl;
#endif
#ifdef RSTIMING
				tCheckConsistency.stop();
				ttCheckConsistency += tCheckConsistency;
#endif

				LiftingContainer lc(_ring, F, At_minor, *Atp_minor_inv, zt, _prime);

				RationalReconstruction<LiftingContainer > re(lc);

				Vector1 short_num(A.field(),rank);
				Integer short_den;

				if (!re.getRational(short_num, short_den,0))
					return SS_FAILED;    // dirty, but should not be called
				// under normal circumstances
#ifdef RSTIMING
				ttConsistencySolve.update(re, lc);
				tCheckConsistency.start();
#endif
				VectorFraction<Ring> cert(_ring, short_num. size());
				cert. numer = short_num;
				cert. denom = short_den;
				cert.numer.resize(b.size());
				_ring.subin(cert.numer[rank], cert.denom);
				_ring.assign(cert.denom, _ring.one);
				BMDI.mulin_left(cert.numer, TAS_P);
#ifdef DEBUG_INC
				cert.write(std::cout << "cert:") << std::endl;
#endif

				bool certifies = true; //check certificate
				BlasVector<Ring> certnumer_A(_ring,A.coldim());
				BAR.applyVTrans(certnumer_A, A_check, cert.numer);
				typename BlasVector<Ring>::iterator cai = certnumer_A.begin();
				for (size_t i=0; certifies && i<A.coldim(); ++i, ++cai)
					certifies &= _ring.isZero(*cai);
#ifdef RSTIMING
				tCheckConsistency.stop();
				ttCheckConsistency += tCheckConsistency;
#endif
				if (certifies) {
					if (level == SL_CERTIFIED) lastCertificate.copy(cert);
					return SS_INCONSISTENT;
				}
				commentator().report (Commentator::LEVEL_IMPORTANT, PARTIAL_RESULT) << "system is suspected to be inconsistent but it was only a bad prime" << std::endl;
				continue; // try new prime. analogous to u.A12 != A22 in Muld.+Storj.
			}

#ifdef RSTIMING
			tMakeConditioner.start();
#endif
			// we now know system is consistent mod p.
			BlasMatrix<Ring> A_minor(_ring, rank, rank);    // -- will have the full rank minor of A
			BlasMatrix<Field> *Ap_minor_inv;          // -- will have inverse mod p of A_minor
			BlasMatrix<Ring> *P = NULL, *B = NULL;   // -- only used in random case

			if (!randomSolution) {
				// use shortcut - transpose Atp_minor_inv to get Ap_minor_inv
				Element _rtmp;
				Ap_minor_inv = Atp_minor_inv;
				for (size_t i=0; i<rank; ++i)
					for (size_t j=0; j<i; ++j) {
						Ap_minor_inv->getEntry(_rtmp, i, j);
						Ap_minor_inv->setEntry(i, j, Ap_minor_inv->refEntry(j, i));
						Ap_minor_inv->setEntry(j, i, _rtmp);
					}

				// permute original entries into A_minor
				for (size_t i=0; i<rank; ++i)
					for (size_t j=0; j<rank; ++j)
						_ring.assign(A_minor.refEntry(i, j), A_check.getEntry(srcRow[i], srcCol[j]));
#ifdef RSTIMING
				tMakeConditioner.stop();
				ttMakeConditioner += tMakeConditioner;
#endif

				if (makeMinDenomCert && level >= SL_LASVEGAS){
					B = new BlasMatrix<Ring>(_ring, rank, A.coldim());
					for (size_t i=0; i<rank; ++i)
						for (size_t j=0; j<A.coldim(); ++j)
							_ring.assign(B->refEntry(i, j), A_check.getEntry(srcRow[i],j));
				}
			}
			else {
				P = new BlasMatrix<Ring>(_ring, A.coldim(), rank);
				B = new BlasMatrix<Ring>(_ring, rank,A.coldim());
				BlasMatrix<Field> Ap_minor(F, rank, rank);
				Ap_minor_inv = new BlasMatrix<Field>(F, rank, rank);
				int nullity;

				LinBox::integer tmp2=0;
				size_t maxBitSize = 0;
				for (size_t i=0; i<rank; ++i)
					for (size_t j=0; j<A.coldim(); ++j){
						_ring.assign(B->refEntry(i, j), A_check.getEntry(srcRow[i], j));
						_ring.convert(tmp2, A_check.getEntry(srcRow[i], j));
						maxBitSize = std::max(maxBitSize, tmp2.bitsize());
					}
#ifdef RSTIMING
				bool firstLoop = true;
#endif
				// prepare B to be preconditionned through BLAS matrix mul
				MatrixApplyDomain<Ring, BlasMatrix<Ring> > MAD(_ring,*B);
				MAD.setup(2);

				do { // O(1) loops of this preconditioner expected
#ifdef RSTIMING
					if (firstLoop)
						firstLoop = false;
					else
						tMakeConditioner.start();
#endif
					// compute P a n*r random matrix of entry in [0,1]
					typename BlasMatrix<Ring>::Iterator iter;
					for (iter = P->Begin(); iter != P->End(); ++iter) {
						if (rand() > RAND_MAX/2)
							_ring.assign(*iter, _ring.one);
						else
							_ring.assign(*iter, _ring.zero);
					}

					// compute A_minor = B.P
#if 0
					if (maxBitSize * log((double)A.coldim()) > 53)
						MD.mul(A_minor, *B, *P);
					else {
						double *B_dbl= new double[rank*A.coldim()];
						double *P_dbl= new double[A.coldim()*rank];
						double *A_minor_dbl = new double[rank*rank];
						for (size_t i=0;i<rank;++i)
							for (size_t j=0;j<A.coldim(); ++j){
								_ring.convert(B_dbl[j+i*A.coldim()], B->getEntry(i,j));
								_ring.convert(P_dbl[i+j*rank], P->getEntry(j,i));
							}
						cblas_dgemm(CblasRowMajor, CblasNoTrans,
							    CblasNoTrans,
							    rank, rank, A.coldim(), 1,
							    B_dbl, A.coldim(), P_dbl, rank, 0,A_minor_dbl, rank);

						for (size_t i=0;i<rank;++i)
							for (size_t j=0;j<rank;++j)
								_ring.init(A_minor.refEntry(i,j),A_minor_dbl[j+i*rank]);

						delete[] B_dbl;
						delete[] P_dbl;
						delete[] A_minor_dbl;
					}
#endif

					MAD.applyM(A_minor,*P);



					// set Ap_minor = A_minor mod p, try to compute inverse
					for (size_t i=0;i<rank;++i)
						for (size_t j=0;j<rank;++j)
							F.init(Ap_minor.refEntry(i,j),
							       _ring.convert(tmp2,A_minor.getEntry(i,j)));
#ifdef RSTIMING
					tMakeConditioner.stop();
					ttMakeConditioner += tMakeConditioner;
					tInvertBP.start();
#endif
					BMDF.inv((BlasMatrix<Field>&)*Ap_minor_inv, (BlasMatrix<Field>&)Ap_minor, nullity);
#ifdef RSTIMING
					tInvertBP.stop();
					ttInvertBP += tInvertBP;
#endif
				} while (nullity > 0);
			}
			// Compute newb = (TAS_P.b)[0..(rank-1)]
			BlasVector<Ring> newb(b);
			BMDI.mulin_right(TAS_P, newb);
			newb.resize(rank);

			BlasMatrix<Ring>  BBA_minor(A_minor);
#if 0
			BlasMatrix<Field> BBA_inv(F,*Ap_minor_inv);
			BlasMatrix<Integer>  BBA_minor(A_minor);
			BlasMatrix<Field> BBA_inv(*Ap_minor_inv);
			LiftingContainer lc(_ring, F, BBA_minor, BBA_inv, newb, _prime);
#endif
			LiftingContainer lc(_ring, F, BBA_minor, *Ap_minor_inv, newb, _prime);

#ifdef DEBUG_DIXON
			std::cout<<"length of lifting: "<<lc.length()<<std::endl;
#endif
			RationalReconstruction<LiftingContainer > re(lc);

			Vector1 short_num(_ring,rank); Integer short_den;

			if (!re.getRational(short_num, short_den,0))
				return SS_FAILED;    // dirty, but should not be called
			// under normal circumstances
#ifdef RSTIMING
			ttSystemSolve.update(re, lc);
			tCheckAnswer.start();
#endif
			VectorFraction<Ring> answer_to_vf(_ring, short_num. size());
			answer_to_vf. numer = short_num;
			answer_to_vf. denom = short_den;

			if (!randomSolution) {
				// short_answer = TAS_Q * short_answer
				answer_to_vf.numer.resize(A.coldim()+1,_ring.zero);
				BMDI.mulin_left(answer_to_vf.numer, TAS_Qt);
				answer_to_vf.numer.resize(A.coldim());
			}
			else {
				// short_answer = P * short_answer
				BlasVector<Ring> newNumer(_ring,A.coldim());
				BAR.applyV(newNumer, *P, answer_to_vf.numer);
				//BAR.applyVspecial(newNumer, *P, answer_to_vf.numer);

				answer_to_vf.numer = newNumer;
			}

			if (level >= SL_LASVEGAS) { //check consistency

				BlasVector<Ring> A_times_xnumer(_ring,b.size());

				BAR.applyV(A_times_xnumer, A_check, answer_to_vf.numer);

				Integer tmpi;

				typename Vector2::const_iterator ib = b.begin();
				typename BlasVector<Ring>::iterator iAx = A_times_xnumer.begin();
				int thisrow = 0;
				bool needNewPrime = false;

				for (; !needNewPrime && ib != b.end(); ++iAx, ++ib, ++thisrow)
					if (!_ring.areEqual(_ring.mul(tmpi, *ib, answer_to_vf.denom), *iAx)) {
						// should attempt to certify inconsistency now
						// as in "if [A31 | A32]y != b3" of step (4)
						needNewPrime = true;
					}

				if (needNewPrime) {
					delete Ap_minor_inv;
					if (randomSolution) {delete P; delete B;}
#ifdef RSTIMING
					tCheckAnswer.stop();
					ttCheckAnswer += tCheckAnswer;
#endif
					continue; //go to start of main loop
				}
			}

			//answer_to_vf.toFVector(answer);
			num = answer_to_vf. numer;
			den = answer_to_vf. denom;
#ifdef RSTIMING
			tCheckAnswer.stop();
			ttCheckAnswer += tCheckAnswer;
#endif
			if (makeMinDenomCert && level >= SL_LASVEGAS)  // && randomSolution
			{
				// To make this certificate we solve with the same matrix as to get the
				// solution, except transposed.
#ifdef RSTIMING
				tCertSetup.start();
#endif
				Integer _rtmp;
				Element _ftmp;
				for (size_t i=0; i<rank; ++i)
					for (size_t j=0; j<i; ++j) {
						Ap_minor_inv->getEntry(_ftmp, i, j);
						Ap_minor_inv->setEntry(i, j, Ap_minor_inv->refEntry(j, i));
						Ap_minor_inv->setEntry(j, i, _ftmp);
					}

				for (size_t i=0; i<rank; ++i)
					for (size_t j=0; j<i; ++j) {
						A_minor.getEntry(_rtmp, i, j);
						A_minor.setEntry(i, j, A_minor.refEntry(j, i));
						A_minor.setEntry(j, i, _rtmp);
					}

				// we then try to create a partial certificate
				// the correspondance with Algorithm MinimalSolution from Mulders/Storjohann:
				// paper | here
				// P     | TAS_P
				// Q     | transpose of TAS_Qt
				// B     | *B (== TAS_P . A,  but only top #rank rows)
				// c     | newb (== TAS_P . b,   but only top #rank rows)
				// P     | P
				// q     | q
				// U     | {0, 1}
				// u     | u
				// z-hat | lastCertificate

				// we multiply the certificate by TAS_Pt at the end
				// so it corresponds to b instead of newb

				//q in {0, 1}^rank
				Givaro::ZRing<Integer> Z;
				BlasVector<Givaro::ZRing<Integer> > q(Z,rank);
				typename BlasVector<Givaro::ZRing<Integer> >::iterator q_iter;

				bool allzero;
				do {
					allzero = true;
					for (q_iter = q.begin(); q_iter != q.end(); ++q_iter) {
						if (rand() > RAND_MAX/2) {
							_ring.assign((*q_iter), _ring.one);
							allzero = false;
						}
						else
							(*q_iter) = _ring.zero;
					}
				} while (allzero);
#ifdef RSTIMING
				tCertSetup.stop();
				ttCertSetup += tCertSetup;
#endif
				//LiftingContainer lc2(_ring, F, BBA_minor, BBA_inv, q, _prime);
				LiftingContainer lc2(_ring, F, A_minor, *Ap_minor_inv, q, _prime);

				RationalReconstruction<LiftingContainer> rere(lc2);
				Vector1 u_num(_ring,rank); Integer u_den;
				if (!rere.getRational(u_num, u_den,0)) return SS_FAILED;

#ifdef RSTIMING
				ttCertSolve.update(rere, lc2);
				tCertMaking.start();
#endif
				// remainder of code does   z <- denom(partial_cert . Mr) * partial_cert * Qt
				VectorFraction<Ring> u_to_vf(_ring, u_num.size());
				u_to_vf. numer = u_num;
				u_to_vf. denom = u_den;
				BlasVector<Ring> uB(_ring,A.coldim());
				BAR.applyVTrans(uB, *B, u_to_vf.numer);

#if 0
				std::cout << "BP: ";
				A_minor.write(std::cout, _ring) << std::endl;
				std::cout << "q: ";
				for (size_t i=0; i<rank; ++i) std::cout << q[i]; std::cout << std::endl;
				u_to_vf.write(std::cout  << "u: ") << std::endl;
#endif

				Integer numergcd = _ring.zero;
				vectorGcdIn(numergcd, _ring, uB);

				// denom(partial_cert . Mr) = partial_cert_to_vf.denom / numergcd
				VectorFraction<Ring> z(_ring, b.size()); //new constructor
				u_to_vf.numer.resize(A.rowdim());

				BMDI.mul(z.numer, u_to_vf.numer, TAS_P);

				z.denom = numergcd;

				// 				z.write(std::cout << "z: ") << std::endl;

				if (level >= SL_CERTIFIED)
					lastCertificate.copy(z);

				// output new certified denom factor
				Integer znumer_b, zbgcd;
				VDR.dotprod(znumer_b, z.numer, b);
				_ring.gcd(zbgcd, znumer_b, z.denom);
				_ring.div(lastCertifiedDenFactor, z.denom, zbgcd);

				if (level >= SL_CERTIFIED)
					_ring.div(lastZBNumer, znumer_b, zbgcd);
#ifdef RSTIMING
				tCertMaking.stop();
				ttCertMaking += tCertMaking;
#endif
			}

			delete Ap_minor_inv;
			delete B;

			if (randomSolution) {delete P;}

			// done making certificate, lets blow this popstand
			return SS_OK;
		}
		return SS_FAILED; //all primes were bad
	}




	/*
	 * Specialization for Block Hankel method
	 */
	// solve non singular system using block Hankel
	template <class Ring, class Field, class RandomPrime>
	template <class IMatrix, class Vector1, class Vector2>
	SolverReturnStatus
	RationalSolver<Ring,Field,RandomPrime,BlockHankelTraits>::solveNonsingular(Vector1& num,
										   Integer& den,
										   const IMatrix& A,
										   const Vector2& b,
										   size_t blocksize,
										   int maxPrimes) const
	{

		linbox_check(A.rowdim() == A.coldim());
		linbox_check(A.rowdim() % blocksize == 0);


		// reduce the matrix mod p
		Field F(_prime);
		typedef typename IMatrix::template rebind<Field>::other FMatrix;
		FMatrix Ap(A, F);

		// precondition Ap  with a random diagonal Matrix
		typename Field::RandIter G(F,0,123456);
		BlasVector<Field> diag(F,Ap.rowdim());

		for(size_t i=0;i<Ap.rowdim();++i){
			do {
				G.random(diag[i]);
			} while(F.isZero(diag[i]));
		}



		Diagonal<Field> D(diag);

		Compose<Diagonal<Field>, FMatrix> DAp(D,Ap);

		size_t n = A.coldim();
		size_t numblock = n/blocksize;

		// generate randomly U and V
		BlasMatrix<Field> U(F,blocksize,A.rowdim()), V(A.coldim(),blocksize);

		for (size_t j=0;j<blocksize; ++j)
			for (size_t i=j*numblock;i<(j+1)*numblock;++i){
				G.random(V.refEntry(i,j));
			}
		for (size_t i=0;i<n;++i)
			G.random(U.refEntry(0,i));

#ifdef RSTIMING
		Timer chrono;
		chrono.clear();
		chrono.start();
#endif

		// compute the block krylov sequence associated to U.A^i.V
		BlackboxBlockContainerRecord<Field, Compose<Diagonal<Field>,FMatrix> >  Seq(&DAp, F, U, V, false);

#ifdef RSTIMING
		chrono.stop();
		std::cout<<"sequence generation: "<<chrono<<"\n";
		chrono.clear();
		chrono.start();
#endif

		// compute the inverse of the Hankel matrix associated with the Krylov Sequence
		BlockHankelInverse<Field> Hinv(F, Seq.getRep());
		BlasVector<Field> y(F,n), x(F,n, F.one);

#ifdef RSTIMING
		chrono.stop();
		std::cout<<"inverse block hankel: "<<chrono<<"\n";
#endif

		typedef BlockHankelLiftingContainer<Ring,Field,IMatrix,Compose<Diagonal<Field>,FMatrix>, BlasMatrix<Field> > LiftingContainer;
		LiftingContainer lc(_ring, F, A, DAp, D, Hinv, U, V, b, _prime);
		RationalReconstruction<LiftingContainer > re(lc);

		if (!re.getRational(num, den, 0)) return SS_FAILED;

#ifdef RSTIMING
		std::cout<<"lifting bound computation : "<<lc.ttSetup<<"\n";
		std::cout<<"residue computation       : "<<lc.ttRingApply<<"\n";
		std::cout<<"rational reconstruction   : "<<re.ttRecon<<"\n";
#endif

		return SS_OK;
	}



	/*
	 * Specialization for Sparse Elimination method
	 */
	// solve non singular system using Sparse LU
	// max prime is not use. only check with one prime
	template <class Ring, class Field, class RandomPrime>
	template <class IMatrix, class Vector1, class Vector2>
	SolverReturnStatus
	RationalSolver<Ring,Field,RandomPrime,SparseEliminationTraits>::solveNonsingular(Vector1& num,
												     Integer& den,
												     const IMatrix& A,
												     const Vector2& b,
												     int maxPrimes) const
	{

		linbox_check(A.rowdim() == A.coldim());

		typedef typename Field::Element Element_t;

		// reduce the matrix mod p
		const Field F(_prime);
		typedef typename IMatrix::template rebind<Field>::other FMatrix;
		FMatrix Ap(A, F);

		// compute LQUP Factorization
		Permutation<Field> P((int)A.coldim(),F),Q((int)A.rowdim(),F);
		FMatrix L(F, A.rowdim(), A.rowdim());
		unsigned long rank;
		Element_t det;

		GaussDomain<Field> GD(F);
		GD.QLUPin(rank,det,Q,L,Ap,P,Ap.rowdim(), Ap.coldim());
		if (rank != A.rowdim()) {
#if 0
			throw LinboxError ("ERROR in DIXON SparseLU: singular matrix or bad prime");
#endif
			// Choose a nonrandom solution with smallest entries:
			// Sets solution values to 0 for coldim()-rank columns
			// Therefore, prune unnecessary elements
			// in those last columns of U
			size_t origNNZ=0,newNNZ=0;
			for(typename FMatrix::RowIterator row=Ap.rowBegin();
			    row != Ap.rowEnd(); ++row) {
				if (row->size()) {
					origNNZ += row->size();
					size_t ns=0;
					for(typename FMatrix::Row::iterator it = row->begin();
					    it != row->end(); ++it, ++ns) {
						if (it->first >= rank) {
							row->resize(ns);
							break;
						}
					}
					newNNZ += row->size();
				}
			}
			commentator().report (Commentator::LEVEL_IMPORTANT, PARTIAL_RESULT) << "Pruned : " << (origNNZ-newNNZ) << " unnecessary elements in upper triangle" << std::endl;
		}


		typedef SparseLULiftingContainer<Ring,Field,IMatrix,FMatrix> LiftingContainer;
		LiftingContainer lc(_ring, F, A, L, Q, Ap, P, rank, b, _prime);
		RationalReconstruction<LiftingContainer > re(lc);

		if (!re.getRational(num, den, 0))
			return SS_FAILED;
		else
			return SS_OK;
	}

} //end of namespace LinBox

//BB : moved the following "guarded" code in a new file, verbatim :
#include "rational-solver2.h"

#endif //__LINBOX_rational_solver_INL


// Local Variables:
// mode: C++
// tab-width: 8
// indent-tabs-mode: nil
// c-basic-offset: 8
// End:
// vim:sts=8:sw=8:ts=8:noet:sr:cino=>s,f0,{0,g0,(0,\:0,t0,+0,=s